ArticlePDF Available

Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging

Authors:

Abstract and Figures

Sparse hyperspectral unmixing aims at finding the sparse fractional abundance vector of a spectral signature present in a mixed pixel. However, there are several types of noise present in the hyperspectral images. These are called mixed noise including stripes, impulse noise and Gaussian noise which deteriorate the performance of sparse unmixing algorithms. In this study, we simultaneously unmix and denoise the hyperspectral image in a unified framework in the presence of mixed noise. In the denoising step, we utilize a low-rank and sparse decomposition based on a nonconvex approach to approximate the rank of hyperspectral data and eliminate the sparse noise terms. In the unmixing part, we employ a semi-supervised sparse unmixing algorithm which uses a nonconvex heuristic similar to denoising step to promote the sparsity of the abundance matrix. We conduct several experiments on synthetic and real hyperspectral data sets to validate the effectiveness of the proposed method in denoising and unmixing processes.
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
Digital Object Identifier 10.1109/ACCESS.2017.DOI
Simultaneous Nonconvex Denoising and
Unmixing for Hyperspectral Imaging
TANER INCE, (Member, IEEE), TUGCAN DUNDAR
Electrical and Electronics Engineering Department, University of Gaziantep, 27310 Gaziantep, Turkey (e-mail: tanerince@gantep.edu.tr, dundar@gantep.edu.tr)
Corresponding author: Taner Ince (e-mail: tanerince@gantep.edu.tr).
ABSTRACT Sparse hyperspectral unmixing aims at finding the sparse fractional abundance vector of
a spectral signature present in a mixed pixel. However, there are several types of noise present in the
hyperspectral images. These are called mixed noise including stripes, impulse noise and Gaussian noise
which deteriorate the performance of sparse unmixing algorithms. In this study, we simultaneously unmix
and denoise the hyperspectral image in a unified framework in the presence of mixed noise. In the denoising
step, we utilize a low-rank and sparse decomposition based on a nonconvex approach to approximate the
rank of hyperspectral data and eliminate the sparse noise terms. In the unmixing part, we employ a semi-
supervised sparse unmixing algorithm which uses a nonconvex heuristic similar to denoising step to promote
the sparsity of the abundance matrix. We conduct several experiments on synthetic and real hyperspectral
data sets to validate the effectiveness of the proposed method in denoising and unmixing processes.
INDEX TERMS Hyperspectral image (HSI), denoising, sparse unmixing, mixed noise, low-rank represen-
tation (LRR), abundance estimation.
I. INTRODUCTION
Hyperspectral imaging is used in various fields of science
such as remote sensing, astronomy, mineralogy and fluores-
cence microscopy. In these fields, a great deal of applications
such as classification [1], noise removal [2], target detection
[3] and super-resolution [4], [5] are studied extensively in the
remote sensing community.
Spectral unmixing [6] is the process of finding the pure
spectral signatures (endmembers) of a mixed pixel with cor-
responding fractions (abundances). Linear spectral unmixing
methods are used frequently in the literature, as they are sim-
ple and provide analytically tractable solutions. These meth-
ods are mainly based on the endmember extraction step in
the scene followed by the abundance estimation at each pixel.
For endmember extraction, many algorithms are developed in
the literature. These algorithms are categorized as geometri-
cal, statistical and sparse regression based approaches. Some
of the geometrical based approaches are N-FINDR [7], pixel
purity index (PPI) [8] and vertex component analysis (VCA)
[9]. These methods require one pure pixel per endmember
assumption which is not usually guaranteed. Minimum vol-
ume based algorithms are proposed such as minimum volume
simplex analysis (MVSA) [10] and simplex identification via
variable splitting and augmented Lagrangian (SISAL) [11]
without enforcing the pure pixel assumption. In addition,
nonnegative matrix factorization (NMF) based methods [12],
[13] are also proposed to identify the endmembers in the
absence of pure pixels. The statistical approaches include
the independent component analysis (ICA) with application
to hyperspectral data [14] and Bayesian approach [15]. In
the abundance estimation step, the fractional abundances of
extracted endmembers are calculated [16].
Sparse unmixing approach assumes that a mixed pixel is a
linear combination of spectral signatures from a priori avail-
able spectral library. The number of endmembers are small
compared to size of the spectral library, therefore only small
number of spectral signatures contributes to mixed pixel
which means that the fractional abundance vector is expected
to be sparse. Then, sparse unmixing approach aims at finding
the sparse abundance vector corresponding to the spectral
signatures in the library [17]–[19]. There are many studies
in the literature for sparse unmixing [20]–[25]. The SUnSAL
(sparse unmixing by variable splitting and augmented La-
grangian) [20] solves an unconstrained optimization problem
to obtain the abundance vector for each pixel. Collaborative
SUnSAL (CLSUnSAL) is introduced in [24] which uses the
idea that neighboring pixels in a hyperspectral scene have the
same set of endmembers. Therefore, the abundance matrix
has a joint-sparse structure [26]. Furthermore, using the prop-
erty of the piecewise smoothness of the abundance map, a
total-variation (TV) [27] based unmixing method is proposed
in [28] which is called SUnSAL-TV. There are also a few
VOLUME 4, 2016 1
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
recent studies for hyperspectral unmixing in the framework
of sparse unmixing approach in the literature [29]–[31].
In order to deal with outliers in the unmixing process,
robust unmixing algorithms are proposed in the literature
[11], [32]–[36]. SISAL [11] solves a nonconvex optimization
problem by solving a sequence of subproblems and a soft
penalty is imposed for outliers that is outside of the simplex.
The volume minimization (VolMin) [32] algorithm uses an
outlier-robust loss function onto the data fitting point and
a modified log-determinant loss function is used as volume
regularizer. The robust affine set fitting (RASF) [33] finds a
robust affine set and at the same time it detects and removes
the outliers. Besides, NMF based approaches are studied for
hyperspectral unmixing in the presence of mixed noise [34]–
[36]. A sparsity regularized robust NMF (RNMF) appears in
[34] that performs hyperspectral unmixing in the presence of
mixed noise. Recently, a TV regularized reweighted sparse
NMF (TV-RSNMF) is proposed for hyperspectral unmixing
that is also robust to noise [36]. Robust collaborative NMF
(R-CoNMF) [35] estimates the number of endmembers,
spectral signatures of endmembers and fractional abundances
of each endmember simultaneously to avoid errors in each
step of the unmixing process.
Besides this, low-rank representation (LRR) [37], [38]
is used for hyperspectral unmixing [39], [40]. A semi-
supervised LRR method for bilinear mixture model is pro-
posed in [39] to exploit the spatial correlation among the
neighboring pixels. Giampouras et. al propose an alternat-
ing direction sparse and low-rank unmixing algorithm (AD-
SpLRU) [40] to minimize the rank and sparsity simultane-
ously.
Since HSI contains noise in different forms, several LRR
based approaches are adapted HSI denoising [41]–[48]. Low-
rank matrix recovery (LRMR) [41] denoises HSI using the
Go Decomposition (GoDec) algorithm [49]. Zhu et. al pro-
pose a low-rank spectral nonlocal approach method to restore
the HSI [42]. The low-rank property is utilized to obtain the
precleaned patches and then they are clustered using spectral
nonlocal method. Global and local redundancy and correla-
tion (RAC) in spatial/spectral dimensions are investigated in
[43] to denoise HSI. Furthermore, since the noise level of the
different bands of the HSI are different, He et. al propose
the noise-adjusted iterative low-rank matrix approximation
(NAILRMA) for Gaussian noise and noise-adjusted iterative
low-rank matrix recovery (NAILRMR) for mixed noise [44].
The spatial smoothness and low-rank property of HSI are
studied in [45], which is termed as TV regularized low-rank
matrix factorization (LRTV), to restore the HSI. Moreover,
several works propose nonconvex low-rank approximation
methods to approximate the rank of HSI better. [46] propose
the weighted Schatten p-norm low-rank matrix approxima-
tion (WSN-LRMA) to approximate the rank of HSI. It ap-
proximates the rank of HSI in an iterative manner. Recently,
nonconvex low-rank matrix approximation (NonLRMA) [47]
is proposed for HSI denoising which approximates the rank
of HSI iteratively.
Moreover, there exists algorithms that performs unmixing
and denoising operations simultaneously [50], [51]. A joint
sparsity and total variation-based unmixing method (JSTV)
approach appears in [50] which removes mixed noise and
unmix HSI simultaneously. The coupled HSI denoising and
unmixing method (CHyDU) is studied in [51] that uses
spectral information as feedback to denoising scheme that
improves the denoising and unmixing results.
In this work, we simultaneously denoise and unmix the
hyperspectral data in the presence of mixed noise to enhance
the denoising and unmixing capability of the algorithm. In
contrast to denoise followed by an unmix framework, we si-
multaneously denoise and unmix the hyperspectral data in the
proposed scheme. The spectral distortion, which is a common
problem in the denoising, deteriorates the performance of the
unmixing algorithms. Therefore, spectral distortion should
be eliminated in the process of denoising. So, in the denois-
ing process, spectrally corrected data can be used after the
unmixing process. In this manner, denoising and unmixing
capability of individual algorithms are increased when they
are solved simultaneously. In the denoising part, we propose
a nonconvex low-rank and sparse decomposition approach to
remove the sparse noise and Gaussian noise. In the unmixing
part, we employ a semi-supervised sparse unmixing method
which uses a known spectral library. In the sparse unmixing
method, we utilize the same nonconvex approach similar to
denoising part to better promote the sparsity of the abundance
matrix.
The rest of the paper is organized as follows. Section
II gives necessary background and formulates the proposed
approach. Section III introduces the proposed method. The
simulated and real data experiments are given in Section IV.
Finally, Section V concludes the paper and some suggestions
and future works are given in this section.
II. PROBLEM FORMULATION
Suppose that a HSI obeys the noise degradation model as
f=u+n+e
where the matrices f,u,nand ehaving dimension of
m×n×Lrepresent the noisy HSI, clean HSI, Gaussian
noise and sparse noise, respectively. Using the patch-based
denoising framework, we first extract a subcube of size
d×d×Lcentered at pixel (i, j). For this subcube, we can
FIGURE 1: Low-rank matrix from subcube extracted from
HSI.
2VOLUME 4, 2016
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
write the noise model as
f(i,j)=u(i,j)+n(i,j)+e(i,j)(1)
Then, we write the matrix form of the observation model (1)
by converting the subcube to a matrix which is illustrated in
Fig. 1
Y=X+N+E(2)
Here Y,X,Nand Ehaving size of L×d2correspond to
f(i,j),u(i,j),n(i,j)and e(i,j), respectively.
Since hyperspectral data has a low-rank structure, robust
principal component analysis (RPCA) [52], [53] is utilized
to restore the HSI. RPCA is formulated as
min
X,E rank(X) + τkEk0s.t. Y =X+E(3)
where τis the regularization parameter and kEk0denotes the
l0norm of Ewhere it represents the number of nonzero terms
in E. However, (3) is difficult to solve due to the nonconvex
nature of rank function and l0norm. Therefore, the problem
is relaxed to a convex formulation by replacing the rank
function with nuclear norm denoted as k·kand l0norm
with l1norm. Then, convex formulation of (3) is written as
min
X,E kXk+τkEk1s.t. Y =X+E
where l1norm of Eand nuclear norm of Xare defined as
kEk1=Pi,j |Eij |and kXk=Piσi(X), respectively.
σi(X)denotes the ith singular value of X. However, this
formulation of RPCA does not consider the Gaussian noise.
Therefore, an additional constraint is added to remove the
Gaussian noise in [54]
min
X,E kXk+τkEk1s.t. kYXEkF
This formulation of RPCA and its variants are applied in
several HSI restoration approaches [44], [45]. By definition,
nuclear norm is defined as the sum of singular values, there-
fore, the rank of the matrix may not be represented efficiently.
For this reason, nonconvex rank approximation methods [46],
[47] are proposed to estimate the rank of HSI and restore the
HSI, which has satisfactory denoising performances. Gener-
ally, nonconvex rank approximation approaches use iterative
approaches to approximate the rank function in (3) which
causes longer computation times compared to nuclear norm
minimization.
Furthermore, based on linear mixture model (LMM), clean
hyperspectral data is composed of linear combination of
endmembers such that
X=MS (4)
where ML×qis the mixing matrix containing qendmem-
bers and Sq×mn is the abundance matrix which satisfy the
abundance non-negativity constraint (ANC): S0and
abundance sum-to-one constraint (ASC): 1TS= 1 due to
the physical considerations.
A. SIMULTANEOUS NONCONVEX DENOISING AND
UNMIXING
In this section, we formulate Simultaneous Nonconvex De-
noising and Unmixing (SNDeUn) method. The proposed
framework is illustrated in Fig. 2. We first extract a patch of
fixed size from HSI and solve the simultaneous denoising and
unmixing algorithm for each extracted patch until conver-
gence and collect the denoised patch and abundance matrix
of the patch. We perform the same operation for all patches
in the HSI. After all patches are done, we average the results
to obtain the clean HSI and abundance maps for all endmem-
bers. In the denoising part, we propose a denoising method
based on low-rank and sparse decomposition to remove the
mixed noise in the data. In the unmixing part, we employ a
sparse unmixing approach. In both denoising and unmixing
parts, we solve a nonconvex heuristic in the form of method
of multipliers which approximates the rank of hyperspectral
data and improves the sparsity of the abundance matrix.
SNDeUn is a unified optimization problem which is for-
mulated as
min
X,S,E
1
2kYXEk2
F+λGλ,p(σ(X)) + γGγ,p (S)
+τkEk1+β
2kXMSk2
F+IR+(S)
(5)
Here, Gλ,p and Gγ,p are nonconvex functions that approx-
imate the rank of Xand sparsity of S, respectively. IR+
is an indicator function meaning that the abundance matrix
has nonnegative values. λ,γ,τand βare the regularization
parameters for each regularizer. In this vein, denoising and
unmixing supports each other to obtain a better denoised
image and abundance map. It can be deduced that if unmix-
ing part is excluded from SNDeUn, optimization problem
becomes
min
X,E
1
2kYXEk2
F+λGλ,p(σ(X)) + τkEk1(6)
We call (6) as nonconvex low-rank denoising (NonLrDe). We
give the detailed analysis of SNDeUn here only. NonLrDe
can be analyzed accordingly. We resort the method of mul-
tipliers to solve the optimization problem. By introducing
auxiliary variables, (5) can be written as
min
X,S,E
1
2kYP1Ek2
F+λGλ,p(σ(P2)) + γGγ,p (P3)
+τkEk1+β
2kP4P5k2
F+IR+(P6)
s.t. P1=X;P2=X;P3=S
P4=X;P5=MS P6=S
(7)
we can arrange (7) in a closed form such that
min
X,S,E f(X, S, E)subject to F1X+F2S+ZP = 0
(8)
VOLUME 4, 2016 3
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
Clean
patch
Patch
abundance
Less
Noise
Spectral
correction
Unmixing
step
Denoising
step
Extract new patch
Average
results
Clean HSI
Abundance
maps
FIGURE 2: Simultaneous Nonconvex Denoising and Unmixing (SNDeUn).
where
f(X, S, E) =1
2kYP1Ek2
F+λGλ,p(σ(P2))
+γGγ ,p(P3) + τkEk1+β
2kP4P5k2
F
+IR+(P6)
P=PT
1, P T
2, P T
3, P T
4, P T
5, P T
6T
and
F1=
I
I
0
I
0
0
, F2=
0
0
I
0
M
I
Z=
I00000
0I0000
0 0 I000
000I0 0
0000I0
00000I
Algorithm 1 Pseudocode of the algorithm
1: Initialization: k= 0,µ,X(0),S(0) ,P(0),Λ(0)
2: repeat:
3: X(k+1) arg minXL(X, S(k), E(k), P (k),Λ(k))
4: S(k+1) arg minSL(X(k+1), S, E (k), P (k),Λ(k))
5: E(k+1) arg minEL(X(k+1), S (k+1), E, P (k),Λ(k))
6: P(k+1) arg minPL(X(k+1), E (k+1), S(k+1) , P, Λ(k))
7: Λ(k+1) Λ(k)F1X(k+1) F2S(k+1) ZP (k+1)
8: Update iteration kk+ 1
9: until some stopping criteria is satisfied
ADMM algorithm for the optimization problem (8) is
given in Algorithm 1 where the augmented Lagrangian for-
mulation is given as
L(X, E, S, P, Λ) =f(X, S, E )
+µ
2kF1X+F2S+ZP Λk2
F
(9)
µis a positive constant called as augmented Lagrangian
penalty parameter and Λis the Lagrange multipliers as-
sociated to the constraint F1X+F2S+ZP = 0. ADMM
minimizes Lsequentially with respect to X,S,Eand
Pat each iteration and then Lagrange multipliers are up-
dated. ADMM algorithm for Algorithm 1 stops either max-
imum iteration number is reached or kF1X(k)+F2S(k)+
ZP (k)k2
F< is satisfied where =p(3q+L)K0.0
is the relative error tolerance which depends on the spectral
library size and image dimensions. The detailed analysis of
Algorithm 1 is similar to works in [20], [24], [55] and it is
given in Appendix.
4VOLUME 4, 2016
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
B. CONNECTION TO EXISTING WORK
In literature, HSI denoising methods based on nonconvex
low-rank approximation use iterative approaches to approx-
imate the rank of HSI [46], [47]. The WSN-LRMA [46]
minimizes the weighted Schatten p-norm to estimate the
rank of hyperspectral data iteratively, which uses the ideas
of robust principal component analysis (RPCA) [52], [53]
to denoise hyperspectral data. NonLRMA [47] uses a non-
convex regularizer in the solution which has been solved
iteratively and proved to have a fast convergence rate. Besides
this, CHyDU [51] solves a coupled denoising and unmixing
problem which uses a dictionary based sparse representation
[56] and nuclear norm approximation to denoise HSI and
does not consider the mixed noise in the HSI. Furthermore,
sparsity of the abundance matrix is elevated by l1norm
regularizer. In our study, we utilize a patch based approach
and a nonconvex heuristic in the solutions of both denoising
and unmixing parts which accounts for mixed noise. The rank
of HSI and the sparsity of abundance maps are provided by
a simple nonconvex shrinkage operations without using an
iterative approach.
III. EXPERIMENTAL RESULTS
In this section, we present the results of proposed algorithms
for both denoising and unmixing. We split the experimental
results into three parts. In the first part of the experiments,
we give denoising results of NonLrDe and compare it to the
recently proposed low-rank based schemes. In the second
part of the experiments, results of SNDeUn are given. The
denoising and unmixing results of SNDeUn are denoted by
SNDeUn-De and SNDeUn-Un, respectively. In the third part,
real data experiments are given. The performance of the
algorithms are measured based on the following metrics. The
peak signal-to-noise ratio (PSNR) index and the structural
similarity (SSIM) index [57] are used to compare the denois-
ing results of the algorithms.
PSNR and SSIM for each band of HSI is defined as
PSNRl(ˆ
Xl,Xl) = 10 log mn
Pm
i=1 Pn
j=1[ˆ
Xl(i, j)X(i, j )]2
SSIMl(ˆ
Xl,Xl) = (2µXlµˆ
Xl+C1)(2σˆ
XlXl+C2)
(µ2
Xl+µ2
ˆ
Xl+C1)(σ2
Xl+σ2
ˆ
Xl+C2)
Moreover, mean PSNR (MPSNR) and mean SSIM (MSSIM)
indices are defined as
MPSNR =1
L
L
X
l=1
PSNRl(ˆ
Xl,Xl)
MSSIM =1
L
L
X
l=1
SSIMl(ˆ
Xl,Xl)
Here, Xland ˆ
Xldenotes the original and restored hyperspec-
tral images in lth band. µXland µˆ
Xlare the mean intensity
values of Xland ˆ
Xl.σ2
Xland σ2
ˆ
Xl
are the variances of Xl
and ˆ
Xl, respectively. σˆ
XlXlis the covariance between Xl
and ˆ
Xl.
The signal to reconstruction error (SRE), root mean square
error (RMSE) and spectral angle mapper (SAM) are used to
evaluate unmixing results of the algorithms. SRE is defined
as
SRE = 10 log10 E[kSk2
2]
E[kSˆ
Sk2
2]
Here, Sis the ground truth abundance map and ˆ
Sdenotes the
estimated abundance map and RMSE is described as
RMSE =v
u
u
t
1
mn
q
X
i=1
kSiˆ
Sik2
where Siand ˆ
Siare the actual and estimated abundance
vectors, respectively.
SAM is the angle in degree between the estimated and
actual spectra and it is defined as
SAM =1
mn
mn
X
i=1
arccos ˆuT
iui
kˆuik2kuik2
where ˆuiand uiare the estimated and actual spectrum of the
individual pixels, respectively.
A. SIMULATED DATA EXPERIMENTS FOR MIXED NOISE
REMOVAL
In this section, we perform several experiments for HSI
denoising under different noise level and scenarios. We com-
pare NonLrDe method with state-of-the-art low-rank based
denoising methods proposed recently in the literature. These
are LRMR [41], [50], NAILRMR [44], NonLRMA [47]
and WSN-LRMA [46]. The codes of these algorithms are
provided by the authors. We use Indian Pines [58] synthetic
data set as simulated data 1 (SD1), which is created similarly
as in [45]. The dimensions of SD1 is 145 ×145 ×224.
The noisy HSI is generated by adding Gaussian noise with
standard deviation σand impulse noise of percentage levels
P. In the experiments, we add constant level of Gaussian and
impulse noise to all bands of the synthetic data set. In order
to simulate mixed noise scenario, we add stripes to selected
bands of the HSI and constant level of Gaussian and impulse
noise to all bands of HSI. Stripes were simulated on 30%
of the bands which were selected randomly. The number of
stripes of each selected band ranges from 3 to 10 lines.
The compared algorithms are solved using the optimal
parameter set for each solver. MPSNR and MSSIM results
of the algorithms are reported in Table 1. The values with
the highest MPSNR and MSSIM are given in bold and the
values with the second highest MPSNR and MSSIM are
underlined. We can observe that at higher noise levels, NonL-
rDe achieves better MPSNR and MSSIM values compared
to other methods, which means it has a robust denoising
performance under moderate noise levels. Fig. 3 shows the
denoising results of the different algorithms visually. LRMR
and NAILRMR have similar results visually. WSN-LRMA,
NonLRMA and NonLrDe have also similar visual qualities
as Table 1 indicates. We also show the PSNR and SSIM
VOLUME 4, 2016 5
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
values of each band in Fig. 4. We can observe that NonLrDe
has slightly higher PSNR and SSIM values for each band.
Furthermore, we evaluate the spectral signatures before and
after denoising. The reflectance values for the pixel (100,100)
are shown in Fig. 5 for all algorithms. WSN-LRMA, NonL-
RMA and NonLrDe have almost same spectrums which are
able to remove the ripples in the spectrums. Furthermore, we
investigate the nonconvexity parameter pand convergence of
NonLrDe. Fig. 6(a) shows the MPSNR values of NonLrDe
with respect to nonconvexity parameter p. It can be observed
that when pincreases towards to 1, PSNR value of NonLrDe
decreases. Fig. 6(b) shows the PSNR values versus iteration
number. PSNR value remains constant after 50 iterations.
Therefore, it is enough for NonLrDe to convergence after 50
iterations.
(a) Clean (b) Noisy (c) LRMR
(d) NAILRMR (e) NonLRMA (f) WSN-LRMA (g) NonLRDe
FIGURE 3: Denoising results on SD1. False-color image
(R:6, G:88, B:221), (σ= 0.1,P= 0.2).
B. SIMULTANEOUS DENOISING AND UNMIXING
RESULTS
In this section, we test denoising and unmixing performance
of SNDeUn. We compare the SNDeUn with the state-of-
the-art sparse unmixing methods proposed in the literature.
These methods are CLSUnSAL [24], ADSpLRU [40], JSTV
[50] and SUnSAL-TV [28]. CLSUnSAL, ADSpLRU and
SUnSAL-TV do not consider mixed noise in their formu-
lations, therefore we consider only Gaussian noise in the
measurements to compare these algorithms. JSTV [50] is
designed to handle mixed noise in the measurements, also
SUnSAL-TV has ability to remove the mixed noise at low
noise levels. Therefore, in the mixed noise scenario, we
compare SNDeUn with JSTV and SUnSAL-TV.
In the simulations, digital spectral library (splib06) [59]
obtained from the U.S. Geological Survey (USGS) is used.
It includes the spectra of 498 materials measured in 224
spectral bands distributed uniformly in the interval 0.4 and
2.5 µm. We create a spectral library Mby selecting spectral
signatures from splib06 whose spectral angles are greater
than 10 which contains 62 endmembers. The simulated data
2 (SD2) for unmixing scenario is a 48 ×48 HSI where the
50 100 150 200
20
25
30
35
Band number
PSNR
LRMR
NAILRMR
WSN−LRMA
NonLRMA
NonLrDe
(a)
50 100 150 200
0.6
0.7
0.8
0.9
1
Band number
SSIM
LRMR
NAILRMR
WSN−LRMA
NonLRMA
NonLrDe
(b)
FIGURE 4: PSNR and SSIM values of each band of the
experimental results for SD1 (σ= 0.1, P = 0.2). (a) PSNR
values. (b) SSIM values.
rank of the data is adjusted to 4 by selecting the endmembers
from M. It is created by following the same procedure given
in [28]. The HSI and the four abundance maps are shown in
Fig. 7.
We compare SNDeUn-Un on simulated data by adding
constant level of Gaussian noise with standard deviation
σto each band of the HSI. We adjusted the parameters
of SNDeUn-Un and all compared algorithms to their best
performances in terms of the RMSE. Table 2 reports the SRE
and RMSE values of SNDeUn-Un. It can be observed clearly
that SNDeUn-Un has best performances at all noise levels.
In order to measure the unmixing performance of
SNDeUn-Un under mixed noise scenarios, we created the
noisy data by adding constant levels of Gaussian noise with
standard deviation σand impulse noise of constant per-
centage Pto all bands of simulated data. For mixed noise
scenario, stripes are added to selected bands of the HSI and
6VOLUME 4, 2016
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
TABLE 1: MPSNR and MSSIM VALUES OF DENOISING RESULTS FOR SD1 IN THE PRESENCE OF MIXED NOISE.
Noise
Type
Evaluation Index LRMR NAILRMR NonLRMA WSN-LRMA NonLrDe
σ= 0.025
P= 0.05
MPSNR
MSSIM
45.80
0.9939
47.45
0.9957
47.75
0.9945
48.64
0.9955
48.61
0.9956
σ= 0.05
P= 0.1
MPSNR
MSSIM
40.94
0.9772
42.44
0.9846
41.59
0.9778
40.29
0.9754
42.16
0.9810
σ= 0.1
P= 0.2
MPSNR
MSSIM
28.40
0.8501
29.71
0.8983
34.52
0.9100
34.10
0.9150
34.96
0.9158
(Mixed noise)
σ= 0.1
P= 0.1
stripes
MPSNR
MSSIM
37.39
0.9464
38.29
0.9573
38.38
0.9653
39.00
0.9778
39.49
0.9683
0 50 100 150 200 250
0
0.2
0.4
0.6
0.8
Band number
Reflectance
(a) Original
0 50 100 150 200 250
0
0.2
0.4
0.6
0.8
1
Band number
Reflectance
(b) Noisy
0 50 100 150 200 250
0
0.2
0.4
0.6
0.8
Band number
Reflectance
(c) LRMR
0 50 100 150 200 250
0
0.2
0.4
0.6
0.8
Band number
Reflectance
(d) NAILRMR
0 50 100 150 200 250
0
0.2
0.4
0.6
0.8
Band number
Reflectance
(e) NonLRMA
0 50 100 150 200 250
0
0.2
0.4
0.6
0.8
Band number
Reflectance
(f) WSN-LRMA
0 50 100 150 200 250
0
0.2
0.4
0.6
0.8
Band number
Reflectance
(g) NonLRDe
FIGURE 5: Spectrum of pixel (100, 100) in the denoised results of SD1 (σ= 0.1, P = 0.2).
constant level of Gaussian noise and impulse noise are added to all bands of HSI. Stripes were simulated on 10% of the
VOLUME 4, 2016 7
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
0 0.2 0.4 0.6 0.8 1
20
25
30
35
p
PSNR
(a)
10 20 30 40 50
10
15
20
25
30
35
Iteration
PSNR
(b)
FIGURE 6: PSNR values versus pand iteration number for
SD1 (σ= 0.1,P= 0.2).
Pixels
Pixels
10 20 30 40
5
10
15
20
25
30
35
40
45
Pure
Materials
Mixtures of 2
Endmembers
Mixtures of 3
Endmembers
Mixtures of 4
Endmembers
(a)
(b) (c) (d) (e)
FIGURE 7: (a) SD2 (b)-(e) Abundances of endmembers
bands which were selected randomly. The number of stripes
of each selected band ranges from 3 to 5 lines.
We compare SNDeUn-Un with SUnSAL-TV and JSTV.
Table 3 reports the unmixing results under mixed noise. It
can be observed that, the performance of SNDeUn-Un has
the best performance at all noise levels. From the results we
can conclude that, SNDeUn is a robust method to unmix
hyperspectral data in the presence of mixed noise.
In Table 4, we report the results of the denoising perfor-
mance of the SNDeUn-De in the presence of mixed noise. We
compare the denoising results of SNDeUn-De by NonLrDe
together with LRMR [41], [50], NAILRMR [44], NonLRMA
[47] and WSN-LRMA [46]. It can be observed that SNDeUn-
De has best PSNR and SSIM values at all noise levels due to
the simultaneous solution of unmixing and denoising steps in
SNDeUn. Moreover, NonLrDe has the second highest PSNR
and SSIM values at higher noise levels.
C. REAL DATA EXPERIMENTS
In real data experiment, we use Hyperspectral Digital Im-
agery Collection Experiment (HYDICE) urban area data set
and related spectral library [60] which can be downloaded at
http://www.tec.army.mil/hypercube. The size of the data set
is 307×307 ×210 and spectral library contains 49 signatures
distributed in the interval 0.35 and 2.5 µm. Due to the atmo-
spheric effects and water absorbtion bands, some bands of the
urban data set have strong noise such as stripes and impulse
noise as well as other types of noise. In [41], [46], noise
effected bands and water absorbtion bands are removed in the
simulations, whereas we use all of the bands of the urban data
set in the simulations to show the robustness of the proposed
algorithms. We present the denoising and unmixing results of
both NonLrDe and SNDeUn-De, respectively. We compare
NonLrDe and SNDeUn with LRMR [41] , NonLRMA [47]
and WSN-LRMA [46]. The patch size, step size, rank and
sparsity parameter of LRMR is set to 20, 4, 4 and 7000,
respectively. For WSN-LRMA, regularization parameter for
low rank and sparse components are set to 0.01 and 1.2,
patch size, step size and nonconvexity parameter are set to
20, 7, 0.7, respectively. For NonLRMA, we use the param-
eters given in [47]. The real data experiment parameters for
NonLRDe and SNDeUn are given in Table 5. We compare
different bands of the Urban data set that have different noise
structure in the individual bands. Fig. 8 shows the denoising
results of the bands 87, 207 and 108 of urban data. Band 87
has slight noise whereas band 207 and 108 have strong noise.
Band 87 has little noise therefore all of the algorithms have
nearly same results visually. For band 108, it can be observed
that SNDeUn-De has best performance visually compared
to other algorithms. Band 207 is polluted by stripes and
impulse noise. NonLrDe and NonLRMA has similar results
visually and stripes are removed in the denoised images.
However, WSN-LRMA and LRMR leave stripes in their
denoised images. SNDeUn-De has the best results visually
in removing the stripes and impulse noise as well as better
image quality visually.
8VOLUME 4, 2016
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
TABLE 2: RMSE AND SRE VALUES OF UNMIXING RESULTS FOR SD2 IN THE PRESENCE OF GAUSSIAN NOISE
(RMSE×103).
Noise
Type
Evaluation Index CLSUnSAL SUnSAL-TV JSTV ADSpLRU SNDeUn-Un
σ= 0.025 SRE
RMSE
SAM
10.58
20.85
0.49
20.89
6.36
0.28
22.24
5.45
0.20
16.78
10.22
0.27
25.93
3.56
0.21
σ= 0.05 SRE
RMSE
SAM
6.65
32.77
0.95
13.04
15.71
0.55
14.93
12.64
0.42
11.81
18.10
0.48
21.27
6.09
0.38
σ= 0.1SRE
RMSE
SAM
3.81
45.50
1.77
10.21
18.34
1.01
11.19
19.45
0.78
7.62
29.31
1.07
12.09
17.52
0.86
TABLE 3: RMSE AND SRE VALUES OF UNMIXING RESULTS FOR SD2 IN THE PRESENCE OF MIXED NOISE
(RMSE×103).
Noise
Type
Evaluation Index SUnSAL-TV JSTV SNDeUn-Un
σ= 0.025
P= 0.05
SRE
RMSE
SAM
10.23
21.71
1.68
21.36
6.03
0.22
24.71
4.10
0.24
σ= 0.05
P= 0.1
SRE
RMSE
SAM
7.28
30.51
2.87
14.37
13.48
0.46
19.64
7.35
0.45
σ= 0.1
P= 0.2
SRE
RMSE
SAM
5.41
37.83
5.53
8.47
26.59
0.96
11.04
19.77
0.88
(Mixed noise)
σ= 0.05
P= 0.05
stripes
SRE
RMSE
SAM
10.69
20.59
1.75
14.82
12.81
0.45
20.15
6.93
0.43
TABLE 4: MPSNR and MSSIM VALUES OF DENOISING RESULTS FOR SD2 IN THE PRESENCE OF MIXED NOISE.
Noise
Type
Evaluation
Index LRMR NAILRMR NonLRMA WSN-LRMA NonLrDe SNDeUn-De
σ= 0.025
P= 0.05
MPSNR
MSSIM
47.23
0.9903
48.87
0.9924
46.80
0.9885
47.03
0.9886
47.73
0.9909
49.86
0.9929
σ= 0.05
P= 0.1
MPSNR
MSSIM
35.36
0.9394
36.28
0.0.9552
41.66
0.9659
39.75
0.9506
42.19
0.9707
43.84
0.9742
σ= 0.1
P= 0.2
MPSNR
MSSIM
28.20
0.7092
31.61
0.0.8054
34.68
0.8652
35.27
0.8730
35.43
0.8811
37.16
0.8921
(Mixed noise)
σ= 0.05
P= 0.05
stripes
MPSNR
MSSIM
41.36
0.9615
42.12
0.9694
41.52
0.9663
39.36
0.9215
42.13
0.9703
44.35
0.9762
In the unmixing experiment, we give the results of Urban
for two cases. In the first case, water absorbtion bands of
the Urban data are removed. We use 189 bands of Urban
data. These bands are 1-104, 110-138 and 152-207. We call
the resultant image as high-SNR image. High-SNR image
also contain stripes and impulse noise. In the second case,
we include atmospheric bands and do not remove any band
of the Urban data in the experiments which we call it as
low-SNR image. Since, there is no ground truth abundance
maps for Urban data, we use abundance maps obtained by
[13], [61] as benchmark abundance maps. Fig. 9 shows the
abundance maps for "Asphalt", "Grass", "Roof" and "Tree"
obtained by SUnSAL-TV, JSTV and SNDeUn as well as
benchmark abundance maps for each spectral signature. It
VOLUME 4, 2016 9
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
can be observed that abundance maps obtained by SNDeUn
and JSTV are visually similar to benchmark abundance maps.
Fig. 10 shows the abundance maps obtained by SUnSAL-
TV, JSTV and SNDeUn for low-SNR image. For this image,
abundance map obtained by SNDeUn is much similar to
benchmark abundance maps. which is also visually similar to
Fig. 9. We can conclude that SNDeUn is a robust unmixing
method even if the data is highly polluted by atmospheric
effects and other kinds of noise sources.
D. SELECTION OF PARAMETERS AND COMPUTATION
TIME
The optimal parameters of NoNLrDe and SNDeUn are ad-
justed by varying the parameters and recording the best
parameter set that gives the minimum RMSE value. The
parameters used in the experiments are given in Table 5.
Although NonLrDe and SNDeUn have many parameters
including regularization parameters, patch size, step size and
noncovnexity parameter, we only change the value of λ
in the experiments based on the noise level change in the
simulated data. The other parameters are fixed as Table 5
reports. Computation times of the algorithms for real data
experiments are given in Table 6. Clearly, NonLRMA is
the fastest of the algorithms under comparison. In real data
experiments, block size and step size of NonLRMA is adjust
to 50 and 16 as it is suggested in the paper [47]. Block size
and step size of NonLrDe and SNDeUn are fixed to 20 and 8,
respectively. Therefore, they have longer computation times.
However, NonLrDe is the second fastest algorithm under
comparison. Also, SNDeUn has a higher computation time
compared to NonLrDe due to the sparse unmixing step in the
solution.
IV. CONCLUSION
In this paper, we have proposed a simultaneous denoising
and sparse unmixing method for HSI in the presence of
mixed noise. In the proposed method, Gaussian and sparse
noise components are removed based on a nonconvex low-
rank and sparse decomposition scheme which do not use
an iterative approach to approximate the rank of the data.
Nonconvex low-rank approximation uses a simple shrinkage
operation that approximates the rank of the data. In the sparse
unmixing part, we use a nonconvex regularizer to enhance
the sparsity of the abundance matrix. Denoising and unmix-
ing parts are solved simultaneously to increase performance
and robustness of the individual denoising and unmixing
methods. Experiments on simulated and real data sets show
that the proposed method is effective in HSI unmixing and
denoising and outperforms the other algorithms proposed in
the literature.
APPENDIX
The detailed algorithm of SnDeUn is presented in Algorithm
2. We first expand the augmented Lagrangian formulation in
Algorithm 2 Algorithm for SNDeUn
1: Initialization: k= 0,µ,X(0),S(0) ,P(0)
1, . . . , P (0)
6,
2: Λ(0)
1, . . . , Λ(0)
6
3:
4: repeat:
5: X(k+1) arg minXL(X, E(k), S(k),
6: P(k)
1, . . . , P (k)
6,
7: Λ(k)
1, . . . , Λ(k)
6)
8: S(k+1) arg minSL(X(k+1), E (k), S,
9: P(k)
1, . . . , P (k)
6,
10: Λ(k)
1, . . . , Λ(k)
6)
11: E(k+1) arg minEL(X(k+1), S (k+1), E,
12: P(k)
1, . . . , P (k)
6)
13: for i=1:6
14: P(k+1)
iarg minPiL(X(k+1), S (k+1),
15: P(k)
1, . . . , Pi, . . . , P (k)
6)
16: end
17: Update Lagrange multipliers:
18: Λ(k+1)
1Λ(k)
1X(k+1) +P(k+1)
1
19: Λ(k+1)
2Λ(k)
2X(k+1) +P(k+1)
2
20: Λ(k+1)
3Λ(k)
3S(k+1) +P(k+1)
3
21: Λ(k+1)
4Λ(k)
4X(k+1) +P(k+1)
4
22: Λ(k+1)
5Λ(k)
5MS(k+1) +P(k+1)
5
23: Λ(k+1)
6Λ(k)
6S(k+1) +P(k+1)
6
24: Update iteration kk+ 1
25: until some stopping criteria is satisfied
(9) as
L(X, E, P1, P2, P3, P4, P5, P6,Λ1,Λ2,Λ3,Λ4,Λ5,Λ6) =
1
2kYP1Ek2
F+λGλ/µ,p(σ(P2)) + γGγ/µ,p (P3)+
τkEk1+β
2kP4P5k2
F+IR+(P6)
+µ
2kXP1Λ1k2
F+µ
2kXP2Λ2k2
F
+µ
2kSP3Λ3k2
F+µ
2kXP4Λ4k2
F
+µ
2kMS P5Λ5k2
F+µ
2kSP6Λ6k2
F
(10)
then the optimization problem is carried out over the variable
Xby ignoring the terms that do not contain the variable Xin
(10) which leads to the following optimization problem
X(k+1) arg min
X
µ
2kXP(k)
1Λ(k)
1k2
F+
µ
2kXP(k)
2Λ(k)
2k2
F+
µ
2kXP(k)
4Λ(k)
4k2
F
(11)
The iterative closed form solution of (11) is
X(k+1) (1/3)[(P(k)
1+ Λ(k)
1)+(P(k)
2+ Λ(k)
2)
+ (P(k)
4+ Λ(k)
4)]
10 VOLUME 4, 2016
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
TABLE 5: PARAMETER SETTING OF PROPOSED ALGORITHMS.
Solver Dataset λ τ γ β µ p Patch Size Step Size
NonLrDe SD1
SD2
Real Data
{1, . . . , 5}
{1, . . . , 5}
5
0.05 -
-
-
-
-
-
0.5
0.5
0.25
0.01 20 8
SNDeUn SD2
Real Data
{1, . . . , 5}
50.05 104
104
50.5
0.25
0.01 20 8
Original LRMR WSN-LRMA NonLRMA NonLrDe SNDeUn-De
Original LRMR WSN-LRMA NonLRMA NonLrDe SNDeUn-De
Original LRMR WSN-LRMA NonLRMA NonLrDe SNDeUn-De
FIGURE 8: From top to bottom row, denoising results of band 87, 108 and 207 in the real data experiment.
TABLE 6: COMPUTATION TIME OF THE ALGO-
RITHMS ON REAL DATA SET.
Solver Computation Time (Hour)
LRMR 0.659
WSN-LRMA 0.839
NonLRMA 0.250
NonLrDe 0.345
SNDeUn-De 0.446
Similarly, the reduced optimization problem for Sis
S(k+1) (MTM+ 2I)1[MT(P(k)
5+ Λ(k)
5)+(P(k)
3+ Λ(k)
3)
+ (P(k)
6+ Λ(k)
6)]
E(k+1) arg min
E
1
2kYP(k)
1Ek2
F+τkEk1(12)
the solution of (12) is
E(k+1) soft(YP(k)
1, τ )
For P1, the reduced optimization problem is
P(k+1)
1arg min
P1
1
2kYP1E(k+1)k2
F+
µ
2kX(k+1) P1Λ(k)
1k2
F
(13)
The solution to (13) is
P(k+1)
11
1 + µ(YE(k+1) +µ(X(k+1) Λ(k)
1))
P2is obtained by solving the reduced optimization problem
P(k+1)
2arg min
P2
λGλ/µ,p(σ(P2))+ µ
2kX(k+1)P2Λ(k)
2k2
F
(14)
VOLUME 4, 2016 11
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
Ground Truth SUnSAL-TV JSTV SNDeUn
Ground Truth SUnSAL-TV JSTV SNDeUn
Ground Truth SUnSAL-TV JSTV SNDeUn
Ground Truth SUnSAL-TV JSTV SNDeUn
FIGURE 9: Abundance maps obtained by SUnSAL-TV, JSTV and SNDeUn for high-SNR image. From left column to right
column, abundance maps corresponding to Asphalt, Grass, Roof and Tree, respectively.
In order to obtain the solution of (14), we use the method
proposed in [62]. Suppose we have an optimization problem
of the form
min
WGδ,p(W) + 1
2δkWVk2
F(15)
The proximal function of Gδ,p(W) = Pi,j gδ,p(wij )is p-
shrinkage operation [62] for each entry of Gδ,p(W)defined
as
shrinkp(v, δ ) = max{0,|v| − δ|v|p1}v/|v|
which is defined as nonconvex shrinkage.
So, the solution for P(k+1)
2is nonconvex shrinkage. First,
SVD of (X(k+1) Λ(k)
2)is obtained as
SV D(X(k+1) Λ(k)
2) = UΣVT
then a nonconvex singular value shrinkage operator denoted
as Dp,λ/µ is applied to eigenvalue matrix Σas
Dp,λ/µ(Σ) =shrinkp, λ/µ)
= max(0,|σi| − λ
µ|σi|p1)σi/|σi|
12 VOLUME 4, 2016
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
Ground Truth SUnSAL-TV JSTV SNDeUn
Ground Truth SUnSAL-TV JSTV SNDeUn
Ground Truth SUnSAL-TV JSTV SNDeUn
Ground Truth SUnSAL-TV JSTV SNDeUn
FIGURE 10: Abundance maps obtained by SUnSAL-TV, JSTV and SNDeUn for low-SNR image. From left column to right
column, abundance maps corresponding to Asphalt, Grass, Roof and Tree, respectively.
where σiis the ith diagonal element of Σ. Then P(k+1)
2is
obtained as
P2(k+1) =UDp,λ/µ(Σ)VT
Similarly, the reduced optimization problem for P3is
P(k+1)
3arg min
P3
γGγ /µ,p(P3) + µ
2kS(k+1) P3Λ(k)
3k2
F
where the solution is is
P(k+1)
3=shrinkp(S(k+1) Λ(k)
3, λ/µ)
where nonconvex shrinkage operator is applied in element-
wise manner. The solution for P4and P5is carried out by
solving
(P(k+1)
4, P (k+1)
5)arg min
P4,P5
β
2kP4P5k2
F+
µ
2kX(k+1) P4Λ(k)
4k2
F+
µ
2kMS(k+1) P5Λ(k)
5k2
F
VOLUME 4, 2016 13
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
whose solutions are
P(k+1)
41
β+µ(βP5+µ(X(k+1) Λ(k)
4)))
P(k+1)
51
β+µ(βP4+µ(M S(k+1) Λ(k)
5))
Finally to compute P6, we solve the following optimiza-
tion problem
P(k+1)
6arg min
P6
IR+(P6) + µ
2kS(k+1) P6Λ(k)
6k2
F
which is the projection of S(k+1) Λ(k)
6onto the nonnegative
orthant defined as
P(k+1)
6max(S(k+1) Λ(k)
6,0)
REFERENCES
[1] T. Dundar and T. Ince, “Sparse representation-based hyperspectral image
classification using multiscale superpixels and guided filter,” IEEE Geosci.
Remote Sens. Lett., vol. 16, no. 2, pp. 246–250, Feb 2019.
[2] T. Ince, “Hyperspectral image denoising using group low-rank and spatial-
spectral total variation,” IEEE Access, vol. 7, pp. 52 095–52 109, 2019.
[3] X. Lu, W. Zhang, and X. Li, “A Hybrid Sparsity and Distance-Based
Discrimination Detector for Hyperspectral Images,” IEEE Trans. Geosci.
Remote Sens., vol. 56, no. 3, pp. 1704–1717, Mar. 2018.
[4] C. Yi, Y. Zhao, J. Yang, J. C. Chan, and S. G. Kong, “Joint hyperspectral
superresolution and unmixing with interactive feedback,” IEEE Trans.
Geosci. Remote Sens., vol. 55, no. 7, pp. 3823–3834, July 2017.
[5] C. Yi, Y. Zhao, and J. C. Chan, “Spectral super-resolution for multispectral
image based on spectral improvement strategy and spatial preservation
strategy,” IEEE Trans. Geosci. Remote Sens., pp. 1–15, 2019.
[6] N. Keshava and J. F. Mustard, “Spectral unmixing,” IEEE Signal Process.
Mag., vol. 19, no. 1, pp. 44–57, Jan 2002.
[7] M. E. Winter, “N-findr: an algorithm for fast autonomous spectral end-
member determination in hyperspectral data,” in Proc. SPIE, vol. 3753,
1999, pp. 266–275.
[8] J. W. Boardman, F. A. Kruse, and R. O. Green, “Mapping target signatures
via partial unmixing of AVIRIS data,” in Fifth JPL Airborne Earth Science
Workshop, vol. 95. JPL Publication, 1995, pp. 23–26.
[9] J. M. P. Nascimento and J. M. B. Dias, “Vertex component analysis: a
fast algorithm to unmix hyperspectral data,” IEEE Trans. Geosci. Remote
Sens., vol. 43, no. 4, pp. 898–910, April 2005.
[10] J. Li, A. Agathos, D. Zaharie, J. M. Bioucas-Dias, A. Plaza, and X. Li,
“Minimum volume simplex analysis: A fast algorithm for linear hyper-
spectral unmixing,” IEEE Trans. Geosci. Remote Sens., vol. 53, no. 9, pp.
5067–5082, Sept 2015.
[11] J. M. Bioucas-Dias, “A variable splitting augmented lagrangian approach
to linear spectral unmixing,” in 2009 First Workshop on Hyperspectral
Image and Signal Processing: Evolution in Remote Sensing, Aug 2009,
pp. 1–4.
[12] L. Miao and H. Qi, “Endmember extraction from highly mixed data using
minimum volume constrained nonnegative matrix factorization,” IEEE
Trans. Geosci. Remote Sens., vol. 45, no. 3, pp. 765–777, March 2007.
[13] S. Jia and Y. Qian, “Constrained nonnegative matrix factorization for
hyperspectral unmixing,” IEEE Trans. Geosci. Remote Sens., vol. 47,
no. 1, pp. 161–173, Jan 2009.
[14] J. M. P. Nascimento and J. M. B. Dias, “Does independent component
analysis play a role in unmixing hyperspectral data?” IEEE Trans. Geosci.
Remote Sens., vol. 43, no. 1, pp. 175–187, Jan 2005.
[15] N. Dobigeon, S. Moussaoui, J.-Y. Tourneret, and C. Carteret, “Bayesian
separation of spectral sources under non-negativity and full additivity
constraints,” Signal Process., vol. 89, no. 12, pp. 2657–2669, Dec. 2009.
[Online]. Available: http://dx.doi.org/10.1016/j.sigpro.2009.05.005
[16] J. M. Bioucas-Dias, A. Plaza, N. Dobigeon, M. Parente, Q. Du, P. Gader,
and J. Chanussot, “Hyperspectral unmixing overview: Geometrical, statis-
tical, and sparse regression-based approaches,” IEEE J. Sel. Topics Appl.
Earth Observ. Remote Sens., vol. 5, no. 2, pp. 354–379, April 2012.
[17] B. K. Natarajan, “Sparse approximate solutions to linear systems,” SIAM
Journal on Computing, vol. 24, no. 2, pp. 227–234, 1995.
[18] S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition
by basis pursuit,” SIAM Journal on Scientific Computing, vol. 20, no. 1,
pp. 33–61, 1998.
[19] E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from
incomplete and inaccurate measurements,” Communications on Pure and
Applied Mathematics, vol. 59, no. 8, pp. 1207–1223, 2006.
[20] J. M. Bioucas-Dias and M. A. T. Figueiredo, “Alternating direction al-
gorithms for constrained sparse regression: Application to hyperspectral
unmixing,” in 2010 2nd Workshop on Hyperspectral Image and Signal
Processing: Evolution in Remote Sensing, June 2010, pp. 1–4.
[21] M. D. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Sparse unmixing of
hyperspectral data,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 6, pp.
2014–2039, June 2011.
[22] N. Dobigeon, J. Y. Tourneret, and C. I. Chang, “Semi-supervised linear
spectral unmixing using a hierarchical bayesian model for hyperspectral
imagery,” IEEE Trans. Signal Process., vol. 56, no. 7, pp. 2684–2695, July
2008.
[23] K. E. Themelis, A. A. Rontogiannis, and K. D. Koutroumbas, “A novel
hierarchical bayesian approach for sparse semisupervised hyperspectral
unmixing,” IEEE Trans. Signal Process., vol. 60, no. 2, pp. 585–599, Feb
2012.
[24] M. D. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Collaborative sparse
regression for hyperspectral unmixing,” IEEE Trans. Geosci. Remote
Sens., vol. 52, no. 1, pp. 341–354, Jan 2014.
[25] F. Chen and Y. Zhang, “Sparse hyperspectral unmixing based on con-
strained lp - l2 optimization,” IEEE Geosci. Remote Sens. Lett., vol. 10,
no. 5, pp. 1142–1146, Sept 2013.
[26] J. A. Tropp, A. C. Gilbert, and M. J. Strauss, “Algorithms for simultaneous
sparse approximation. part i: Greedy pursuit,” Signal Processing, vol. 86,
no. 3, pp. 572 – 588, 2006.
[27] L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise
removal algorithms,” Phys. D, vol. 60, no. 1-4, pp. 259–268, Nov. 1992.
[28] M. D. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Total variation spatial
regularization for sparse hyperspectral unmixing,” IEEE Trans. Geosci.
Remote Sens., vol. 50, no. 11, pp. 4484–4502, Nov 2012.
[29] S. Zhang, J. Li, K. Liu, C. Deng, L. Liu, and A. Plaza, “Hyperspectral
unmixing based on local collaborative sparse regression,” IEEE Geosci.
Remote Sens. Lett., vol. 13, no. 5, pp. 631–635, May 2016.
[30] S. Zhang, J. Li, Z. Wu, and A. Plaza, “Spatial discontinuity-weighted
sparse unmixing of hyperspectral images,” IEEE Trans. Geosci. Remote
Sens., vol. 56, no. 10, pp. 5767–5779, Oct 2018.
[31] S. Zhang, J. Li, H. Li, C. Deng, and A. Plaza, “Spectral-spatial weighted
sparse regression for hyperspectral image unmixing,” IEEE Trans. Geosci.
Remote Sens., vol. 56, no. 6, pp. 3265–3276, June 2018.
[32] X. Fu, K. Huang, B. Yang, W. K. Ma, and N. D. Sidiropoulos, “Robust
volume minimization-based matrix factorization for remote sensing and
document clustering,” IEEE Trans. Signal Process., vol. 64, no. 23, pp.
6254–6268, Dec 2016.
[33] T. H. Chan, A. Ambikapathi, W. K. Ma, and C. Y. Chi, “Robust affine
set fitting and fast simplex volume max-min for hyperspectral endmember
extraction,” IEEE Trans. Geosci. Remote Sens., vol. 51, no. 7, pp. 3982–
3997, July 2013.
[34] W. He, H. Zhang, and L. Zhang, “Sparsity-regularized robust non-negative
matrix factorization for hyperspectral unmixing,” IEEE J. Sel. Topics
Appl. Earth Observ. Remote Sens., vol. 9, no. 9, pp. 4267–4279, Sept 2016.
[35] J. Li, J. M. Bioucas-Dias, A. Plaza, and L. Liu, “Robust collaborative
nonnegative matrix factorization for hyperspectral unmixing,” IEEE Trans.
Geosci. Remote Sens., vol. 54, no. 10, pp. 6076–6090, Oct 2016.
[36] W. He, H. Zhang, and L. Zhang, “Total variation regularized reweighted
sparse nonnegative matrix factorization for hyperspectral unmixing,” IEEE
Trans. Geosci. Remote Sens., vol. 55, no. 7, pp. 3909–3921, July 2017.
[37] J. Chen and J. Yang, “Robust subspace segmentation via low-rank repre-
sentation,” IEEE Trans. Cybern., vol. 44, no. 8, pp. 1432–1445, Aug 2014.
[38] G. Liu, Z. Lin, S. Yan, J. Sun, Y. Yu, and Y. Ma, “Robust recovery of
subspace structures by low-rank representation,”IEEE Trans. Pattern Anal.
Mach. Intell., vol. 35, no. 1, pp. 171–184, Jan 2013.
[39] Q. Qu, N. M. Nasrabadi, and T. D. Tran, “Abundance estimation for
bilinear mixture models via joint sparse and low-rank representation,”
IEEE Trans. Geosci. Remote Sens., vol. 52, no. 7, pp. 4404–4423, July
2014.
[40] P. V. Giampouras, K. E. Themelis, A. A. Rontogiannis, and K. D.
Koutroumbas, “Simultaneously sparse and low-rank abundance matrix es-
timation for hyperspectral image unmixing,” IEEE Trans. Geosci. Remote
Sens., vol. 54, no. 8, pp. 4775–4789, Aug 2016.
14 VOLUME 4, 2016
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2019.2938633, IEEE Access
T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
[41] H. Zhang, W. He, L. Zhang, H. Shen, and Q. Yuan, “Hyperspectral image
restoration using low-rank matrix recovery,” IEEE Trans. Geosci. Remote
Sens., vol. 52, no. 8, pp. 4729–4743, Aug 2014.
[42] R. Zhu, M. Dong, and J. H. Xue, “Spectral nonlocal restoration of
hyperspectral images with low-rank property,” IEEE J. Sel. Topics Appl.
Earth Observ. Remote Sens., vol. 8, no. 6, pp. 3062–3067, June 2015.
[43] Y. Q. Zhao and J. Yang, “Hyperspectral image denoising via sparse rep-
resentation and low-rank constraint,” IEEE Trans. Geosci. Remote Sens.,
vol. 53, no. 1, pp. 296–308, Jan 2015.
[44] W. He, H. Zhang, L. Zhang, and H. Shen, “Hyperspectral image denoising
via noise-adjusted iterative low-rank matrix approximation,” IEEE J. Sel.
Topics Appl. Earth Observ. Remote Sens., vol. 8, no. 6, pp. 3050–3061,
June 2015.
[45] ——, “Total-variation-regularized low-rank matrix factorization for hyper-
spectral image restoration,” IEEE Trans. Geosci. Remote Sens., vol. 54,
no. 1, pp. 178–188, Jan 2016.
[46] Y. Xie, Y. Qu, D. Tao, W. Wu, Q. Yuan, and W. Zhang, “Hyperspectral
image restoration via iteratively regularized weighted schatten p-norm
minimization,” IEEE Trans. Geosci. Remote Sens., vol. 54, no. 8, pp.
4642–4659, Aug 2016.
[47] Y. Chen, Y. Guo, Y. Wang, D. Wang, C. Peng, and G. He, “Denoising of
hyperspectral images using nonconvex low rank matrix approximation,”
IEEE Trans. Geosci. Remote Sens., vol. 55, no. 9, pp. 5366–5380, Sept
2017.
[48] X. Cao, Z. Xu, and D. Meng, “Spectral-spatial hyperspectral image classi-
fication via robust low-rank feature extraction and markov random field,
Remote Sensing, vol. 11, no. 13, 2019.
[49] T. Zhou and D. Tao, “Godec: Randomized low-rank & sparse matrix
decomposition in noisy case,” in Proceedings of the 28th International
Conference on Machine Learning (ICML-11), ser. ICML ’11, L. Getoor
and T. Scheffer, Eds. New York, NY, USA: ACM, June 2011, pp. 33–40.
[50] H. K. Aggarwal and A. Majumdar, “Hyperspectral unmixing in the pres-
ence of mixed noise using joint-sparsity and total variation,” IEEE J. Sel.
Topics Appl. Earth Observ. Remote Sens., vol. 9, no. 9, pp. 4257–4266,
Sept 2016.
[51] J. Yang, Y. Q. Zhao, J. C. W. Chan, and S. G. Kong, “Coupled sparse de-
noising and unmixing with low-rank constraint for hyperspectral image,”
IEEE Trans. Geosci. Remote Sens., vol. 54, no. 3, pp. 1818–1833, March
2016.
[52] E. J. Candès, X. Li, Y. Ma, and J. Wright, “Robust principal component
analysis?” J. ACM, vol. 58, no. 3, pp. 11:1–11:37, Jun. 2011.
[53] J. Wright, A. Ganesh, S. Rao, Y. Peng, and Y. Ma, “Robust principal com-
ponent analysis: Exact recovery of corrupted low-rank matrices via convex
optimization,” in Advances in Neural Information Processing Systems 22.
Curran Associates, Inc., 2009, pp. 2080–2088.
[54] Z. Zhou, X. Li, J. Wright, E. Candes, and Y. Ma, “Stable principal com-
ponent pursuit,” in 2010 IEEE International Symposium on Information
Theory, June 2010, pp. 1518–1522.
[55] M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image
recovery using variable splitting and constrained optimization,” IEEE
Trans. Image Process., vol. 19, no. 9, pp. 2345–2356, Sept 2010.
[56] M. Aharon, M. Elad, and A. Bruckstein, “rmk-svd: An algorithm for de-
signing overcomplete dictionaries for sparse representation,” IEEE Trans.
Signal Process., vol. 54, no. 11, pp. 4311–4322, Nov 2006.
[57] Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality
assessment: from error visibility to structural similarity,” IEEE Trans.
Image Process., vol. 13, no. 4, pp. 600–612, April 2004.
[58] [Online]. Available: https://engineering.purdue.edu/~biehl/MultiSpec/
hyperspectral.html
[59] R. N. C. [et al.], USGS digital spectral library splib06a. U.S. Geological
Survey Denver, CO, 2007.
[60] [Online]. Available: http://www.agc.army.mil/
[61] F. Zhu, Y. Wang, B. Fan, S. Xiang, G. Meng, and C. Pan, “Spectral
unmixing via data-guided sparsity,” IEEE Trans. Image Process., vol. 23,
no. 12, pp. 5412–5427, Dec 2014.
[62] R. Chartrand, “Nonconvex splitting for regularized low-rank + sparse
decomposition,” IEEE Trans. Signal Process., vol. 60, no. 11, pp. 5810–
5819, Nov 2012.
TANER INCE (M’05) received the B.Sc., M.Sc.
and Ph.D. degrees in electrical and electron-
ics engineering from University of Gaziantep,
Gaziantep, Turkey, in 2003, 2006 and 2012, re-
spectively. He was a Visiting Researcher at Uni-
versity of California, Los Angeles, United States,
during August-April 2014, supported by the Inter-
national Research Fellowship Program (2219) of
The Scientific and Technological Research Coun-
cil of Turkey (TUBITAK).
From 2004 to 2012, he was a Research Assistant at University of
Gaziantep where he is currently an Assistant Professor. His research interests
are in the area of compressed sensing, sparse representation and machine
learning with applications to remote sensing image analysis.
TUGCAN DUNDAR received the B.Sc. and
M.Sc. degrees in electrical and electronics engi-
neering from University of Gaziantep, Gaziantep,
Turkey, in 2016 and 2019, respectively.
He is currently a Research Assistant in the
University of Gaziantep. His research interests are
in the area of sparse representation and machine
learning with applications to remote sensing image
analysis.
VOLUME 4, 2016 15
... Some algorithms were developed to perform denoising and unmixing jointly in a unified framework, with the purpose of improving both. Both in [39,40], sparse representation frameworks were developed, where denoising and unmixing acted as constraints of each other. In most of the aforementioned studies, the performed experiments were small scale and rather anecdotal in nature. ...
... The combination of unmixing and denoising has been treated in some recent studies. Several research works [35,39,40] developed specific methods that could cope with noise and perform denoising and unmixing simultaneously. In [32][33][34], it was shown that denoising may help to boost the endmember extraction and the unmixing performance. ...
Article
Full-text available
Hyperspectral linear unmixing and denoising are highly related hyperspectral image (HSI) analysis tasks. In particular, with the assumption of Gaussian noise, the linear model assumed for the HSI in the case of low-rank denoising is often the same as the one used in HSI unmixing. However, the optimization criterion and the assumptions on the constraints are different. Additionally, noise reduction as a preprocessing step in hyperspectral data analysis is often ignored. The main goal of this paper is to study experimentally the influence of noise on the process of hyperspectral unmixing by: (1) investigating the effect of noise reduction as a preprocessing step on the performance of hyperspectral unmixing; (2) studying the relation between noise and different endmember selection strategies; (3) investigating the performance of HSI unmixing as an HSI denoiser; (4) comparing the denoising performance of spectral unmixing, state-of-the-art HSI denoising techniques, and the combination of both. All experiments are performed on simulated and real datasets.
... Ince et. al propose a coupled unmixing and denoising method [17] to enhance the unmixing performance. Joint-sparse blocks and low-rank unmixing (JSpBLRU) [18] divides the abundance matrix into blocks and solves a joint-sparse regression problem to enhance the sparsity of each block. ...
Preprint
We propose a superpixel weighted low-rank and sparse unmixing (SWLRSU) method for sparse unmixing. The proposed method consists of two steps. In the first step, we segment hyperspectral image into superpixels which are defined as the homogeneous regions with different shape and sizes according to the spatial structure. Then, an efficient method is proposed to obtain a spatial weight term using superpixels to capture the spatial structure of hyperspectral data. In the second step, we solve a superpixel guided low-rank and spatially weighted sparse approximation problem in which spatial weight term obtained in the first step is used as a weight term in sparsity promoting norm. This formulation exploits the spatial correlation of the pixels in the hyperspectral image efficiently, which yields satisfactory unmixing results. The experiments are conducted on simulated and real data sets to show the effectiveness of the proposed method.
... Therefore, some techniques extract endmembers from class averages defined by ground truth information [83] or from a library of endmembers [84]. Some methods were developed for performing the denoising and unmixing in a unified framework for boosting the performance of each other [85], [86]. Recently, Block-Gaussian-Mixture Priors have been proposed for both hyperspectral denoising and inpainting [87]. ...
Preprint
Full-text available
Remote sensing provides valuable information about objects or areas from a distance in either active (e.g., RADAR and LiDAR) or passive (e.g., multispectral and hyperspectral) modes. The quality of data acquired by remotely sensed imaging sensors (both active and passive) is often degraded by a variety of noise types and artifacts. Image restoration, which is a vibrant field of research in the remote sensing community, is the task of recovering the true unknown image from the degraded observed image. Each imaging sensor induces unique noise types and artifacts into the observed image. This fact has led to the expansion of restoration techniques in different paths according to each sensor type. This review paper brings together the advances of image restoration techniques with particular focuses on synthetic aperture radar and hyperspectral images as the most active sub-fields of image restoration in the remote sensing community. We, therefore, provide a comprehensive, discipline-specific starting point for researchers at different levels (i.e., students, researchers, and senior researchers) willing to investigate the vibrant topic of data restoration by supplying sufficient detail and references. Additionally, this review paper accompanies a toolbox to provide a platform to encourage interested students and researchers in the field to further explore the restoration techniques and fast-forward the community. The toolboxes are provided in https://github.com/ImageRestorationToolbox.
... Such data, however, are often contaminated by stripe noise, which is mainly due to differences in the nonuniform response of individual detectors, calibration error, dark currents, and so on [4]- [6]. Stripe noise not only degrades visual quality but also seriously affects subsequent processing, such as unmixing [7], [8], classification [9], [10], and target recognition [11]. Therefore, stripe noise removal, i.e., destriping, has been an important research topic in remote sensing and related fields. ...
Preprint
Full-text available
This paper proposes an effective and efficient destriping method based on zero-gradient constraints, which are compatible with various regularization functions. Removing stripe noise, i.e., destriping, from three-dimensional (3D) imaging data is an essential task in terms of visual quality and subsequent processing. Stripe noise has flat structures in the vertical and temporal directions, meaning that the vertical and temporal gradients are equal to zero. Exploiting this fact, we first propose a new model for characterizing stripe noise. Our model constrains the stripe noise gradient to be zero, which we name the zero-gradient constraint, leading to effective destriping regardless of what regularization is applied to imaging data. Then, we formulate two types of convex optimization problems involving the zero-gradient constraints for destriping and develop efficient solvers for the problems based on a diagonally preconditioned primal-dual splitting algorithm (DP-PDS). We demonstrate the advantages of our model through destriping experiments using hyperspectral images (HSI) and infrared (IR) videos.
Article
Full-text available
In this paper, a new supervised classification algorithm which simultaneously considers spectral and spatial information of a hyperspectral image (HSI) is proposed. Since HSI always contains complex noise (such as mixture of Gaussian and sparse noise), the quality of the extracted feature inclines to be decreased. To tackle this issue, we utilize the low-rank property of local three-dimensional, patch and adopt complex noise strategy to model the noise embedded in each local patch. Specifically, we firstly use the mixture of Gaussian (MoG) based low-rank matrix factorization (LRMF) method to simultaneously extract the feature and remove noise from each local matrix unfolded from the local patch. Then, a classification map is obtained by applying some classifier to the extracted low-rank feature. Finally, the classification map is processed by Markov random field (MRF) in order to further utilize the smoothness property of the labels. To ease experimental comparison for different HSI classification methods, we built an open package to make the comparison fairly and efficiently. By using this package, the proposed classification method is verified to obtain better performance compared with other state-of-the-art methods.
Article
Full-text available
Hyperspectral image (HSI) denoising is challenging not only because of the difficulty in preserving both spectral and spatial structures simultaneously, but also due to the requirement of removing various noises, which are often mixed together. In this paper, we present a nonconvex low rank matrix approximation (NonLRMA) model and the corresponding HSI denoising method by reformulating the approximation problem using nonconvex regularizer instead of the traditional nuclear norm, resulting in a tighter approximation of the original sparsity-regularised rank function. NonLRMA aims to decompose the degraded HSI, represented in the form of a matrix, into a low rank component and a sparse term with a more robust and less biased formulation. In addition, we develop an iterative algorithm based on the augmented Lagrangian multipliers method and derive the closed-form solution of the resulting subproblems benefiting from the special property of the nonconvex surrogate function. We prove that our iterative optimization converges easily. Extensive experiments on both simulated and real HSIs indicate that our approach can not only suppress noise in both severely and slightly noised bands but also preserve large-scale image structures and small-scale details well. Comparisons against state-of-the-art LRMA-based HSI denoising approaches show our superior performance.
Article
While hyperspectral (HS) images play a significant role in many applications, they often suffer from issues such as low spatial resolution, low temporal resolution, and some of the acquired spectral bands are either with low signal-to-noise ratio (SNR) or invalid because of the very high-noise level. To address this issue, a spectral super-resolution method is proposed in this paper to recover a high-spectral-resolution HS image from multispectral (MS) images. The reconstructed HS image will have the same spatial resolution and coverage as the input MS image. The proposed method involves spectral improvement strategy and spatial preservation strategy. For spectral improvement strategy, auxiliary MS/HS image pairs of different landscapes are exploited to estimate spectral response relationship so that an HS image is obtained as an intermediate result. Then, spectral dictionary learning is exploited to recover a more accurate spectral reconstruction result. Spatial preservation strategy is used as a spatial constraint to ensure spatial consistency. In addition, the low-rank property of HS image is also introduced to make the use of global spectral coherence among HS bands. Experiments are conducted on both simulated and real datasets including spectral enhancement of RGB image and the MS image generated by AVIRIS data and real MS/HS data (ALI and Hyperion) captured by Earth Observing-1 (EO-1) satellite. Experiment results demonstrate the superiority of our proposed method to other state-of-the-art methods.
Article
Hyperspectral images (HSIs) are frequently corrupted by various types of noise, such as Gaussian noise, impulse noise, stripes, and deadlines due to the atmospheric conditions or imperfect hyperspectral imaging sensors. These types of noise, which are also called mixed noise, severely degrade the HSI and limit the performance of post-processing operations, such as classification, unmixing, target recognition, and so on. The patch-based low-rank and sparse based approaches have shown their ability to remove these types of noise to some extent. In order to remove the mixed noise further, total variation (TV)-based methods are utilized to denoise HSI. In this paper, we propose a group low-rank and spatial-spectral TV (GLSSTV) to denoise HSI. Here, the advantage is twofold. First, group low-rank exploits the local similarity inside patches and non-local similarity between patches which brings extra structural information. Second, SSTV helps in removing Gaussian and sparse noise using the spatial and spectral smoothness of HSI. The extensive simulations show that GLSSTV is effective in removing mixed noise both quantitatively and qualitatively and it outperforms the state-of-the-art low-rank and TV-based methods.
Article
We propose a spatial-spectral hyperspectral image classification method based on multiscale superpixels and guided filter (MSS-GF). In order to use spatial information effectively, MSSs are used to get local information from different region scales. Sparse representation classifier is used to generate classification maps for each region scale. Then, multiple binary probability maps are obtained for each of the classification maps. Adding GF denoises the classification results and then improves the classification accuracy. Finally, the class label of each pixel is determined by majority voting rule.
Article
Spectral unmixing is an important technique for remotely sensed hyperspectral image interpretation, of which the goal is to decompose the image into a set of pure spectral components (endmembers) and their abundance fractions in each pixel of the scene. Sparse-representation-based approaches have been widely studied for remotely sensed hyperspectral unmixing. A recent trend is to incorporate the spatial information to improve the spectral unmixing results. Those methods generally assume that the abundances of the pixels are piecewise smooth and fall into a homogeneous region occupied by the same endmembers and their corresponding fractional abundances. However, in real scenarios, abundances may vary abruptly from pixel to pixel. Therefore, the former assumption in most spatial models does not hold. To address this limitation, we propose a new strategy to preserve the spatial details in the abundance maps via a spatial discontinuity weight. Our experimental results, conducted with both simulated and real hyperspectral data sets, illustrate the good potential of our discontinuity-preserving strategy for sparse unmixing, which can greatly improve the abundance estimation results.
Article
Spectral unmixing aims at estimating the fractional abundances of a set of pure spectral materials (endmembers) in each pixel of a hyperspectral image. The wide availability of large spectral libraries has fostered the role of sparse regression techniques in the task of characterizing mixed pixels in remotely sensed hyperspectral images. A general solution for sparse unmixing methods consists of using the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> regularizer to control the sparsity, resulting in a very promising performance but also suffering from sensitivity to large and small sparse coefficients. A recent trend to address this issue is to introduce weighting factors to penalize the nonzero coefficients in the unmixing solution. While most methods for this purpose focus on analyzing the hyperspectral data by considering the pixels as independent entities, it is known that there exists a strong spatial correlation among features in hyperspectral images. This information can be naturally exploited in order to improve the representation of pixels in the scene. In order to take advantage of the spatial information for hyperspectral unmixing, in this paper, we develop a new spectral-spatial weighted sparse unmixing (S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> WSU) framework, which uses both spectral and spatial weighting factors, further imposing sparsity on the solution. Our experimental results, conducted using both simulated and real hyperspectral data sets, illustrate the good potential of the proposed S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> WSU, which can greatly improve the abundance estimation results when compared with other advanced spectral unmixing methods.
Article
Hyperspectral target detection is an approach which tries to locate targets in a hyperspectral image on the condition of given targets spectrum. Many classical target detectors are based on the linear mixing model (LMM) and sparsity model. The LMM has a poor performance in dealing with the spectral variability. Therefore, more studies focus on the sparsity-based detectors, most of which are based on residual reconstruction. Owing to the fact that the impure dictionary for the test pixel weakens the detection performance and the discrimination ability of residual function has direct influence on the detecting accuracy, the dictionary purity and discriminative residual function are two most important factors affecting the accuracy of sparsity-based target detectors. In order to obtain more purified dictionary and discriminative residual function, this paper proposes a novel sparsity-based detector named the hybrid sparsity and distance-based discrimination (HSDD) detector for target detection in hyperspectral imagery. The residual function is constrained by the discrimination information during the dictionary construction, which enhances the dictionary purification. Only background samples are used to construct the dictionary because it is easier to remove the target pixel than to select it on the condition that majority of pixels are the background pixels. Hence, a purification process is applied for background training samples in order to construct an effective competition between the residual term and discriminative term. Extensive experimental results with four hyperspectral data sets demonstrate that the proposed HSDD algorithm has a better performance than the state-of-the-art algorithms.
Article
Blind hyperspectral unmixing (HU), which includes the estimation of endmembers and their corresponding fractional abundances, is an important task for hyperspectral analysis. Recently, nonnegative matrix factorization (NMF) and its extensions have been widely used in HU. Unfortunately, most of the NMF-based methods can easily lead to an unsuitable solution, due to the nonconvexity of the NMF model and the influence of noise. To overcome this limitation, we make the best use of the structure of the abundance maps, and propose a new blind HU method named total variation regularized reweighted sparse NMF (TV-RSNMF). First, the abundance matrix is assumed to be sparse, and a weighted sparse regularizer is incorporated into the NMF model. The weights of the weighted sparse regularizer are adaptively updated related to the abundance matrix. Second, the abundance map corresponding to a single fixed endmember should be piecewise smooth. Therefore, the TV regularizer is adopted to capture the piecewise smooth structure of each abundance map. In our multiplicative iterative solution to the proposed TV-RSNMF model, the TV regularizer can be regarded as an abundance map denoising procedure, which improves the robustness of TV-RSNMF to noise. A number of experiments were conducted in both simulated and real-data conditions to illustrate the advantage of the proposed TV-RSNMF method for blind HU.
Article
This paper presents an interactive feedback scheme of spatial resolution enhancement and spectral unmixing in hyperspectral imaging. Traditionally spatial resolution enhancement and spectral unmixing operations have been carried out separately, often in series. In such sequential processing, spatially enhanced hyperspectral images (HSIs) may introduce distortion in spectral fidelity making spectral unmixing results unreliable, or vice versa. Since both high- and low-resolution HSIs have the same endmembers, the deviation in spectral unmixing between targets and estimated high-resolution HSIs can be used as feedback to control spatial resolution enhancement. The spatial difference before and after unmixing can also be used as feedback to enhance spectral unmixing. Therefore, spectral unmixing is utilized as a constraint to spatial resolution enhancement, while spatial resolution enhancement helps improve spectral unmixing results. The performance of spatial resolution enhancement and spectral unmixing can be improved since one behaves like a prior to the other. Experimental results on both simulated and real HSI data sets demonstrate that the proposed interactive feedback scheme simultaneously achieved spatial resolution enhancement and spectral unmixing fidelity. This paper is an extended version of the previous work.