Post on 29-Jul-2020
Hyperspectral Unmixing: Geometrical, Statistical,
and
Sparse Regression-Based Approaches
José M. Bioucas-Dias
Instituto de Telecomunicações
Instituto Superior Técnico
Universidade de Lisboa
Portugal
SUPELEC, SONDRA - 2014
Hyperspectral imaging (and mixing)
c:
o ----·
Mixed pixel
( oil + rocks)
Pure pi.xel
(water)
4 ----------------
3000 J /'-
j::i: -r --
300 eoo 900 t 200 1500 ·aoc 210024DO
Wa\'elength (om) u 4000
c<J
5 l000 <J
2..0.00
c:: 1000
0 ... ' t
300 eoo soo ·200 1500 ·aoo 2HlO 21&00
Wa\·el ngth (nm)
Mixed pi.xel
(·'egetation + soil)
5000
... 4000 Col
5 30!)() Col
4.:.0. 00
c:: 1000
lOO eoo 900 1200 1500 taoo 2100 2400
Wavelengtb (om)
Hyperspectral unmixing
AVIRIS of Cuprite,
Nevada, USA
R – ch. 183 (2.10 m)
G – ch. 193 (2.20 m)
B – ch. 207 (2.34 m)
Alunite Kaolinite Kaolinite #2
VCA [Nascimento, B-D, 2005]
Outline
Mixing models
Linear
Nonlinear
Signal subspace identification
Unmixing
Geometrical-based
Statistical-based
Sparse regression-based
J. M. Bioucas-Dias, A. Plaza, N. Dobigeon, M. Parente, Q. Du, P. Gader, and J. Chanussot,
"Hyperspectral unmixing overview: geometrical, statistical, and sparse regression-based approaches",
IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing,
vol. 5, no. 2, pp. 354-379, 2012.
W.-K. Ma, J. Bioucas-Dias, T.-H. Chan, N. Gillis, P. Gader, A. Plaza, A. Ambikapathi and C.-Y. Chi,
"A signal processing perspective on hyperspectral unmixing", IEEE Signal Processing Magazine, Jan.,
2014.
4
Linear mixing model (LMM)
6 i .
7
2
m = 6
6 4
½1i 3
½2i 7
7 5
½Li
Incident radiation interacts only
with one component
(checkerboard type scenes)
Hyperspectral linear
unmixing Estimate
5
Nonlinear mixing model
Intimate mixture (particulate media) Two-layers: canopies+ground
Radiative transfer theory
material fractions media parameters single scattering double scattering
6
CUHK - 2012 7
Schematic view of the unmixing process
Spectral linear unmixing (SLU)
Given N spectral vectors of dimension L:
Subject to the LMM:
ANC: abundance
nonnegative
constraint
Determine:
The mixing matrix M (endmember spectra)
The fractional abundance vectors
ASC: abundance
sum-to-one
constraint
SLU is a blind source separation problem (BSS)
8
Subspace identification
Problem: Identify
the subspace generated by the columns of
Reasoning underlying DR
1. Lightens the computational
complexity
2. Attenuates the noise power
by a factor of
Subspace identification
11
Subspace identification algorithms
refe
cta
nce
Exact ML solution [Scharf, 91] (known p, i.i.d. Gaussian noise)
PCA - Principal component analysis (unknown p, i.i.d. noise)
NAPC - Noise adjusted principal components [Lee et al., 90]
MNF - Maximum noise fraction [Green et al., 88]
HFC - Harsanyi-Farrand-Chang [Harsanyi et al., 93]
NWHFC - [Chang, Du, 94]
HySime - Hyperspectral signal identification by minimum error [B-D, Nascimento, 08]
Hypothesis testing and random matrix theory [Kritchman, Nadler, 2009],
[Cawse, Robin, Sears, 11]
1
0.8
x
y
yA Example (HySime)
0.6
0.4
0.2
0 0 50 100 150 200 250
band 12
Unsupervised unmixing frameworks
Geometrical
Exploits parallelisms between the linear mixing
model and properties of convex sets
Statistical
Approaches linear unmixing as a statistical
inference problem
Sparse regression
Approaches linear unmixing as a sparse
regression problem
13
Minimum volume simplex
Methodology: Find the simplex of
minimum volume containing the data ([Craig, 1990, 1994])
Equivalent formulation: find such that
is minimized and all data points are inside the
simplex defined by the columns of
14
Minimum volume simplex (MVS)
Pure pixels No pure pixels No pure pixels
MVS works MVS works MVS does not work
15
Minimum volume simplex algorithms
Hard assumption
The data set contains at least one pure pixel of each material
Search endmembers in the data set
Computationally lighter
PPI - [Boardman, 93]; N-FINDR - [Winter, 99]; IEA - [Neville et al., 99];
AMEE – [Plaza et al, 02]; SMACC – [Gruninger et al., 04]
VCA - [Nascimento, B-D, 03, 05]; SGA - [Chang et al., 06]
AVMAX, SVMAX - [Chan, Ma, Ambikapathi, Chi, 2011];
RNMF- [Gillis, 2013];
SD-SOMP, SD-ReOMP - [Fu, Ma, Chan, B-D, Iordche, 2013];
14
Pure pixel based algorithms
N-FINDR
AVMAX
VCA
SVMAX
iteratively increase Idet( m1, ... , mp) 1
rn 1, . . . ,mp E {y1, ... ,y N}
max
st: m1, ... , mp E conv{y1, ... , YN}
for k == 1 : p
rn:== argmax II(Pifç)TYi li Yi
M :== [M, rn]
for k == 1 : p
rn:== argmax II(Pifyi ll Yi
M :== [M, rn]
15
Unmixing example
HYDICE sensor M E 21o x 6 N == 500 x 307 resolution == O.75m
• Data
0 N-FINDR
VCA
16
Minimum volume simplex algorithms
Minimum-volume constrained nonnegative matrix
factorization (MVC-NMF) (inspired by NMF [Lee, Seung, 01])
volume regularizer
DPFT - [Craig, 90]; CCA - [Perczel et al., 89] (seminal works on MVC)
ICE - [Breman et al., 2004] ( );
MVC-NMF - [Miao,Qi, 07] ( );
SPICE - [Zare, Gader, 2007] ( )
L1/2 - NMF - [Qian, Jia, Zhou, Robles-Kelly, 11] ( )
CoNMF - [Li, B-D, Plaza, 12] ( )
17
Minimum volume simplex algorithms
MVSA – Minimum volume simplex analysis [Li, B-D, 08]
Optimization variable
ASC ANC
MVSA solves a sequence of quadratic programs
MVES [Chan, Chi, Huang, Ma, 2009]
Solves a sequence of linear programs by exploiting the cofactor
expansion of det (Q)
The existence of pure pixels is a sufficient condition for exact
identification of the true endmembers
18
Robust minimum volume simplex algorithms: outliers
SISAL – Simplex identification via split augmented Lagrangian [B-D,09]
ASC
soft ANC
SISAL solves a sequence of convex subproblems using ADMM
19
Robust minimum volume simplex algorithms: noise
RMVES – Robust MVES [Ambikapathi, Chan, Ma, Chi, 12]
ANC - chance constraint
probit function
Solves a sequence of quadratic programs
exploiting the cofactor expansion of det(Q)
20
Example: data set contains pure pixels
a)
Y(1
202,:
)
0.8
data points 1
true sisal
0.7
0.6
0.5
0.4
0.3
0.2
0.1
estimated
true
initial (vc
(sisal)
0.8
0.6
0.4
0.2
0
vca
0 0.2 0.4 0.6 0.8 Y(100,:)
0 50 100 150 200 250
Time:
VCA 0.5 sec
SISAL 2 sec
IWMSDF, SYSU- 2014 21
Example: Data set does not contain pure pixels
(mvca)
Y(1
202,:
)
0.8
0.7
0.6
0.5
data points
estimated - (sisal)
true
initial - (vca)
1
0.8
0.6
true sisal vca
0.4 0.4
0.3
0.2
0.1 0 0.2 0.4 0.6 0.8
Y(100,:)
0.2
0 0 50 100 150 200 250
22
No pure pixels and outliers
v2'*
Y
4
3.5
3
2.5
2
1.5
1
Endmembers and data points (2D projection)
data points
true
SISAL
VCA
NFINDR
MVC
ERROR(mse):
SISAL = 0.03
VCA = 0.88
NFINDR = 0.88
MVC-NMF = 0.90
TIMES (sec):
SISAL = 0.61,
VCA = 0.20
NFINDR = 0.25
MVC-NMF = 25
0.5 -6 -5 -4 -3 -2 -1 0
v1'*Y
23
Real data: ArtImageDataA (converted into absorbance)
oil
Tiles 1,2 (‘Prussian blue’ + oil) 2
Tiles 15,16 (‘Ultramarine blue’ + oil) 1
Projection on [e , e ] 1 2
0 Estimate endmember signatures by SISAL
2
-1
U. blue
1.5 -2
1 -3 -20 -15 -10 -5 0
0.5
0
P. blue
SISAL
solution
-0.5 0 50 100 150 200
24
Geometrical Approaches: Limitations
PPI, N FINDR VCA, SGA AMEE, SVMAX AVMAX
depend on the existence of pure pixels in the data
Do not work in highly mixed scenarios
v2'*
Y
Limitations of geometrical methods Inference determined by a small number of pixels;
some may be outliers
MVS – Computationally heavy
- , , ,
3.5
3
2.5
Endmembers and data points (2D projection)
data points
true
SISAL
VCA
NFINDR
MVC-NMF
2
25 1.5
Statistical approaches
observation model
prior (Bayesian framework)
posterior density
inference
26
Spectral linear unmixing and ICA/IFA
Formally, SLU is a linear source separation problem
Independent Component Analysis (ICA) come to mind
ICA Fastica, [Hyvarinen & Oja, 2000]
Jade, [Cardoso, 1997]
Bell and Sejnowski, [Bell and Sejnowski , 1995]
IFA IFA, [Moulines et al., 1997], [Attias, 1999]
27
Statistical approaches: ICA
Assumptions
1. Fractional abundances (sources) are independent
2. Non-Gaussian sources
Endmembers compete for the same area
Sources are Dependent
ICA does not apply [Nascimento, B-D, 2005]
IWMSDF, SYSU- 2014 28
Representative Bayesian approaches
[Parra et al., 00]
[Moussaoui et al., 06,a,b], [Dobigeon et al., 09,a,b], [Dobigeon et al., 09,b],
[Arngreen, 11]
data term associated with the LMM
Uniform on the simplex conjugate prior distributions for some unknown parameters
Infer MMSE estimates by Markov chain Monte Carlo algorithms
DECA - [Nascimento, B-D 09, 12]
data term associated with a noiseless LMM
Dirichlet mixture model
MDL based inference of the number of Dirichlet modes
MAP inference (GEM - algorithm)
29
DECA – Dependent component analysis
parameters of the Dirichlet mixture
simulated data
minimum volume term
30
DECA – Results on Cuprite
[Mittelman, Dobigeon, Hero, 12]
data term associated with the LMM
Latent label process enforcing
adjacent pixels to have the same label
spatial prior: tree-structured sticky
hierarchical Dirichlet process (SHDP)
MMSE inference by MCMC
Model order inference
(number of endmembers)
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
[Iordache, B-D, Plaza, 11, 12]
0
0
0 0
0
0
Unmixing: given y and A, find the sparsest solution of
Advantage: sidesteps endmember estimation
Disadvantage: Combinatorial problem !!!
32
Sparse regression-based SLU
Problem – P0
(library, , undetermined system)
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
IHT (see [Blumensath, Davis, 11], [Kyrillidis, Cevher, 12])
33
Convex approximations to P0
CBPDN – Constrained basis pursuit denoising
Equivalent problem
Striking result: In given circumstances, related with the
coherence among the columns of matrix A, BP(DN)
yields the sparsest solution ([Donoho 06], [Candès et al. 06]).
Efficient solvers for CBPDN: SUNSAL, CSUNSAL [B-D, Figueiredo, 10]
35
Real data- AVIRIS Cuprite
50
100
150
.200
250
300
"
SELECTED "'- ... • IMAGE [ \t'
[Iordache, B-D, Plaza, 11, 12] 35
Real data- AVIRIS Cuprite
200 400 600 800 1000 1200 1400 1600 1800 2000
[Iordache, B-D, Plaza, 11, 12] 36
Sparse reconstruction of hyperspectral data: Summary
Bad news: Hyperspectral libraries have poor RI constants
Good news: Hyperspectral mixtures are highly sparse, very often p · 5
Surprising fact: Convex programs (BP, BPDN, LASSO, …) yield much
better empirical performance than non-convex state-of-the-art
competitors
Directions to improve hyperspectral sparse reconstruction
Structured sparsity + subspace structure
(pixels in a give data set share the same support)
Spatial contextual information
(pixels belong to an image)
37
Constrained total variation sparse regression (CTVSR)
[Iordache, B-D, Plaza, 11]
Total Variation of
i-th band
Related work [Zhao, Wang, Huang, Ng, Plemmons, 12]
Other Regularizers:
vector total variation (VTV)! promotes piecewise smoo vectors [Bresson, Chan, 02], [Goldluecke et al., 12], [Yuan, Zhang, Shen, 12]
convex generalizations of Total Variation based on the Structure Tensor [Lefkimmiatis et al., 13]
sparse representation (2D, 3D) in the wavelet domain
38
Ilustrative examples with simulated data : SUnSAL-TV
Original data cube
Original abundance of EM5
SUnSAL estimate
SUnSAL-TV estimate
41
Constrained colaborative sparse regression (CCSR)
0 0
0 0
0 0
0 0
0 0
0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0
0
0 0
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0 0 0
0 0 0 0
0 0 0 0
0
[Iordache, B-D, Plaza, 11, 12] [Turlach, Venables, Wright, 2004]
=
multiple measurements
share the same support
Theoretical guaranties (superiority of multichanel) : the probability of recovery
failure decays exponentially in the number of channels. [Eldar, Rauhut, 11]
40
Ilustrative examples with simulated data : CSUnSAL
Original x
50
Without collaborative sparseness
50
With collaborative sparseness
50
100 100 100
150 150 150
200 200 200
250 250 250
300 300 300
20 40 60 80 100
20 40 60 80 100
43 20 40 60 80 100
MUSIC – Colaborative SR algorithm
MUSIC-CSR algorithm [Iordahe, B-D, Plaza, 2013]
1) Estimate the signal subspace using, e.g.
the HySime algorithm.
2) Compute and define
the index set
3) Solve the colaborative sparse regression optimization
[B-D, Figueiredo, 2012]
Related work: CS-MUSIC
(N < k and iid noise)
[Kim, Lee, Ye, 2012]
44
MUSIC – CSR results
A – USGS , Gaussian shaped noise, SNR = 25 dB, k = 5, m = 300,
acummulated
abundances
50
100
150
200
250
300
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
MUSIC-CSR SNR = 11.7 dB
computation time ' 10 sec true
endmembers
CSR SNR = 0 dB
computation time ' 600 sec 45
Brief Concluding remarks
HU is a hard inverse problem (noise, bad-conditioned direct
operators, nonlinear mixing phenomena)
HU calls for sophisticated math tools and framework (statistical
inference, optimization, machine learning)
The research efforts devoted to non-linear mixing models are
increasing
Linear mixing
Apply geometrical approaches when there are data vectors near or
over the simplex facets
Apply statistical methods in highly mixed data sets
Apply sparse regression methods, if there exits a spectral library
for the problem in hand
44
Spectral nonlinear unmixing (SNLU). Just a few topics
Detecting nonlinear mixtures in polynomial post-nonlinear mixing
model, [Altmann, Dobigeon, Tourneret, 11,13]
hypothesis test
Bilinear unmixing model, [Fan, Hu, Miller, Li, 09], [Nascimento, B-D, 09],
[Halimi, Altmann, Dobigeon, Tourneret, 11,11]
Kernel-based unmixing algorithms to specifically account for intimate
mixtures [Broadwater, Chellappa, Burlina, 07], [Broadwater, Banerjee, 09,10, 11], [Chen,
Richard, Ferrari, Honeine, 13]
N. Dobigeon, J.-Y. Tourneret, C. Richard, J. C. M. Bermudez, S. McLaughlin and A. O. Hero, "Nonlinear
unmixing of hyperspectral images: models and algorithms," IEEE Signal Process. Magazine, vol. 31, no 1,
pp. 82-94, 2014
R. Heylen, M.Parente, and P. Gader, "A review of nonlinear hyperspectral unmixing methods,"
Selected Topics in Applied Earth Observations and Remote Sensing, IEEE Journal of , vol.7, no.6,
pp.1844-1868, 2014 45