Review of Spectral Unmixingfor Hyperspectral Imagery

Lidan Miao

Sept. 29, 2005

Motivation

• The wide existence of mixed signals– In hyperspectral image, the measurement of a single pixel is

usually a contribution from several materials (endmembers).

• What is spectral unmixing?– A process by which mixed pixel spectra are decomposed into

endmember signatures and their fractional abundances.

• Applications – Subpixel detection. – Classification– Material quantification

Data Mixing Model

• Linear mixing– Mixing scale is macroscopic and there is negligible interaction

among distinct endmembers.

• Nonlinear mixing– Mixing scale is microscopic. The incident radiations scattered

through multiple bounces involves several endmembers.

Linear model Nonlinear model

Linear Mixing Model

• Measurement model

– Observation vector – Material signature matrix– Abundance fractions

• Nonnegative and sum-to-one constraints

p p p x Ms n

1 2[ ]l c c M m m m

1 1 2[ ]Tc cs s s s

1 1 2[ ]Tl lx x x x

1

0, 1c

j jj

s s

Unmixing Algorithms

• Supervised – Obtain M from laboratory data or training samples.– With known M, unmixing is a least square problem.– Not reliable due to spatial and temporal variation in illumination

and atmospheric conditions.

• Unsupervised – Linear spectral mixture analysis (LSMA)

• Two steps: estimate M from given data + inversing.

– Independent component analysis (ICA)• How to interpret the system model?

– Lagrange constrained neural network (LCNN)• Pixel-based algorithm.

LSMA

• Assumptions– There exist at least one pure pixel for each class.– The material signature matrix is the same for all image pixels in

the scene.– The number of endmembers can be determined.

• Two-step process– Endmember detection

• Convex geometry-based approach (MVT, PPI, N-FINDR, VCA)• LSE-based approach (UFCLS, USCLS, UNCLS)

– Abundance estimation• Least square method (LSE, SCLS, NCLS, FCLS)• Orthogonal subspace projection (OPS)• Quadratic programming (QP)• Target constrained method (CEM)

Convex Geometry

• Convex hull– The set of all convex combinations of point in C.

• Simplex– Convex hull of k+1 affinely independent points.

• Strong parallelism between LSMA and CG.– Endmember detection is

equivalent to identifying vertices.

1 1 2 2 1conv | , 0, 1, , , 1k k i i kC C i k x x x x

Convex hull in R2 Simplex in R2

1 0 0, , are linearly independent.k x x x x

LSE-based Approach

• Minimize the LSE between the linear mixture model and estimated measurements.

• Select the brightest pixel as the first endmember,after each iteration, select the most distinct pixel as new endmember.

• Use currently selected endmembers to unmix and calculate LSE.

ˆ( )LSE x x x

ICA

• System model

• Application– Blind source separation (BSS) and deconvolution.

• Assumption– Mutually independent sources.

• Permutation and scaling problem:– This compromises its application in spectral unmixing

1 11 1 1

1

l N l c c N

T Tc

T Tl l lc c

m m

m m

X M S

x s

x s

WM PC

ICA in Spectral Unmixing

• Two interpretations of system model

1 11 1 1

1

l N l c c N

T Tc

T Tl l lc c

m m

m m

X M S

x s

x s

1 11 1 1

1

N l N c c l

T Tc

T TN N Nc c

s s

s s

X S M

x m

x m

–Vector xi is the stack notation of image of band i.

–Column of M is endmember signature.

–Vector si is the abundance of endmember i at all pixel positions.

–Sources are not mutually independent.

–Vector xi is the spectrum of pixel i.

–Vector mi is endmember signature.

–Row vector si is the abundance of pixel i.

–Sources are mutually independent.

LCNN

• System model– Pixel-by-pixel processing

• Two principles– Maximum entropy

• Given incomplete information, maximum entropy is the least bias estimate• Closed system, no energy exchange

– Minimum Helmholtz free energy • The minimum free energy is achieved at thermal equilibrium state.• Open system, nonzero energy exchange

p p p p x M s n1

0, 1, 0c

j j ijj

s s m

Minimum Energy-based LCNN

• Objective function

• Nonlinear programming formulation

• The problem in objective function cannot be solved by learning algorithm.

0 0 0,

1 1

min ( , ) ln( ) ( ) 1c c

TB j j B j

j j

H K T s s K T s

W sW s Wx s

1

1

minimize: ln( )

subject to:

1, 0, 1, ,

c

j jj

c

j jj

s s

s s j c

Wx s 0

Evaluation SystemSelection of spectral

signatures

Generation of simulated scene

Unmixing algorithm

Ground truth

Abundancemap

Endmembersignatures

Estimated abundance

Extractedendmembers

•1. Evaluation of abundance fraction

–Root mean square error (RMSE)

–Fractional abundance angle distance (FAAD)

•2. Evaluation of endmember signature

–Spectral angle distance (SAD)

–Spectral information divergence (SID)

1 2

Simulated Data

0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Spectral signature of four endmembers

( m)

Re

flect

an

ce (

%)

BiotiteAmmonio-jarositeChloriteHalloysite

Unmixing Results (1)

Unmixing Result (2)

0 100 200 3000

0.5

1True end-members

0 100 200 3000

0.5

1Estimated end-membersTrue abundance

20 40 60

20

40

60

Estimated abundance

20 40 60

20

40

60

0 100 200 3000

0.5

1

0 100 200 3000

0.5

1

20 40 60

20

40

6020 40 60

20

40

60

0 100 200 3000

0.5

1

0 100 200 3000

0.5

1

20 40 60

20

40

6020 40 60

20

40

60

0 100 200 3000

0.5

1

0 100 200 3000

0.5

1

20 40 60

20

40

6020 40 60

20

40

60

0 100 200 3000

0.5

1True end-members

0 100 200 3000

0.5

1Estimated end-membersTrue abundance

20 40 60

20

40

60

Estimated abundance

20 40 60

20

40

60

0 100 200 3000

0.5

1

0 100 200 3000

0.5

1

20 40 60

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40

6020 40 60

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40

60

0 100 200 3000

0.5

1

0 100 200 3000

0.5

1

20 40 60

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6020 40 60

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60

0 100 200 3000

0.5

1

0 100 200 3000

0.5

1

20 40 60

20

40

6020 40 60

20

40

60

VCA USCLS

• There exist pure pixels in the scene (SNR=30dB)

Unmixing Result (3)

0 100 200 3000

0.5

1True end-members

0 100 200 3000

0.5

1Estimated end-membersTrue abundance

20 40 60

20

40

60

Estimated abundance

20 40 60

20

40

60

0 100 200 3000

0.5

1

0 100 200 3000

0.5

1

20 40 60

20

40

6020 40 60

20

40

60

0 100 200 3000

0.5

1

0 100 200 3000

0.5

1

20 40 60

20

40

6020 40 60

20

40

60

0 100 200 3000

0.5

1

0 100 200 3000

0.5

1

20 40 60

20

40

6020 40 60

20

40

60

0 100 200 3000

0.5

1True end-members

0 100 200 3000

0.5

1Estimated end-membersTrue abundance

20 40 60

20

40

60

Estimated abundance

20 40 60

20

40

60

0 100 200 3000

0.5

1

0 100 200 3000

0.5

1

20 40 60

20

40

6020 40 60

20

40

60

0 100 200 3000

0.5

1

0 100 200 3000

0.5

1

20 40 60

20

40

6020 40 60

20

40

60

0 100 200 3000

0.5

1

0 100 200 3000

0.5

1

20 40 60

20

40

6020 40 60

20

40

60

VCA USCLS

• There are no pure pixels in the scene (SNR=30dB)

Unmixing Result (4)

10 20 30 40 50-0.01

0

0.01

0.02

0.03

0.04RMSE of abundance

SNR(dB)

RM

SE

PPINFINDRVCA

10 20 30 40 500

10

20

30

40

50

60

70Angle between abundance

SNR(dB)

AA

D (

de

gre

e)

PPINFINDRVCA

10 20 30 40 500

5

10

15

20Angle between endmembers

SNR(dB)

SA

D (

de

gre

e)

PPINFINDRVCA

10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25Spectral information divergence

SNR(dB)

SID

PPINFINDRVCA

Experiments on Real Data (1)

component 1 component 2 component 3

component 4 component 5 N-FINDR

N-FINDR

component 1 component 2 component 3

component 4 component 5 component 6

VCA

Experiments on Real Data (2)

component 1 component 2

component 3 component 4

FastICA

component 1 component 2 component 3

component 4 component 5

USCLS

Conclusion

• Convex geometry-based methods can successfully extract endmembers.

• ICA is not a robust algorithm for spectral unmixing.• More works on LCNN.

– Spatial information?– Redefine the objective function?