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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 (email: tanerince@gantep.edu.tr, dundar@gantep.edu.tr)
Corresponding author: Taner Ince (email: 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 lowrank 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, lowrank 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 superresolution [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 NFINDR [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 jointsparse structure [26]. Furthermore, using the prop
erty of the piecewise smoothness of the abundance map, a
totalvariation (TV) [27] based unmixing method is proposed
in [28] which is called SUnSALTV. There are also a few
VOLUME 4, 2016 1
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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
outlierrobust loss function onto the data ﬁtting point and
a modiﬁed logdeterminant 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 (TVRSNMF) is proposed for hyperspectral unmixing
that is also robust to noise [36]. Robust collaborative NMF
(RCoNMF) [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, lowrank 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 lowrank 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 lowrank spectral nonlocal approach method to restore
the HSI [42]. The lowrank 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 noiseadjusted iterative lowrank matrix approximation
(NAILRMA) for Gaussian noise and noiseadjusted iterative
lowrank matrix recovery (NAILRMR) for mixed noise [44].
The spatial smoothness and lowrank property of HSI are
studied in [45], which is termed as TV regularized lowrank
matrix factorization (LRTV), to restore the HSI. Moreover,
several works propose nonconvex lowrank approximation
methods to approximate the rank of HSI better. [46] propose
the weighted Schatten pnorm lowrank matrix approxima
tion (WSNLRMA) to approximate the rank of HSI. It ap
proximates the rank of HSI in an iterative manner. Recently,
nonconvex lowrank 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 variationbased 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 lowrank and sparse decomposition approach to
remove the sparse noise and Gaussian noise. In the unmixing
part, we employ a semisupervised 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 patchbased
denoising framework, we ﬁrst extract a subcube of size
d×d×Lcentered at pixel (i, j). For this subcube, we can
FIGURE 1: Lowrank matrix from subcube extracted from
HSI.
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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 lowrank 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·k∗and 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. kY−X−EkF≤
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 nonnegativity constraint (ANC): S≥0and
abundance sumtoone 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 lowrank 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
2kY−X−Ek2
F+λGλ,p(σ(X)) + γGγ,p (S)
+τkEk1+β
2kX−MSk2
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
2kY−X−Ek2
F+λGλ,p(σ(X)) + τkEk1(6)
We call (6) as nonconvex lowrank 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
2kY−P1−Ek2
F+λGλ,p(σ(P2)) + γGγ,p (P3)
+τkEk1+β
2kP4−P5k2
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)
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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
2kY−P1−Ek2
F+λGλ,p(σ(P2))
+γGγ ,p(P3) + τkEk1+β
2kP4−P5k2
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
0−I0000
0 0 −I000
000−I0 0
0000−I0
00000−I
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 k←k+ 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 Λ/µ is 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)K0.0
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.
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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
lowrank approximation use iterative approaches to approx
imate the rank of HSI [46], [47]. The WSNLRMA [46]
minimizes the weighted Schatten pnorm 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 lowrank based schemes. In the second
part of the experiments, results of SNDeUn are given. The
denoising and unmixing results of SNDeUn are denoted by
SNDeUnDe and SNDeUnUn, respectively. In the third part,
real data experiments are given. The performance of the
algorithms are measured based on the following metrics. The
peak signaltonoise 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[kS−ˆ
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
kSi−ˆ
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 stateoftheart lowrank based
denoising methods proposed recently in the literature. These
are LRMR [41], [50], NAILRMR [44], NonLRMA [47]
and WSNLRMA [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. WSNLRMA,
NonLRMA and NonLrDe have also similar visual qualities
as Table 1 indicates. We also show the PSNR and SSIM
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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. WSNLRMA, 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) WSNLRMA (g) NonLRDe
FIGURE 3: Denoising results on SD1. Falsecolor 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 stateof
theart sparse unmixing methods proposed in the literature.
These methods are CLSUnSAL [24], ADSpLRU [40], JSTV
[50] and SUnSALTV [28]. CLSUnSAL, ADSpLRU and
SUnSALTV 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
SUnSALTV has ability to remove the mixed noise at low
noise levels. Therefore, in the mixed noise scenario, we
compare SNDeUn with JSTV and SUnSALTV.
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 SNDeUnUn on simulated data by adding
constant level of Gaussian noise with standard deviation
σto each band of the HSI. We adjusted the parameters
of SNDeUnUn and all compared algorithms to their best
performances in terms of the RMSE. Table 2 reports the SRE
and RMSE values of SNDeUnUn. It can be observed clearly
that SNDeUnUn has best performances at all noise levels.
In order to measure the unmixing performance of
SNDeUnUn 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
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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 WSNLRMA 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) WSNLRMA
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
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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 SNDeUnUn with SUnSALTV and JSTV.
Table 3 reports the unmixing results under mixed noise. It
can be observed that, the performance of SNDeUnUn 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 SNDeUnDe in the presence of mixed noise. We
compare the denoising results of SNDeUnDe by NonLrDe
together with LRMR [41], [50], NAILRMR [44], NonLRMA
[47] and WSNLRMA [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 SNDeUnDe, respectively. We compare
NonLrDe and SNDeUn with LRMR [41] , NonLRMA [47]
and WSNLRMA [46]. The patch size, step size, rank and
sparsity parameter of LRMR is set to 20, 4, 4 and 7000,
respectively. For WSNLRMA, 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 SNDeUnDe 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, WSNLRMA and LRMR leave stripes in their
denoised images. SNDeUnDe has the best results visually
in removing the stripes and impulse noise as well as better
image quality visually.
8VOLUME 4, 2016
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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×10−3).
Noise
Type
Evaluation Index CLSUnSAL SUnSALTV JSTV ADSpLRU SNDeUnUn
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×10−3).
Noise
Type
Evaluation Index SUnSALTV JSTV SNDeUnUn
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 WSNLRMA NonLrDe SNDeUnDe
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 1104, 110138 and 152207. We call
the resultant image as highSNR image. HighSNR 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
lowSNR 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 SUnSALTV, JSTV and SNDeUn as well as
benchmark abundance maps for each spectral signature. It
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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 lowSNR 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 lowrank 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)
i←arg 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)
1←Λ(k)
1−X(k+1) +P(k+1)
1
19: Λ(k+1)
2←Λ(k)
2−X(k+1) +P(k+1)
2
20: Λ(k+1)
3←Λ(k)
3−S(k+1) +P(k+1)
3
21: Λ(k+1)
4←Λ(k)
4−X(k+1) +P(k+1)
4
22: Λ(k+1)
5←Λ(k)
5−MS(k+1) +P(k+1)
5
23: Λ(k+1)
6←Λ(k)
6−S(k+1) +P(k+1)
6
24: Update iteration k←k+ 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
2kY−P1−Ek2
F+λGλ/µ,p(σ(P2)) + γGγ/µ,p (P3)+
τkEk1+β
2kP4−P5k2
F+IR+(P6)
+µ
2kX−P1−Λ1k2
F+µ
2kX−P2−Λ2k2
F
+µ
2kS−P3−Λ3k2
F+µ
2kX−P4−Λ4k2
F
+µ
2kMS −P5−Λ5k2
F+µ
2kS−P6−Λ6k2
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
2kX−P(k)
1−Λ(k)
1k2
F+
µ
2kX−P(k)
2−Λ(k)
2k2
F+
µ
2kX−P(k)
4−Λ(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)]
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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 10−4
10−4
50.5
0.25
0.01 20 8
Original LRMR WSNLRMA NonLRMA NonLrDe SNDeUnDe
Original LRMR WSNLRMA NonLRMA NonLrDe SNDeUnDe
Original LRMR WSNLRMA NonLRMA NonLrDe SNDeUnDe
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
WSNLRMA 0.839
NonLRMA 0.250
NonLrDe 0.345
SNDeUnDe 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
2kY−P(k)
1−Ek2
F+τkEk1(12)
the solution of (12) is
E(k+1) ←soft(Y−P(k)
1, τ )
For P1, the reduced optimization problem is
P(k+1)
1←arg min
P1
1
2kY−P1−E(k+1)k2
F+
µ
2kX(k+1) −P1−Λ(k)
1k2
F
(13)
The solution to (13) is
P(k+1)
1←1
1 + µ(Y−E(k+1) +µ(X(k+1) −Λ(k)
1))
P2is obtained by solving the reduced optimization problem
P(k+1)
2←arg min
P2
λGλ/µ,p(σ(P2))+ µ
2kX(k+1)−P2−Λ(k)
2k2
F
(14)
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T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
Ground Truth SUnSALTV JSTV SNDeUn
Ground Truth SUnSALTV JSTV SNDeUn
Ground Truth SUnSALTV JSTV SNDeUn
Ground Truth SUnSALTV JSTV SNDeUn
FIGURE 9: Abundance maps obtained by SUnSALTV, JSTV and SNDeUn for highSNR 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
2δkW−Vk2
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 − δvp−1}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(Σ, λ/µ)
= max(0,σi − λ
µσip−1)σi/σi
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T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
Ground Truth SUnSALTV JSTV SNDeUn
Ground Truth SUnSALTV JSTV SNDeUn
Ground Truth SUnSALTV JSTV SNDeUn
Ground Truth SUnSALTV JSTV SNDeUn
FIGURE 10: Abundance maps obtained by SUnSALTV, JSTV and SNDeUn for lowSNR 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)
3←arg min
P3
γGγ /µ,p(P3) + µ
2kS(k+1) −P3−Λ(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
2kP4−P5k2
F+
µ
2kX(k+1) −P4−Λ(k)
4k2
F+
µ
2kMS(k+1) −P5−Λ(k)
5k2
F
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T. Ince et al.: Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging
whose solutions are
P(k+1)
4←1
β+µ(βP5+µ(X(k+1) −Λ(k)
4)))
P(k+1)
5←1
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)
6←arg min
P6
IR+(P6) + µ
2kS(k+1) −P6−Λ(k)
6k2
F
which is the projection of S(k+1) −Λ(k)
6onto the nonnegative
orthant deﬁned as
P(k+1)
6←max(S(k+1) −Λ(k)
6,0)
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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 AugustApril 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