BAYESIAN HYPERSPECTRAL UNMIXING WITH MULTIVARIATE BETA ... ddranish/publications/   bayesian...

Click here to load reader

download BAYESIAN HYPERSPECTRAL UNMIXING WITH MULTIVARIATE BETA ... ddranish/publications/   bayesian hyperspectral

of 112

  • date post

    27-Jun-2018
  • Category

    Documents

  • view

    212
  • download

    0

Embed Size (px)

Transcript of BAYESIAN HYPERSPECTRAL UNMIXING WITH MULTIVARIATE BETA ... ddranish/publications/   bayesian...

  • BAYESIAN HYPERSPECTRAL UNMIXING WITH MULTIVARIATE BETADISTRIBUTIONS

    By

    DMITRI DRANISHNIKOV

    A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

    OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

    UNIVERSITY OF FLORIDA

    2014

  • c 2014 Dmitri Dranishnikov

    2

  • To my family, for encouraging me to pursue my dreams

    3

  • ACKNOWLEDGMENTS

    I would like to thank my advisor, Dr. Paul Gader, for all of his guidance and support

    throughout my studies and research. I would also like to thank my committee members,

    Dr. Sergei Shabanov, Dr. Anand Rangarajan, Dr. Yuli Rudyak, and Dr. Joseph Wilson,

    for all of their help and valuable suggestions.

    Thank you as well to my many former and current lab-mates and friends, for

    providing valuable criticism of my work. I am particularly grateful to my friends Rin

    Azrak, Marie Mendoza, and Diana Petrukhina for encouraging me to research and

    to write. Words are not sufficient to express my thanks to Rin Azrak in particular, for

    her boundless kindness and inspiration without which this work would not have been

    possible.

    Above all, thank you to my family, my parents Alex and Anna Dranishnikov, and my

    brother Peter Dranishnikov for their love, support, and understanding.

    4

  • TABLE OF CONTENTS

    page

    ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    CHAPTER

    1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    1.1 Linear Mixing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Normal Compositional Model . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Statement of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Overview of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    2.1 Geometric Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.1 Pure Pixel Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 Minimum Volume Based Methods . . . . . . . . . . . . . . . . . . . 20

    2.2 Statistical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 Two General Approaches . . . . . . . . . . . . . . . . . . . . . . . 232.2.2 Bayesian Source Separation . . . . . . . . . . . . . . . . . . . . . . 26

    2.2.2.1 Dependent component analysis . . . . . . . . . . . . . . 262.2.2.2 Bayesian positive source separation . . . . . . . . . . . . 272.2.2.3 BSS : methods . . . . . . . . . . . . . . . . . . . . . . . . 28

    2.2.3 Normal Compositional Model . . . . . . . . . . . . . . . . . . . . . 322.2.3.1 Maximum likelihood for NCM-based models . . . . . . . . 332.2.3.2 Bayesian NCM-based models . . . . . . . . . . . . . . . 342.2.3.3 Summary of NCM-based models . . . . . . . . . . . . . . 39

    2.3 Evaluation Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.3.1 Synthetic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.3.2 Remotely Sensed Images . . . . . . . . . . . . . . . . . . . . . . . 42

    2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    3 TECHNICAL APPROACH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    3.1 Beta Compositional Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1.2 Choice of Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 48

    3.2 Review of Markov Chain Monte Carlo Methods . . . . . . . . . . . . . . . 483.2.1 Metropolis Hastings . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.2 Gibbs Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    5

  • 3.2.3 Metropolis within Gibbs . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Review of Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

    3.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.2 Sklars Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3.3 Gaussian Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3.4 Archimedian Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . 55

    3.4 BBCM : A Bayesian Unmixing of the Beta Compositional Model . . . . . . 563.4.1 Sum of Betas Approximation . . . . . . . . . . . . . . . . . . . . . 563.4.2 Bayesian Proportion Estimation . . . . . . . . . . . . . . . . . . . . 583.4.3 Bayesian Endmember Distribution Estimation . . . . . . . . . . . . 593.4.4 BBCM : A Gibbs Sampler for Full Bayesian Unmixing of the BCM . 63

    3.5 BCBCM : Unmixing the Copula-based Beta Compositional Model . . . . . 643.5.1 Likelihood Approximation . . . . . . . . . . . . . . . . . . . . . . . 663.5.2 Covariance and Copula . . . . . . . . . . . . . . . . . . . . . . . . 673.5.3 Copula Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.5.4 BCBCM : Metropolis Hastings . . . . . . . . . . . . . . . . . . . . . 71

    3.6 A New Theorem on Copulas and Covariance . . . . . . . . . . . . . . . . 72

    4 RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

    4.1 Synthetically Generated Data . . . . . . . . . . . . . . . . . . . . . . . . . 804.1.1 Unmixing Proportions . . . . . . . . . . . . . . . . . . . . . . . . . 814.1.2 Endmember Distribution Estimation . . . . . . . . . . . . . . . . . . 824.1.3 Full Unmixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    4.2 Experiments with the Gulfport Dataset . . . . . . . . . . . . . . . . . . . . 834.2.1 Comparison with NCM . . . . . . . . . . . . . . . . . . . . . . . . . 85

    4.3 BCBCM Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3.1 Covariance Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 864.3.2 Synthetic Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.3.3 Mixture of True Distributions . . . . . . . . . . . . . . . . . . . . . . 884.3.4 Comparison with NCM, LMM, and BCM . . . . . . . . . . . . . . . 89

    5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

    REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

    BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

    6

  • LIST OF TABLES

    Table page

    4-1 BBCM Synthetic Data : Proportion Estimation . . . . . . . . . . . . . . . . . . . 94

    4-2 BBCM Synthetic Data : ED Estimation . . . . . . . . . . . . . . . . . . . . . . . 94

    4-3 BBCM Synthetic Data : Full Estimation . . . . . . . . . . . . . . . . . . . . . . . 95

    4-4 BCM, Mean Distance to Truth and Labelings . . . . . . . . . . . . . . . . . . . 95

    4-5 CBCM Synthetic Data : Full Estimation . . . . . . . . . . . . . . . . . . . . . . 95

    4-6 CBCM True Data : Full Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 95

    7

  • LIST OF FIGURES

    Figure page

    3-1 Histogram of labeled HSI Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

    3-2 The Independence Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

    3-3 The Gaussian Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

    3-4 PDF of the Gaussian Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

    4-1 Distribution of Asphalt - Gulfport Data . . . . . . . . . . . . . . . . . . . . . . . 91

    4-2 Distribution of Dirt - Gulfport Data . . . . . . . . . . . . . . . . . . . . . . . . . 92

    4-3 Distribution of Tree - Gulfport Data . . . . . . . . . . . . . . . . . . . . . . . . . 92

    4-4 Spectra from Synthetic Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

    4-5 Spectra from Copula-Based Synthetic Dataset . . . . . . . . . . . . . . . . . . 93

    4-6 Mapping from Covariance to Copula . . . . . . . . . . . . . . . . . . . . . . . . 94

    4-7 KL Divergence of Gaussian and Beta Distributions from Hand-labeled distributionsin Gulfport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

    4-8 Estimated and True Mean values with Synthetic Data . . . . . . . . . . . . . . 97

    4-9 Estimated and True Sample Size values with Synthetic Data . . . . . . . . . . . 97

    4-10 Gulfport Mississippi Subimage . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

    4-11 Gulfport Mississippi Subimage Class Partition . . . . . . . . . . . . . . . . . . . 98

    4-12 BBCM : Estimated Means - Gulfport . . . . . . . . . . . . . . . . . . . . . . . . 99

    4-13 BBCM : Estimated Proportions - Gulfport . . . . . . . . . . . . . . . . . . . . . 100

    4-14 BBCM : Estimated Distributions - Gulfport . . . . . . . . . . . . . . . . . . . . . 101

    4-15 BCM Estimate of the Tree Distribution . . . . . . . . . . . . . . . . . . . . . . . 102

    4-16 NCM Estimate of the Tree Distribution . . . . . . . . . . . . . . . . . . . . . . . 102

    4-17 Ground Truth for Tree Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 103

    8

  • Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of