Hyperspectral Unmixing · 2020. 7. 22. · DECLARATION I declare that the thesis entitled...

Hyperspectral Unmixing

A Thesis submitted to Gujarat Technological University

for the Award of

Doctor of Philosophyin

Electronics and Communication Engineering

by

Nareshkumar Mohanlal Patel(Enrollment No. : 139997111008 )

under supervision of

Dr. Himanshu B. Soni

GUJARAT TECHNOLOGICAL UNIVERSITYAHMEDABAD

July - 2020

c©[Nareshkumar Mohanlal Patel]

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DECLARATION

I declare that the thesis entitled Hyperspectral Unmixing, submitted by me for the degreeof Doctor of Philosophy is the record of research work carried out by me during the periodfrom October 2013 to June 2019 under the supervision of Dr Himanshu B. Soni and this hasnot formed the basis for the award of any degree, diploma, associateship, fellowship, titlesin this or any other University or other institution of higher learning.

I further declare that the material obtained from other sources has been duly acknowledgedin the thesis. I shall be solely responsible for any plagiarism or other irregularities, if noticedin the thesis.

Signature of the Research Scholar: . . . . . . . . . . . . . . . . . . . . . . Date: . . . . . . . . . . . . . . . . . . . . . .

Name of Research Scholar: Nareshkumar Mohanlal Patel

Place: Vallabh Vidyanagar

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CERTIFICATE

I certify that the work incorporated in the thesis Hyperspectral Unmixing submitted byNareshkumar Mohanlal Patel was carried out by the candidate under my supervision/guidance.To the best of my knowledge: (i) the candidate has not submitted the same research work toany other institution for any degree/diploma, Associateship, Fellowship or other similar titles(ii) the thesis submitted is a record of original research work done by the Research Scholarduring the period of study under my supervision, and (iii) the thesis represents independentresearch work on the part of the Research Scholar.

Signature of Supervisor: . . . . . . . . . . . . . . . . . . . . . . . . . . . Date: . . . . . . . . . . . . . . . . . . . . . . . . . . .

Name of Supervisor: Dr. Himanshu B. Soni


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Course-work Completion Certificate

This is to certify that Nareshkumar Mohanlal Patel Trapasiya no. 139997111008 is a PhDscholar enrolled for PhD program in the branch Electronics and Communication of GujaratTechnological University, Ahmedabad

(Please tick the relevant option(s))

� He/She has been exempted from the course-work (successfully completed duringM.Phil Course).

� He/She has been exempted from Research Methodology Course only (successfullycompleted during M.Phil Course).

� He/She has successfully completed the PhD course work for the partial requirementfor the award of PhD Degree. His/ Her performance in the course work is as follows-

Grade Obtained in Research Methodol-ogy (PH001)

Grade Obtained in Self Study Course(Core Subject) (PH002)

AB AB

Signature of Supervisor: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


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Originality Report Certificate

It is certified that PhD thesis titled Hyperspectral Unmixing, by Nareshkumar MohanlalPatel has been examined by us. We undertake the following:

(a) Thesis has significant new work knowledge as compared already published or areunder consideration to be published elsewhere. No sentence, equation, diagram, table,paragraph or section has been copied verbatim from previous work unless it is placedunder quotation marks and duly referenced.

(b) The work presented is original and own work of the author (i.e. there is no plagiarism).No ideas, processes, results or words of others have been presented as author’s ownwork.

(c) There is no fabrication of data or results which have been compiled analyzed.

(d) There is no falsification by manipulating research materials, equipment or processes,or changing or omitting data or results such that the research is not accurately repre-sented in the research record.

(e) The thesis has been checked using Turnitin (copy of originality report attached) andfound within limits as per GTU Plagiarism Policy and instructions issued from time totime (i.e. permitted similarity index ≤ 25%).

Signature of the Research Scholar: . . . . . . . . . . . . . . . . . . . . . . Date: . . . . . . . . . . . . . . . . . . . . . .



Signature of Supervisor: . . . . . . . . . . . . . . . . . . . . . . . . . . . Date: . . . . . . . . . . . . . . . . . . . . . . . . . . .



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PhD THESIS Non-Exclusive License toGUJARAT TECHNOLOGICAL UNIVERSITY

In consideration of being a PhD Research Scholar at GTU and in the interests of the fa-cilitation of research at GTU and elsewhere, I, Nareshkumar Mohanlal Patel havingEnrollment No. 139997111008 hereby grant a non-exclusive, royalty free and perpetuallicense to GTU on the following terms:

(a) GTU is permitted to archive, reproduce and distribute my thesis, in whole or in part,and or my abstract, in whole or in part ( referred to collectively as the "Work") any-where in the world, for non-commercial purposes, in all forms of media;

(b) GTU is permitted to authorize, sub-lease, sub-contract or procure any of the acts men-tioned in paragraph (a);

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(d) The Universal Copyright Notice ( c©) shall appear on all copies made under the author-ity of this license;

(e) I undertake to submit my thesis, through my University, to any Library and Archives.Any abstract submitted with the thesis will be considered to form part of the thesis.

(f) I represent that my thesis is my original work, does not infringe any rights of others,including privacy rights, and that I have the right to make the grant conferred by thisnon-exclusive license.

(g) If third party copyrighted material was included in my thesis for which, under theterms of the Copyright Act, written permission from the copyright owners is required,I have obtained such permission from the copyright owners to do the acts mentionedin paragraph (a) above for the full term of copyright protection.

(h) I retain copyright ownership and moral rights in my thesis, and may deal with the copy-right in my thesis, in any way consistent with rights granted by me to my Universityin this non-exclusive license.

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(i) I further promise to inform any person to whom I may hereafter assign or licensemy copyright in my thesis of the rights granted by me to my University in this non-exclusive license.

(j) I am aware of and agree to accept the conditions and regulations of PhD including alpolicy matters related to authorship and plagiarism.

Signature of the Research Scholar: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Date: . . . . . . . . . . . . . . . . . . . . . . . . . . Place: Vallabh Vidyanagar

Signature of Supervisor: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Date: . . . . . . . . . . . . . . . . . . . . . . . . . . Place: Vallabh Vidyanagar

Seal

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Thesis Approval Form

The viva-voce of the PhD Thesis submitted by Nareshkumar Mohanlal Patel(EnrollmentNo. 139997111008) entitled Hyperspectral Unmixing was conducted on . . . . . . . . . . . . . . . . . . . . . . . . .(day and date) at Gujarat Technological University. (Please tick any one of the fol-lowing option)

� The performance of the candidate was satisfactory. We recommend that he/she be awardedthe PhD degree.

� Any further modifications in research work recommended by the panel after 3 monthsfrom the date of first viva-voce upon request of the Supervisor or request of IndependentResearch Scholar after which viva-voce can be re-conducted by the same panel again (brieflyspecify the modification suggested by the panel).

� The performance of the candidate was unsatisfactory. We recommend that he/she shouldnot be awarded the PhD degree (The panel must give Justifications for rejecting the researchwork ). .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Name and Signature of Supervisor with Seal 1) (External Examiner 1) Name and Signature

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ABSTRACT

Remote Sensing (RS) is the field of science that includes all those activities necessary for theobservation, acquisition and interpretation of information related to objects, events, phenom-ena or any other item under investigation, without making physical contact with the object,event, or phenomenon under investigation. Remote Sensing has captured the attention ofresearch communities across the globe for research and development due to advancementin imaging technology. Specifically over the past a few decades, hyperspectral imaginghas drawn significant attention and became an important scientific tool for various fields ofreal-world applications. Hyperspectral imaging sensors captures electromagnetic radiationin the portion of spectrum extending from the visible region through the near-infrared andmid-infrared (wavelengths between 0.3µm and 2.5µm), in hundreds of narrow (on the orderof 10nm) contiguous bands. Reflection, absorption and emitting characteristics of all sub-stances available on earth surface are different at specific wavelengths of electromagneticspectrum and they are related to their molecular composition. The plot of measured radia-tion verses wavelength is known as spectral signature of the material, which will be usefulto uniquely characterize and identify the given substance. Due to poor spatial resolution ofhyperspectral image, very often, a single pixel of an image may contain mixing of severalsubstances. So captured spectra is mixing of spectra of the endmembers. The number ofsubstances present in a captured scene is called endmember and their proportion in a pixelis known as fractional abundance. In the field hyperspectral image analysis, the process ofaccurate estimation of number of materials, their spectral signatures and abundance map isknown as hyperspectral unmixing. The steps involved in the process of hyperspectral areatmospheric correction, dimensionality reduction, and unmixing.

Hyperspectral unmixing enables variety of applications like anomaly detection, change de-tection, mineral exploitation, manmade material identification and detection and target detec-tion. Large data size, poor spatial resolution, nonavailability of pure endmember signaturesin data set, mixing of materials at various scales and variability in spectral signature makeslinear spectral unmixing a challenging and inverse-ill posed task. Researchers have devisedand investigated many models searching for robust, stable, tractable, and accurate unmixing.Mainly there are three basic approaches to manage the linear spectral unmixing problem:Geometrical, Statistical and Sparse regression. Geometrical approach exploits the fact thatlinearly mixed vectors are in simplex set.

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In last decade, several algorithms have been proposed based on this approach like Pixel pu-rity index(PPI), N-Finder, Vertex Component Analysis(VCA), Minimum Volume SimplexAnalysis (MVSA), Minimum Volume Enclosed Simplex (MVES) etc. Statistical approachfocuses on parameter estimation techniques. Nonnegative Matrix Factorization (NMF) isused for parameter estimation with some additional constraints of hyperspectral imagery.Geometrical and statistical approaches assume the availability of only hyperspectral datacube, so these are known as Blind Source Separation (BSS) techniques. Sparse Regression(SR) approach formulates unmixing as a linear sparse regression problem, in a fashion sim-ilar to that of compressive sensing. SR assumes the availability of some standard publicallyavailable spectral libraries, which contains spectral signatures of many materials measuredon the earth surface using advance spectro radiometer. The problem of linear spectral un-mixing is now simplified to finding the optimal subset of spectral signatures from the libraryinstead of direct endmember extraction from data cube. Spectral signatures estimation isby project of sparse regression approach. United States Geological Survey(USGS) and Ad-vanced Spaceborne Thermal Emission Reflection Radiometer (ASTER) are publically avail-able spectral libraries, which make SR approach as prominent technique for LSU. SpectralUnmixing using variable Splitting Augmented Lagrangian (SUnSAL) is a basic approach tosparse regression. Later innovatice contribution in the field of sparse unmixing is the jointconsideration of groups of pixels and groups of materials, using the Collaborative Hierarchi-cal Lasso (CHL). Another important contribution is the inclusion of spatial information insparse unmixing, which is achieved in this work by means of a Total Variation (TV) regular-izer.

Main contribution of the thesis constitutes following: (a) Consideration of existence of fewmaterials out of many present in the spectral library (via line sparsity) along with spatialregularization, makes our approach more powerful. (b) Our proposed work automaticallyextracts required dominant endmembers from HSI. Our hybrid algorithm exploits advantagesof our proposed (CSUnSAL-TV) and HySime algorithms. In addition, we have addressed(i) An application of LSU, an anomaly detection, using VCA algorithm and multi-temporalhyperspectral images and (ii) identification of changes in material proportion over the timeusing multi-temporal hyper spectral images via SUnSUL. Number of experiments have beenperformed on synthetic data cubes as well as on real HSI to validate our contribution.

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Acknowledgment

Firstly, I would like to express my sincere gratitude to my Ph.D. supervisor, Dr. HimanshuSoni, Principal, G H Patel College of Engineering and Technology, Vallabh Vidyanagar forhis continuous support and kind guidance throughout the tenure of my research. He alwaysraised his bar high, to expand the horizon of my research endeavor and meet the desiredobjectives within a stipulated amount of time. He always remains as a source of inspirationand provided a good environment to cultivate new ideas and explore them in the field ofinterest. I am very much obliged to him for his profound approach, motivation and spendingvaluable time to mold this work and bring a hidden aspect of research in a light. It has beenremaining a great and memorable experience to follow the learning curve formed by hisexpertise and immense knowledge.

I extend the special thanks to my Doctorate Progress Committee (DPC) members, Dr. Up-ena Dalal, Associate Professor, Electronics and Communication Engineering Department,SVNIT, Surat and Dr. Tanmay Pawar, Head of Electronics Engineering Department, BVM,Vallabh Vidyanagar, for their valuable comments, useful suggestions and encouragement tovisualize the problem from the different perspective. Their humble approach and the way ofappreciation for good work have always created the amenable environment and boost-up myconfidence to push the limit.

I am very much thankful to Charutar Vidya Mandal [CVM] for facilitating me enough spaceand resources to complete my work with pleasant experience and ease of comfort.

I have no words to express my feeling for my wife Nita for standing with me all the timeand carried out all the social responsibilities on her shoulders and made me relieve to spendample amount of time for my research. I am very much grateful for my Father, brother andall the family members for their best wishes and support all the time.

The man, who has almost all the answers to many of my doubt is Prof. Sameer Trapasia,Assistant Professor of Electronics and Communication Engineering Department, G H PatelCollege of Engineering and Technology. I am indebted for his support and help to developa good insight for optimization theory and related problems. I appreciate his spellboundknowledge in mathematics and philosophy, his spiritual depth and unparallel thinking to-wards any situation of life.

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I am very much thankful to all my relatives, friends, colleagues, faculty members of GCET,and my students for all their support and help during my research.

Finally, I express my broad sense of gratitude to the almighty for his grace and blessing.

Nareshkumar Mohanlal Patel

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Table of Content

DECLARATION iii

CERTIFICATE iv

Originality Report Certificate vi

PhD THESIS Non-Exclusive License viii

Thesis Approval Form x

ABSTRACT xi

Acknowledgment xiii

List of Abbreviations xviii

List of Figures xx

List of Tables xxiv

1 Introduction 11.1 History and Overview of Remote Sensing . . . . . . . . . . . . . . . . . . 11.2 Introduction to Hyperspectral Images . . . . . . . . . . . . . . . . . . . . . 31.3 Hyperspectral Unmixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Applications of Hyperspectral unmixing . . . . . . . . . . . . . . . . . . . 81.5 Research Gaps and Problem Statement . . . . . . . . . . . . . . . . . . . 101.6 Objective and Scope of work . . . . . . . . . . . . . . . . . . . . . . . . . 111.7 Organization of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Literature Survey 132.1 Mixing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Linear Mixing Model . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Nonlinear Mixing Model . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Physics of Imaging Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 19

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2.2.1 Reflectance Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Hyperspectral unmixing processing steps . . . . . . . . . . . . . . . . . . . 212.4 Unmixing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Geometrical Approaches . . . . . . . . . . . . . . . . . . . . . . . 222.4.2 Statistical based approaches . . . . . . . . . . . . . . . . . . . . . 302.4.3 Sparse regression approach . . . . . . . . . . . . . . . . . . . . . . 33

3 Performance analysis of VCA Algorithm and Anomaly Detection using VCAalgorithm 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 VCA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3 Description of working VCA Algorithm with synthetically generated hyper-

spectral datacube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Anomaly detection using VCA algorithm . . . . . . . . . . . . . . . . . . 473.4.1 Simulation Results with synthetic data. . . . . . . . . . . . . . . . . 473.4.2 Simulation Results with Real hyperspectral Image . . . . . . . . . . 51

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Proposed Collaborative Sparse Unmixing using Variable Spiting and aug-mented lagrangian with Total Variation (CSUnSAL-TV) 564.1 Introduction to Sparse Unmixing . . . . . . . . . . . . . . . . . . . . . . . 564.2 Spectral Unmixing using Variable Splitting and Augmented Lagrangian (SUn-

SAL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.3 Proposed Collaborative Sparse Unmixing using Variable Splitting and Aug-

mented Lagrangian with Total Variation (CSUnSAL-TV) . . . . . . . . . . 654.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4.1 Simulation results for synthetic Data set . . . . . . . . . . . . . . . 684.4.2 Simulation results with real data . . . . . . . . . . . . . . . . . . . 80

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Hyperspectral Change Detection using Multi-temporal Hyperspectral imagesand Sparse Unmixing algorithm 825.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2 Block diagram of Change Detection mechanism using sparse unmixing al-

gorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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5.3 Simulation Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 855.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6 Proposed Automated extraction of dominant endmembers from hyperspec-tral image using CSUnSAL-TV and HySime 906.1 Hyperspectral Subspace identification using single error . . . . . . . . . . . 90

6.1.1 Simulation Results of HySime . . . . . . . . . . . . . . . . . . . . 936.2 Proposed mechanism of automatic extraction of dominant endmembers using

SUnSAL and Hysime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.3 Simulation Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 97

6.3.1 Synthetic Data Cube . . . . . . . . . . . . . . . . . . . . . . . . . 976.3.2 Real hyperspectral data cube . . . . . . . . . . . . . . . . . . . . . 99

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7 Conclusion and Future Scope 103

List of Publications 113

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List of Abbreviations

HSI Hyper Spectral ImageSU Spectral UnmixingHU Hyperspectral UnmixingHSC Hyper Spectral CemeraRS Remote SensingHySime Hyperspectral Subspace identification using minimum errorVCA Vertex Component AnalysisMVSA Minimum Volume Simplex AnalysisSUnSAL Spectral Unmixing using Variable Spliting and Augmented LagrangianTV Total VariationRADAR RAdio Detection And RangingLiDAR Light Detection And RangingJPL Jet Propulsion Lingle Depot terminationUSGS United States Geological SurveyNMF Nonnegative Matrix FactorizationNP Non deterministic PolynomialMVC Minimum Volume ConstraintALS Alternate Least SquareADMM Alternate Directional Method of MultiplierLMM Linear Mixing ModelANC Abundance Non negativity ConstraintASC Abundance to Sum One ConstraintSRE Signal to Reconstruction ErrorSID Spectral Information DivergenceAFAE Abundance Fraction Angle ErrorSAE Signature Angle ErrorRMS Root Mean SquareBSS Blind Source SeperatorSR Sparse RegressionRMSE Root Mean Square ErrorHYDRA Hyperspectral Data Retrieval and AnalysisSNR Signal to Noise RatioMP Matching PursuitBP Basis PursuitOMP Orothogonal Matching PursuitPCA Principal Component Analysis

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MNF Minimum Noise FractionSVD Singular Value DecompositionHCD Hyperspectral Change DetectionLSU Linear Spectral UnmixingDC Data Cube

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List of Figures

1.1 Types of Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Types of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Data acquisition principal for hyperspectral sensors [68] . . . . . . . . . . . 61.4 An illustration of hyperspectral data cube [63] . . . . . . . . . . . . . . . . 71.5 Spectral signatures at two different pixel in HSI[89] . . . . . . . . . . . . . 81.6 Toy hyperspectral image to illustrate the concept of Pure and Mixed Pixels . 91.7 Fractional Abundance maps for toy HSI . . . . . . . . . . . . . . . . . . . 91.8 An illustration of Hyperspectral imaging and unmixing process [3] . . . . . 10

2.1 Linear Mixing : The measured radiance at a pixel is a weighted average ofthe radiances of the materials present at the pixel [9] . . . . . . . . . . . . . 14

2.2 Graphical interpretation of LMM[78] . . . . . . . . . . . . . . . . . . . . . 162.3 Geometrical Interpretation of LMM[79] . . . . . . . . . . . . . . . . . . . 172.4 Nonlinear Mixing Models[9] . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Plot of transmittance, absorption and reflectance of a green vegetation.Blue

color indicates transmittance, Green Color indicates Reflectance and whitearea between the green and blue represents the energy absorbed by the greenvegetation [84] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Comparison or radiance and reflectance spectrum of grass. The shape of theradiance spectrum is strongly influenced by the absorption features from theatmosphere [84] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 Schematic diagram of the hyperspectral unmixing process [9] . . . . . . . . 212.8 (a)A triangle is 2-D simplex with three vertices representing the pure pixels

of the image (b) A tetrahedron is 3-D simplex with four vertices representingthe pure pixels of the image. Triangle and tetrahedron enclosed all the picturevectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.9 Toy example illustrating the performance of the PPI endmember extractionalgorithm in a 2-dimensional space [82] . . . . . . . . . . . . . . . . . . . 25

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2.10 Graphical interpretation of the N-FINDR algorithm in a 3-D space. (a) N-FINDR initialized randomly (p = 4). (b) Final volume estimation by N-FINDR [32]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.11 Illustration of the VCA algorithm [70]. . . . . . . . . . . . . . . . . . . . . 282.12 Illustration of a mixing model with sparse prior. A hyperspectral pixel can be

described as a mixture of only few endmembers from a spectral dic- tionaryweighted by a sparse abundance vector [6]. . . . . . . . . . . . . . . . . . . 35

3.1 Abundance maps of Almandine, Brucite, Chlorite for endmembers for syn-thetic hyperspectral datacube . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Reflectance spectra of Almandine, Brucite and Chlorite . . . . . . . . . . . 393.3 Scatter Plot ( band = 150 and band = 50) all pixels of p − 1 dimensional

projection o synthetic Hyperspectral image of size 5 × 5, p = 3,γ = 1,SNR = 60dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Illustration of the VCA algorithm [70] . . . . . . . . . . . . . . . . . . . . 413.5 Scatter Plot ( band = 80 and band = 122) all pixels ofp − 1 dimensional

projection of synthetic Hyperspectral image of size 10 × 10, p = 3,γ = 1,SNR = 40dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6 True and Estimated Spectral signatures . . . . . . . . . . . . . . . . . . . 463.7 True and Estimated abundance map . . . . . . . . . . . . . . . . . . . . . 463.8 (a) Pixels arrangement of 5× 5 synthetic DCs (b) Original synthetic DC#1

without anomaly ( pixel number 16 , 2 and 4 are pure pixels correspondsto material 1,2 and 3 (c) Synthetic DC#2 with anomaly (d) 2-Dimensionalrepresentation of Synthetic DC#1 (e) 3-Dimensional representation of syn-thetic DC#2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.9 Simulation result of VCA Algorithm forDC#1 of first experiment. First andthird columns show true endmember signatures and abundances respectively.Second and fourth column shows spectral signatures and abundances for theestimated endmembers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.10 Simulation results of VCA algorithm for DC#2 of first experiment . . . . 503.11 (a) Portion of real hyperspectral image (Cuprite data ) of size 250 × 191

(b) Anomaly is added to real hyperspectral image in pixel number 30001 to30100, 30251 to 30350, 30751 to 30850 . . . . . . . . . . . . . . . . . . . 52

3.12 Spectral Signatures of Six dominant materials present in Real HyperspectralDC#1 of second experiment . . . . . . . . . . . . . . . . . . . . . . . . . 52

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3.13 Abundance maps of six dominant materials present in Real HyperspectralDC#1 of second experiment . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.14 Spectral Signatures of Seven dominant materials present in Real Hyperspec-tral DC#2 of second experiment . . . . . . . . . . . . . . . . . . . . . . . 53

3.15 Abundance maps of seven dominant materials present in HyperspectralDC#2

of second experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1 (a) Mixed pixel (b) Linear Mixing Model for single mixed pixel . . . . . . 574.2 Sparse Regression based approach for Hyperspectral Unmixing . . . . . . . 604.3 Concept of line sparsity of fractional abundances for HSI . . . . . . . . . . 654.4 True and Estimated abundance map for DC#1 . . . . . . . . . . . . . . . 704.5 SRE as a function of SNR for DC#1 . . . . . . . . . . . . . . . . . . . . 714.6 SRE as a function of SNR for DC#1 . . . . . . . . . . . . . . . . . . . . 724.7 True and Estimated abundance map for DC#2 . . . . . . . . . . . . . . . 734.8 SRE as a function of SNR for DC#2 . . . . . . . . . . . . . . . . . . . . 744.9 SRE as a function of SNR for DC#2 . . . . . . . . . . . . . . . . . . . . 754.10 True and Estimated abundance map for DC#3 . . . . . . . . . . . . . . . 764.11 SRE as a function of SNR for DC#3 . . . . . . . . . . . . . . . . . . . . 774.12 SRE as a function of SNR for DC#3 . . . . . . . . . . . . . . . . . . . . 784.13 USGS map showing the location of different minerals in the Cuprite mining

district in Nevada. The map is available online at: . . . . . . . . . . . . . . 794.14 Comparison of estimated abundance map of three endmembers for real cuprite

data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.1 Proposed Change Detection Mechanism using CSUnSAL-TV . . . . . . . . 845.2 True Abundance map for DC#1 and DC#2 . . . . . . . . . . . . . . . . 865.3 Concept of line sparsity of fractional abundances for HSI . . . . . . . . . . 875.4 Estimated Abundance map for DC#1 and DC#2 . . . . . . . . . . . . . . 88

6.1 MSE as a function of the parameter k, for SNR = 35 dB for DC#1 . . . . . 946.2 MSE as a function of the parameter k, for SNR = 35 dB for DC#2 . . . . . 946.3 MSE as a function of the parameter k, for SNR = 35 dB for DC#3 . . . . . 956.4 Proposed mechanism for automated extraction of dominant endmembrs from

hyperspectral image using CSUnSAL-TV and HySime . . . . . . . . . . . 966.5 True Abundance map used to generate synthetic hyperspectral data cube . . 986.6 Sum of contribution of all endmembers of spectral library . . . . . . . . . . 99

xxii

6.7 Abundance map of Estimated Dominant endmembers . . . . . . . . . . . . 1006.8 Estimated abundance map dominant endmembers of real hyper-spectral im-

age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.9 Spectral Signatures of dominant endmembers of real hyper-spectral image . 102

xxiii

List of Tables

1.1 Specifications of Hyperspectral Sensors . . . . . . . . . . . . . . . . . . . 5

3.1 rmsSID, rmsSAE and rmsAEFE as a function SNR for N = 100, p = 3,L = 224,γ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1 Specification of spectral library used for simulation . . . . . . . . . . . . . 68

6.1 Signal subspace dimension as a function of SNR for three different Data cubegenerated with white noise . . . . . . . . . . . . . . . . . . . . . . . . . . 95

xxiv

CHAPTER 1

Introduction

1.1 History and Overview of Remote Sensing

The desire originated from the human mind to explore and understand the world withoutphysical contact and seating from the office pushes the boundaries of the scientific and tech-nical limits, and that made the field of today’s science, Remote Sensing (RS). In his workDe Anima, Aristotle inferred about the nature of light as an actual transparent state trav-eling through a completely transparent medium, thus signifying the required condition forvisualization. The term "camera obscura" was first used in 1604 and it was one of the mostinteresting optical inventions by Leonardo da Vinci. Later in 1666, Sir Isaac Newton hadshown that using a prism, a ray of white light could be decomposed into a number of raysof different colors and using the second prism the decomposed rays could be re-combinedto form original white light. This phenomenon gave birth to the science and art of “drawingwith light”, broadly known as “photography”. In 1827, Niepce had to shoot the first pho-tograph in the history of humanity. In 1958, Gaspard-Félix Tournachon (Nadar) has takenthe first aerial photograph from an altitude of 1200 feet over Paris using a balloon. Theinnovative technologies required for RS and imaging of the earth surface necessitates the re-quirement of a comprehensive terminology. The Remote Sensing was coined by Ms. EvelynPruitt, US Office, and Naval Research in 1960. Before 1960 the field was known as “AerialPhotography". Later in 1909, photography from an airplane is started.

Remote Sensing is the field of science that includes all those activities necessary for the ob-servation, acquisition and interpretation of information related to objects, events, phenomenaor any other item under investigation, without making physical contact with the object, event,or phenomenon under investigation [16],[57]. Remote Sensing deals with the collection ofdata through the detection of electromagnetic radiation radiated from an entity under con-sideration. Remote sensing imagery has many applications in mapping land-use and cover,

1

CHAPTER 1. Introduction

(a) Passive Remote Sensing (b) Active Remote Sensing

Figure 1.1: Types of Remote Sensing

agriculture, soils mapping, forestry, city planning, archaeological investigations, militaryobservation, and geomorphological surveying, among other uses[16].

Considering the electromagnetic radiation as the principal physical carrier of information,the main differentiation of remote sensing systems is based on the typology of the sourceof energy exploited. Depending on whether these systems measure the radiation that isnaturally available, or the energy used to illuminate the target under investigation is emittedby the sensor, are defined as passive or active, respectively. The pictorial representationsof active and passive remote sensing are shown in Figure 1.1. The reflected sunlight isfound to be the most abundant radiant source that is detected by the passive sensors. Passivesensors rely on the energy provided by the sun, which is either reflected or absorbed andthen re-emitted from the earth’s surface. The visible radiation is available only during solarillumination, but on the other hand, thermal IR radiation could be detected at any instantprovided that it is adequate to be recorded. Examples of the most popular passive sensorsare cameras, scanning sensors, and microwave radiometers. Active sensors instead, emit theenergy required to illuminate the target under investigation and then detect the backscattered

2

1.2. Introduction to Hyperspectral Images

radiation. RAdio Detection And Ranging (RADAR) and Light Detection And Ranging (Li-DAR) are the best examples of broadly used active systems. Active sensors itself are thesource of radiation so the data acquisition can be performed at any time. Mostly sensorsprovide information either in the form of image or signal formats, which allows tackling alarge number of applications with remarkable advantages [57],[80],[16].

1.2 Introduction to Hyperspectral Images

According to the number of wavelengths at which radiation or reflectance is measured bysensors, the acquired images in remote sensing are classified it into four categories like grayscale, color, multispectral, hyperspectral images as shown in Figure 1.2.

Figure 1.2: Types of Images

In a gray scale image, also known as a panchromatic image, each pixel represents an inte-gration of light intensity received by sensors over the visible spectrum. Color images, alsoknown as Red, Green, and Blue (RGB) images are recorded at three different wavelengthsof visible spectrum 485nm, 550nm and 645nm respectively. A human being can see in thevisible spectrum, goldfish can see in the infrared spectrum and bumble bees can see in theultraviolet region of the electromagnetic spectrum. Multi-spectral and hyperspectral imageryprovide us the power to see as goldfish (infrared), bumble bees (ultraviolet) and humans (red,green and blue). Multispectral images are a record of the electromagnetic radiation at tensof spectral bands and give the higher spectral resolution compared with the gray scale andcolor images. In 1980, a scientist from Jet Propulsion Laboratory (JPL) had developed an

3


instrument that could sense an object with an unprecedented spectral resolution [16]. Hy-perspectral imaging, sometimes known as imaging spectroscopy, is the big milestone in thefield of remote sensing.

Imaging spectroscopy links the Signal processing and Remote Sensing[60]. HyperspectralImage (HSI), a three-dimensional datacube, has rich spectral and spatial information to solveand understand the real world problems. Nowadays, due to the availability of hyperspectralimages, the hyperspectral signal processing has become the fastest growing topic in the fieldof remote sensing for researchers. The meaning of “hyper” is "too many” and hyperspectralrefers to the too many narrow and contiguous bands. This imaging sensor uses sunlight as asource of radiation so it is a passive sensor. Hyperspectral sensor captures emitted, scatteredor radiated electromagnetic wave in the specific portion of the electromagnetic spectrum. Itcovers the visible region through the near infrared and mid-infrared (0.3µm−2.5µm) in hun-dreds of bands with a spectral resolution of 10nm [34]. The hyperspectral image is the recordof electromagnetic reflection from an object for more than a hundred bands of electromag-netic spectrum. In the last decades, various hyperspectral sensors are available with differentspecifications. Table 1.1 presents examples of aircraft and satellite hyperspectral sensors andtheir principal characteristics. In addition to this, the Indian Space Research Organization(ISRO), India has launched two extra-terrestrial hyperspectral sensors, namely Hyperspec-tral Imager(HySI) [47] and Moon Mineralogy Mapper(M3)[74], for Chandrayaan-I in 2008.HySI is used to explore the mineralogical content of the lunar surface. From an altitude of100km HySI acquires information of lunar surface with a swath width of 20km and spatialresolution of 80m × 80m. It covers 0.42µm − 0.96µm region of the electromagnetic spec-trum with 64 narrow and continuous spectral bands with a spectral resolution of < 20nm.For details specification refer [47],[74].

The hyperspectral image acquisition principal is illustrated in Figure 1.3. AVIRIS [34], Prob-I [53] and Hymap [27] instruments perform the collection of data in a whisk-broom modeto the cross-track direction by mechanical scanning and in the along-track direction by themovement of the platform. Hyperion [91] and HyDICE [81] instruments use a push broomimaging sensor, which acquires data in a cross-track line without any mechanical scanning.

The spectral and spatial characteristics of the ground surface are described with the helpof three-dimensional data cube. It has one spectral and two spatial dimensions. The plotof wavelength vs reflectance is known as a spectral signature. The spectral signature willbe unique for an object because each object available on the earth surface absorbs, emits

4

1.2. Introduction to Hyperspectral Images

Table 1.1: Specifications of Hyperspectral Sensors

AVIRIS HyDICE Hymap Prob-1 Hyperion

Introduction Year airborne airborne airborne airborne spaceborn

Platform 20 6 5 2.5 705

Nominal Altitute (km) 20 3 10 5 30

Spatial Resolution (m) 10 10 17 10 10

Spectral Coverage ( um) 0.4-2.5 0.4-2.5 0.4-2.5 0.4-2.5 0.4-2.5

Number of Channels 224 210 128 128 220

Swath Width (km) 12 0.9 6 3 7.7

and reflects electromagnetic wave according to its molecular structure. Depending on thisfact HSI is used in several applications like environmental monitoring, atmospheric studies,wildfire tracking, crop assessment, land cover classifications etc. The spectral signature ofthe highlighted pixel of HSI is shown to the left side and the two-dimensional image at asingle wavelength is shown to the right side of Figure 1.4. Hyperspectral imagery providesmore accurate and detailed information than human eyes and any other type of remotelysensed data. This fact is explained with the example of an image of two pots of plants asshown in Figure 1.5. It shows two pots of plants; one has natural plant and second hasartificial plant made of plastic material. For a human being, it is not possible to distinguishthe plant without touch but the spectral signatures of two plants are different as shown inFigure 1.5. The spectral response of natural plant is more in magnitude than plastic materialplant in the infrared region.

Various tasks are being performed with the help of hyperspectral imagery [76]. The algo-rithms developed during the last few years are organized as per the following task [64].

• Dimensionality Reduction : It consists of reducing the unwanted and noisy bandsfrom the observed image to reduce the processing burden of subsequent tasks [65],[55].

• Anomaly detection : Anomaly can be considered as a rare and unwanted object in thescene. Anomaly detection is the process of searching the rare pixel from the hyper-spectral data [86].

• Change Detection : The process of detecting sub pixel level variations from the sceneunder observation using multi-temporal hyperspectral images is known as Hyperspec-tral Change Detection in the field of Remote Sensing. The changes in the scene may

5


Figure 1.3: Data acquisition principal for hyperspectral sensors [68]

be due to seasonal variation, natural disaster or as time [58],[83].

• Hyperspectral Classification : It is the process of assigning the label to each pixel ofhyperspectral data according to spectral signature [30][73].

• Spectral Unmixing : It is the process of accurate estimation of a number of endmem-bers, their spectral signatures and fractional abundance maps [9],[46].

1.3 Hyperspectral Unmixing

Modern hyperspectral sensors offer very high spectral resolution because they sample elec-tromagnetic spectrum at more than 200 bands but they offer very poor spatial resolutionbecause they are deployed far away from the area that they are imaging. The spatial resolu-tion of AVIRIS images is very poor, in order of 20m× 20m. So, a single pixel may containmore than one materials. Figure 1.6. shows the toy hyperspectral image to understand theconcept of the pure pixel and mixed pixel

6

1.3. Hyperspectral Unmixing

Figure 1.4: An illustration of hyperspectral data cube [63]

In a hyperspectral image, the pixels containing only one material are known as pure pixelsand the pixels containing more than one materials are known as mixed pixels. The total num-ber of materials present in the image are known as endmembers, their fractional proportionin the image is known as abundance map. The plot of reflectance vs wavelength is known asthe spectral signature of the material. Fractional abundance map corresponds to the materialspresent in the image shown in Figure 1.6 are shown in figure 1.7. The fractional abundancemap shows the distribution of a material in a region under observation. Value 1 in fractionalabundance map indicates the presence of only that material and value between 0 and 1 indi-cates the fractional proportion of that material in the corresponding pixel. Figure 1.8 showsthe linear mixing and unmixing process for mixed pixels of the hyperspectral image. Theobserved spectral signature of the mixed pixel will be the weighted sum of individual spectraof materials present in that pixel as shown in Figure 1.8.

The process of separating constituent spectra from the observed spectral signature and esti-mating their fractional proportion in the pixel is known as hyperspectral unmixing. In thecontext of hyperspectral images, it is known Spectral Unmixing (SU). It is very much usefulin certain application area like speech processing, medical application, remote sensing etc.In short, the process of estimating the number of endmembers present in the scene, theirspectral signatures and fraction abundance map is known as Hyperspectral Unmixing. Largedata size, poor spatial resolution, not the availability of pure endmember signatures in dataset, mixing of materials at various scales and variability in spectral signature makes SU as achallenging and inverse-ill posed task. SU is discussed in details in chapter II.

7


Figure 1.5: Spectral signatures at two different pixel in HSI[89]

1.4 Applications of Hyperspectral unmixing

Hyperspectral was limited to primary applications such as ores mining but with the adventof technology, it is used in vast fields as exemplified below:

• Mineralogy: Hyperspectral unmixing was first used in mineralogy exploring its in-herent traits. Various minerals could be identified from their crude form using theirunique spectral signatures.

• Surveillance: As a spy agent, it could be used against enemy campsites to detect theadvancement in technology with the help of drones and satellite imagery.

• Agriculture: As a part of the ecosystem, crop health could be monitored in variousgeographical regions with its recording for disease analysis.

• Ophthalmology: The oxygen level in the retina could be monitored using hyperspec-tral imaging to monitor the human-eye vision.

• Astronomy and space surveillance: Properties of HSI fits well for the astronomywhere material detection and composition is important for understanding the featuresof distant objects. As spectral signatures are unique for every material, HSI is a pow-erful tool to detect them.

8

1.4. Applications of Hyperspectral unmixing

Figure 1.6: Toy hyperspectral image to illustrate the concept of Pure andMixed Pixels

Figure 1.7: Fractional Abundance maps for toy HSI

• Food and pharmaceutical processing: HSI is highly used to identify faults, defectsor presence of foreign bodies/microbes present in food material which are not detectedby high spatial-resolution cameras.

• Environment: HSI is highly used to identify faults, defects or presence of foreign bod-ies/ microbes present in food material which are not detected by high spatial-resolutioncameras.

The list of applications of HSI is immense but a common theme is the use of measuredspectra for identifying particular materials with the abundances within the imagery. Thistask referred to as unmixing has been described in the following section

9


Figure 1.8: An illustration of Hyperspectral imaging and unmixing process[3]

1.5 Research Gaps and Problem Statement

Hyperspectral unmixing is discussed in detials chapter 2. The research gaps have been iden-tified as follow:

1. Sparse regression based algorithm assumes the availability of spectral libraries andit estimates the fractional abundance map for all the materials present in the library.Normally the library contains more than thousand spectral signatures. So identifica-tion of fractional abundance map becomes a time-consuming and complex task. Asper the literature survey, number of endmembers are estimated by HySime Algorithmand CSUnSAL-TV estimates the fractional abundance map of all the materials whose

10

1.6. Objective and Scope of work

spectral signatures are available in the library. There should be some standard mech-anism for automatic detection of dominant endmembers from the spectral library andtheir abundance map.

2. Performance of endmember extraction algorithm is evaluated and comparative anal-ysis is done with synthetically generated hyperspectral images as well as real data.Inliterature nobody have discussee the specific applications of the unmixing algorithms.

3. Performance of recently proposed sparse unmixing algorithms like SUnSAL and SUnSAL-TV could be improved by incorporating a collaborative approach.

Hyperspectral Camera (HSC) is generally mounted on satellite or on an airborne vehicle.Spatial Resolution of images taken by the hyperspectral camera is poor. A single pixel ofHSI may contain more than one material. With normal images, it is not possible to determinethe number of endmembers, their spectral signatures and their fractional abundances. Byassuming Linear Mixing Model for HSI formation, the problem of LSU is defined as follow.

"Given only hyperspectral data cube, determine the number of materials present in the image,known as endmembers, their spectral signatures and their proportion, known as abundancemap".

1.6 Objective and Scope of work

The objective is to explore the spatial and spectral characteristics of a hyperspectral image forfilling the identified research gaps. The main objective is to incorporate the concept of linesparsity with Total Variation (TV) to achieve more powerful spectral unmixing algorithm.

The scope of work includes,

1. Improvement in the knowledge of different approaches of spectral unmixing.

2. Evaluation of the performance of existing algorithms and perform one or two applica-tion of spectral unmixing using existing and proposed algorithm.

3. Acquisition of the knowledge about spectral libraries used for sparse unmixing.

4. Development of methodology for the automatic extraction of dominant materials presentin the scene.

5. Improvement of existing algorithm by incorporating constraints of HSI.

11


1.7 Organization of thesis

The thesis is organized as follow:

Chapter 1 discussed the overview of remote sensing, introduction to hyperspectral imag-ing, Hyperspectral Unmixing and its applications, objectives of the research work and thesisorganization.

Chapter 2 gives a theoretical background of the linear mixing model used for hyperspec-tral image formation. Moreover, processing steps and current state of art for three basicapproaches for hyperspectral unmixing is discussed in detail.

Chapter 3 presents a detail description of the vertex component analysis algorithm and itsperformance analysis using synthetic as well as real data set. Application of anomaly detec-tion is discussed with using VCA algorithm for multi-temporal hyperspectral data.

Chapter 4 describes a proposed methodology of automatic extraction of dominant endmem-ber from hyperspectral data using SUnSAL and HySime algorithms.

Chapter 5 introduces the novel algorithm for hyperspectral unmixing. Performance of theproposed algorithm is verified with the help of synthetically generated images as well as realhyperspectral data cube.

Chapter 6 describes the application of change detection using a semi-blind approach usingmulti-temporal hyperspectral images.

Finally, Chapter 7 concludes the Ph.D. thesis stating the outcome of research work and dis-cussing the future research scope.

12

CHAPTER 2

Literature Survey

This chapter introduces the hyperspectral image formation models, physics of imaging spec-troscopy, processing steps for unmixing and comprehensive literature review of current stateof art of unmixing algorithms. Several atmospherically corrected real hyperspectral datacubes are available for research work, discussed at the end of the chapter.

2.1 Mixing Model

In remote sensing Hyper Spectral Cameras(HSCs) are mounted on satellite or on airbornevehicles. HSC measures the electromagnetic radiation reflected from an object at more than100 wavelengths. The spatial resolution of the image taken by HSC is poor and it is in orderof 4m× 4m to 20m× 20m. It is possible that more than one material may be present in onepixel. Therefore the observed spectral signature of a single pixel may contain the spectralsignatures of materials present in that pixel. Mainly there are two mixing models found asper literature : the Linear Mixing Model (LMM) and the Non-linear Mixing Model (NMM)[46]. Linear mixing model is the most widely and popular hyperspectral image formationmodel. In LMM internal interference is ignored while NMM considers internal interferencebetween spectral signatures of materials [36].

2.1.1 Linear Mixing Model

In LMM, it is assumed that light is scattered by only one material before it reaches the sensor.It means that light signal is not reflected from multiple objects [85]. The mixing of spectralsignatures occurs only due to the low spatial resolution of the image. If a single materialoccupies the complete pixel area of an image taken by hyperspectral sensor then there wouldnot be any kind of mixing of spectral signatures. Figure 2.1 shows the linear mixing model

13

CHAPTER 2. Literature Survey

scenario for a single pixel of a hyperspectral image [9]. It is seen that observed spectralsignature of a pixel is mixed of three materials with spectral signatures m1,m2 and m3 withrespective fractional proportion of α1, α2 , α3.

For every single pixel of the three-dimensional hyperspectral data cube, mathematically theLinear mixing model (LMM) can be expressed as

yi =

p∑j=1

mijαj + ni (2.1)

where the subscript i represents the spectral band number and subscript j represents theendmember number from endmember matrix M . The length of observed vector y is L, i.e.,i = 1, 2, ..., L and number of endmembers in endmember matrix are p, i.e. j = 1, 2,......, p.mij represents the reflectance at spectral band i of jth endmember, the fractional proportionof jth endmember in a pixel is given by αj . ni represents the error term for the spectral bandi.

Figure 2.1: Linear Mixing : The measured radiance at a pixel is a weightedaverage of the radiances of the materials present at the pixel [9]

In general, mathematically LMM can be written in compact matrix form as

y = Mα + n (2.2)

14

2.1. Mixing Model

Where y ∈ RL is observed spectral vector, M ∈ RL×p is endmember matrix containing ppure spectral signatures ( m1,m2,m3, ......,mp ) of length L, α ∈ Rp is fractional abundancesof the endmembers for a given pixel and n ∈ RL is the measurement and observational erroraffecting at each spectral band.

The above mentioned linear mixing model can be extended for complete hyperspectral imagecontaining N pixels. The LMM can be written as

Y = MX +N

where matrix Y ∈ RL×N ,M ∈ RL×p, X ∈ Rp×N and N ∈ RL×N represents the hyperspec-tral image, measurement matrix, the abundance matrix and the additive noise respectively.Row vectors of the abundance matrix shows the proportion of respective endmember in allthe pixels of an image. Here abundance matrix X is given as

X =

α11 α12 · · · α1N

α21 α22 · · · α2N

...... . . . ...

αp1 αp2 · · · αpN

Here αij represents the proportion of endmember i in the jth pixel. In order to properly rep-resent a physically realizable scene, two constraints are normally imposed on the abundancematrix. They are Abundance Nonnegative Constraint (ANC) and the Abundance sum to oneConstraint (ASC) defined in equations 2.3 and 2.4.

αij ≥ 0, i = 1, 2,...., p and j = 1, 2,....,N (2.3)

p∑i=1

αij = 1 (2.4)

The value of fractional abundance is always nonnegative, lieing in the range of 0 to 1 and sumof its values for a single pixel is always one. These are known as Abundance Non-negativityConstraint (ANC) and Abundance Sum-to-one Constraint (ASC), which are represented incompact form as ANC indicates that the value of fractional abundance is always nonnegative,lie in the range of 0 to 1 . ASC means that the sum of fractional abundances for a single pixel

15


is always one. A material can not absorb light more than incident light so the value of Mwould be non negative [38].

Figure 2.2: Graphical interpretation of LMM[78]

The Linear Spectral Unmixing (LSU) process considering Abundance Non-negativity Con-straints (ANC) and Abundance Sum-to-one Constraints (ASC) is known as Fully ConstraintsLinear Spectral Unmixing (FCLSU). The linear mixture model can be interpreted in graphicfashion by using a scatter plot between two bands or, more generally, between two non-collinear projections of the spectral vectors. For illustrative purposes, figure 2.2 provides asimple graphical interpretation in which the endmembers are the most extreme pixels defin-ing a simplex which encloses all the other pixels in the data, so that we can express everypixel inside the simplex as a linear combination of the endmembers. As a result, a key aspectwhen considering the linear mixture model is the correct identification of the endmembers,which are extreme points in the l-dimensional space. The extreme points of simplex formedin scatter plot are known as pure pixels. Every pixel present inside the simplex can be repre-sented as linear combination of vertices of the simplex.

Two basic requirements for the solution of Linear Spectral Unmixing algorithms are as fol-lows:

1. Number of endmembers , p, present in hyperspectral Image, Y .

16

2.1. Mixing Model

2. There should be at least one pure pixel per endmember present in the image.

Now, we need to interpret the LMM as per convex geometry after defining LMM. Assumethat the columns of measurement matrix, M , are affinely independent. (m2 − m1,m2 −m1,m3 −m1, ......,mp −m1) are linearly independent , with the ANC constraint the set

C ≡ {Y = MX|X ∈ Rp×N}

is a positive simplicial cone. If the ASC constraint is met, all the pixels lie in an affine hulldefined as

A ≡ {Y = MX|1TNX = 1}

This means that , if ANC and ASC are both satisfied, all the pixels are in a convex hull(Simplex) defined as equation

Figure 2.3: Geometrical Interpretation of LMM[79]

S ≡ {Y = MX|X ∈ Rp×N , 1TNX = 1}

The relationship for cone, affine hull and simplex is showed in Figure 2.3. This geometricalpoint of view for hyperspectral unmixing will be further discussed in section 2.4.1.

17


2.1.2 Nonlinear Mixing Model

In contrast to LMM, the Nonlinear Mixing Model (NLMM) assumes that the incident lightscatters multiple times from the material under observation before reaching the sensor[36].Thus the observed spectral signature of a single pixel for an image received by the sensorwill be the nonlinear mixing of spectral signatures of constituent materials of the observedpixel. Depending on the reflection level, nonlinear mixing models are broadly classifiedin two levels, viz. a Classical (Multilevel) level and microscopic (Intimate) level. Whenlight is reflected from different objects other than the target or scattered from one or moreobjects, then it is multilayer mixing as shown in Figure 2.4. When light is reflected fromhomogeneous mixture of materials then mixing is known as intimate mixture as shown inFigure 2.4. In this mixture, the emitted photons from the molecules of one material areabsorbed by the other molecules, which may emit additional photon as shown in Figure 2.4.

(a) Microscopic (Intimate level) mixing (b) Classical (Multilevel) mixing

Figure 2.4: Nonlinear Mixing Models[9]

It is quite difficult to model nonlinear mixture of materials [12],[37]. However, some method-ologies are available, which describes the nonlinear mixing model[29]. Several Kernel basedmethods [15],[14] like Radial Basis Function (RBF)[2] and polynomial expansion [24] are

18

2.2. Physics of Imaging Spectroscopy

available to solve NLMM based spectral unmixing problems. A bilinear model is proposedspecifically for multilayer Mixing[35]. Recently several researchers have applied neural net-work based approaches for unmixing problem based on nonlinear mixing [77],[56].

2.2 Physics of Imaging Spectroscopy

Imaging spectroscopy is better known as Hyperspectral Imaging. The name has been derivedfrom the spectroscopic measurement ability, with spatial assignment in an image. Inter-atomic and inter-molecular interactions within the electromagnetic spectrum are studied inspectroscopy. The interactions which include reflection, absorption, transmission, refractionetc could be represented in-terms of wavelength of the wave under propagation.

Figure 2.5: Plot of transmittance, absorption and reflectance of a green veg-etation.Blue color indicates transmittance, Green Color indicates Reflectanceand white area between the green and blue represents the energy absorbed by

the green vegetation [84]

2.2.1 Reflectance Spectrum

Ratio of the intensity of reflected light to that of the incident light is parameterized as re-flectance of the light. It’s a dimensionless quantity[66]. The plot of reflectance as a functionof wavelength is known as the spectrum of reflectance. The reciprocal of reflectance is called

19


transmittance. The degree of light from the surface transmission normalized by the incidentlight intensity, provides the magnitude of transmittance. The fraction of light excludingtransmission and reflection which is absorbed is termed as absorption.

The parametric relationship between the aforementioned quantities is given by:

τλ +Rλ + αλ = 1

where τλ is transmittance, Rλ is reflectance and αλ is absorption.

The amount of radiation incident on the optical sensor, known as sensor radiance is measuredby a remote sensing optical sensor. The reflection obtained from the earth’s surface and theatmospheric influence forms the composite radiation. The light energy is modified with theinteractions of ionized gaseous particles and the aerosols found in the atmosphere. Hencethe surface reflectance and the atmospheric effects directly influence the sensor radiance.Characterization of the atmospheric effects along with the inverse processing of the equation,helps to recover the reflectance spectrum from the at sensor radiance and the correspondingprocess is better known as atmospheric correction.

Figure 2.6: Comparison or radiance and reflectance spectrum of grass. Theshape of the radiance spectrum is strongly influenced by the absorption fea-

tures from the atmosphere [84]

20

2.3. Hyperspectral unmixing processing steps

2.3 Hyperspectral unmixing processing steps

Spectral Unmixing processing steps for hyperspectral images are shown in Figure 2.7. Itinvolves atmospheric correction, dimensionality reduction, and unmixing, which may betackled via the classical endmember determination plus inversion, or via sparse regressionor sparse coding approaches [9]. Often, endmember determination and inversion are im-plemented simultaneously. A brief characterization of each of these steps has been shownbelow:

Figure 2.7: Schematic diagram of the hyperspectral unmixing process [9]

1. Atmospheric correction. The atmospheric attenuation and scattering of the light af-fects the radiance at a sensor. The atmospheric correction compensates by convertingradiance into reflectance, which is an intrinsic property of a material. Atmosphericcorrection is optional because few spectral unmixing algorithms directly works withRadiance data, means no need to convert in Reflectance..

21


2. Dimensionality Reduction. The space spanned by the image spectra is generallymuch lesser than existing number of bands. This identification of appropriate sub-spaces is inevitable for reducing the dimensionality, improving the performance, com-putation and data storage. Moreover, with the improvement of accuracy in the linearmixture model, the signal subspace dimension turns to be one less than equal to thenumber of endmember. It is considered as a vital Figure in hyperspectral unmixing[68].

3. Unmixing. It involves the identification of the endmembers in the scene and the frac-tional abundances at each pixel. Three general methodologies involved will be fol-lowed in the corresponding section.

2.4 Unmixing Methods

They are basically categorized into geometrical, statistical, and sparse regression based meth-ods:

• Geometrical approaches exploits the the concept of convexity. It assumes that thevolume formed by all data points of image is simplex and all mixed pixels are interiorto the simplex.

• Statistical approaches estimation techniques for extraction of endmembers spectral sig-natures and later calculates the fractional abundance map of endmembers.

• Sparse regression methods exploits the availability of spectral library. In this approachfractional abundances map are represented as sparse with respect to the library andformulates the problem as sparse recovery task.

2.4.1 Geometrical Approaches

A set of vectors, Y = (y1, y2, y3, ......, yN) ⊂ RL, is known as convex set, if and only if astraight line drawn between any two points lies inside the search space (set Y). For {ai} ⊂ R

a set of scalar for all i ∈ K = {1, 2, ...., k} the weighted sum of vectors for set Y is given ask∑i=1

aiyi. Then set Y is called linearly independent set, if there exists unique solution to the

equationk∑i=1

aiyi = 0 is given by {ai} = 0 for i ∈ K. With respect to geometrical view point,

22

2.4. Unmixing Methods

an Affine combination is found to be linear grouping of Y sufficing the conditionk∑i=1

aiyi =

1, ai ≥ 0,∀i ∈ K. The set is hence referred to a vectorial set of convex combinations.The set of each combination obtained from the various elements of Y is denoted as C(Y)and is referred to be the convex hull of Y. Let the index set be denoted by Kη = K\{η}

Figure 2.8: (a)A triangle is 2-D simplex with three vertices representing thepure pixels of the image (b) A tetrahedron is 3-D simplex with four verticesrepresenting the pure pixels of the image. Triangle and tetrahedron enclosed

all the picture vectors.

denote the index set from which index η has been deleted. If the set of vector differences,Y ′ = {yi − yη : i ∈ Kη} is linearly independent for some η ∈ K it can be shown that, Y ′

is a linearly independent set ∀η ∈ K. Therefore, the set Y = (y1, y2, y3, ......, yN) ⊂ RL issaid to be affine independent if and only if the set Y ′ = {yi− yη : i ∈ Kη} ⊂ RLis a linearlyindependent set for some η ∈ K .It’s crucial to note that vectors y1, y2, y3, ......, yN are affine

independent provided that the unique solution to the set of simultaneous equationk∑i=1

aiyi = 0

andk∑i=1

ai = 0 is provided by the expression ai = 0 for all i ∈ K. The Convex Hull of the

affinely independent points form a simplex that is the minimum convex set formed by n+ 1

vertices. For X to be affine independent, then its convex hull C(X) should be a simplex withm dimensions or an m-simplex. Hence a 0-simplex implies a point, a 1-simplex refers to a

23


segment obtained from two affine independent points. A 2-simplex infers a triangle formedby three affine independent points while a 3-simplex could be considered as a tetrahedrondemarcated by four affine independent points.

Existence of at least one pure pixel per endmember in the image implies at least one the spec-tral vector on each vertices formed by data simples. The aforementioned assumption enablesthe design of computationally efficient algorithms which is inevitable, but does not hold truein many datasets. However the set of most pure pixels are found in the data using thesealgorithms. Due to their low computational requirements and clear conceptual gist, they areoften used in linear hyperspectral unmixing applications. Following shows the representativealgorithms belonging to this class:

Pixel Purity Index (PPI) algorithm : PPI algorithm is openly available in ENVI (ENvi-ronment for Visualizing Images) tool that has been developed by research System, so it isconsidered as a yardstick and a baseband in hyperspectral analysis of image for the extrac-tion of endmember. PPI is convex geometry based algorithm [10],[11]. In this approach, theMNF (Minimum Noise Fraction) is used to augment the SNR (Signal to Noise Ratio) andshrink the data dimensions. The complete approach of PPI algorithm has been shown in Fig-ure 2.9. The dark dots denote the pixels from hyperspectral images while axes shown refer tothe random vectors. The random sequence of pixel vectors is called skewers. The hyperspec-tral data is projected onto the skewers using the PPI approach. The extreme points pertainingto each skewer direction is saved and the several times of each extreme pixel is presumed tobe the better candidates as endmembers. The Figure 2.9 indicates that the number of timesof extreme points for point A,B,C and D are respectively 1,1,1 and 2. Hence point D gainsbetter consideration as candidate for the endmember. Since PPI technique only generates thecandidate endmembers, it is upto the user to decide the endmembers by selecting from thecandidates. PPI is the first algorithm for endmembers extraction so several drawbacks areobserved from PPI. Various versions of PPI have been published till date for example, an It-erative PPI(IPPI)[20], A Fast Iterative PPI (FIPPI)[19], Random PPI (RPPI)[21], AutomaticPPI (APPI)[23]. For real time applications several researchers have published their workrelated to FPGA implementation of PPI [33],[50] and GPU implementation of PPI [95],[82].

24


Figure 2.9: Toy example illustrating the performance of the PPI endmemberextraction algorithm in a 2-dimensional space [82]

Algorithm 1 Pseudo code of Pixel Purity Algorithm [23]1: Preprocessing

1. Find the VD to estimate the number of bands required for dimensionality reduction.2. Apply the MNF or PCA transform to reduce dimensionality.

2: Initialization: Let k be a sufficiently large positive integer and produce a set of k unitvectors called “skewers”,{skewerj}kj=1.

3: For each skewerj , all the data sample vectors are projected onto skewerj to find samplevectors at its extreme positions to form an extrema set for the skewer skewerj , denotedby Sextrema(skewerj). Despite the fact that a different skewerj generates a differentextrema set Sextrema(skewerj), it is very likely that some sample vectors may appear inmore than one extrema set. Define an indicator function of a set S, IS(r).

IS(r) =

{1; ifr ∈ S0; ifr /∈ S

NPPI(r) =∑j

ISSkewer(skewerj)(r) (2.5)

where NPPI(r) is defined as the PPI score of the sample vector r.

25


Algorithm 2 Pseudo-code for PPI4: Find the PPI scores NPPI(r) for all the sample vectors defined by eq 2.5.5: Let t be a threshold value set for the PPI score. Extract all the sample vectors withNPPI(r) ≥ 0 which will be the desired set of final endmembers. Usually, the thresholdt can be set to 1.

N-FINDR N-FINDER is one of the most popular and widely used EEA developed by M.E.Winter in 2004[94]. It is used as benchmark and base for modification for many EEA be-cause of its simplicity. It is an iterative volume maximization algorithm. The N-FINDRis based on the assumption that a simplex with (p-1) dimensional volume having verticesidentified by the purest pixels is generally greater than that formed by additional pixels com-bination. The N-FINDR procedure is initiated with an arbitrary selection of pixels as shownin 2.10(a), followed by the calculation of simplex volume formed by the pixels. The pro-cess is repeated by replacing pixel and observing for the collection of pixels maximizing thevolume of the simplex defined by the selected endmembers. The associated volume of eachpixel undergoes calculation within each endmember position by replacing the endmemberand obtaining the resultant volume. If such replacements have consequential increase in vol-ume, the endmember is replaced by the pixel. The procedure undergoes repetition until thereare no more endmember replacements found as shown in Figure 2.10(b). Pseudo code forN-finder algorithm is shown in [22].

Figure 2.10: Graphical interpretation of the N-FINDR algorithm in a 3-Dspace. (a) N-FINDR initialized randomly (p = 4). (b) Final volume estimation

by N-FINDR [32].

26


Algorithm 3 Pseudo-code for N-FINDER [22]1: Preprocessing

1. Estimate number of endmembers p using Vector Decomposition(VD)2. Perform Dimensionality reduction using techniques such as MNF, PCA to reduce

dimension of data set from L to p, where L indicates number of spectral bands.2: Initialization: Randomly initialize the set of endmembers E(0) from the input data

set.E(0) = {E(0)1 , E

(0)2 , ....., E

(0)p } and k = 0

3: Exhaustive Search1. Volume Calculation: Calculate the volume of the current endmember set as fol-

lows: V (E(k)) =

∣∣∣∣∣∣∣∣∣∣∣det

1 . . . 1... . . . ...

E(k)1 . . . E

(k)p

∣∣∣∣∣∣∣∣∣∣∣(p−1)!

2. Replacement and recalculation of volume: Take each vector from data set in allendmember positions in the E(k) set and recalculate the volume. if new volume isgreaten than previous one then update data set with, then update the volume andupdate the E(k) set to be E(k+1), and increment k by one. If the new volume is notlarger than the previous one, then no replacement is required.

4: Repeat steps 3 until the endmember set converges and no new replacement takes place.

If number of pixels in dataset are N, then to complete the exhaustive search step there are(N

p

)= N !

p!(N−p)! vertex simplexes needed. Algorithm will search for all data points avail-

able which may cause very high cost in terms of time. Several version of N-FINDER havebeen proposed to increase the speed and to reduce the computation complexity [75],[61],[97].

Vertex Component Analysis Vertex Component analysis if fully automatic method for un-supervised endmember extraction from hyperspectral data [70]. The algorithm exploits twofacts

• The endmembers are the vertices of a simplex

• The affine transformation of a simplex is also a simplex.

It works with or without dimensionality reduction. Computation complexity of VCA is lessthe other algorithms from geometrical approach. VCA estimates the endmember one by one

27


Figure 2.11: Illustration of the VCA algorithm [70].

by projecting the data to orthogonal subspaces. The new endmember is considered as theextreme projection. The process of algorithm is shown in following steps.

Algorithm 4 Pseudo-code for VCA [70]1: INPUT :Y = (y1, y2, y3, ......, yN) ,where Y ∈ RL×N

2: Number of endmembers, p, are identified by HySime algorithm.3: Projecting data onto a vector orthogonal to the subspace spanned by the endmembers

already determined.4: Finding pixel with extreme projection5: Incorporating this pixel into endmember.6: Generating orthogonal vector and projecting data to new vector.7: Repeat step3 to 6 until all endmembers are identified.

Figure 2.11 shows two iterations of the VCA algorithm. It shows simplex Sp, projection ofconvex cone Cp which is defined by the mixture of two endmembers: black dots representthe data in the hyperspectral image. Here f1 and f2 are vector orthogonal to the subspacespanned by the endmembers already determined. ma and mb are extreme point on the direc-tion f1 and f2 after projecting all the points. In the first iteration, data is projected onto the

28


first direction f1. The extreme of the projection corresponds to endmember ma. In the nextiteration, the endmember is found by projecting data onto direction f2, which is orthogonalto ma and the extreme of the projection corresponds to mb. Therefore ma and mb are thetwo endmembers of the data estimated by the algorithm [70]. Real Time implementation ofVCA on GPU is discussed in [5]. The modified VCA is proposed in [59]

The Iterative Error Analysis (IEA) It is performed directly on the spectral data rather thanthe principle component transformations. It is known for its performance on a series oflinear constrained unmixing. Endmembers are selected as those pixels which minimize theremaining error in the unmixed image. The termination of algorithm takes place when theunmixing error falls beneath a threshold or reaches the total number of endmembers. Theprocedural steps of IEA are shown below: to initiate the procedure, an initial vector which isusually the mean spectrum of data is calculated.

Minimum volume based algorithms do not assume the availability of pure pixels in scene.Hence it is known as the non-convex optimization problem. To hunt for the smallest simplexcontaining the points in the hyperspectral data is the main objective of MV based algorithms.The vertices of a simplex are referred to its endmembers. Due to inadequacy of points inthe dataset, the outcome of the estimated endmembers may vary slightly form the basicendmembers.

Craig’s Minimum Volume transform CMVT is a mirror version of N-FINDER. It triesto build the smallest possible simplex comprising the complete dataset (while N-FINDERefforts to build the largest simplex within the data), treating the vertices of simplex as theendmembers. To begin with MVT algorithm recognizes the subspace, thereby applying pro-jective projection to the data called dark point Fixed transform. Next it minimizes the volumeof simplex by equation using all spectral vectors fitting the simplex.

V (M0) =abs(

∣∣∣M ∣∣∣)k!

Where M0 = [0,M ] is for obtaining non-zero volume and |·| is the determinant operator.At every instant the algorithm alters one facet of the simplex, keeping the other fixed. Theprocedure of the algorithm is as follows:

Minimum Volume Simplex Analysis Minimum Volume Simplex Analysis (MVSA) is anunsupervised, linear unmixing algorithm belonging to the minimum volume class. It is ableto unmix hyperspectral image having the pure pixel supposition despoiled. Trying to fit a

29


simplex of minimum volume in to hyperspectral data set, many times land into local minimaas it is a hard non-convex optimization problem. To avoid clinging towards local minima, aworthy initialization is of vital significance. MVSA could be initialized with an overstatedsimplex version provided by VCA. Though this type of initialization may be far away fromthe global-optimum, it lies in the systematic attraction basin of a worth quality local min-imum. On the other hand, since VCA produces a simplex demarcated by spectral vectorsexisting in the given data set, all the spectral vectors could be discarded that are lying in-side the simplex, accelerating the algorithm. In addition, by careful selection of the inflatingfactor, the majority of constraints associated with the abundance source fractions becomeinactive. This also contributes towards the acceleration of the algorithm.

Minimum volume simplex algorithms are quite vulnerable towards outliers. Hence to obtainrobustness towards outliers and noise, a final step is executed by replacing the abundancefraction positivity hard constraint with a pivot (hinge) type soft constraint. The step is em-ployed after obtaining the minimum volume simplex and it is found to preserve the goodquality of local minima.

2.4.2 Statistical based approaches

In some cases, the spectral data is highly mixed, and the geometrical based methods may notbe applied because they lack enough spectral vectors in the simplex facet. In these cases,the statistical method is an alternative way for the unmixing problem, however, usually withhigher computational complexity. Following are some representative approaches.

Independent Component Analysis (ICA)

Originally ICA [40],[39] was proposed to solve the "Cocktail Problem" but now it is used inhyperspectral unmixing [92],[48]. It attempts to separate different sound signals by search-ing for statistically independent directions within the data, given that each sound is generatedindependently of the other sound in the room. In unmixing problem if sounds are indepen-dent, ICA could provide the correct result. However, due to the ASC constraint, which isstatistical dependence , ICA could not be applied to hyperspectral unmixing directly. There-fore ,dependence compromised ICA methods [69] have been proposed to solve the problemby multiplying the spectral vectors with an unmixing matrix which minimizes the mutual

30


information among channels. In this way some endmembers could be approximately esti-mated. Recently, abundance constraints are also considered in the ICA model to enhance theperformance [93].

Nonnegative Matrix Factorization

For a non-negative matrix Y ∈ RL×N and a positive integer r < min(m,n) the non-negativematrix factorization is generally used to obtain two non-negative matrices M ∈ RL×p andX ∈ Rp×N such that

Y ≈MX

or equivalently, the column {yj}Nj=1 are expressed as

yj ≈Mxj

where yj ∈ RL and xj ∈ Rp [51].

Here the parameter p is referred to the desired rank of matrix M . It is presumed to be knownin advance or it could be determined based on the given data Y . Presumably, the column ofM characterizes the latent variables, i.e it refers to the physical meaningful non-negative partof underlying data. Due to this characteristic, NMF finds a vast range of utility in dimen-sionality suppression, data-analytics, target recolonization, feature abstraction and so on. Acomparative analysis of various linear mixing models has been discussed in the section forhyperspectral image formation with equations. It indicates the potential of NMF to disinte-grate the mixed pixels. Formulation of an optimization problem is an approach for solvingthe NMF problem, minimizing the Euclidean distance between Y and MX

minimize f (M,X) = 12‖Y −MX‖2F

s.t.M ≥ 0, X ≥ 0

where the symbol ≥ denotes component wise inequality, i.e., M ≥ 0 means mij ≥ 0 fori = 1, 2, ...., L and j = 1, 2, ...., p The operator ‖‖F represents the Frobenius norm given by

‖Y −MX‖2F =L∑i=1

p∑j=1

(Yij − (MX)ij)2

31


For the above cost function, many learning strategies have been proposed till date. NMFalgorithm are broadly classified in three classes : Multiplicative Update (MU) algorithms,Gradient Descent(GD) algorithms, and Alternating Least Squares (ALS) algorithms [51].The most popular method to solve the optimization problem of (5) is the multiplicative rule[52]. The algorithm starts from two Nonnegative matrices and multiplies the elements ofM and X by some positive factors within each iteration. It has been proved that under themultiplicative rule, the distance objective function is monotonically non increasing. Thebasic multiplicative update rule for algorithm is shown below.

Algorithm 5 Basic Multiplicative Update algorithm for NMF [52]1: M = rand(L, p) % Initialize M as dense random matrix.2: X = rand(p,N) % Initialize X as dense random matrix.3: for i = 1 : maxiter

4: (MU): M← M. ∗ (YXT)./(MXXT + 10−9)

5: (MU): X← X. ∗ (MTY )./(MTMX + 10−9)

6: end

The gradient descent algorithm was first proposed by Chue et al in 2004 [51]. It is the fact thatthe above Multiplicative update algorithm can be considered a gradient descent method. Theupdate rule for basic gradient descent algorithm for NMF is shown below. In this algorithmεM and εX denotes the step size parameters.

Algorithm 6 Basic Gradient Descent algorithm for NMF (Chue et al, 2004)1: M = rand(L, p) % Initialize M as dense random matrix.2: X = rand(p,N) % Initialize X as dense random matrix.3: for i = 1 : maxiter

4: M = M − εM ∂f∂M

5: X = X − εX ∂f∂X

6: end

ALS class is known as the last class of NMF algorithms. It comprises of least square stepsfollowed by another least square steps in an alternating pattern, thus signifying the nameALS. Paatero and Tapper in 1994 are known to use ALS algorithms for the first time in 1994[72]. It exploits the fact that, when the complete optimization problem given in Eq. (1) isnon convex in M and X simultaneously, it is convex either in M or in X .

32


Algorithm 7 Basic ALS algorithm for NMF (Paatero and Tapper, 1994)1: M = rand(L, p) % Initialize M as dense random matrix.2: for i = 1 : maxiter

3: (LS) : Solve for M in matrix equation MTMX = MTY .4: (Non Negative) : Set all negative elements in M to 0.5: (LS) : Solve for X in matrix equation XXTMT = XY T

6: (Nonnegative) : Set all negative elements in X to 0.7: end

In recent years, Nonnegative Matrix Factorization (NMF) has attracted attention of researcheras one of the prominent techniques of hyperspectral unmixing. NMF treats hyperspectralunmixing as Blind source separation and estimates endmembers and their abundance simul-taneously. The non negative matrices M and X estimated by NMF algorithms approximatethe endmembers and their abundance maps respectively. However, the standard NMF algo-rithms do not impose any constraint on these bases except for non-negativity, which is notsufficient enough to lead to a well-defined problem. Several algorithms have been proposedwith some additional constraints along with standard NMF algorithm. Minimum VolumeConstraint (MVC) is incorporated with basic NMF algorithm for estimating endmembersfrom highly mixed hyperspectral data without assumption of pure pixel [96],[67]. NMFwith prior knowledge of some spectral signatures is proposed in [87]. NMF with piecewisesmoothness and sparseness constraints is proposed in [45].

2.4.3 Sparse regression approach

The classical linear spectral problem has been revisited and reformulated in this section. It isa semi-supervised technique using Sparse Regression (SR). The SR optimization problemshave been reviewed with relevant to our unmixing problem.

In geometrical and Statistical approach, the tradition of hyperspectral unmixing is to esti-mate endmembers spectral signatures from the image and then approximates the fractionalabundance map using Least Square approach. Several Statistical based algorithms estimateendmembers signatures and their abundance maps simultaneously. Geometrical and Statisti-cal approaches are considered Blind Source Separation technique because they assume onlyhyperspectral data cube. In Sparse regression approach, the spectral signature of mixed pixel

33


is assumed to be linear mixing of several spectral signatures available in standard spectrallibrary in advance.

Assume linear mixing model for hyperspectral image formation. The mixed pixel in hyper-spectral image can be represented as

yj = Axj (2.6)

where yj ∈ RL represent the spectral signature of jth pixel, A ∈ RL×p represents theendmember matrix, xj ∈ Rp represents fractional abundance vector correspond to jth pixel.When A contains more endmembers compared to spectral channels (over-complete) p > L oreven p » L, x could be expected as sparse. Since hyperspectral sensors comprise of hundredsof channels with spatial resolution up to 30 m, the supposition of x being sparse could berational. It gives rise to a new horizon in the unmixing problem where, in place of extractingendmembers from an image or manually looking for endmembers, researchers may use vastnumber of predefined endmembers.

In sparse unmixing, the problem of linear spectral unmixing is handled in a semi-supervisedmanner using spectral libraries [44]. Spectral library contains the spectral signatures mea-sured by advanced field Spectro-radiometers of various materials on earth surface. The mainassumption is carried with the mixed pixels that could be expressed as a linear combinationof a number of pure spectral signatures that are well known in advance and also available inlibrary. One such is available from U.S. Geological Survey (USGS) [26], containing around1300 mineral signatures, or the NASA Jet Propulsion Laboratory’s Advanced Space borneThermal Emission and Reflection Radiometer (ASTER) spectral library[4], with assemblingof over 2400 spectra of natural and artificial materials. The abundance estimation processneither depends on the obtainability of pure spectral signatures in the input data, nor on thecapability of a certain endmember extraction algorithm for identifying such pure signatures;when the unmixing based problem is approached with the spectral libraries. In contrast, theprocess is decreased to find the library having an optimal subset of signatures, that couldmodel each mixed pixel of the scene in a best way.

With reference to Figure 2.12 , let xj ∈ Rp denote the fractional abundance vector of pixelyj with regards to the library A. Now we can say that vector xj is now k-sparse because only

34


Figure 2.12: Illustration of a mixing model with sparse prior. A hyperspectralpixel can be described as a mixture of only few endmembers from a spectral

dictionary weighted by a sparse abundance vector [6].

k elements are non zero. With this definition SR problem can be written as

minx‖x‖0 subject to ‖Y − Ax‖2 ≤ δ, x ≥ 0, 1Tx = 1 (2.7)

where ‖x‖0 denotes the number of non-zero components of x. and denotes the number ofnon-zero components of x and δ ≥ 0 is the error tolerance due to noise and modeling errors.

A resolution of problem equation, belongs to the set of sparsest signals belonging to the(m− 1) probability simplex which satisfies the error tolerance inequality ‖y − Ax‖2 ≤ δ.

The problem shown in the equation, is NP-hard (non-deterministic Polynomial Hard) [71],so leaves less hope for solving it directly. In such cases, Greedy algorithms, such as theMatching Pursuits(MP) [62], Orthogonal Matching Pursuit (OMP)[90], and Basis Pursuit(BP) [25] are the two alternative methods used for computing the sparsest solution. The

35


basis pursuit is used to replace the l0 norm in equation with the l1 norm.

minx‖x‖1 subject to ‖y − Ax‖2 ≤ δ, x ≥ 0, 1Tx = 1 (2.8)

when extended from one pixel to complete image

minx‖X‖1 subject to ‖Y − AX‖2 ≤ δ,X ≥ 0, 1TX = 1 (2.9)

The constrained optimization problems can be converted into unconstrained versions by min-imizing the respective Lagrangian. Now the problem is equivalent to:

minx

1

2‖AX − Y ‖2 + λ ‖X‖1 , X ≥ 0, 1TX = 1 (2.10)

Different from equation (2.9 ), equation (2.10) relates to a convex problem that could besolved by well-known algorithm called sparse unmixing via variable splitting and augmentedlagrangian (SUnSAL)[7]. It employs the alternating directional approach of multiplier (ADMM)[13] algorithm for solving the given problem. In recent years, a number of constraints havealso been added to sparse regression method to cope up with different circumstances. Re-searcher have included the spatial information on sparse unmixing formulation by means ofTotal Variation regularize with an assumption that two neighboring pixels have alike frac-tional abundances relating the same endmember[43]. Based on the fact that out of manyspectral signatures present in library only few appears in the scene, a new methodology withcollaborative approach is proposed in [41]. Robust Collaborative sparse regression based onrobust linear mixing model(rLMM) is proposed in [54]. Sparse regression approaches havealso used spectral constraints with the prior knowledge of the endmembers [88]. Moreover, aself dictionary based sparse regression method was also proposed which estimates endmem-bers and abundances using hyperspectral images with pure pixels without over completeddictionary [31]. Hyperspectral unmixing technique based on sparse filtering approach ispresented in [1].

36

CHAPTER 3

Performance analysis of VCA Algorithm andAnomaly Detection using VCA algorithm

3.1 Introduction

Vertex Component Analysis (VCA) is the fully automatic unsupervised method to extractendmember spectral signatures from the datacube [70]. Depending on the relationship be-tween linear mixing model and convex geometry the algorithm works on the two facts : (1)Vertices of the simplex are endmembers and (2) the affine transformation of a simplex is alsoa simplex. VCA is iterative procedure to find the endmembers present in the scene. VCAalgorithm iteratively projects the spectral vectors onto a direction orthogonal to the subspacespanned by the endmembers already determined. The extreme of the projection is consideredas the new endmember. The procedure is repeated until all constituents are estimated. VCAassumes the presence of at least one pure pixel per endmember present in the scene. VCA ismost powerful algorithm amongst the all endmember extraction algorithms based on convexgeometry. Computation complexity is also near about half than best convexity based method.Computational complexity of VCA can be reduced by reducing the number of band with lit-tle loss of performance. Based on convex geometry and linear mixing model, the observedmixed pixel vectors are in simplex and vertices of the simplex are endmembers. Fractionalabundance map is determined according to the position of mixed pixel in the simplex.

3.2 VCA Algorithm

Mathematically the linear mixing model for single pixel of hyperspectral datacube, as dis-cussed in section, is given by

y = s+ n

37

CHAPTER 3. Performance analysis of VCA Algorithm and Anomaly Detection usingVCA algorithm

y = Mx+ n

where M is the endmembers matrix , i.e. M ≡ [m1,m2, ....,mp] (spectral signature ofjth endmember is denoted with mj , number of endmembers present in the pixel are p ),x ≡ γα (variation in illumination due to surface topology is modeled with scaling fac-tor γ), α is the abundance vector represents the fractional proportion of endmembers, i.e.α = [α1, α2, ...., αp]

T (αj represents the proportion of jth material in the observed pixel), nrepresents the additive noise.

The value of fractional abundance is always non-negative, lie in the range of 0 to 1 and sumof its values for single pixel is always one. These are known as abundance non-negativityconstraint (ANC) abundance sum-to-one constraint (ASC) , α ∈ ∆p. ASC and ANC im-poses each pixel as vector in L-dimensional Euclidean space, where axis of the spacesare bands of pixel vector). The set of abundance vector is simplex and it implies the setSs =

{s ∈ RL : s = Mα,α ∈ ∆p

}to be a simplex. If we assume n = 0 then the observed

vector will be from the convex cone Cp ={y ∈ RL : y = Mγα, α ∈ ∆p, γ > 0

}which is a

convex cone, owing to scale factor γ . To illustrate the concept of simplex and convex conesynthetic datacube is generated as per linear mixing model assuming n = 0 .

Synthetic data cube of size 5 × 5 is generated with a mixture of three endmembers andfractional abundance map as shown in Figure 3.1. The spectral signatures of 3 minerals(Almadine, Brucite and Chlorite) as shown in Figure 3.2 were selected from USGS spectrallibrary. Figure 3.3 shows the projection of pixel vector in 2D-subspace. The shape of simplexis triangle and three vertices of simplex are the endmember spectral signature. Small dots aremixed spectra belonging to the simplex Sx(γ = 1) and to the cone Cp(γ > 1), respectively.

Figure 3.1: Abundance maps of Almandine, Brucite, Chlorite for endmembersfor synthetic hyperspectral datacube

38

3.3. Description of working VCA Algorithm with synthetically generated hyperspectraldatacube

Figure 3.2: Reflectance spectra of Almandine, Brucite and Chlorite

The further steps of VCA algorithm can be better visualized graphically if we take datacubecontaining only two endmembers. The further process is illustrated graphically with help ofFigure 3.4 for datacube containing two endmembers. All points of simplex Sx are projectedonto the properly chosen hyperplane, yTu = 1. The group of projected points on hyperplaneis also simplex and denoted with Sp, Sp =

{r ∈ RL : r = y

/yTu,y ∈ Cp

}. Here the choice

of u , discussed later in pseudo code, ensure that there are no observed vectors orthogonalto it. Once simplex of projected data point, Sp, is identified the VCA algorithm iterativelyprojects data onto a direction orthogonal to the subspace spanned by the endmembers alreadydetermined. The new endmember signature corresponds to the extreme of the projection.Figure 3.4 illustrates the two iterations of the VCA algorithm applied to the simplex Sp

defined by the mixture of two endmembers. In the first iteration, data is projected onto thefirst direction f1. The extreme of the projection corresponds to endmember ma. In the nextiteration, endmember mb is found by projecting data onto direction f2, which is orthogonalto ma. The algorithm iterates until the number of endmembers is exhausted.

3.3 Description of working VCA Algorithm with synthetically gener-ated hyperspectral datacube

Algorithm illustrates the pseudo-code of the VCA algorithm. Lets first understand the mean-ing of the symbols used in pseudo-code.

39


Figure 3.3: Scatter Plot ( band = 150 and band = 50) all pixels of p − 1dimensional projection o synthetic Hyperspectral image of size 5 × 5, p =

3,γ = 1, SNR = 60dB

[M]:,j

shows the the jth column of the matrix M[M]:,i:k

shows the ith to kth columns of the matrix M

p indicates the number of endmembers

N is the number of pixels in the image.

L is the number of spectral bands of the image.

Y original hyperspectral image

VCA is the initialized by providing original hyperspectral image Y as input. The numberof endmember present in the image is estimated by Hyperspectral Subspace Identificationusing Minimum Error (HySime) algorithm, discussed in details with simulation results in

40


Figure 3.4: Illustration of the VCA algorithm [70]

chapter 4. If number of endmembers presents in an image is known then it can be directinput to the algorithm. Decision regarding the subspace dimension is taken by comparingSNR with its threshold value. If SNR of input datacube is greater than calculated thresholdvalue then data is projected on P dimensional subspace using SVD otherwise onto p − 1

dimensional subspace using PCA. Matrix X is hyperspectral datacube after dimensionalityreduction. Each column of X represents the spectral signature of observed image at only pbands. Here X is nonnegative matrix because it contains reflactance so the mean vector u ofX is also nonnegative. In next step, an empty auxiliary matrix, A, is initialized to store theprojection of estimated spectral signatures. It is our assumption that there is an existence ofone pure pixel per endmember in the input image Y. In each iteration of for loop, a vector forthonormal to the space spanned by the columns of A is generated and R is projected on f.Extreme projection is stored as endmember signature in the matrix A. In next iteration of forloop , a vector f is is now orthonormal to the subspace spanned by updated column of matrixA. Finally, spectral signatures of endmembers, M , are estimated as per steps 28 and 30.

41


Algorithm 8 Original VCA algorithm(Nascimento and Biscous)1: INPUT :Y = [y1, y2, y3, ......, yN ] ,Y ∈ RL×N , where N shows number of pixels and L

shows number of bands in image.2: p = HySime(Y ); The number of endmembers (p) is estimated with HySime algorithm.3: Set threshold SNR : SNRTH = 15 + 10 log10(p)

Dimensionality Reduction (DR) is performed using SVD or PCA to transform L dimen-sional pixel vectors to p or p − 1 dimensional pixel vectors depending on the estimatedSNR.

4: If SNR > SNRTH then5: d := p;

6: Ud = Y Y T/N ;ProjectionmatrixUd is obtained by SVD

7: X := UTd Y ;

8: u := mean (X) ; u is 1× d vector.9: [R]:,j := [X]:,j

/([X]T:,j u

); it indicates projective projection of the data on d- dimension

vector10: else11: d := p− 1;

12: Ud = ([Y ]− y) ([Y ]− y)T/N ; Projection Matrix Ud is obtained by PCA and the sam-

ple mean of [Y ]:,i is denoted with y, where i = 1, 2, ..., N .

13: [X]:,j := UTd

([Y ]:,j − y

);

14: k := arg maxj=1,2,...,N

∥∥∥[X]:,j

∥∥∥ ;

15: κ := [κ|κ|...|κ] ; κ is 1×N vector.

16: R :=

[X

κ

]; it shows the projective projection of data using κ vector

17: end if;18: Generate an auxiliary matrix A = [eu|0|...|0], where eu = [0, ...., 0, 1]T is p-dimensional

vector. A ∈ Rp×p stores the estimated endmembers signatures. Assume that at least onepure pixel is present in input data Y .

Various experiments have been performed to estimates the abundance map and spectral sig-natures. For qualitative performance analysis of VCA algorithm, we use following measures;Signature Angle Error (SAE), Spectral Information Divergence(SID) and Abundance Frac-tion Angle Error (AFAE). Columns of Measurement matrix M ≡ [m1,m2, ....,mp] are spec-tral signatures. The abundance matrix is given as S ≡ [S1, S2, ...., SN ], where each column

42


Algorithm 9 Original VCA algorithm(Nascimento and Biscous)19: For i := 1 : p do20: w := randn(0, Ip); w is a zero-mean random Gaussian vector of covariance Ip21: f := (Ip−AA#)w

‖(Ip−AA#)w‖ ; f is a vector orthonormal to the subspace spanned by [A]:,1:i.

22: v := fTR v is a vector which belongs to only pure pixels.23: k := arg maxj=1,.....,N

∣∣∣[v]:,j

∣∣∣ ; finds projection extreme24: [A]:,i := [R]:,k ;25: [indice]i := k; stores the pixel index.26: end for.

M is a L × p estimated mixing matrix, column contains the estimated endmember sig-nature.

27: If SNR > SNRTH then28: M := Ud [X]:,indice ;29: else30: M := Ud [X]:,indice + r;31: end if.

of matrix S indicates the abundance vector of corresponding pixel.

The estimated endmembers are the columns of M ≡ [m1, m2, ..., mp] and the estimatedabundance fractions are given by S = M# [r1, r2, ..., rN ].

The angle between ith endmember, mi and its estimates, mi is given as

εmi≡ arccos

((mi)

T mi

‖mi‖ ‖mi‖

)

The angle between abundance vector of ith material, [S]i,: , and its estimates ,[S]i,:, is givenas

εSi≡ arccos

([S]i,:)T [S]i,:

‖[S]i,:‖∥∥∥[S]i,:

∥∥∥

The vector containing the angle between original spectral signature and its estimate is de-noted as εm and given by εm ≡

[εm1 , εm2 , ..., εmp

]T . Similarly the vector containing theangle between original abundance map and its estimate is denoted by εS and given as εS ≡[εS1 , εS2 , ..., εSp

]T .

43


The root means square value for SAE and AEFE is calculated as follow.

rmsSAE =

(1

pE[‖εm‖2

])1/2

rmsAFAE =

(1

pE[‖εS‖2

])1/2

The symmetric Kullback distance [49], a relative entropy-based distance, is another errormeasure used to compare the similarity between signatures, also known as spectral informa-tion divergence (SID) [18]. SID is given

SID(mi,mi) = DKL(mi|mi) +DKL(mi|mi)

where DKL(mi|mi) is the relative entropy of mi with respect to mi given by

DKL(mi|mi) ≡L∑j=1

pj log

(pjqj

)

and pj = mij

/∑Lk=1mik and qj = mij

/∑Lk=1 mik.

εSID ≡[SIDm1,m1 , SIDm2,m2 , ..., SIDmp,mp

]TThe rms value of SID is calculated as

rmsSID =

(1

pE[‖εSID‖2

])1/2

3.3.1 Simulation Results

All the experiments are performed using synthetically generated hyperspectral images. Forconstructing the data, we used the subset of USGS library containing 21 spectral signatures,each of 224 contiguous band in the range of 400−2500nm. We have selected three materialsnamely, Almandine WS478, Brucite HS247.3B and Corundum HS283.3B from the subset of

44


Figure 3.5: Scatter Plot ( band = 80 and band = 122) all pixels ofp − 1dimensional projection of synthetic Hyperspectral image of size 10× 10, p =

3,γ = 1, SNR = 40dB

library. The signatures of these three materials form an endmember matrix M of size 224×3

where the three column vectors correspond to the spectral signatures. In this experiment wehave taken image of size 10 × 10 pixels for each band. Fractional abundance vector for apixel will be of size 3 × 1. The observed pixel vector of size 224 × 1 is constructed usingmeasurement matrix M of size 224 × 3 and fractional abundance vector of size 3 × 1 fora given pixel as per linear mixing model. Total number of 10 × 10 = 100 pixel vectorsare generated to form synthetic hyperspectral image. The original abundance map used togenerate image is known as ground truth for the image.

The ground truth and estimated fractional abundance map is shown in Figure 3.7. Groundtruth and estimated spectral signatures using VCA algorithm is shown in Figure 3.6. Figure3.5 show the projection of all the data points on 2 dimensional subspace. The shape formedby all the data points in 2-D subspace is triangle and vertices of the simplex are pure pix-els means endmember spectral signatures present in the image. The synthetic data cube isthen corrupted with different noise level i.e 20dB, 30dB, 40dB, 50dB and 60 dB. The perfor-mance evaluation parameters like rms values of SID, SAE and AEFE are calculated as perexpression discussed previously in this section.

45


(a) True spectral signatures for the three endmembers of synthetic data

(b) Estimated spectral signtures for the three endmembers of synthetic data using VCA algorithmfor SNR = 40 dB

Figure 3.6: True and Estimated Spectral signatures

(a) Ground truth abundance maps for the three endmembers of synthetic data

(b) Estimated Abundance map using VCA algorithm for SNR = 40 dB

Figure 3.7: True and Estimated abundance map

46

3.4. Anomaly detection using VCA algorithm

Table 3.1: rmsSID, rmsSAE and rmsAEFE as a function SNR forN = 100,p = 3, L = 224,γ = 1

SNR (dB) rmsSID rmsSAE rmsAEFE

20 2.05× 10−3 2.3◦ 32.80◦

30 2.26× 10−4 0.733◦ 28.99◦

40 2.56× 10−5 0.257◦ 26.64◦

50 1.97× 10−6 0.0705◦ 27.40◦

60 1.82× 10−7 0.0219◦ 28.00◦

3.4 Anomaly detection using VCA algorithm

Multi-temporal data makes anomaly detection possible for hyperspectral image. In this sec-tion anomaly detection application for hyperspectral image is performed using synthetic andreal hyperspectral image.

3.4.1 Simulation Results with synthetic data.

For our first simulation two synthetic data sets are generated using selected spectral signa-tures from library, containing 21 spectral signatures of 224 bands and is a subset of USGSspectral library. First data set of size 5 × 5 is generated as per LMM using manually gen-erated abundance as shown in Figure 3.8 (b) and spectral signatures taken from library. InFigure. 3.8 (b) and (c), colors are used to differentiate endmembers and their proportion.Figure 3.8 (d) and (e) shows (P-1) dimensional representation of all pixels, where P is thenumber of endmembers present in image. Figure 3.8 (d) shows the volume formed by allpixels of DC#1 is triangle and is simplex in 2−D. Vertices of simplex are the pure materi-als present in the image. Any point inside simplex can be represented as linear combinationof vertices of simplex. Unmixing is performed on first data set using VCA algorithm. InFigure 3.9 simulated results are shown in terms of original and estimated spectral signaturesand abundances. Second data set is generated by adding anomaly as spectral signature ofunknown material in pixel numbers 15, 19, 20, 24 and 25. Pixel number 19 is pure pixel foradded anomaly. Figure 3.8 (e) shows the volume formed by all pixels of DC#2 is tetragonaland is simplex in 3−D.

47


Figure 3.8: (a) Pixels arrangement of 5 × 5 synthetic DCs (b) Original syn-thetic DC#1 without anomaly ( pixel number 16 , 2 and 4 are pure pixelscorresponds to material 1,2 and 3 (c) Synthetic DC#2 with anomaly (d) 2-Dimensional representation of Synthetic DC#1 (e) 3-Dimensional represen-

tation of synthetic DC#2.

48


Figure 3.9: Simulation result of VCA Algorithm for DC#1 of first exper-iment. First and third columns show true endmember signatures and abun-dances respectively. Second and fourth column shows spectral signatures and

abundances for the estimated endmembers.

49


Figure 3.10: Simulation results of VCA algorithm for DC#2 of first experi-ment

50


Again the second data set is unmixed using VCA algorithm. Figure 3.10 shows the originaland estimated spectral signature and abundance map of DC#2. Fourth row of Figure 3.10shows the estimated spectral signature and abundance of added anomaly. After calculat-ing correction of estimated spectral signature with spectral signatures available in standardspectral libraries, like USGS ( United States Geological Survey)[26] and ASTER (AdvancedSpace borne Thermal Emission and Reflection Radiometer)[4], user can determine the typeof added anomaly.

3.4.2 Simulation Results with Real hyperspectral Image

Multi-temporal real hyperspectral images are not easily available with ground truth knownto researcher. So researcher randomly adds anomaly or changes some parts of image forresearch work. Atmospherically corrected and refined cuprite image is easily available toresearchers with ground truth to validate their work. It is most widely used dataset for hy-perspectral unmixing. This image covers the Cuprite in Las Vegas, USA. It is recorded at224 channel with spectral resolution of 10nm from 370nm to 2480nm. As a part of di-mensionality reduction and atmospheric correction channel number 1 − −2, 221 − −224,104−−113 and 148−−167 are removed. A small portion of size 250× 190 pixels and 188

channels is used in our experimental work. There are 14 types of minerals present in thissite and spectral signatures of all available in the USGS spectral library, which are sum-marizes as "Alunite", "Andradite", "Buddingtonite", "Dumortierite", "Kaolinite1","Kaolinite2", "Muscovite", "Montmorillonite", "Nontronite", "Pyrope", "Sphene","Chalcedony". Data size of original cuprite is very large and it covers large geographi-cal area so in this simulation, cropped cuprite image with data size is used as DC#1 asshown in Figure 3.11(a). As per literature the cropped region contains only 6 minerals.

As per literature survey there are six dominant materials presents in this region. Figure 3.12and 3.13 shows the estimated spectral signatures and their abundance maps of six dominantmaterials in a given geographical area of cuprite image. Results are obtained using VCAalgorithm by taking number of endmembers present in image are six. Anomaly is addedin such a way to make it as one of the dominant material in a given geographical area.Simulation results of VCA algorithm for DC#2 of second experiment are shown in Figure3.14 and 3.15. Estimated abundance map of added anomaly is shown in Figure 3.14 andspectral signature in figure 3.15 with red circles.

51


Figure 3.11: (a) Portion of real hyperspectral image (Cuprite data ) of size250 × 191 (b) Anomaly is added to real hyperspectral image in pixel number

30001 to 30100, 30251 to 30350, 30751 to 30850

Figure 3.12: Spectral Signatures of Six dominant materials present in RealHyperspectral DC#1 of second experiment

52


Figure 3.13: Abundance maps of six dominant materials present in Real Hy-perspectral DC#1 of second experiment

Figure 3.14: Spectral Signatures of Seven dominant materials present in RealHyperspectral DC#2 of second experiment

53


Figure 3.15: Abundance maps of seven dominant materials present in Hyper-spectral DC#2 of second experiment

54

3.5. Summary

3.5 Summary

Vertex Component Analysis algorithm is a blind source separation technique based on con-vex geometry. It works on the fact that pure materials are the vertices of the simplex formedby all the pixels of scene. Anomaly can be easily detected from hyperspectral image, if pro-portion of anomaly is in proportion with other dominant materials of geographical area andmulti-temporal images of scene are available. The detected spectral signature of material isused to determine the type of anomaly and fractional abundance is used to locate the positionand proportion of anomaly.

55

CHAPTER 4

Proposed Collaborative Sparse Unmixing usingVariable Spiting and augmented lagrangian with

Total Variation (CSUnSAL-TV)

4.1 Introduction to Sparse Unmixing

In typical linear spectral unmixing problem, we are given only hyperspectral data cube Y,and our objective is to estimate endmember matrix M and fractional abundance map X foreach pixel of the image. This is tackled as Blind Source Separation (BSS) problem in geo-metrical and statistical based approaches. In these approaches, algorithms first extract end-members spectral signatures and then it estimates the fractional abundance map using someleast square approach. Due to large data size, endmember variability, non availability of purepixel image Hyperspectral unmixing is treated as inverse ill posed problem. Recently, dueto increasing availability of spectral libraries, researchers have tried to represent a spectralsignature of mixed pixel as a linear combination of several spectral signatures available inadvance in spectral library. In this work, we have assumed linear mixing model for hyper-spectral image formation.

Now assume that due to poor spatial resolution of HSI, a mixed pixel in image contain threematerials as shown in Figure 4.1 (a). The captured spectral signature of mixed pixel will bethe linear mixing of constituent spectra as shown in Figure 4.1(b).

As discussed in Section 2.5, mixed pixel of hyprepstral image can be expressed mathemati-cally as per LMM as

yj = Mxj (4.1)

56

4.1. Introduction to Sparse Unmixing

Figure 4.1: (a) Mixed pixel (b) Linear Mixing Model for single mixed pixel

where yj ∈ RL represent the spectral signature of jth pixel, M ∈ RL×p represents theendmember matrix, xj ∈ Rp represents fractional abundance vector corresponds to jth pixel.

Specifically, Sparse unmixing is a new strategy to model mixed pixel observations as linearcombinations of spectra from a library collected on the ground by a field Spectro-radiometer,thus avoiding the endmember extraction step. Most algorithm of unmixing discussed in chap-ter II assume the availability pure pixels in the data but the fact is quite different. Due to verylow spatial resolution of data, hyperspectral pixels are alway mixed and the most commonsituation is that the hyperspectral image has no pure pixels at all. Sparse unmixing techniquedoes not assume the presence of pure pixels. Sparse Unmixing algorithm does not extractendmember signature, instead they are selected from spectral library after estimation of frac-tional abundance map. In Sparsity based approaches fractional proportion of all the spectralsignatures present in the library is estimated first then spectral signatures with dominant pro-portion is selected from the library. USGS and ASTER are most widely used spectral libraryin the process of sparse unmixing because of freely availability in public domain. USGS andASTER contain 1300 and 2400 spectral signatures respectively.

Now assume that endmember matrix M is replaced with spectral library A containing m

spectral signature, where value if m is very large compared to L. Here A is over complete,i.e,. it contain more signatures than spectral bands m > L. Now one can consider x as k-sparse because most elements of x are zero and only k elements are nonzero. This propertybrings a new look into the unmixing problem where, instead of extracting endmembers from

57

CHAPTER 4. Proposed Collaborative Sparse Unmixing using Variable Spiting andaugmented lagrangian with Total Variation (CSUnSAL-TV)

an image or looking for endmembers manually, researchers can use hundreds of predefinedendmembers. With reference to A, equation 4.1 can written as

yj = Axj (4.2)

where A ∈ RL×m is spectral library containing m spectral signatures of L bands each.

The value of fractional abundance is always nonnegative, lie in the range of 0 to 1 and sum ofits values for single pixel is always one. These are known as Abundance Non-negativity Con-straint (ANC), Abundance Aum-to-one Constraint (ASC), which are represented in compactform as

ANC : xj ≥ 0 (4.3)

ASC :∑

xj = 1 (4.4)

In general, mathematically LMM can be written in compact matrix form for complete Hy-perspectral image as

Y = AX + n (4.5)

Where, Y ∈ RL×N is a observed image, A ∈ RL×m is spectral library containing m spectralsignatures of length L, X ∈ Rm×N is fractional abundance map corresponds to spectrallibrary A. Each column of Matrix Y represent the spectral signature of corresponding pixelcaptured by hyperspectral camera[9],[85],[42].

In the problem of linear spectral unmixing, given a hyperspectral data cube Y and our objec-tive is to estimate the endmembers signatures and their fractional abundances, denoted by Mand x for each pixel of the image respectively. Semi supervised approach of linear spectralunmixing is also subjected to few potential drawbacks[42].

• Spectral signatures of library are rarely captured same as airborne or satellite bornedata. Advanced atmospheric correction step is performed that converts measured radi-ance into reflectance.

• Second, mutual coherence between two columns of spectral library is very poor. Onlyfew materials are presents, in order of 3 to 5, in a pixel. So highly sparse signal mitigate

58

4.2. Spectral Unmixing using Variable Splitting and Augmented Lagrangian (SUnSAL)

the disadvantage of low mutual coherence.

• Third, standard optimization tools can not be used for sparse solution of under deter-mined system. In simple word number of unknown are more than number of equa-tions so normal optimization algorithms cants be used to solve problem. Thousands ofspectral signatures of image make problem of spectral unmixing even more computa-tionally complex. To cope with this computationally complex task a method based onAugmented Lagrangian and Alternative directional method of multiplier (ADMM) isused.

4.2 Spectral Unmixing using Variable Splitting and Augmented La-grangian (SUnSAL)

In Sparse Unmixing (SU), endmembers are not extracted from hyperspectral data cube tosolve linear spectral unmixing problem but endmembers are selected from spectral librarycontaining large number of spectral samples available in advance. For sparse unmixingsearching operation is conducted in a potentially very large library denoted by A ∈ RL×m,where L is the number of bands and m is the number of endmembers in the library. Num-ber of materials available in the library are much larger than number of bands, i.e.,L < m.With this assumption in mind, let x ∈ Rm denote the fraction abundance vector for givenpixel related to the spectral library A. Let us assume that out of m materials available inA, less than k materials are present in the scene, where k << m. The fractional abundancevector for a single pixel, yij , can be expressed as k-sparse vector because most of componentof vector are zero. The problem of linear spectral unmixing is considered as semi-blind ap-proach under the assumption that the observed spectral signature can be represented as linearcombination of pure spectral signatures available in library, In practice, it is a combinatorialproblem that induces Sparsity effective techniques and linear regression with the regulators,since the number of final members in a mixed pixel is usually very small compared to thedimensionality. Figure 4.2 shows the sparse regression based approach for Hyperspectralunmixing [42]. Sparse regression approach is based on the assumption that the mixed pixelcan be expressed in the form of linear combination of a number of spectral signature knownin advance and available in standard library.

59


Figure 4.2: Sparse Regression based approach for Hyperspectral Unmixing

As discussed in chapter II, we can now write our SR problem for single mixed pixel shownin Figure 4.1 as follows

minx‖x‖1 subject to Ax = y, x ≥ 0, 1Tx = 1 (4.6)

With the use of lagrangian multiplier, λ> 0, constraint optimization problem is converted tounconstrained problem

minx

1

2‖Ax− y‖22 + λ ‖x‖1 , x ≥ 0 (4.7)

If optimization problem shown in equation is extended to full image than equation become

minx

1

2‖AX − Y ‖22 + λ ‖X‖1 , X ≥ 0 (4.8)

ANC constraint is also incorporated using indicator function as the following equivalentform:

minX

1

2‖AX − Y ‖2F + λ ‖X‖1,1 + lR+(X) (4.9)

Where lR+(X) =∑n

i=1 lR+(xi) is the indicator function xi represents the ith column ofX and lR+(xi) is zero if xi belongs to the nonnegative orthant and +∞ otherwise). Withthe use of variable splitting concept discussed in by Jose M. Bioucas-Dias and Mario A. T.

60


Figueiredo the above expression can be written as

minU,V1,V2,V3

12‖V1 − Y ‖2F + λ ‖V2‖1,1 + lR+(V3)

subject to V1 = AU, V2 = U, V3 = U(4.10)

In compact form

minU,V

g(V ) subject to GU +BV = 0 (4.11)

where, V ≡ (V1,V2,V3)

g(V ) ≡ 1

2‖V1 − Y ‖2F + λ ‖V2‖1,1 + lR+(V3)

G =

A

I

I

, B =

−I 0 0

0 −I 0

0 0 −I

Augmented Lagrangian of objective function is given by.

L(U,V,D) = g(U,V)+µ

2‖GU + BV −D‖2F

The expansion of Augmented Lagrangian is

L(U,V1,V2,V3,D1,D2,D3) =12‖V1 − Y ‖2F + λ ‖V2‖1,1

+lR+(V3) + µ2‖AU − V1 −D1‖2F + µ

2‖U − V2 −D2‖2F

+µ2‖U − V3 −D3‖2F

61


Algorithm 10 ADMM Pseudo code for SUnSAL Problem [7]1: Initialization : Set k = 0, Choose2: Repeat :3: Uk+1= (ATA + 2I)−1(AT(V(k)

1 +D(k)1 ) + (V(k)

2 +D(k)2 ) + (V(k)

3 +D(k)3 ))

4: Vk+11 = 1

1+µ[Y+µ(AU(k)−D

(k)1 )]

5: Vk+12 = soft((U(k+1)+D

(k)2 ),λ

µ)

6: Vk+13 = max((U (k) −D(k)), 0)

7: Dk+11 = D

(k)1 − AUK+1 + V k+1

1

8: Dk+12 = D

(k)2 − AUK+1 + V k+1

2

9: Dk+13 = D

(k)3 − AUK+1 + V k+1

3

10: Until Stopping criteria is satisfied.

In mathematics, statistics, and computer science, particularly in machine learning and inverseproblems, regularization is the process of adding information in order to solve an ill-posedproblem or to prevent over fitting.

The ability to obtain useful sparse solutions for an under determined system of equationsdepends on the degree of the coherence between the column of the system matrix and degreeof sparseness of the original signals. The most favourable scenarios corresponds to highalysparse signals and the system matrices with low coherence. Unfortunately in hyper spectralapplications the spectral signatures of the materials are highly correlated. On the other handthe number of materials present in the scene is often small. Therefore to mitigate the un-desirable high coherence of hyper spectral library total variation constraint is added. Thisapproach is known as sparse unmixing via variable splitting augmented Lagrangian and totalvariation (SUnSAL-TV). This method includes spatial information on the sparse unmixingformulation by means of the Total Variation (TV) regularizer. This regularizer accounts forspatial homogeneity: it is very likely that two neighboring pixels have similar fractionalabundances for the same endmember. The TV regularizer acts as a priori information, whichimproves the conditioning of the underlying inverse problem.

The objective function with additional constraint of total variation is given as

minX‖AX − Y ‖2F + λ ‖X‖1,1 + λTV TV (X), X ≥ 0 (4.12)

62


WhereTV (X) =

∑{i,j}∈ε

‖xi − xj‖1 (4.13)

is a vector extension of the non-isotropic TV, which promotes piecewise constant (or smooth)transitions in the fractional abundance of the same endmember among neighboring pixels,and ε denotes the set of horizontal and vertical neighbors in the image [43].

Horizontal difference operator Hh and vertical difference operator Hv are defined as follow[43] :

HhX = [d1, d2, ..., dn] (4.14)

where di = xi − xih, with i and ih denoting a pixel and its horizontal neighbor.

HvX = [v1, v2, ..., vn] (4.15)

vi = xi − xiv, with i and iv denoting a pixel and its vertical neighbor.

With this, we define

HX =

[HhX

HVX

]

Now the objective function can be written as

minx

1

2‖AX − Y ‖2F + λ ‖X‖1,1 + λTV ‖HX‖1,1 + iR+ (X) (4.16)

Using variable splitting the above equation can be rewritten as

minU,V1,V2,V3,V4,V5

12‖V1 − Y ‖2F + λ ‖V2‖1,1 + λTV TV ‖V4‖1,1 + lR+(V5)

subject to V1 = AU, V2 = U, V3 = U, V4 = HU, V5 = U(4.17)

In short objective function can be written as

minU,V


63


where, V ≡ (V1,V2,V3,V4,V5)

g(V ) ≡ 1

2‖V1 − Y ‖2F + λ ‖V2‖2,1 + λTV TV ‖V4‖1,1 + lR+(V5)

Augmented Lagrangian of the objective function is given by equation (4.19).µ > 0 is apositive constant, and D/µ denotes the Lagrange multipliers associated with the constraintGU + BV = 0. Augmented Lagrangian of the objective function is given by.


2‖GU + BV −D‖2F (4.19)


L(U,V1,V2,V3,V4,V5,D1,D2,D3,D4,D5) =12‖V1 − Y ‖2F + λ ‖V2‖1,1

+λTV TV ‖V4‖1,1 + lR+(V5) + µ2‖AU − V1 −D1‖2F

+µ2‖U − V2 −D2‖2F + µ

2‖U − V3 −D3‖2F + µ

2‖HU − V4 −D4‖2F

+µ2‖U − V5 −D5‖2F

(4.20)

The SUnSAL algorithm optimizes the objective function with respect to U (step 3), V (step4 to 8) and then updates the Lagrange multipliers (step 9 to 13).

Algorithm 11 Pseudo-code for the SUnSAL-TV [43]1: Initialization : Set k = 0 , choose u > 0, U (0) > 0,V (0) > 0 and D(0) > 0.2: Repeat :3: Uk+1= (ATA + 2I)−1(AT(V(k)

1 +D(k)1 ) + (V(k)

2 +D(k)2 ) + (V(k)

3 +D(k)3 ) + (V(k)

4 +D(k)4 ) + (V(k)

5 +D(k)5 ))

4: Vk+11 = 1

1+µ[Y+µ(AU(k)−D

(k)1 )]

5: Vk+12 = soft((U(k+1)+D

(k)2 ),λ

µ)

6: V k+13 = (HTH + I)−1(U(k)−D

(k)3 +H(V(k)

4 +D(k)4 ))

7: Vk+14 = soft((D(k)

4 +HV(k)3 ),λTV

µ)

8: Vk+15 = max((U (k) −D(k)

5 ), 0)

9: Dk+11 = D

(k)1 − AUK+1 + V k+1

1

10: Dk+12 = D

(k)2 − UK+1 + V k+1

2

11: Dk+13 = D

(k)3 − UK+1 + V k+1

3

12: Dk+14 = D

(k)4 −HUK+1 + V k+1

4

13: Dk+15 = D

(k)5 − UK+1 + V k+1

5

14: Until stopping criteria is satisfied.

64

4.3. Proposed Collaborative Sparse Unmixing using Variable Splitting and AugmentedLagrangian with Total Variation (CSUnSAL-TV)

4.3 Proposed Collaborative Sparse Unmixing using Variable Splittingand Augmented Lagrangian with Total Variation (CSUnSAL-TV)

Hyperspectral images cover a large geographic area and usually, it is found that only a fewmaterials are found from this area, in order of 5 to 10. As we know the spectral librariescontain signatures of more than thousand materials. As per formulation of sparse regression,the dimension of abundance matrix will be m× N, where m is the number of materialswhose spectral signatures are recorded and available in the library and N is the number ofpixels. It is observed that out of m materials from the library only a few are present in thescene. So only a few rows of the matrix contain non-zero elements as shown in Fig.??. Sowe have added the concepts of line sparsity along with total variation regularization, whichpromotes the existence of materials to its neighboring pixels. The line sparsity is indicatedwith the use of mixed norm l2,1.

Figure 4.3: Concept of line sparsity of fractional abundances for HSI

The modified objective function is given by

minX‖AX − Y ‖2F + λ ‖X‖2,1 + λTV TV (X), X ≥ 0 (4.21)

Where the‖X‖2,1 =

∑m

k=1

∥∥xk∥∥1

(4.22)

andTV (X) =

∑{i,j}∈ε

‖xi − xj‖1 (4.23)

65


xk indicates the kth line of X and positivity of abundance matrix is indicated by X ≥ 0.The second term is a convex term in our objective function and is called l2,1 norm. It forcesthe sparsity along the row of matrix X , means minimum number of nonzero rows. For moredetail regarding TV Regularization please refer [43].The objective function after applyingADMM and with use of variable splitting and augmented lagrangian becomes as

minU,V1,V2,V3,V4,V5

12‖V1 − Y ‖2F + λ ‖V2‖2,1 + λTV TV ‖V4‖1,1 + lR+(V5)

subject to V1 = AU, V2 = U, V3 = U, V4 = HU, V5 = U(4.24)

In short objective function can be written as

minU,V


where, V ≡ (V1,V2,V3,V4,V5)

g(V ) ≡ 1

2‖V1 − Y ‖2F + λ ‖V2‖2,1 + λTV TV ‖V4‖1,1 + lR+(V5)

Augmented Lagrangian of the objective function is given by equation (4.26).µ > 0 is apositive constant, and D/µ denotes the Lagrange multipliers associated with the constraintGU + BV = 0. Augmented Lagrangian of the objective function is given by.


2‖GU + BV −D‖2F (4.26)


L(U,V1,V2,V3,V4,V5,D1,D2,D3,D4,D5) =12‖V1 − Y ‖2F + λ ‖V2‖2,1

+λTV TV ‖V4‖1,1 + lR+(V5) + µ2‖AU − V1 −D1‖2F

+µ2‖U − V2 −D2‖2F + µ

2‖U − V3 −D3‖2F + µ

2‖HU − V4 −D4‖2F

+µ2‖U − V5 −D5‖2F

(4.27)

The SUnSAL algorithm optimizes the objective function with respect to U (step 3), V (step4 to 8) and then updates the Lagrange multipliers (step 9 to 13).

66

4.4. Results and discussion

Algorithm 12 Pseudo-code for the CSUnSAL-TV1: Initialization : Set k = 0 , choose u > 0, U (0) > 0,V (0) > 0 and D(0) > 0.2: Repeat :3: Uk+1= (ATA + 2I)−1(AT(V(k)

1 +D(k)1 ) + (V(k)

2 +D(k)2 ) + (V(k)

3 +D(k)3 ) + (V(k)

4 +D(k)4 ) + (V(k)

5 +D(k)5 ))

4: Vk+11 = 1

1+µ[Y+µ(AU(k)−D

(k)1 )]

5: Vk+12 = vect− soft((U(k+1)+D

(k)2 ),λ

µ)

6: V k+13 = (HTH + I)−1(U(k)−D

(k)3 +H(V(k)

4 +D(k)4 ))

7: Vk+14 = soft((D(k)

4 +HV(k)3 ),λTV

µ)

8: Vk+15 = max((U (k) −D(k)

5 ), 0)

9: Dk+11 = D

(k)1 − AUK+1 + V k+1

1

10: Dk+12 = D

(k)2 − UK+1 + V k+1

2

11: Dk+13 = D

(k)3 − UK+1 + V k+1

3

12: Dk+14 = D

(k)4 −HUK+1 + V k+1

4

13: Dk+15 = D

(k)5 − UK+1 + V k+1

5

14: Until stopping criteria is satisfied.

4.4 Results and discussion

In this section, we show the improvement in the performance of unmixing algorithm byincluding l2,1 norm instead of l1,1 along with total variation regularization. The main idea isto analyze the effect of inclusion of collaborative approach in the results of unmixing. In thissection first we shall analyze our work with the help of synthetically generated data cubesand then with real hyperspectral data cube. We have used Signal to Reconstruction Error(SRE) as the performance evaluation parameter given as

SRE =E[‖X‖22

]E

[∥∥∥X − X∥∥∥22

] (4.28)

and SRE in dBSRE(dB) = 10log10(SRE) (4.29)

SRE gives more information regarding error power compared signal so here SRE is used inplace of classical Root-Mean-Square-Error (RMSE).

67


We have used following two spectral libraries in our experiments with synthetic data cubeand real data set for performance discrimination. Table characterize the library A1 and A2.

• Spectral library A1 ∈ R224×498 is generated by randomly selecting spectral signaturesof 498 materials from USGS library. Spectral signature of each material is a record ofreflectance at 224 different wavelengths in the range of 0.4− 2.5um

• Spectral library A2 ∈ R224×240 Subset of A1, where the angle between any two differ-ent columns is larger than 4.44◦ . We have made this pruning because there are manysignatures in A1 which correspond to very small variations, including scalings, of thesame material.

Table 4.1: Specification of spectral library used for simulation

Spectral Library A

Description USGSNumber of Spectra 498Number of Spectral Band 224Minimum Wavelength in µm 0.4Maximum Wavelength in µm 2.5Spark (A) ( Upper bound) 21Mutual Coherence µ (S) 0.99998

Mutual coherence between columns of the spectral library is very high. Here we have used asubset of USGS library to validates the result of algorithms. Here we have used highly sparsesynthetic images, as per scenario of HSI, that mitigates the problem of mutual coherencebetween the column of a library. We draw attention on the very high values of the coherencefor the spectral library.

4.4.1 Simulation results for synthetic Data set

In this section, to prove improvement in existing algorithm, fractional abundances are gen-erated using publically available synthesis tool package Hyperspectral Data Retrieval andAnalysis (HYDRA) [28] and other synthetic data used in [1]. The ANC and ASC constraintsare taken into consideration while synthetic data cube generation. Three data cubes are gen-erated for performance evaluation of proposed algorithm. The distribution of all endmembersis smooth with a sharp transition.

68


• Synthetic Data cube 1 (DC#1) of size 35× 35 is generated using randomly selecting3 spectral signatures from spectral library A. The fractional abundances generated bysynthesis tool HYDRA satisfies the ASC and ANC constraints. In simulation, the ran-dom spectral signature numbers are fixed to [21, 394, 487]. After DC#1 generation,the scene is corrupted with low-pass filtering i.i.d. Gaussian noise for four levels ofthe SNR, i.e., 30 dB, 40 dB, 50 dB and 60 dB.

• Synthetic Data cube 2 (DC#2) of size 50× 50 is generated using randomly selecting5 spectral signatures from spectral library A and the fractional abundances are takenfrom satisfies the ASC and ANC constraints. In simulation, the random spectral sig-nature numbers are fixed to [21, 176, 394, 419, 487]. The scene is corrupted with noiseas DC#1.

• Synthetic Data cube 3 (DC#3) of size 100×100 is generated using randomly selecting9 spectral signatures from spectral library A and the fractional abundances generatedby same synthesis tool, HYDRA. In simulation, the random spectral signature numbersare fixed to [21, 31, 129, 135, 176, 377, 394, 419, 487]. Again the scene is corruptedwith noise like first two data sets.

A Large number of experiments have been performed on these data cubes to evaluate theperformance of our algorithm. For comparative analysis, we included the simulation resultsof SUnSAL and SUnSAL-TV. All the algorithms were tested for three data sets with differentcombination of constants, like λ, λTV and µ. Performance of proposed unmixing algorithmis compared for different noise level. Specifically, we consider SNR levels of 30dB, 40dB,50dB and 60dB. This experiment is performed for three sets DC#1, DC#2 and DC#3.Figure.4.4 shows the true and estimated abundance map of DC#1 for SNR= 40dB andλ = 0.0005 and λTV = 0.005. Figure.4.5 and figure 4.7 shows the plot of SRE (dB) as afunction of SNR for different combination of λ and λTV for DC#1. From figure 4.5 and4.6, we can conclude that the performance of proposed algorithm is better than SUnSALand SUnSAL-TV for a lower value of SNR upto 40dB. It is worth mentioning that we notonly evaluated the performance of proposed method with different data sets but also witha combination of λ and λTV . Similarly, true and estimated abundance map for DC#2 andDC#3 are shown in figure4.7 and 4.10 respectively. In figure4.10 forDC#3 we have shownabundance map of only three endmembers. The plot of SRE versus SNR for different valueof λ and λTV for data cube DC#2 are shown in Figure4.8 and 4.9. The plot of SRE versusSNR for different value of λ and λTV for data cube DC#3 are shown in Figure 4.11 and

69


(a) True Abundance map for DC#1

(b) Estimated Abundance map using SUnSAL for DC#1

(c) Estimated Abundance map using SUnSAL-TV for DC#1

(d) Estimated Abundance map using CSUnSAL-TV

Figure 4.4: True and Estimated abundance map for DC#1

70


(a) λ = 0.0005 and λTV = 0.001

(b) λ = 0.0005 and λTV = 0.005

Figure 4.5: SRE as a function of SNR for DC#1

71


(a) λ = 0.005 and λTV = 0.001

(b) λ = 0.005 and λTV = 0.005


72





(d) Estimated Abundance map using CSUnSAL-TV for DC#2


73


(a) λ = 0.0005 and λTV = 0.001

(b) λ = 0.0005 and λTV = 0.005


74


(a) λ = 0.005 and λTV = 0.001

(b) λ = 0.005 and λTV = 0.005


75





(d) Estimated Abundance map using CSUnSAL-TV for DC#3


76


(a) λ = 0.0005 and λTV = 0.001

(b) λ = 0.0005 and λTV = 0.005


77


(a) λ = 0.005 and λTV = 0.001

(b) λ = 0.005 and λTV = 0.005


78


Figure 4.13: USGS map showing the location of different minerals in theCuprite mining district in Nevada. The map is available online at:

4.12. In most of cases, the performance of the proposed algorithm is better than SUnSALand SUnSAL-TV. It is interesting to note that the performance of proposed approach is muchbetter for a lower value of λ and λTV for all data cubes.

79


4.4.2 Simulation results with real data

Figure ?? shows the oiginal AVIRIS Cuprite data set used in our simulation. The data setis available in reflectance on-line for research work and most widely used by researchers tovalidate the performance of their work. In our experiment, we have used a very small portionof size 250 × 190 pixels of an original image, which is a subset of on line data. The origi-nal image consists of 224 bands captured between 0.4µm and 2.5µm, with 10nm nominalspectral resolution. To avoid the effect of noise, and due to low and water absorption bands1 − 2, 105 − 115, 150 − 170, and 223 − 224 are removed prior to analysis, leaving a totalof 188 spectral bands. The signatures of all the materials present in cuprite site are well un-derstood and available in USGS spectra library, denoted splib065 and released in September2007. Figure 4.14 shows the qualitative comparision comparison between the classificationmaps produced by the USGS Tetracorder algorithm and the fractional abundances inferredby SUnSAL, SUnSAL-TV and CSUnSAL-TV. It is very difficult to analyse with perfor-mance evaluation parameter because of nonavailability of original data. To show at a largerscale with proper aspect ratio, we have taken only three abundance map out of nine.

4.5 Summary

Recently introduced sparse unmixing is a powerful tool in the field of RS to solve the problemof LSU due to the availability of libraries. Sparse unmixing neither rely on endmember ex-traction algorithms nor require pure pixel in the scene. To improve the performance of sparsunmixing based algorithm we have added the concept of line sparsity along with the spatialcorrelation between the image feature. In this paper we have revisited the basic mathematicalformulation of sparse unmixing and introduced line sparsity l2,1 along with a TV regularizeterm. A new algorithm known as Collaborative Sparse Unmixing via variable Splitting Aug-mented Lagrangian with Total Variation (CSUnSAL-TV) has been proposed. Experimentalresults, discussed in the previous sections, prove that the inclusion of line sparsity with totalvariation regularizes improves the performance of unmixing at the same computational cost.

80

4.5. Summary

(a) Estimated abundance map of real cuprite data set using SUnSAL

(b) Estimated abundance map of real cuprite data set using SUnSAL-TV

(c) Estimated abundance map of real cuprite data set using CSUnSAL-TV

Figure 4.14: Comparison of estimated abundance map of three endmembersfor real cuprite data set

81

CHAPTER 5

Hyperspectral Change Detection usingMulti-temporal Hyperspectral images and Sparse

Unmixing algorithm

5.1 Introduction

Hyperspectral Change Detection (HCD) is the process of obtaining information about thechanges in the abundance map of materials present in scene using multi-temporal Hyper-spectral Images of Instantaneous Field of View (IFoV) under observation. Multi-temporalImages are taken by a Hyperspectral Camera (HSC) at different time and may be under dif-ferent environmental conditions. The information regarding changes in their IFOV couldbe useful for variety of applications like environmental monitoring, city planning, militarysurveillance and disaster management and etc. HSC measures the Electro-Magnetic (EM)radiation at more than 200 narrow and contiguous bands in their IFOV. Three dimensionalimage captured by HSC has two spatial and one spectral dimension. Spectral dimensioncovers the visible, near-infrared, and shortwave infrared spectral bands (0.4µm− 2.5µm) ofelectromagnetic spectrum with very high spectral resolution of 10nm. Linear Spectral Un-mixing (LSU) is widely used technique in the field of Remote Sensing (RS) for the estimationof number of materials, known as endmembers, their fractional proportion, known as abun-dance maps and their spectral signatures. The Sparse Regression (SR) based approach, thebest solution for LSU in semi-supervised manner, assumes the availability of some standardpublically available spectral libraries, which contains spectral signatures of many materialsmeasured on the earth surface using advance Spectro-radiometer. In SR based approach theproblem of LSU is simplified to finding the optimal subset of spectral signatures from thespectral library known in advance. High spectral resolution of HSI and advent of Sparse Un-mixing (HU) algorithms are capable of finding small variations in Hyperspectral images. In

82

5.2. Block diagram of Change Detection mechanism using sparse unmixing algorithm

this paper, Sparse Unmixing using variable Splitting and Augmented Lagrangian (SUnSAL)is used for change detection for multi-temporal Hyperspectral images. Simulation resultsconducted with the help of synthetically generated data cubes, using HYDRA tool, validatesthe performance of SUnSAL for change detection application.

5.2 Block diagram of Change Detection mechanism using sparse un-mixing algorithm

The process of detecting sub pixel level variations from the scene under observation usingmulti-temporal hyperspectral images is known as Hyperspectral Change Detection in thefield of Remote Sensing (RS). The changes in scene may be due to seasonal variation, natu-ral disaster or as time. HCD enables variety of applications like natural disasters monitoringon earth surface, military target movement detection, crop assessment, disaster management,urban planning [1][2]. Reflection, emission and absorption characteristics of all substancesare function of wavelength of Electromagnetic (EM) spectrum. The reflection vs.wavelengthplot is known as spectral signature of substance and it is used to identify the materials. [2]. InHSI, the number of substances present is known as endmembers. The fractional proportionof endmembers in image is known as abundance map. The process of estimating number ofendmembers, their spectral signatures and their abundance map is challenging task. Variousalgorithms have been proposed till date to manage the problem of Linear Spectral Unmix-ing (LSU)[4][5]. Geometrical and Statistical approach based algorithms assumes that onlyhyperspectral data cube is available for LSU[5]. In Sparse Regression (SR) approach basedalgorithm assumes that observed spectral signature of pixel may be represented as linearmixing of few spectral signatures from Spectral Library [6]. Several traditional methods likeChronochrome (CC), Covariance Equalization (CE) were founds in literature do not provideintra pixel information [1][3]. In Hyperspectral images changes occur within pixel due tovery poor spatial resolution so traditional methods do not provide accurate results for changedetection. LSU has emerged as powerful technique in the field of RS to get information up tosub pixel level. Recently several researchers have proposed different techniques for changedetection or anomaly detection from Hyperspectral images using various spectral unmixingalgorithms. Recently the techniques for anomaly detection using Vertex Component Al-gorithm (VCA) for multi-temporal hyperspectral images have been proposed [2]. Varioussparse unmixing based algorithms have been proposed in last few years [9] [10]. In thispaper for change detection, sparse unmixing is performed by SUnSAL [11] on horizontally

83

CHAPTER 5. Hyperspectral Change Detection using Multi-temporal Hyperspectralimages and Sparse Unmixing algorithm

Figure 5.1: Proposed Change Detection Mechanism using CSUnSAL-TV

concatenated hyperspectral images. The performance is verified with the help of syntheti-cally generated multi temporal hyperspectral image using HYDRA tool. LSU is more robustand gives accurate results when there are changes within pixels[12].

In this work, we assume that both images are atmospherically corrected and captured under

84

5.3. Simulation Results and Discussion

the same circumstances. Two images, taken by single camera but at different time, are hori-zontally concatenated to form a single image with doubled pixels. Now the sparse unmixingis performed on a single image to estimate the fractional abundance map as described inpseudo code. Then estimated fraction abundance map is separated into fractional abundancemaps for original image and changed image as shown in Figure 5.1.

5.3 Simulation Results and Discussion

In this section, to demonstrate the application of change detection using multi-temporal hy-perspectral data we have used synthetically generated Hyperspectral images taken at differ-ent time due to non-availability of real data set of multi temporal Hyperspectral images. Theresultant data cubes are then corrupted with Gaussian Noise with different values of SNR.Three spectral signatures, namely Alunite GDS84 NaO3, Diaspore HS416.3B and Dry longgrass AV87-2, are selected from library database to create hyperspectral data cube. Thesubset of the USGS spectral library contains 498 spectral signatures and denoted by A. Se-lected spectral signatures are stored in a measurement matrix M. The size of data cube iskept 50 × 50 and we have assumed that only three materials from the measurement matrixare present in data cube. To show the variation in scene two sets of abundance map are gen-erated. Figure 5.2 (a) and (b) shows the fractional abundance map for DC#1 and DC#2

respectively. Fractional abundance maps are generated using the spherical Gaussian field asavailable in the HYDRA package [14]. As per Linear Mixing Model (LMM) for hyperspec-tral image formation, as discussed in section 2, DC#1 of size 50 × 50 × 224 is generatedusing measurement matrix M and fractional abundance maps as shown in 5.2(a). In order togenerate the data corresponding to different time instant, we use the same set of endmem-bers but with different abundances as shown in figure 5.2(b). Similarly another data cube ofsame size, denoted as DC#2, is generated. DC#1 and DC#2 show the possible temporalvariation in its IFOV of HSC over a given period of time.

In sparse unmixing of hyperspectral data first fractional abundance map of materials areestimated and then spectral signatures are selected from the spectral library. As shown inFigure 5.1, each DC is converted into L × N Matrix, where N is the number of pixelsin DC and each column of matrix represents the spectral signature of corresponding pixel.According to the size of DC, we assume the size of the abundance map as P × N matrix.The jth row of abundance map matrix indicates the proportion of jth material from spectrallibrary in all pixels of the scene. Change detection is performed using synthetic data cubes

85


(a) True abundance map for DC#1

(b) True abundance map for DC#2

Figure 5.2: True Abundance map for DC#1 and DC#2

(DC#1 and DC#2) and spectral A, as per the methodology discussed in section 5.2. Forchange detection using sparse unmixing, two images are horizontally concatenated as shownin Figure 5.1. Now the number of pixels will be doubled, 2N , in input image for sparseunmixing process. Now the size of fractional abundance map X used for sparse unmixingwill be P × 2N . Figure 5.4 (a) and (b) shows the estimated fractional abundance for DC#1

and DC#2. After comparison of estimated abundance map for both DCs user can knowabout the changes occur in fractional proportion of material in the scene.

Fractional abundance map for horizontally concatenated DCs are shown in Figure 5.3. Thevertical axis indicates the spectral signature number with respect to spectra library and hori-zontal axis indicates the pixel number of horizontally concatenated data cubes. In this simu-lation we have taken two data cube of size 50 X 50. So, total number of pixels in horizontally

86


Figure 5.3: Concept of line sparsity of fractional abundances for HSI

concatenated data cubes will be 5000. First 2500 pixels are corresponds to first data cubesand remaining 2500 pixels corresponds to second data cube. From the fig (4), it is seenthat only three materials are present in both DCs. It is also seen that the fractional abun-dance maps are not same for these three materials in two data cubes. Difference in fractionalabundance map of material is considered as change in hyperspectral data cube and it is high-lighted with red color circles. Different sets of data cube are generated using HYDRA toolto validate the performance of SUnSAL algorithm for change detection application. Othersets of data cubes are generated with same size but endmembers are taken as 5 and 9.

87


(a) Estimated abundance map for DC#1

(b) Estimated abundance map for DC#2

Figure 5.4: Estimated Abundance map for DC#1 and DC#2

88

5.4. Summary

5.4 Summary

Due to very poor spatial resolution of hyperspectral image the change in material fractionalproportion will occur at sub pixel level. So change detection algorithms used for normalimages cannot be used to detect change from multi-temporal hyperspectral images. LSU isthe most efficiently performed in sparse regression based technique to get information up tosub pixel level. In SU of Hyperspectral data approach unmixing is simplified to finding theoptimal subset of endmember signature from the library. Complexity of Sparse Unmixingvia variable splitting and augmented lagrangian (SUnSAL) is reduced with help of ADMMand Augmented Lagrangian. SUnSAL gives more accurate results due to the availabilityof spectral libraries. Direct comparison of algorithm from SR based approaches is not pos-sible with other algorithm. Simulation results show that SUnSAL can be used for Changedetection and it enables variety of application.

89

CHAPTER 6

Proposed Automated extraction of dominantendmembers from hyperspectral image using

CSUnSAL-TV and HySime

Linear Spectral Unmixing (LSU) is a widely used technique, in the field of Remote Sensing(RS), for the accurate estimation of number of endmembers, their spectral signatures andfractional abundances. Sparse regression based approach assumes the availability of somestandard publicly available spectral libraries, which contain signatures of many materialsmeasured on the earth surface using advance Spectra radiometer. As per SR approach theproblem of linear spectral unmixing is simplified to finding the optimal subset of spectralsignatures from the library known in advance. USGS and ASTER are standard libraries usedfor sparse unmixing contains thousands of spectral signatures. In SR approach the abundancemap is estimated first for all the spectral signatures present in the library. In actual scenarioonly few materials are present in the scene, so manually it is very difficult to determine thedominant materials present in the scene. In this chapter we have proposed an automatedextraction of dominant endmembers from the hyperspectral image using CSUnSAL-TV andHySime algorithm . CSUnSAL-TV is discussed in details in chapter 5. HySime algorithm[8], discussed in next section, is used to estimate the number of endmembers present in thescene. Our simulation results conducted for both standard publicly available synthetic fractaldataset and real hyperspectral dataset, like cuprite image, shows procedural improvement inspectral unmixing.

6.1 Hyperspectral Subspace identification using single error

Hyperspectral sensors provide more accurate and more detailed data because they acquirespectral vectors with hundreds of bands, yielding large amount of data. For example, AVIRIS

90

6.1. Hyperspectral Subspace identification using single error

collects a 512 (along track) × 614 (across track) × 224 (bands) × 12 (bits) data cube in 43seconds, corresponding to more than 700 Mbits, and Hyperion collects 4 Mbits in 3 seconds,corresponding to 366Kbytes/Km2 [68]. Such huge data volumes put stringent requirementsin what concerns communications, storage, and processing.

Each pixel of an Hyperspectral image can be represented as a vector in the Euclidean spaceRL, where L is the number of bands and each channel is assigned to one axis of space.Under the linear mixing scenario, the spectral vectors are a linear combination of endmembersignatures. The number of endmembers present in a given scene is, very often, much lessthan the number of bands L. Therefore, hyperspectral vectors lie in a low dimensional linearsubspace. The identification of this subspace enables the representation spectral vectors in alow dimensional subspace, thus yielding gains in computational time and complexity and indata storage. In this section we have discussed the Minimum Mean Squared Error (MMSE)based approach to determine the signal subspace in hyperspectral imagery. This techniqueis known as Hyperspectral Subspace Identification using minimum error(HySime).It startsby estimating the signal and the noise correlation matrices using multiple regressions. Theeigenvectors of the signal correlation matrix are then used to build a sequence of nestedsubspaces. Eigen decomposition is used to calculate eigenvectors and unitary matrix. Thesignal subspace is inferred by minimizing the sum of the projection error power with thenoise power, which are, respectively, decreasing and increasing functions of the subspacedimension. Therefore, if the subspace dimension is overestimated the noise power term isdominant, whereas if the subspace dimension is underestimated the projection error powerterm is the dominant. The overall scheme is adaptive in the sense that it does not depend onany tuning parameters.

Let us assume the observed spectral vector for single pixel is

r = x+ n (6.1)

where x ∈ RL is signal vector and n ∈ RL is noise vector. We assume that the signal vectoris in unknown p dimensional subspace, i.e.,

x = Ms (6.2)

where M is full rank L × p matrix. As per LMM, M ≡ [m1,m2, .......,mp] is endmem-ber matrix and s is abundance vector for given pixel. s shows the fractional proportion of

91

CHAPTER 6. Proposed Automated extraction of dominant endmembers fromhyperspectral image using CSUnSAL-TV and HySime

endmembers in given pixel and it follows the ANC and ASC constricts as discussed in sec-tion.Here L is number of bands of spectral signature and p is number of endmembers spectralsignature to constitute the spectral signature of a single pixel as per LMM. So always L > p.

Let us assume that the noise is zero mean Gaussian iid with variance σ2n per band. In this

condition,the maximum likelihood (ML) estimate of the signal subspace is spanned by thep-dominant eigenvectors of the sample correlation matrix of r, i.e.,〈M〉 = 〈[e1, e2, ....., ep]〉,(the notation 〈M〉 represents the subspace spanned by the columns of M ) where ei, fori = 1, ....., p, are the p-dominant eigenvectors of the sample correlation matrix Kr.

The pseudo-code for HySime is shown in the Algorithm 14. Inputs to the algorithms is ob-served hyperspectral image R = [r1, r2, ......., rN ], where R ∈ RL×N and ri ∈ RL is thespectral vector of ith pixel. For initialization of algorithm sample correlation matrix of ob-served image Kr is calculated as shown in step 1. Noise estimation is a classical problemin data analysis and particularly in remote sensing. In Hyperspectral imagery, the simplestnoise estimation is the shift difference method discussed in detail in [68]. In step 2 of thealgorithm the noise vector ni is using matlab fuction taken from [68]. Initially signal corre-lation matrix and noise correlation matrix are estimated as shown in step 2 and step 3. Formore details regarding noise estimation and Noise correlation matrix generation refer [8].The eigen decomposition of Kx ∈ RL×L is given as

Kx = E∑

ET

where E ≡ [e1, ...., ek, ek+1, ....., eL] is known as unitary matrix and it it contains the eigen-vector corresponds to eigenvalue matrix

∑. Eigenvectors are arranged in descending order

in magnitude with respect to eigenvalues. The Eigenvector matrix is decomposed in two or-thogonal sub matrices known as orthogonal subspaces: 〈EK〉 spanned by EK ≡ [e1, ...., ek]

and 〈EK〉⊥ spanned by E⊥K ≡ [ek+1, ...., eL]. 〈EK〉⊥ is the orthogonal subspace of 〈EK〉 andk is order of signal subspace.

As shown in step 4 projection matrix on to 〈EK〉 is calculated as Uk = EkETk . Let xk = Ukr

be the projection of the observed spectral vector r onto the subspace 〈EK〉 .

The mean squared error between x and xk is given as

mse(k) = tr(U⊥k Kr

)+ 2tr

(UkKn

)+ c

92

6.1. Hyperspectral Subspace identification using single error

where c is an irrelevant constant. Step 5 calculates the Signal subspace. Signal subspace isthe minimization of mse(k) given as

k = arg mink

{tr(U⊥k Kr

)+ 2tr

(UkKn

)}

Algorithm 13 Pseudo code of HySime (Nascimento and Biscous, 2008)[8]

1: INPUT: R ≡ [r1, r2, ....., rN ], Kr ≡(RRT

)/N

2: Kn := 1N

∑i (nin

Ti ); Kn is the noise correlation matrix estimates.

3: Kx := 1N

∑i

((ri − ni)

(ri − nTi

)); Kx is the signal correlation matrix estimates.

4: Uk = EkETk ;, where Ek are the eigenvectors of Kx

5: k = arg mink

{tr(U⊥k Kr

)+ 2tr

(UkKn

)}

6.1.1 Simulation Results of HySime

To Validate the performance of HySime, three toy hyperspectral data cubes are generated.The spectral signatures are selected randomly from the USGS digital spectral library A asdiscussed in previous section. The fractional abundances generated by synthesis tool HY-DRA satisfies the ASC and ANC constraints. Each DC is corrupted with white noise withdifferent SNR value, i.e., 30 dB, 40 dB, 50 dB and 60 dB.

• DC#1 of size 35 × 35 is generated using randomly selecting 3 spectral signatures(21, 135, 394 ) from spectral library A.

• DC#2 of size 50 × 50 is generated using randomly selecting 5 spectral signatures(21, 129, 176, 394, 487 ) from spectral library A.

• DC#3 of size 100 × 100 is generated using randomly selecting 9 spectral signatures(21, 31, 129, 135, 176, 377, 394, 419, 487 ) from spectral library A.

Table 6.1 shows the estimated signal subspace dimension for toy hyperspectral data cubesas a function of SNR. Figure 6.1,6.2 and 6.3 shows the mse for HySime as a function of theparameter k, for SNR = 35 dB for three different data cubes.

93


Figure 6.1: MSE as a function of the parameter k, for SNR = 35 dB forDC#1


94

6.2. Proposed mechanism of automatic extraction of dominant endmembers usingSUnSAL and Hysime

Table 6.1: Signal subspace dimension as a function of SNR for three differentData cube generated with white noise

SNR DC#1 DC#2 DC#320 dB 14 5 830 dB 13 5 940 dB 6 5 950 dB 3 5 960 dB 3 5 9


6.2 Proposed mechanism of automatic extraction of dominant endmem-bers using SUnSAL and Hysime

Spectral Unmixing using variable Splitting and Augmented Lagrangian (SUnSAL) is basicand less computationally complex algorithm of sparse regression approach [7]. In sparsitybased unmixing, it is assumed that the spectral signatures of materials present in scene arefrom library itself. In general size of spectral library is huge means they contain spectral sig-natures of thousands of materials present on earth surface. SUnSAL estimates the abundancemap for all materials available in spectral library. So for user it is very difficult to determinethe fractional abundance map of materials which dominates the region of observation. Usersare not aware about the ground truth of image so they have to check abundance map of eachand every material present in the library manually. It is very tedious and time consumingtask.

95


Figure 6.4: Proposed mechanism for automated extraction of dominant end-membrs from hyperspectral image using CSUnSAL-TV and HySime

96


To cope up with this problem we have proposed a new mechanism which extracts dominantendmembers from the scene. In proposed mechanism of automatic extraction of dominantendmembers, we have used the HySime algorithms to determine the number of materialspresent in the scene. HySime is discussed in detail in previous section. Figure 6.4 shows theproposed mechanism for automatic extraction of dominant endmembers from Hyperspectralimage using SUnSAL and Hysime. This mechanism uses the outcome of two algorithms,SUnSAL and hySime, and spectral library. It is performed in two steps. In first step, sum offractional abundances is calculated for each material present in spectral library of all pixelsof image. In second step, according to contribution in image, materials are arranged indescending order and first select k-dominant materials according to the contribution in image,where k is calculated with the powerful HySime algorithm. HySime is well proven andtested for hyperspectral subspace identification as per literatures. Most of the algorithms ofgeometrical and statistical approaches uses the outcome of HySime to unmix hyperspectralimage.

6.3 Simulation Results and Discussion

Performance of proposed mechanism is verified with the help of standard synthetic data setsas well as real hyperspectral image. For the simulation, we have used the outcome of hySimefor our mechanism. As we know USGS spectral library contains 1300 spectral signatures,but here we have used only 498 spectral signatures

6.3.1 Synthetic Data Cube

Synthetic Data Cube (DC) is generated using randomly selecting 9 spectral signatures fromthe subset A of publically available USGS spectral library. The size of image is 100 × 100

and the dimension of A is 224× 498. Out of 498 spectral signatures of A, we have selectedrandomly 9 spectral signatures for synthetic data cube generation. Figure 6.5 shows the trueabundance map for 9 materials of 100 × 100 image. For DC, the random spectral signaturenumbers are fixed to [21, 31, 129, 135, 176, 377, 394, 419, 487]. Synthetic data cube is cor-rupted with noise with different value of SNR. Performance of CSUnSAL-TV is evaluatedwith standard performance evaluation parameter Signal to Reconstruction Error (SRE).

97


Figure 6.5: True Abundance map used to generate synthetic hyperspectraldata cube

98


Figure 6.6: Sum of contribution of all endmembers of spectral library

Abundance map of all materials for given synthetic data cube is estimated using SUnSAL andendmembers are sorted in descending order according to their contribution in synthetic datacube. After this number of endmembers of DC is estimated using HySime. The subspaceestimated by HySime for data cube is 9. Estimated abundance map of 9 dominant materialsis shown in figure 6.7. As we know spectral signal generation is by product of sparsity basedunmixing. Spectral signatures are selected from spectral library of dominant materials.

6.3.2 Real hyperspectral data cube

Well known and publically available AVIRIS Cuprite Data set is used in our simulationresults. The data set of this scene is publically available real hyperspectral data cube forresearch work. Number of researchers have used scene of cuprite for development and vali-date their work. This scene covers Cuprite in Las Vegas, NV, U.S. and contains 224 channels,ranging from 370 nm to 2480 nm. Prior to simulation, some bands, 1–2, 105–115, 150–170,and 223–224, are removed due to low signal to noise ratio and water absorption. The cupritedata set used in our simulation has only 188 bands. A region of 250 x 190 pixels of cupritedata set is considered for simulation, where there are 14 types of minerals as per the ground

99


Figure 6.7: Abundance map of Estimated Dominant endmembers

100

6.4. Summary

Figure 6.8: Estimated abundance map dominant endmembers of real hyper-spectral image

truth. The Cuprite site contains minerals of interest, all included in the USGS library con-sidered in experiments, denoted splib067 and released in September 2007.Same steps areperformed on real data cube as synthetic data cube. The simulation results are shown inFigure 6.8 and 6.9.

6.4 Summary

Sparse Regression based algorithms estimate the fractional abundance map of all the ma-terials whose spectral signatures are within library. Thousands of spectral signatures areavailable in library. In real scenario few materials are preset in the hyperspectral image.The hyperspectral data set require automatic selection of endmembers from potentially verylarge library. Hyperspectral subspace identification by minimum error is efficient algorithmto estimates the number of endmembers of hyperspectral image. By combining the outcomeof CSUnSAL-TV and HySime algorithm , the fractional abundance maps of dominant end-members can be estimated. The proposed mechanism for automated extraction of dominantendmembers from the scene is verified with the help of synthetically generated as well asreal hyperspectral data set.

101


Figure 6.9: Spectral Signatures of dominant endmembers of real hyper-spectral image

102

CHAPTER 7

Conclusion and Future Scope

Vertex Component Analysis (VCA) algorithm is a blind source separation technique basedon convex geometry. It works on the fact that pure materials are the vertices of the simplexgeometry formed by all the pixels of the scene. An anomaly could be easily detected fromthe hyperspectral image, if the quantity of anomaly is in proportion with other dominantmaterials of geographical area and multi-temporal images of the scene are available. Thedetected spectral signature of material is used to determine the type of anomaly and fractionalabundance is used to locate the position and proportion of anomaly.

Recently introduced sparse unmixing is a powerful tool in the field of RS to solve the prob-lem of LSU due to the availability of libraries. Sparse unmixing neither rely on endmemberextraction algorithms nor require pure pixel in the scene. To improve the performance of thesparse unmixing based algorithm, we have exploited the concept of line sparsity along withthe Total Variation regularization. We have revisited the basic mathematical formulation ofsparse unmixing and introduced line sparsity l2,1 along with a TV regularizer term. A new al-gorithm called Collaborative Sparse Unmixing via variable Splitting Augmented Lagrangianand Total Variation (CSUnSAL-TV) has been specifically developed. Experimental resultsprove that the inclusion of line sparsity with total variation improves the performance ofunmixing at the cost of computation complexity.

Sparse Regression based algorithms estimates the fractional abundances for all the materi-als whose spectral signatures are present in the library. We need to extract the fractionalabundance map of only dominant endmembers. HySime is a powerful algorithm to estimatethe number of dominant endmembers present in the scene. We have exploited the outcomeof both algorithms to device a new mechanism for extracting fractional abundance maps ofonly dominant endmembers present in the scene. In our work we have used the linear mixingmodel for hyperspectral image formation. The accuracy of sparsity based algorithm could

103

CHAPTER 7. Conclusion and Future Scope

be increased by incorporating the constraints of nonlinear mixing model as future develop-ment. In addition many directions are there in the field of spectral unmixing like refinementof libraries, consideration of endmember signature variability to enhance the performance ofexisting algorithms.

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List of Publications

[1] N. Patel and H. Soni, "Anomaly detection using VCA algorithm for multi-temporalhyperspectral images," 2017 International Conference on Wireless Communications,Signal Processing and Networking (WiSPNET), Chennai, 2017, pp. 2248-2252. doi:10.1109/WiSPNET.2017.8300159.

[2] N. Patel and H. Soni, " Hyperspectral Change Detection using Multi-temporal Hyper-spectral images and Sparse Unmixing algorithm”, International Journal of Applied En-gineering Research (IJAER), ISSN Online : 0973-9769, ISSN Print : 0973-4562, UGCJournal Number : 64529

[3] N. Patel and H. Soni, "Automated extraction of dominant endmembers from hyperspec-tral Image using SUnSAL and HySime", International Journal of Computer Aided Engi-neering and Technology, Print ISSN: 1757-2657 Online ISSN: 1757-2665, Vol. 12, no.3,DOI: 10.1504/IJCAET.2020.106210

[4] N. Patel and H. Soni, (in press) "Collaborative Sparse Unmixing using Variable Split-ting and Augmented Lagrangian with Total Variation”, International Journal of ComputerApplication in Technology, ISSN Online : 1741-5047, ISSN Print : 0952-8091, UGCJournal Number : 2768, SCOPUS INDEXED

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Hyperspectral Unmixing · 2020. 7. 22. · DECLARATION I declare that the thesis entitled...

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