1 Information Extraction Principles for Hyperspectral Data David Landgrebe Professor of Electrical &...

Information Extraction Principles for Hyperspectral Data

David LandgrebeProfessor of Electrical & Computer Engineering

Purdue Universitylandgreb@ecn.purdue.edu

• A Historical Perspective• Data and Analysis Factors• Hyperspectral Data Characteristics• Examples• Summary of Key Factors

Outline

1957 - Sputnik

REMOTE SENSING OF THE EARTH

Atmosphere - Oceans - Land

Brief History

1958 - National Space Act - NASA formed

1960 - TIROS I

1960 - 1980 Some 40 Earth Observational Satellites Flown

Image Pixels

Enlarged 10 Times

Thematic Mapper Image

Three Generations of Sensors

Band No.

101520253035404550

1 2 3 4

Green Veg.

Bare Soil6-bit data

MSS1968

Band No.

1 2 3 4 5 6 7

Green Veg.

Bare Soil

8-bit data

TM1975

Wavelength (µm)

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Emerging Crop

10-bit data

Hyperspectral1986

Systems View

Sensor On-BoardProcessing

PreprocessingData

AnalysisInformationUtilization

Human Participationwith Ancillary Data

Ephemeris,Calibration, etc.

Scene Effects on Pixel

Data Representations

Image Space Spectral Space

0.40 0.80 1.20 1.60 2.00 2.40

Wavelength (µm)

Water Trees Soil

Feature Space

0 500 1000 1500 2000

Band 0.60

Sample

• Image Space - Geographic Orientation

• Feature Space - For Use in Pattern Analysis• Spectral Space - Relate to Physical Basis for Response

Data Classes

SCATTER PLOT FOR TYPICAL DATA

17 34 51 68 85 102

BiPlot of Channels 4 vs 3

Channel 3

++ +++

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+++++++++ +++ +

+ + +++++ +++

+ ++++++++++++++ +

+ +++ +

+++ +++

++ +++++++

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++++++++

+++ ++ ++++

++++ +++++

+ + ++++ ++

++++++

+ + + +++

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++++++

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++++ ++

+ ++++++++ ++ ++

++++++ ++++++++

++ +++++++

+++++++++

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++ + +

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++ ++++

++++++

+++ ++

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+ ++++++++++ + +

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++++ ++

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+ ++ ++

++ + +

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++ +++++ ++ + +

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+++ +++

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+++ ++

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+++ + ++

+ + ++

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+ ++++++

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+ ++++

+ +++ +

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+ +++ + +

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+ + ++

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+ + + ++

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+ ++ +

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+ +++ ++

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+ ++ +

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+ + ++ + + ++++

+ ++++ + ++ +++

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+ + ++++

+ +++++

BHATTACHARYYA DISTANCE

Mean Difference Term Covariance Term

Vegetation in Spectral Space

Laboratory Data: Two classes of vegetation

Scatter Plots of Reflectance

0.720.710.700.690.680.670.660.6510

Class 1 - 0.67 µm

Class 2 - 0.67 µmClass 1 - 0.69 µm

Class 2 - 0.69 µm

Scatter of 2-Class Data

Wavelength - µm

Vegetation in Feature Space

15141312111016

Class 1

Class 2

Samples from Two Classes

% Reflectance at 0.67 µm

Hughes Effect

1 100050020010050201052

MEASUREMENT COMPLEXITY n (Total Discrete Values)

G.F. Hughes, "On the mean accuracy of statistical pattern recognizers," IEEE Trans. Inform. Theory., Vol IT-14, pp. 55-63, 1968.

A Simple Measurement Complexity Example

Classifiers of Varying Complexity

• Quadratic Form

gi(X) = 1

2(X i )

T i 1(X i )

• Fisher Linear Discriminant - Common class covariance

gi(X) = 1

2(X i )

T 1(X i )

• Minimum Distance to Means - Ignores second moment

gi(X) = 1

2(X i )

T (X i )

Classifier Complexity - con’t• Correlation Classifier

gi(X) XT i

XTX iT i

• Spectral Angle Mapper

gi(X) cos 1 XTiXTX i

• Matched Filter - Constrained Energy Minimization

gi(X) XTCb

1i iTCb

1 i• Other types - “Nonparametric”

Parzen Window Estimators Fuzzy Set - based Neural Network implementations K Nearest Neighbor - K-NN etc.

Covariance Coefficients to be Estimated

• Assume a 5 class problem in 6 dimensions

• Normal maximum likelihood - estimate coefficients a and b• Ignore correlation between bands - estimate coefficients b

• Ignore correlation between bands - estimate coefficients d

Class 1 Class 2 Class 3 Class 4 Class 5b b b b ba b a b a b a b a ba a b a a b a a b a a b a a ba a a b a a a b a a a b a a a b a a a ba a a a b a a a a b a a a a b a a a a b a a a a ba a a a a b a a a a a b a a a a a b a a a a a b a a a a a b

• Assume common covariance - estimate coefficients c and d

Common Covar.dc dc c dc c c dc c c c dc c c c c d

EXAMPLE SOURCES OFCLASSIFICATION ERROR

Decision boundary defined by the

diagonal covariance classifier

class 2

class 1

Decision boundary defined by Gaussian ML classifier

Number of Coefficients to be Estimated

• Assume 5 classes and p features

No. ofFeatures p

Class Covar.

(a & b above)5{{ p+1)p/2}

Diagonal ClassCommon Covar.

(b above)5p

CommonCovar.

(c & d above){ p+1)p/2}

Diagonal CommonCovar.

(d above)p

5 75 25 15 510 275 50 55 1020 1050 100 210 2050 6375 250 1275 50

200 100,500 1000 20,100 200

Intuition and Higher Dimensional Space

Borsuk’s Conjecture: If you break a stick in two, both pieces are shorter than the original.

Keller’s Conjecture: It is possible to use cubes (hypercubes) of equal size to fill an n-dimensional space, leaving no overlaps nor underlaps.

Science, Vol. 259, 1 Jan 1993, pp 26-27

Counter-examples to both have been found for higher dimensional spaces.

The Geometry of High Dimensional Space

The Volume of a Hypercube concentrates in the corners

1 2 3 4 5 6 70

dimension d

The Volume of a Hypersphereconcentrates in the outer shell

1 2 3 4 5 6 7 8 9 10 110

dimension d

Vd (r ) Vd (r )

Vd (r)rd (r )d

V hypersphere

Vhypercube

d2d 1 d2

Some Implications

High dimensional space is mostly empty. Data in high dimensional space is mostly in a lower dimensional structure.

Normally distributed data will have a tendency to concentrate in the tails; Uniformly distributed data will concentrate in the corners.

Volume of a hypersphere =2rd

dd / 2

(d / 2)

How can that be?

2d / 2

(d / 2)r (d 1)

Differential Volume at r =

0 1 2 3 4 50

Distance from Class Mean, r

Surface of Hypersphere

Volumn of shell

How can that be? (continued)

rd 1e r

2d2 1d2

The Probability Mass at r =

0 1 2 3 4 50

Distance from Class Mean, r

2 3 4 5 10 15 20

Probability Density of Distance r

Probability mass in shell

1 Information Extraction Principles for Hyperspectral Data David Landgrebe Professor of Electrical &...

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