1. Sparse Regression-based Hyperspectral Unmixing
Marian-Daniel Iordache1,2
José M. Bioucas-Dias2
Antonio Plaza1 1
Department of Technology of Computers and Communications, University of Extremadura, Caceres
Spain 2
Instituto de Telecomunicações, Instituto Superior Técnico, Technical University of Lisbon,
Lisbon
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2. Hyperspectral imaging concept
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3. Outline Linear mixing model Spectral unmixing
Sparse regression-based unmixing
Sparsity-inducing regularizers ( )
Algorithms
Results
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4. Linear mixing model (LMM)
Incident radiation interacts only
with one component
(checkerboard type scenes)
Hyperspectral linear
unmixing
Estimate
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5. Algorithms for SLU Three step approach Sparse regression
Dimensionality reduction
(Identify the subspace spanned by the columns of )
Sparse regression
Endmember determination
(Identify the columns of )
Inversion
(For each pixel, identify the vector of proportions )
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6. 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
Unmixing: given y and A, find the sparsest solution of
Advantage: sidesteps endmember estimation
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7. Sparse regression-based SLU
(library, , undetermined system)
Problem – P0
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
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8. Convex approximations to P0
CBPDN – Constrained basis pursuit denoising
Equivalent problem
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]).
Efficient solvers for CBPDN: SUNSAL, CSUNSAL [Bioucas-Dias, Figueiredo, 2010]
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9. 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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10. Hyperspectral libraries
Bad news: hyperspectral libraries exhibits high mutual coherence
Good news: hyperspectral mixtures are sparse (k· 5 very often)
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11. Reconstruction errors (SNR = 30 dB)
ISMA [Rogge et al, 2006]
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12. Real data – AVIRIS Cuprite
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13. Real data – AVIRIS Cuprite
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14. 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)
Matrix with all vectors of fractions
TV regularizer
l1 regularizer
GL regularizer
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15. 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)
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16. GLTV_SUnSAL for hyperspectral unmixing
Criterion:
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
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17. GLTV_SUnSAL results: l1 and GL regularizers
GLTV_SUnSAL (l1)
Library A2
2 groups active
SRE = 5.2 dB
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 9
6.7
14.0 4 12
4.8
12.3
MC runs = 20
SNR = 1
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18. GLTV_SUnSAL results: l1 and GL regularizers
Library
SNR = 20 dB, l1
SNR = 20 dB, l1+TV
Endmember #5
SNR = 30 dB, l1
SNR = 30 dB, l1+TV
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19. Real data – AVIRIS Cuprite
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20. 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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