Instituto de Telecomunicações, Instituto Superior Técnico, Technical University of Lisbon ,...
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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
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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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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
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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
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trueestimated
GLTV_SUnSAL (l1)
Library A2 2 groups active
SRE = 5.2 dB
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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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