Instituto de Telecomunicações,Instituto Superior Técnico,Technical University of Lisbon,

Lisbon

Sparse Regression-basedHyperspectral Unmixing

IGARSS 2011

Antonio Plaza1Marian-Daniel Iordache1,2

Department of Technology of Computers and Communications,University of Extremadura, Caceres

Spain

José M. Bioucas-Dias2

1 2

Hyperspectral imaging concept

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Outline

Sparse regression-based unmixing

Linear mixing model

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Spectral unmixing

Algorithms

Results

Sparsity-inducing regularizers ( )

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Linear mixing model (LMM)

Incident radiation interacts only with one component(checkerboard type scenes)

Hyperspectral linear unmixing

Estimate

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2. Endmember determination(Identify the columns of )

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Algorithms for SLU

3. Inversion(For each pixel, identify the vector of proportions )

1. Dimensionality reduction (Identify the subspace spanned by the columns of )

Sparse regression

Three step approach

Sparse regression-based SLU Spectral vectors can be expressed as linear combinations of a few pure spectral signatures obtained from a (potentially very large) spectral library

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0

0

00

0

Unmixing: given y and A, find the sparsest solution of

Advantage: sidesteps endmember estimation

Very difficult (NP-hard)

Approximations to P0: OMP – orthogonal matching pursuit [Pati et al., 2003] BP – basis pursuit [Chen et al., 2003]BPDN – basis pursuit denoising

Problem – P0

(library, , undetermined system)

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Sparse regression-based SLU

Striking result: In given circumstances, related with the coherence of among the columns of matrix A, BP(DN) yields the sparsest solution ([Donoho 06], [Candès et al. 06]).

Convex approximations to P0

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CBPDN – Constrained basis pursuit denoising

Efficient solvers for CBPDN: SUNSAL, CSUNSAL [Bioucas-Dias, Figueiredo, 2010]

Equivalent problem

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Application of CBPDN to SLU

Extensively studied in [Iordache et al.,10,11]

Six libraries (A1, …, A6 )

Simulated data Endmembers random selected from the libraries Fractional abundances uniformely distributed over the simplex

Real data AVIRIS Cuprite Library: calibrated version of USGS (A1)

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Bad news: hiperspectral libraries exhibits high mutual coherence

Good news: hiperspectral mixtures are sparse (k· 5 very often)

Hyperspectral libraries

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Reconstruction errors (SNR = 30 dB)

ISMA [Rogge et al, 2006]

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Real data – AVIRIS Cuprite

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Real data – AVIRIS Cuprite

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Beyond l1 regularization

Rationale: introduce new sparsity-inducing regularizers to counter the sparse regression limits imposed by the high coherence of the hyperspectral libraries.

New regularizers: Total variation (TV ) and group lasso (GL)

l1 regularizer GL regularizer

TV regularizer

Matrix with all vectors of fractions

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Total variation and group lasso regularizers

i-th image band

promotes similarity between neighboring fractions

i-th pixel

promotes groups of atoms of A (group sparsity)IGARSS 2011

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GLTV_SUnSAL for hyperspectral unmixing

GLTV_SUnSAL algorithm: based on CSALSA [Afonso et al., 11]. Applies the augmented Lagrangian method and alternating optimization to decompose the initial problem into a sequence of simper optimizations

Criterion:

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GLTV_SUnSAL results: l1 and GL regularizers

0 50 100 150 200 250 300 350-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

trueestimated

GLTV_SUnSAL (l1)

Library A2 2 groups active

SRE = 5.2 dB

0 50 100 150 200 250 300 3500

0.05

0.1

0.15

0.2

0.25

0.3

trueestimated

GLTV_SUnSAL (l1+GL)

SRE = 15.4 dB

k (no. act. groups)

no. endmembers

SRE (l1) dB SRE (l1+GL) dB

1 3 9.7 16.3

2 6 7.8 14.5

3 9 6.7 14.0

4 12 4.8 12.3

MC runs = 20SNR = 1

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SNR = 20 dB, l1

GLTV_SUnSAL results: l1 and GL regularizers

SNR = 20 dB, l1+TV

SNR = 30 dB, l1 SNR = 30 dB, l1+TV

Library

Endmember #5

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Real data – AVIRIS Cuprite

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Concluding remarks

Shown that the sparse regression framework has a strong potential for linear hyperspectral unmixing

Tailored new regression criteria to cope with the high coherence of hyperspectral libraries Developed optimization algorithms for the above criteria To be done: reseach ditionary learning techniques

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