Low Rank Representation For Bilinear Abundance Estimation ...Joint Sparse and Low Rank...

Joint Sparse and Low Rank Representation For Bilinear Abundance Estimation Problem

Low Rank Representation For BilinearAbundance Estimation Problem

Qing Qu∗, Xiaoxia Sun∗, Nasser Nasrabadi†, Trac D. Tran∗

∗ The Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD, 21218† U.S. Army Research Laboratory, 2800 Powder Mill Road, Adelphi, MD, 20783

IEEE WHISPERS , 25-28 June 2013, Gainesville, USA


Outline

1 Introduction

2 Abundance Estimation for GBM

3 Joint Sparse and Low Rank Representation Model

4 Experiment Results

5 Conclusion


Introduction

Motivations and Proposals

The spectral unmixing problem consists of two major steps1:Extraction of pure endmembers;Estimations of the abundance maps.

Main contributionsIn this work, we assume that the pure endmembers are knownor can be extracted automatically, our contributions are

We introduced a simple but very effective abundanceestimation method for bilinear mixture models;We proposed joint sparse and low rank models tocapture the spatial information in the hyperspectraldata.

1 N. Keshava, and J. F. Mustard,“Spectral unmixing,” IEEE Signal Process. Mag.,vol. 19, no. 1, pp. 44-57, Jan. 2002.


Introduction

Linear Mixture Model

Suppose we have

A hyperspectral data cube Y = [y1,y2, . . . ,yN ] ∈ RL×N with Lspectral bands;

A pure endmember dictionary A = [a1,a2, ...,aR] ∈ RL×R .

Each pixel sepctrum yk (1 ≤ k ≤ N) is a linear combination of R pureendmembers

yk =R∑

i=1

aixik + nk = Axk + nk , (1)

where nk is the Gaussian white noise. xi is the abundance mapassumed to satisfy

(A1) Abundance nonnegative constraint: xik ≥ 0, ∀i ∈ 1,2, ...,R,

(A2) Abundance sum-to-one constraint:R∑

i=1

xik = 1.


Introduction

l1-regression for LMM

Recent research showed that the abundance estimationproblem could be formulated in a sparse regression framework

minxk||xk ||0, s.t . ||yk − Axk ||2 ≤ ε1, xk ≥ 0, 1T xk = 1, (2)

Instead of minimizing the l0-norm, we solve the following lassoproblem

minxk

12||yk − Axk ||22 + λ1||xk ||1, s.t . xk ≥ 0, (3)

where the sum-to-one constraint cannot be enforced becauseof the nonnegativity constraint and l1-norm.


Introduction

Guo et. al. 2 proposed supervised method by use a smalldictionary A by some endmember detection method so that thesystem is over-determined, and the solution is known to be morestable and robust.

Iordache et. al. 3 proposed a semi-supervised method with ahuge over-complete dictionary A from the standard USGSlibrary, bypassing the endmember extraction step. Because themutual coherence of dictionary A is too high (near unitary), theuniqueness of the solution cannot be guaranteed.

They solve the l1 minimization problem by the nonnegativeconstraint alternating direction method of multipliers (ADMM)4

These methods are based on the LMM so that they cannothandle the wide existing non-linearities in the mixingscheme.

2Z. Guo et. al., ” L1 unmixing and its application to hyperspectral image enhancement”, in Proc. SPIE Conf.Algorithms Technol. Multispectral, Hyperspectral, Ultraspectral Imaging XV, Orlando, Florida, 2009.

3Iordache et. al., “Sparse unmixing of hyperspectral data,” IEEE Trans. Geosci. and Remote Sens., vol. 49, no.6, pp. 2014-2039, Jun. 2011.

4S. Boyd et. al., “Distributed optimization and statistical learning via the alternating direction method of multipli-ers,” Foundations and Trends in Machine Learning, vol. 3, no. 1, pp. 1-122, Nov. 2010.


Abundance Estimation for GBM

Generalized Bilinear Mixture Model

To model the nonlinearities in hyperpsectral images, the generalizedbilinear model (GBM) 5 has been proposed which assumes that eachobserved pixel yk (1 ≤ k ≤ N) can be expressed as

yk = Axk +R−1∑i=1

R∑j=i+1

γijk xik xjk ai � aj + nk , (4)

where 0 ≤ γijk ≤ 1 (1 ≤ i < j ≤ R) is an interaction parameter, and �denotes the Hadamard product operation.

GBM introduces a bilinear term to LMM;

GBM takes the scattering effects into account by thesecond-order bilinear components.

5 Halimi et. al. , “Nonlinear unmixing of hyperspectral images using a generalizedbilinear model,” IEEE Trans. Geosci. and Remote Sens., vol. 49, no. 11, pp.4153-4162, Nov. 2011.



l1-regression for GBM

By taking a closer look, the GBM could be viewed as

yk =R∑

i=1

xikai +R∗∑l=1

elkbl + nk , (5)

where R∗ = 12R(R + 1) and

elk = γijkxikxjk , bl = ai � aj ,

l = j +(2R − i − 2)(i − 1)

2, 1 ≤ i < j ≤ R.

(6)



l1-regression for GBM

Therefore, we transform the bilinear problem into a linearproblem by

yk = Axk + Bek + nk ,

= [A,B]

[xkek

]+ nk ,

= Mφk + nk . (7)

where M and φk are the composite dictionary and compositeabundance, respectively. Similarly, the bilinear abundanceestimation problem could be solved by

minφk

12||yk −Mφk ||22 + λ′1||φk ||1, s.t . φk ≥ 0, (8)



Enforcing the sum-to-one constraint

Though the sum-to-one constraint cannot be enforced in theLMM, it can be enforced in our framework because we areminimizing the l1-norm of φk rather than xk .

minφk

12||yk −Mφk ||22 + λ′1||φk ||1,

s.t . φk ≥ 0, kTφk = 1,(9)

where k = [1T ,0T ]T (1 ∈ RR and 0 ∈ RR∗).



Instead of solving the equality constraint problem exactly, weapproximately solve the following problem

minφk≥0

12||[ykδ1

]−[

Mδ1kT

]φk ||22 + λ′1||φk ||1, (10)

where δ1 > 0 is a regularization parameter and it is still anonnegative constraint sparse regression problem that can beefficiently solved by a nonnegative constraint l1-minimizationalgorithm (i.e., nonnegative constraint ADMM).


Joint Sparse and Low Rank Representation Model

Joint Sparse Regression

Given the data Y contain N pixels in a sliding window,these pixels which are in a small neighbourhood aresupposed to share very similar endmembers (or supportset).Thus, we further propose to solve the following joint sparseregression problem as

minΦ≥0

12||[

Yδ21T

]−[

Mδ2kT

]Φ||2F + λ2||Φ||1,2, (11)

where δ2 > 0, λ2 > 0, || · ||F is the Frobenius norm, andΦ = [φ1, · · · ,φN ]. ||Φ||1,2 =

∑R+R∗i=1 ||φi ||2 and φi ∈ RN is

the i-th row of the matrix Φ.



For the single measurement vector (SMV) problem

minφk≥0

12||[ykδ1

]−[

Mδ1kT

]φk ||22 + λ′1||φk ||1, (12)

we use a nonnegative constraint SMV-ADMM to solve theproblem;For the multiple measurement vector (MMV) problem

minΦ≥0

12||[

Yδ21T

]−[

Mδ2kT

]Φ||2F + λ2||Φ||1,2, (13)

we propose an nonnegative constraint MMV-ADMM to findthe solution 6;

6 Q. Qu, N. Nasrabadi, and T. Tran, “Hyperspectral Abundance Estimation for theGeneralized Bilinear Model with Joint Sparsity Constraint”, ICASSP 2013, Vancou-ver, May 2013, to be appear.



Joint Sparse Regression

The proposed joint-sparsity assumption forces all thepixels in the sliding window to share the same support set.The joint-sparsity constraint is very effective for unmixinghomogenous regions, where most of the pixels consist ofvery similar types of endmembers.However, when the sliding window contains sub-pixeltargets or noncontiguous materials, such a strict spatialconstraint might limit the ability to accurately estimate theproportions of endmembers.



(a) Original Image (b) SMV−ADMM

(c) Joint MMV−ADMM (d) Low Rank Recovery

Mixture of 2 endmembers

Pure Pixel

Background




Boundaryaliasing

Figure : The original and the reconstructed synthetic false-colorhyperspectral images of the 1st, 112nd and 224th bands



Low Rank Representation Model

Theorem

Assume matrices Y ∈ RL×N , A ∈ RL×R and X ∈ RR×N whichsatisfy Y = AX. If rank(Y) = r ≤ N, rank(A) = R, then we have

rank(X) = rank(Y) = r . (14)

The theorem tells us that if the data Y is highly correlated,the rank of the abundance matrix X is low.Instead of joint sparse regression, we propose to minimizethe rank of the abundance matrix according to the theorem.



Low Rank Representation Model

For the GBM, we need to solve the following problem

minX≥0,E≥0

rank(X) + λ3||E||0,

s.t . ||Y− AX− BE||F ≤ ε, 1T X = 1T ,(15)

Because the problem is NP-hard, we solve the followingproblem

minX≥0,E≥0

||X||∗ + λ3||E||1,

s.t . ||[

Yδ31T

]−[

Aδ31T

]X−

[B0T

]E||F ≤ ε,

(16)

where || · ||∗ denotes the nuclear norm of a matrix.



For the low rank recovery (LRR) problem,

minX≥0,E≥0

||X||∗ + λ3||E||1,

s.t . ||[

Yδ31T

]−[

Aδ31T

]X−

[B0T

]E||F ≤ ε,

(17)

we proposed an inexact augmented Lagrangian multipliersalgorithm to solve this problem7

7 Q. Qu, X. Sun, N. Nasrabdai, T. Tran, “Low Rank Representation for bilinear abun-dance Estimation problem”, WHISPERS 2013, Gainsville.


Experiment Results

Metrics for Performance Evaluation

RMSE =

√√√√ 1nR

n∑i=1

||xi − xi ||2, (18)

SRE = 10 log10

(E[||X||22]

E[||X− X||22]

), (19)

RE =

√√√√ 1nL

n∑i=1

||yi − yi ||22, (20)

SAM =1n

n∑i=1

θ[yi , yi ], (21)


Experiment Results

Experiment Setting for Synthetic Data

Pick 12 pure endmembers from USGS library to form thedictionary A.

Generate each pixel by mixing 3 or 4 randomly selectedendmembers in A following Dirichlet distribution and sum-to-oneconstraint.

Add Gaussian white noise with SNR = 40dB to the syntheticdata.

We compare our proposed algorithms with FCLS 8,CSUnSAL+6, KFCLS 9, SK-Hypo 10, GDA 11 .

8J. M. Bioucas-Dias et. al. “Alternating direction algorithms for constrained sparse regression: application tohyperspectral unmixing,” WHISPERS, Reykjavik, Iceland, pp. 1-4, Jun. 2010.

9J. Broadwater et. al., “Kernel fully constrained least squares abundance estimates,” IGARSS, Barcelona,Spain, pp. 4041-4044, Jul. 2007.

10Chen et. al. , “Nonlinear unmixing of hyperspectral data based on a linear-mixture/nonlinear-fluctuation model,”IEEE Trans. Signal Process., vol. 61, no. 2, pp. 480-492, Jan. 2013.

11Halimi et. al., “Nonlinear unmixing of hyperspectral images using a generalized bilinear model,” IEEE Trans.Geosci. and Remote Sens., vol. 49, no. 11, pp. 4153-4162, Nov. 2011.


Experiment Results

Table : Comparison on Synthetic Hyperspectral Image

Criteria Model Estimation AlgorithmFCLS KFCLS CSUnSAL+ GDA SMV-ADMM

SRE (dB) LMM 41.4160 8.7507 38.9119 41.3122 37.8415GBM 11.6985 4.0998 13.1667 10.6583 22.4512

RE (×10−2)LMM 0.0684 N.A. 0.0733 0.0674 0.1031GBM 3.5478 N.A. 0.9300 3.7415 0.5513


Experiment Results

Bilinear Abundance Estimation

Figure : The comparison of estimated abundance X and the bilinearcoefficient E for the proposed methods


Experiment Results

Bilinear Abundance Estimation

Pixel

Endm

ember

Original

2 4 6 8

2

4

6

8

10

12

Pixel

Endm

ember

SMV−ADMM

2 4 6 8

2

4

6

8

10

12

Pixel

Endm

ember

MMV−ADMM

2 4 6 8

2

4

6

8

10

12

Pixel

Endm

ember

LRR

2 4 6 8

2

4

6

8

10

12

Pixel

Endm

ember

2 4 6 8

10

20

30

40

50

60

70

Pixel

Endm

ember

2 4 6 8

10

20

30

40

50

60

70

Pixel

Endm

ember

2 4 6 8

10

20

30

40

50

60

70

Pixel

Endm

ember

2 4 6 8

10

20

30

40

50

60

70

Figure : The comparison of estimated abundance X and the bilinearcoefficient E for the proposed methods


Experiment Results

Figure : A comparison of abundance estimation result on a syntheticimage.


Experiment Results

Table : Comparison of Unmixing Results for the synthetic image

AlgorithmSMV MMV Low Rank

-ADMM -ADMM RecoverySRE (dB) 12.9585 14.1834 20.2845

RMSE (10−2) 1.0680 0.9276 0.4595RE (10−3) 5.4922 5.4531 5.2966

SAM (10−2) 1.0492 1.0268 1.0051


Experiment Results

Experiments on Cuprite Image

Figure : The target scene and the pure endmembers.

The pure endmembers are extracted by Vertex ComponentAnalysis (VCA)1212 Nascimento et. al. , “Vertex component analysis: A fast algorithm to unmix hy-

perspectral data,” IEEE Trans. Geosci. and Remote Sens., vol. 43, no. 4, pp.898-910, Apr. 2005.


Experiment Results


Experiment Results

Table : Comparison of the reconstruction error for all algorithms onthe real hyperspectral image

Model Algorithm µRE (×10−3) σRE (×10−7) µSAM (×10−2) σSAM (×10−6)

LMM FCLS 7.5587 12.4031 2.2560 16.4715CSUnSAL+ 6.4522 7.7908 1.9820 9.8041

PPNMM GDA 6.3501 7.8927 1.9085 9.2313Non-linear SK-Hypo 6.0827 1.4760 1.8863 2.0917

GBMSMV-ADMM 5.7820 11.3906 1.7974 12.6686MMV-ADMM 5.6314 11.9591 1.7559 13.1640LRR 4.1707 1.3997 1.3023 1.7020


Conclusion

Conclusion

In this work, we have proposed several simple but effectivesparsity and low rank based abundance estimationalgorithms demonstrated by extensive simulations.The proposed LRR algorithm is currently computationallyexpensive. We can solve it more efficiently by thelinearized augmented Lagrangian method13

Without the ground truth abundance, the comparisons onreal dataset are currently based on signal reconstructionerror and somehow heuristic. More demonstrations on realdataset needs further investigation.

13 Z. Lin et. al. “Linearized Alternating Direction Method with AdaptivePenalty for Low-Rank Representation”, NIPS, Spain, 2011.


Conclusion

Thanks!

Low Rank Representation For Bilinear Abundance Estimation ...Joint Sparse and Low Rank...

Documents

Transcript of Low Rank Representation For Bilinear Abundance Estimation ...Joint Sparse and Low Rank...