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Low-rank Bayesian Tensor Factorization for Hyperspectral Image Denoising
Kaixuan Wei, Ying Fu∗
School of Computer Science and Technology, Beijing Institute of Technology, Beijing, China
Abstract
In this paper, we present a low-rank Bayesian tensor factorization approach for hyperspectral image (HSI) denoising problem, where
zero-mean white and homogeneous Gaussian additive noise is removed from a given HSI. The approach is based on two intrinsic
properties underlying a HSI, i.e., the global correlation along spectrum (GCS) and nonlocal self-similarity across space (NSS).
We first adaptively construct the patch-based tensor representation for the HSI to extract the NSS knowledge while preserving
the property of GCS. Then, we employ the low rank property in this representation to design a hierarchical probabilistic model
based on Bayesian tensor factorization to capture the inherent spatial-spectral correlation of HSI, which can be effectively solved
under the variational Bayesian framework. Furthermore, through incorporating these two procedures in an iterative manner, we
build an effective HSI denoising model to recover HSI from its corruption. This leads to a state-of-the-art denoising performance,
consistently surpassing recently published leading HSI denoising methods in terms of both comprehensive quantitative assessments
and subjective visual quality.
Keywords: Hyperspectral image denoising, full Bayesian CP factorization, nonlocal self-similarity, global correlation along
spectrum, variational Bayesian inference, tensor rank auto determination.
1. Introduction
Hyperspectral image (HSI) is made up of massive contigu-
ous wavebands for each spatial position of real scenes and pro-
vides much richer information about scenes than multiple/RGB
images. It has been widely used for remote sensing, including
mineral identification [1, 2], land cover classification [3], vege-
tation studies [4], and atmospheric studies [5]. Besides, in the
computer vision field, the availability of detailed physical repre-
sentation of HSI has been substantiated to significantly enhance
the performance of numerous computer vision tasks, such as
inpainting [6], tracking [7], unmixing [8], super-resolution [9],
and face recognition [10].
However, in real cases, a HSI is always corrupted by noise,
which severely degrades the quality of the imagery, and nega-
tively impacts all subsequent HSI processing tasks aforemen-
tioned. Noise is inevitable during the acquisition, and caused
at different stages in both the optics and photodetector [11].
Therefore, HSI denoising plays a vital role in the typical work-
flow of HSI analysis and processing.
From our observations of several state-of-the-art HSI de-
noising methods [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22],
we find that the essence of successful design of HSI denois-
ing algorithm is to reasonably extract useful prior structures
knowledge underlying a HSI. The most commonly employed
prior structures for HSI recovery including its global correlation
along spectrum (GCS) and nonlocal self-similarity across space
∗Corresponding author
Email address: kaixuan_wei@outlook.com, fuying@bit.edu.cn
(Kaixuan Wei, Ying Fu)
(NSS). More specific, the GCS prior denotes a huge amount of
redundancy across the spectral dimension. The high correlation
can be observed among images located in adjacent bands of a
HSI generally. And the NSS prior indicates the enhancement
of sparsity can be achieved by grouping similar images frag-
ments (i.e. blocks), which can further improve the performance
of various HSI recovery methods [6, 16, 20].
Since the traditional 2D image denoising is a well stud-
ied yet still active topic, the simplest way of denoising a HSI
is to apply these off-the-shelf techniques [23, 24, 25, 26, 27]
band by band. However, this kind of coarsely extended meth-
ods ignores the GCS prior completely, which leads to a rela-
tively low-quality result. To address this issue, carefully de-
signed extension of several high performance 2D image denois-
ing methods was proposed recently [12, 15, 14]. One notable
example is a nonlocal transform-domain filter, generally re-
ferred to BM4D [12], which is a non-trivial but straightforward
extension of well-known image denoising method BM3D [24].
Besides, a sparse representation based reconstruction method
[15] which jointly utilize the global and local redundancy and
correlation in spatial/spectral domain, is inspired by previous
outstanding work 3D cubic K-SVD [23]. Similarly, a spec-
tral/spatial adaptive hyperspectral total variance (SSAHTV) de-
noising algorithm [14], in which the spectral noise differences
and spatial information differences are both considered in the
process of noise reduction, is a HSI-oriented variation of the
spatially adaptive TV (SATV) model [28].
Instead of constructing method directly from existed one,
low-rank matrix recovery (LRMR) techniques are employed for
HSI denoising, including convex relaxation based approaches
Preprint submitted to Journal of Neurocomputing March 9, 2020
[29, 30, 31, 32] and Bayesian inference approaches [33, 34, 35,
36, 37]. For example, Zhang et al. [19] directly adopt the Go
Decomposition (GoDec) algorithm [32], which is a optimiza-
tion algorithm aimed to solve LRMR model, to estimate the
low-rank HSI patch. Chen et al. [38] employed a modified
version of robust principal component analysis (RPCA) algo-
rithm which models noise with mixture of Gaussian in the con-
text of LRMR model, originally proposed in [37] to deal with
HSI with non-i.i.d. noise structure. Recently, He et al. [21]
proposed a local matrix recovery method while considering the
global spatial-spectral smoothness through TV regularization,
which achieves great performance especially in complex noise
removal of hyperspectral remote sensing images.
Albeit these LRMR-based approaches are effective to cer-
tain HSI denoising cases, but they only consider the GCS prior
knowledge. Since the HSI can be naturally represented as a
3D tensor instead of a 2D matrix, one obvious extension of
these LRMR-based approaches is to facilitate the power of ten-
sor decomposition [39, 40] which attracts growing attention re-
cent years. The two popular tensor factorization framework,
named as Tucker and CANDECOMP/PARAFAC (CP), are used
to denoise HSI and permit to appreciate the denoising efficiency
respectively[41, 17]. Nevertheless, the input tensor of their
methods is just the original form of the HSI, including two
spatial dimensions and one spectral dimension. This definitely
degrades their denoising performance, since the two separated
spatial dimensions of the HSI are generally not low-rank rep-
resentable. This intuition can be easily comprehended by Fig-
ure 1, which exhibits typical output bands by directly applying
LRTF method and our integrated model. It can be seen in the
LRTF recovery (top right corner) that some vertical or hori-
zontal artifact textures are introduced due to the unsuitable as-
sumption that the two spatial dimensions of a HSI are low-rank
representable.
Figure 1: Typical output bands by the LRTF based methods, which show it is
unreasonable to directly apply LRTF based methods on HSI denoising.
This property of HSI limits the way to straightly apply the
low-rank tensor factorization (LRTF) based methods. To alle-
viate this problem, one question has been raised naturally: Can
we transfer the initial form of a HSI into a more low-rank rep-
resentable one without destroying the spatial and spectral struc-
ture informations? Fortunately, this issue can be efficiently ad-
dressed by a remarkable technique called block matching, orig-
inally introduced by the benchmark BM3D [24] for image de-
noising. There are two approaches to extend the block match-
ing strategy into volumetric data in general. One approach is to
impose the NSS prior on the spectral dimension which further
stacks the cubes of similar voxels into 4D ”groups” respectively
[12]. Another approach, which considers the GCS prior, is to
simply extend a normal 2-D patch to a full band patch (FBP),
then follows the same block matching procedure as the BM3D
[16, 13]. Given the spectral property of a HSI, the FBP based
approach is more efficient than the voxel based approach while
preserving effectiveness.
Once the FBP clusters, which can be viewed as a set of 3D
tensors, have been constructed using the extended block match-
ing strategy, the family of tensor decomposition methods can
be employed to maximize the use of underlying knowledge in-
cluding these aforementioned priors. Xie et al. built a new HSI
denoising model ITS-Reg[13] by applying LRTF based method
on tensors formed by nonlocal similar patches within the HSI.
Zhuang et al. [22] combined the power of LRMR and LRTF
based techniques to construct a global local factorization model
called GLF. However, as the noise level varies drastically, these
convex relaxation based low rank approximation approaches
are prone to overfitting due to the incorrect specified regular-
ization parameters, resulting in severe deterioration of recovery
performance. It is also worth noting that the rank minimization
based on convex optimization of nuclear norm is affected by
the tuning parameters, which may tend to over/underestimate
the HSI.
In this paper, we present a hierarchical probabilistic model
for HSI denoising based on full Bayesian CP tensor factoriza-
tion (LBTF)[42], which can not only fit the underlying noise
adaptively without knowing the specific noise intensity, but also
determine the tensor rank automatically to address the over-
fitting issues. We first adaptively transform the original HSI
into patch-based tensor representations (clusters) to extract the
NSS knowledge while preserving the property of GCS in these
new representations. Then we regard each cluster as a low-rank
noisy observation in our hierarchical probability model in or-
der to obtain the inherent spatial-spectral correlation of HSI.
This model is effectively solved by an elegant deterministic al-
gorithm based on variational inference. The empirical study
demonstrates the superiority of our method, which consistently
outperforms other state-of-the-art HSI denoising methods both
quantitatively and visually.
The rest of this paper is organized as follows. Section 2
presents preliminary multilinear operators and notations. Sec-
tion 3 introduces our Bayesian tensor factorization approach for
HSI denoising. The extensive experimental results on both syn-
thetic and real data are presented in Section 4, followed by con-
clusions in Section 5.
2
2. Notions and Preliminaries
The order of a tensor is the number of dimensions (a.k.a
ways or modes). For clarification, scalars (zero order tensors)
are denoted by lowercase letters, e.g., a. Vectors (first order
tensors) are denoted by boldface lowercase letters, e.g., a. Ma-
trices (second order tensors) are denoted by boldface capital let-
ters, e.g., A. And the k-th row or column vector are denoted by
Ak·,A·krespectively. General tensors (without order constraint)
are denoted by boldface calligraphic letters, e.g., A. Given a
N-th order tensor A∈RI1×···×IN, its (i1,· · · ,×iN)-th entry is
denoted by Ai1···iNwithout boldface.
The denotations of tensor operations follow [40]. The Kro-
necker product of matrices A∈RI×Jand B∈RM×Nis a matrix
of size I M ×JN , denoted by A⊗B. The Khatri-Rao product
of matrices A∈RI×Kand B∈RJ×Kis a matrix of size I J ×K
defined by a columnwise Kronecker product, and denoted by
AB. For convenience, the Khatri-Rao product of a set of
matrices {A(n)|n=1,· · · ,N}in a reverse order is defined by
K
n
A(n)=A(N)A(N−1) · · · A(1) (1)
while the Khatri-Rao product of a set of matrices except the k-th
matrix, is defined by
K
n,k
A(n)=A(N) · · · A(k+1) A(k−1) · · · A(1) (2)
The Hadamard product (a.k.a the Schur product or the en-
trywise product) of two tensors {A,B} ∈ RI1×···×INis denoted
by A◦B. It is a tensor with same dimensions as {A,B}where
each element indexed by i1i2· · · iNis the product of elements
indexed by i1i2· · · iNof the original two tensors.
The inner product of two tensors {A,B}is defined by hA,Bi=
Pi1,··· ,iNAi1,··· ,iNBi1,··· ,iN. For a more general case, we define
DA(1),· · · ,A(N)E=X
i1,··· ,iNY
n
A(n)
i1,··· ,iN(3)
Our framework for LRTF is based on the CP decomposition
which can be viewed as a higher-order generalization of the
widely used matrix singular value decomposition (SVD). Given
a tensor X∈RI1×···×IN, it can be exactly factorized by a CP
model, giving by
X=
R
X
r=1
A(1)
·r}· · · }A(N)
·r=[[A(1) ,· · · ,A(N)]] (4)
where }denotes the outer product of vectors, [[· · · ]] denotes a
Kruskal operator of a set of matrices having the same number
of columns, A(n)is a mode-nfactor matrix of size In×Rand R
is assumed to be the upper bound the rank of tensor X. Then
each element of the tensor can be described as
Xi1,··· ,iN=
R
X
r=1
A(1)
i1rA(2)
i2r· · · A(N)
iNr(5)
3. The Proposed Method
3.1. Formulation
The HSI can be mathematically viewed as a 3D tensor. Given
the additive noise, the noisy observation Ycan be described as
Y=X+(6)
where Xis the original, unknown clean HSI, and ∼ N(0, τ−1)
is the additive noise. Y∈RH×W×B, where H,W,Bare stand for
the corresponding spatial height, spatial width and the number
of spectral band of this HSI respectively. Xand have the
same dimension with Y. In this paper, we mainly consider the
independent and identically distributed (i.i.d.) Gaussian noise
with unknown noise intensity.
For clarification and simplicity, we denote the proposed low-
rank Bayesian tensor factorization technique as LBTF, and the
whole HSI denoising algorithm as LBTF-HSI. The objective of
the proposed LBTF-HSI is to provide an estimation ˆ
Xof the
original Xfrom the noisy observation Y, so that ˆ
Xshould be
similar to Xas much as possible under a commonly adopted
error measure (e.g. l2norm).
3.2. Iterative Denoising Framework
The LBTF-HSI is implemented in an iterative denoising
framework motivated by [24], generally consisted of one to
five near duplicated stages. Each stage comprises three steps:
grouping, low-rank tensor recovery, and aggregation. The group-
ing and aggregation steps are required by the block matching
technique, and the low-rank tensor recovery is what the step we
apply the proposed LBTF algorithm. The flow-diagram of the
LBTF-HSI implementation is illustrated in Figure 2.
In the grouping step, we first separate the noisy HSI Yinto
a set of FBPs with overlap. Then for each local FBP, we con-
struct a FBP cluster by performing block matching. To be spe-
cific, given a reference FBP, the Pnmost similar FBP over all
FBPs will be matched then form together into a FBP cluster,
where Pndenotes the number of patches in a cluster, and the
similarity is measured in a l2norm form for simplicity. Note
this procedure is highly related to the GCS and NSS prior, and
after this operation, both GCS and NSS knowledge are well
preserved and reflected by such new representation, along its
spectral and nonlocal-patch-number modes, respectively.
In the recovery step, we first initialize two zero-entries ten-
sors denoted by Cand Wwith the same size of the noisy ob-
servation Y, then straightly apply the LBTF technique on each
of FBP clusters to exploit the intrinsic low-rank property un-
derlying this new representation. After we reconstruct all clean
FBP clusters, we map every estimation to Clike the original
form Yusing a cumulative scheme. It also does deserve to be
noticed that Wcan be explained as weight with respect to the
cumulative C, which is obtained during the calculation process
of C.
In the aggregation step, like a inverse Hadamard product,
we simply divide the cumulative Cby its corresponding weight
Welementwise to generate an estimate ˆ
Xof the original X.
To regard this estimate ˆ
Xas a regularization, we can construct
3
Figure 2: Flowchart of the proposed LBTF-HSI method. An noisy HSI is firstly splitted into a set of overlapping full bands patch, after grouping by each reference
FBP (step 1), each FBP cluster is fed into the LBTF model to acquire its clean estimation (step 2), then each clean estimation is mapped onto the corresponding
locations as it in the original noisy HSI by an accumulative scheme (step 3). Then either follow an iterative regularization to repeat this process or output the
estimated clean HSI as a final result.
a updated noisy observation Yby iterative regularization then
repeat the same steps described above. This produces an iter-
ative denoising framework. The whole detained procedures of
this framework are summarized in Algorithm 1.
Algorithm 1 Iterative Denoising Framework
Input: Noisy HSI Y;
1: Initialize ˆ
X(0) =Y;
2: for k=1 : Kdo
3: Iterative regularization Y(k)=ˆ
X(k−1) +δ(Y−ˆ
X(k−1));
4: Construct the entire FBP set Ωk;
5: Group matching FBP clusters {Yi}L
i=1;
6: for Each FBP cluster Yido
7: Recover Xifrom Yiby LBTF;
8: end for
9: Aggregate {Xi}L
i=1to form the clean estimate ˆ
X(l);
10: end for
11: Assign ˆ
X=ˆ
X(K);
Output: Estimation ˆ
Xof X;
3.3. Hierarchical Probabilistic Model
Now we present the LBTF algorithm used in the recovery
step, which is the key component of our method. In this step,
each observation is a noisy FBP cluster formed by block match-
ing. Given such noisy observation, we apply the LBTF to infer
the underlying clean cluster.
From a Bayesian perspective, the CP tensor factorization
can be formulated by a hierarchical probabilistic model which
is actually an instance of probabilistic graphical model (PGM)
[43].
The CP generative model in Equation (4) together with the
observation model in Equation (6) directly gives rise to the fol-
lowing hierarchical probabilistic model. As we discussed in
iterative denoising framework, after we acquire a set of FBP
clusters {Yi}L
i=1, for each cluster Y∈RI1×I2×I3(for brevity, the
subscript iis omitted), its probability density can be derived
through being factorized over tensor elements
pY{A(n)}3
n=1, τ =
I1
Y
i1=1
I2
Y
i2=1
I3
Y
i3=1
NYi1i2i3DA(1)
i1·,A(2)
i2·,A(3)
i3·E, τ−1
(7)
where A(n)is the latent mode-nfactor matrix of size In×R,
we note n=1,2,3 for spatial, spectral, and nonlocal-patch-
number modes respectively. Rt≤Rdenotes the ground-truth
rank of tensor X.N(y|µ, τ−1) denotes a Gaussian distribution
of the form
N(y|µ, τ−1)=(τ
2π)1
2ex p −τ
2(y−µ)2(8)
In order to further build our hierarchical probabilistic model,
we need to enforce a suitable probabilistic structure on the un-
derlying factor matrices {A(n)}3
n=1. From the CP model in Equa-
tion (4), notice that each outer product contributes at most one
to the rank of X. Since a low-rank estimation of Xis sought,
our goal is to achieve column sparsity in A(n), such that most
column in A(n)are set equal to zero. To enforce this constraint,
we associate the columns of A(n)with Gaussian priors of preci-
sions (inverse variances) λr, that is,
pA(n)|λ=Y
in
NA(n)
in·0,Λ−1,∀n∈[1,3](9)
4
where Λ=diag(λ) denotes an inverse variance matrix and is
shared by latent factor matrices in all modes. Thus, the r-th
columns of {A(n)}3
n=1have the same sparsity profile enforced by
the common precisions λr. As shown later, many of the preci-
sions λrwill assume very large values during inference, which
effectively removes the corresponding outer-products from X,
and hence reduces the rank of the estimation. We can further
define a hyperprior over hyperparameter λ, which is factorized
over latent dimensionality due to the independent assumption
p(λ)=
R
Y
r=1
Gam(λr|cr
0,dr
0) (10)
where Gam(x|a,b) denotes a Gamma distribution of the form
Gam(x|a,b)=baxa−1e−b x
Γ(a)(11)
Γ(·) is the Gamma function.
Using the similar technique, we also place a hyperprior over
the noise precision τ, i.e.
p(τ)=Gam(τ|a0,b0) (12)
Combining Equations (7) and (9) to (12) together, we can
complete our hierarchical probabilistic model as a PGM, the
whole graph representation is illustrated in Figure 3.
For brevity of notations, we denotes all unknowns including
both latent variable and hyperparameters by Θ = {A(n)}3
n=1,λ,τ.
From Figure 3, we can write the the joint distribution of ob-
served data and all model parameters as
p(Y,Θ)=p(YΩ|{A(n)}3
n=1,τ)
3
Y
n=1
p(A(n)|λ)p(λ)p(τ).(13)
The goal turns to infer the posterior of all involved parameters,
which can be done by maximizing Equation (13) without loss
of generality.
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Figure 3: The probabilistic graphical model representation of Bayesian CP ten-
sor factorization.
3.4. Variational Inference
However, in contrast to the point estimation, we aim to
compute the full posterior distribution of all parameters in Θ.
Since that, a deterministic approximate inference method under
the variational Bayesian (VB) framework [43] is developed to
learn the aforementioned hierarchical probabilistic model. To
achieve this goal, we therefore seek a distribution q(Θ) to ap-
proximate the true posterior distribution p(Θ|Y) by solving the
following optimization problem
min
qKL (q(Θ)|| p(Θ|Y))=−Zq(Θ)ln (p(Θ|Y)
q(Θ))dΘ(14)
where KL(q||p) represents the KL divergence between two dis-
tribution qand p. Since the posterior distribution p(Θ|Y) is
computational intractable in our model, it makes our problem
that cannot be reduced from the VB framework into the expec-
tation maximization (EM) framework. Thus, some constraints
need to be imposed on the variational distribution q(Θ) to make
this optimization feasible. Specifically, it will be assumed that
the variational distribution is factorized w.r.t. each parameter
Θj, so that
q(Θ)=q(λ)q(τ)
3
Y
n=1
qA(n).(15)
This factorized form of variational inference corresponds to an
approximation framework developed in physics called mean
field theory [44]. After that, the closed-form optimal solution
q∗
j(Θj) can be obtained by
ln q∗
j(Θj)=hln p(Y,Θ)iΘ\Θj+const (16)
where h·iis a unary operator denoting expectation and Θ\Θj
denotes the set of Θwith Θjremoved. Since the distributions of
all variables are drawn from the distributions over their parent
variables, we can analytically infer the posterior distributions
of model parameters using Equations (13), (15) and (16).
Estimation of mode-n factors A(n).
q∗(A(n))=
In
Y
in=1
NA(n)
in·DA(n)
in·E,Σ(n)
in∀n∈[1,3] (17)
where the posterior parameters can be updated by
DA(n)
in·E=hτiΣ(n)
inDB(\n)TEvec Y·in·(18)
Σ(n)
in=hτiDB(\n)TB(\n)E+hΛi−1(19)
B(\n)=K
k,n
A(k)(20)
The most complex term is related to B(\n), which is of size
Qk,nIk×R, and denotes the Khatri-Rao product of latent factors
in all modes except nth-mode. vec Y·in·denotes the vectorized
FBP cluster of size Qk,nIk, whose mode-n index is in.
Estimation of hyperparameters λ.
q∗(λ)=
R
Y
r=1
Gam(λr|cr,dr) (21)
5
where
cr=cr
0+1
2
3
X
n=1
In(22)
dr=dr
0+1
2
3
X
n=1DA(n)T
·rA(n)
·rE(23)
Estimation of hyperparameter τ.
q∗(τ)=Gam(λr|a,b) (24)
where
a=a0+1
2
3
Y
n=1
In(25)
b=b0+1
2
Y−[[ A(1) ,A(2),A(3)]]
2
F(26)
Algorithm 2 Low-rank Bayesian Tensor Factorization
Input: A FBP cluster Yi;
1: Initialize factor matrices and their covariance A(n)
in·,Σ(n)
in, hy-
perpriors a0,b0,c0,d0and hyperparameters τ=a0
b0, λr=
cr
0
dr
0
;
2: while not converge do
3: for n=1 to 3 do
4: Update the posterior q(A(n)) using Equations (18)
to (20);
5: end for
6: Update the posterior q(λ) using Equations (22) and (23);
7: Update the posterior q(τ) using Equations (25) and (26);
8: Update the estimated Rank R by maxnRank(A(n))
9: end while
Output: Estimate FBP cluster ˆ
Xiand Rank R;
The whole procedure of model inference is summarized in
Algorithm 2, It’s worth noting that tensor rank is determined
automatically and implicitly. To be specific, during inference,
most of the hyperparameters λiare driven to very large values,
which will force the posterior means of the columns to go to
zero, effectively removing them from the model and reducing
the rank. For implementation of the algorithm, we keep the size
of {A(n)}unchanged during iterations, while an alternative way
is to remove the zero components of {A(n)}after each iteration.
4. Experiment and Analysis
In this section, extensive simulated and real data experi-
ments are conducted to validate the denoising capabilities of
the proposed LBTF-HSI algorithm, and qualitative and visual
results are illustrated. The detailed analysis about our method
is presented in final.
Figure 4: Simulated pseudo color images from Columbia Dataset
4.1. Simulated HSI Denoising
Columbia Dataset. The Columbia HSI Dataset [46]1is em-
ployed in our simulated experiment, which is commonly used
in other algorithms verification [13, 16]. This dataset consists
of 32 real-world scenes of a wide variety of real-world materi-
als and objects, with spatial resolution 512 ×512 and spectral
bands 31. Each HSI includes full spectral resolution reflectance
data collected from 400 nm to 700 nm with 10 nm interval. The
simulated pseudo color images from this dataset are shown in
Figure 4. In our experiments, the intensity of these HSIs is
scaled into [0,1].
Implementation Details. Additive white Gaussian noise (AWGN),
which comes from many natural sources, is added into these
testing HSIs to generate Ycorresponding to our observation
model with noise intensity ranging from 15 to 100 (It’s need to
be clarified we denote the noise intensity with a base 255, i.e.
15 means the standard deviation of Gaussian noise is 15
255 , simi-
larly hereinafter). Unlike other methods, which require specific
noise intensity as a input parameter, we do not feed this infor-
mation into our method since the internal noise intensity can
be automatically learned during its denoising process. Con-
sequently, except particularly mentioned, we provide the real
noise intensity to comparison methods while our method learns
the noise model automatically.
For parameters setting, we need to care about the initializa-
tion strategy in LBTF (Algorithm 2). There are two parame-
ters which are closely relevant to initialization. One is a binary
parameter which can choose the low-rank components initial-
ization scheme between SVD and random generation (follow a
standard normal distribution). Though the theory of VB frame-
work [43] can guarantee every initialized point converges to a
local minimum, we find using random generation rather than
SVD will achieve better performance in the context of HSI de-
noising. This phenomenon can be interpreted by grouping and
aggregation operations involved in our method, which appreci-
ate miscellaneous initialized points rather relatively stable ones.
Another parameter which dominantly affects the denoising ca-
pability of LBTF is the upper bound of rank Rof the low-rank
components. It’s worth noting that we only need to provide a
roughly estimation of the upper bound of objective rank rather
1http://www1.cs.columbia.edu/CAVE/databases/multispectral
6
(a) Clean image
(PSNR, SSIM)
(b) Noisy image
(20.17, 0.19)
(c) BM3D
(34.91, 0.92)
(d) BM4D
(38.61, 0.95)
(e) LRMR
(33.27, 0.72)
(f) LRTV
(29.74, 0.89)
(g) LRTA
(34.53, 0.87)
(h) LLRGTV
(35.35, 0.90)
(i) GLF
(40.29, 0.96)
(j) TDL
(38.07, 0.96)
(k) ITSReg
(39.78, 0.95)
(l) Ours
(40.44,0.97)
Figure 5: The images at band 590 nm of chart and stuffed toy under noise level σ=25 on CAVE dataset. Two demarcated areas in each image are amplified at a 3
times larger scale for easy observation of details.
(a) Clean image
(PSNR, SSIM)
(b) Noisy image
(14.15, 0.11)
(c) BM3D
(28.89, 0.82)
(d) BM4D
(32.16, 0.89)
(e) LRMR
(27.20, 0.56)
(f) LRTV
(26.13, 0.77)
(g) LRTA
(29.63, 0.78)
(h) LLRGTV
(30.72, 0.85)
(i) GLF
(33.80, 0.91)
(j) TDL
(31.79, 0.88)
(k) ITSReg
(33.67, 0.93)
(l) Ours
(34.05,0.93)
Figure 6: The images at band 490 nm of watercolors under noise level σ=50 on CAVE dataset. Two demarcated areas in each image are amplified at a 6 times
larger scale for easy observation of details.
than the indeed objective rank required by other low-rank based
methods [19, 45]. After one iteration of our algorithm, the truth
rank can be automatically estimated. We simply set Rin the
first iteration to 15, and keep track of mean of the truth rank of
all clusters as Rof the next iteration in all of our experiments.
Comparison Methods. The comparison methods include: band-
wise BM3D [24]2, which represents the state-of-the-art for the
2D extended band-wise approach; BM4D [12]2, which repre-
sents state-of-the-arts for the 2D extended 3D-cube-based ap-
proach; LRMR [19], LRTV [45] and LLRGTV [21] which rep-
resent state-of-the-arts for the low-rank matrix-based approach;
LRTA [41], GLF [22], TDL [16]3and ITS-Reg [13]3, which
2http://www.cs.tut.fi/foi/GCF-BM3D/
3http://gr.xjtu.edu.cn/web/dymeng/2
represent state-of-the-arts for the tensor-based approach. All
parameters involved in the competing algorithms were manu-
ally tuned optimally or automatically chosen as described in
the reference papers.
Performance Metrics. To comprehensively assess the perfor-
mance of all competing methods, we employ five quantitative
picture quality indices (PQI) for performance evaluation, in-
cluding peak signal-to-noise ratio (PSNR), structure similar-
ity (SSIM [47]), feature similarity (FSIM [48]), erreur relative
globale adimensionnelle de synthe‘se (ERGAS [49]) and spec-
tral angle map (SAM [50]). PSNR and SSIM are two conven-
tional PQIs in image processing and computer vision. They
evaluate the similarity between the target image and reference
image based on MSE and structural consistency, respectively.
FSIM emphasizes the perceptual consistency with the reference
7
(a) Clean image
(PSNR, SSIM)
(b) Noisy image
(10.63, 0.02)
(c) BM3D
(31.66, 0.79)
(d) BM4D
(33.59, 0.74)
(e) LRMR
(25.93, 0.32)
(f) LRTV
(28.29, 0.70)
(g) LRTA
(30.99, 0.69)
(h) LLRGTV
(32.59, 0.76)
(i) GLF
(35.39, 0.83)
(j) TDL
(34.16, 0.85)
(k) ITSReg
(34.26, 0.82)
(l) Ours
(36.35,0.89)
Figure 7: The images at band 640 nm of flowers under noise level σ=75 on CAVE dataset. One demarcated areas in each image is amplified at a 1.5 times larger
scale for easy observation of details.
(a) Clean image
(PSNR, SSIM)
(b) Noisy image
(8.13, 0.04)
(c) BM3D
(23.89, 0.50)
(d) BM4D
(26.23, 0.64)
(e) LRMR
(21.69, 0.40)
(f) LRTV
(22.79, 0.46)
(g) LRTA
(24.09, 0.46)
(h) LLRGTV
(25.31, 0.65)
(i) GLF
(27.59, 0.74)
(j) TDL
(26.08, 0.65)
(k) ITSReg
(26.69, 0.69)
(l) Ours
(27.85,0.75)
Figure 8: The images at band 590 nm of cloth under noise level σ=100 on CAVE dataset. Two demarcated areas in each image are amplified at a 6 times larger
scale for easy observation of details.
image. The larger these three measures are, the closer the target
HSI is to the reference one. ERGAS measures fidelity of the re-
stored image based on the weighted sum of MSE in each band.
SAM measures the spectral fidelity between the restored image
and the reference image across all spatial positions. Different
from the former three measures, the smaller these two measures
are, the better does the target HSI estimate the reference one.
Performance Evaluation. For each noise setting, all of the five
PQI values for each competing HSI denoising methods on all
32 scenes have been calculated and recorded. Table 1 lists the
average performance over different scenes under noise settings
of all methods. From these quantitative comparison, the advan-
tage of the proposed method can be evidently observed. Par-
ticularly, with the increase of noise intensity, our method sur-
passes the second best ITS-Reg under the measure of PSNR
by a large margin (e.g. 0.96 dB under σ=75, 2.5dB under
σ=75). This is due to the overfitting issue commonly ex-
isted in state-of-the-art methods. Our method successfully ad-
dress this issue by automatically determining the tensor rank,
consequently achieving great performance especially in severe
pollution case. Figures 5 to 8 illustrate the visual results of dif-
ferent methods under different noise levels. It can be seen that
our method consistently outperform other methods as we mea-
sured in Table 1. Specifically, in Figure 6, we can see except
GLF and our method, none of the competing methods can suc-
cessfully recover the exact edge shape of cloud exhibited in the
green box. In Figure 8, only GLF, ITS-Reg and our method
produce smooth and noise-free results, while the fine-grained
details of ours are much clearer and shaper than ITS-Reg’s. We
also compute the PSNR value of each bands in these four HSIs
8
Table 1: Average performance of 10 competing methods w.r.t. 5 PQIs. For each specific noise intensity setting, the results are obtained by averaging through the 32
scenes. The best results of each case among these methods are denoted by boldface.
Sigma Index
Methods
Noisy BM3D BM4D LRMR LRTV LRTA LLRGTV GLF TDL ITSReg Ours
[24] [12] [19] [45] [41] [21] [22] [16] [13]
15
PSNR 24.61 39.81 42.38 37.21 33.54 39.21 38.46 43.41 42.30 43.43 43.46
SSIM 0.291 0.951 0.968 0.869 0.912 0.930 0.948 0.977 0.972 0.972 0.976
FSIM 0.794 0.973 0.981 0.974 0.938 0.971 0.978 0.989 0.987 0.989 0.988
ERGAS 325.24 56.41 41.35 76.49 124.88 60.89 71.05 38.49 41.98 37.26 36.72
SAM 0.785 0.157 0.151 0.391 0.204 0.183 0.175 0.128 0.101 0.138 0.103
25
PSNR 20.17 37.03 39.59 33.49 32.42 36.67 36.63 40.96 39.72 40.57 41.21
SSIM 0.148 0.919 0.943 0.736 0.895 0.893 0.913 0.957 0.957 0.945 0.964
FSIM 0.661 0.955 0.968 0.952 0.922 0.953 0.969 0.984 0.979 0.980 0.982
ERGAS 542.05 77.50 57.16 115.39 136.84 81.21 84.89 50.50 56.39 51.45 48.26
SAM 0.933 0.208 0.215 0.569 0.234 0.218 0.254 0.167 0.123 0.242 0.118
50
PSNR 14.15 33.49 35.65 28.35 29.82 33.16 33.45 37.15 36.16 37.55 37.83
SSIM 0.052 0.862 0.870 0.470 0.846 0.819 0.812 0.890 0.918 0.919 0.927
FSIM 0.465 0.922 0.938 0.890 0.891 0.919 0.944 0.970 0.956 0.963 0.966
ERGAS 1084.15 116.60 90.13 204.78 183.40 120.99 118.41 77.24 84.58 72.85 71.07
SAM 1.124 0.277 0.340 0.797 0.350 0.278 0.433 0.263 0.186 0.243 0.173
75
PSNR 10.63 31.36 33.28 25.27 27.98 31.17 31.28 34.75 34.08 34.78 35.74
SSIM 0.026 0.810 0.794 0.310 0.787 0.762 0.716 0.812 0.875 0.881 0.889
FSIM 0.362 0.894 0.908 0.826 0.870 0.892 0.921 0.957 0.934 0.945 0.951
ERGAS 1626.14 147.89 118.14 290.62 224.31 152.38 149.57 101.80 107.73 100.36 90.47
SAM 1.225 0.338 0.429 0.913 0.477 0.318 0.585 0.357 0.243 0.297 0.224
100
PSNR 8.13 29.83 31.56 23.03 26.50 29.69 29.64 33.03 32.56 31.77 34.26
SSIM 0.015 0.767 0.723 0.214 0.751 0.712 0.635 0.747 0.826 0.835 0.855
FSIM 0.299 0.871 0.879 0.766 0.853 0.869 0.899 0.944 0.911 0.914 0.938
ERGAS 2168.26 175.21 143.73 375.29 267.94 180.21 178.72 123.92 128.06 143.74 107.08
SAM 1.290 0.383 0.496 0.995 0.540 0.350 0.695 0.432 0.299 0.306 0.263
Table 2: Average performance of 10 competing methods w.r.t. 5 PQIs under unkowen Gaussian noise level. The results are obtained by averaging through the 32
scenes. The best results of each case among these methods are denoted by boldface.
PSNR SSIM FSIM ERGAS SAM
None 14.03 ±4.62 0.079 ±0.108 0.462 ±0.197 1235.75±613.62 1.124 ±0.276
BM3D 33.36 ±3.31 0.857 ±0.052 0.919 ±0.030 119.26 ±35.93 0.292 ±0.112
BM4D 35.73 ±3.02 0.877 ±0.055 0.934 ±0.033 92.11 ±27.73 0.320 ±0.145
LRMR 29.35 ±3.77 0.603 ±0.170 0.893 ±0.061 194.22 ±71.11 0.610 ±0.228
LRTV 29.38 ±3.03 0.841 ±0.072 0.894 ±0.043 194.92 ±63.90 0.361 ±0.157
LRTA 33.34 ±3.21 0.844 ±0.065 0.924 ±0.029 119.97 ±37.56 0.236 ±0.080
LLRGTV 32.38 ±3.14 0.783 ±0.092 0.931 ±0.031 135.66 ±42.20 0.410 ±0.200
GLF 36.88 ±3.06 0.859 ±0.086 0.967 ±0.015 82.62 ±27.99 0.287 ±0.168
TDL 36.20 ±3.09 0.915 ±0.035 0.952 ±0.022 85.57 ±24.05 0.183 ±0.085
ITSReg 37.17 ±3.17 0.916 ±0.042 0.959 ±0.023 78.21 ±25.31 0.218 ±0.150
Ours 37.70 ±2.98 0.924 ±0.034 0.965 ±0.017 73.20 ±21.36 0.174 ±0.084
(i.e. watercolors,cloth, etc.). It can be seen in Figure 9, the
PSNR values of all bands obtained by LBTF-HSI are signifi-
cantly higher than those of the other methods.
Denoising under Unknown Noise Level. Motivated by appeal-
ing noise intensity self-adaptive property aforementioned of our
method, we conduct experiments under unknown Gaussian noise
level for further demonstrating the advantages of the proposed
method. Here, we still adopt 32 real-world scenes HSIs from
the Columbia Dataset described above. Unlike former exper-
iment, which recurrently adds Gaussian noise with intensity
from 15 to 100 into 32 clean HSIs to generate 160 corrupted
HSIs, we only generate 32 corrupted HSIs with noise intensi-
ties randomly sampled from a uniform distribution of range [15,
100] in this experiment. Notice the true noise intensities are not
provided, we use an off-the-shelf noise estimation method [51]
to estimate it, which is set as the input parameter for all com-
pared methods except ours. Table 2 summarizes the qualitative
9
400 450 500 550 600 650 700
25
30
35
40
45 BM3D
BM4D
LRMR
LRTV
LRTA
LLRGTV
GLF
TDL
ITSReg
Ours
(a) chart and stuffed toy
400 450 500 550 600 650 700
26
28
30
32
34
36 BM3D
BM4D
LRMR
LRTV
LRTA
LLRGTV
GLF
TDL
ITSReg
Ours
(b) watercolors
400 450 500 550 600 650 700
20
25
30
35
40 BM3D
BM4D
LRMR
LRTV
LRTA
LLRGTV
GLF
TDL
ITSReg
Ours
(c) flowers
400 450 500 550 600 650 700
20
22
24
26
28
30
32 BM3D
BM4D
LRMR
LRTV
LRTA
LLRGTV
GLF
TDL
ITSReg
Ours
(d) cloth
Figure 9: PSNR values across the spectrum corresponding to chart and stuffed
toy (Fig. 5), watercolors (Fig. 6), flowers (Fig. 7) and cloth (Fig. 8) respec-
tively.
results of this experiment, which shows our method surpasses
about 0.48 dB than previous best-performance method ITS-Reg
under the measure of PNSR while with the best stability (less
variance) among all the competing methods.
Run Time. In addition to visual quality, another important as-
pect for an HSI denoising method is the run time. We then
compare the speed of all competing methods. All experiments
are run under the Matlab2016a environment on a machine with
Intel(R) Core(TM) i7-7700K CPU of 4.2GHz and 16 GB RAM.
Figure 10 shows the Time v.s. PSNR of different methods for de-
noising HSIs of size 512 ×512×31. The results are obtained by
10 0
10 1
10 2
10 3
10 4
Time (sec)
29
30
31
32
33
34
35
36
37
38
PSNR (dB)
BM3D
LRMR
LRTV
TDL
BM4D
ITSReg
BCTF-HSI
LRTA
LLRGTV
GLF
Figure 10: Time (second) v.s. PSNR (dB) of all competing method for HSI
denoising.
averaging all 32 scenes with variety of noise intensity. We can
see that effectiveness potentially often sacrifices efficiency. Our
method is relatively slower than TDL, BM4D and GLF. How-
ever, taking the great enhancement in denoising effectiveness
into account, our method is still highly completable with these
two state-of-the-art methods. On the other hand, our method
typically achieves 2 times speed even with better denoising ca-
pability compared with ITS-Reg.
4.2. Real HSI Denoising
Here, the Hyperspectral Digital Imagery Collection Exper-
iment (HYDICE) urban dataset4and the Harvard real-world
hyperspectral datasets (HHD)[52] are utilized to evaluate our
method in real-world noise context. The original HSI in HY-
DICE is of size 304 ×304 ×210. As the bands 139-155, 201-
210 are seriously polluted by the atmosphere and water absorp-
tion, and can provide little useful information, we manually
remove them and leave the remaining test data with a size of
304 ×304 ×183 like [13]. The whole HHD dataset consisting
of 50 noisy hyperspectral images of size 1040 ×1392 ×31 are
captured with the wave-lengths in the range of 420-720 nm at
an interval of 10. We scale these HSIs into the interval [0, 1],
and employ the similar implementation strategies and param-
eter settings for all competing methods as previous simulated
experiments. Noise estimation method [51] used before is also
applied in this setting. We illustrate the experimental results in
Figure 11 and Figure 12 respectively.
Figure 11 includes the restorations of bands 1, 109 of the
urban HSI. We finely choose two demarcated area with spe-
cific semantics to conveniently compare the denoising capabil-
ity of different methods. Specifically, The red box area of band
1 represents the housing estate in urban area. It can be obvi-
ously observed that most of competing methods (e.g. BM3D,
BM4D, LRTA, TDL, ITS-Reg) cannot remove the inappropri-
ate stripes existed in this area, while some methods (i.e. LRMR,
LRTV) produce oversmooth results, in some degree destroying
4http://www.tec.army.mil/hypercube
10
(a) Noisy image (b) BM3D (c) BM4D (d) LRMR (e) LRTV (f) LRTA
(g) LLRGTV (h) GLF (i) TDL (j) ITSReg (k) Ours
Figure 11: Real complex noise removal results results at two bands (indexed by 1, 109 respectively) of HYDICE urban HSI. Two demarcated areas in each image
are amplified at a 6 times larger scale for easy observation of details.
(a) Noisy image (b) BM3D (c) BM4D (d) LRMR (e) LRTV
(f) LLRGTV (g) GLF (h) TDL (i) ITSReg (j) Ours
Figure 12: Real random noise removal results on HHD dataset. One demarcated area in each image is amplified at a 2 times larger scale for easy observation of
details.
the original structure of objects of this housing estate. LL-
RGTV, GLF and our LBTF-HSI successfully gets rid of the
stripe noise while preserving the topology structure of this hous-
ing estate. At band 109, the image is highly corrupted by mis-
cellaneous complex noise. Obvious artefacts are still remained
in the results of many competing methods (i.e. BM3D, BM4D,
LRTA, TDL, ITS-Reg). While LLRGTV and GLF do produce
appealing results with good perceptual quality, these results ap-
parent deviate from the underlying ground truth (see green box
region at band 109). This phenomena may be caused by the
incorrect specified subspace dimension (i.e. objective rank re-
quired by their low rank approximation techniques). As a com-
11
2 4 6 8 10 12
Number of the Nonlocal Patches
43.6
43.8
44
44.2
44.4
44.6
PSNR Values
2 4 6 8 10 12
Number of the Nonlocal Patches
0.976
0.978
0.98
0.982
0.984
0.986
SSIM Values
Figure 13: Effects of patch sizes on denoising performance.
0 50 100
Number of the Nonlocal Patches
43
43.5
44
44.5
45
PSNR Values
0 50 100
Number of the Nonlocal Patches
0
500
1000
1500
Times (s)
Figure 14: Effects of the number of nonlocal patches on denoising performance.
10 20 30
Number of Bands
38
40
42
44
PSNR Values
10 20 30
Number of Bands
0.92
0.94
0.96
0.98
1
SSIM Values
Figure 15: Effects of the number of bands on denoising performance.
parison, Our method does not suffer from the rank determina-
tion issue, thus it not only recovers the de facto semantics of the
demarcated area (i.e. the scene of neighbourhood of highway),
but also produces results with high fidelity.
Figure 12 displays the real random noise removal results
On HHD dataset. From the demarcated window, we can ob-
serve that our LBTF-HSI method obtains artifact-free image
with clearer texture and line pattern. In summary, LBTF-HSI
has obtained better performance in terms of noise suppression,
detail preserving, visual pleasure and PSNR value under differ-
ent noise level, even in the real-world unknown noise context.
4.3. Discussion
Besides the initialization strategy aforementioned, there are
other parameters introduced by different stages of our model,
i.e. patch size, numbers of nonlocal patches (for grouping)
and numbers of iterations (for iterative framework). Figure 13
shows the PSNR/SSIM values with respect to different patch
size. Patch size 6 (6x6) and 7 achieve best PSNR and SSIM
values respectively, among all candidates. Figure 14 illustrates
how PSNR/Times value varies with respect to the number of
nonlocal patches. We can see the denoising results become
gradually better with large number of nonlocal patches, infer-
ring the nonlocal self-similarity could be sufficiently utilized by
our model, even in a relaxed condition. Nevertheless, given the
computational cost and marginal enhancement through increas-
12345
Number of Iterations
32
34
36
38
40
42
44
46
PSNR Values
σ=15 σ=25 σ=50 σ=75 σ=100
Figure 16: Effects of the number of iterations on denoising performance with
respect to different noise levels.
ing the number of nonlocal patches, we set it to 50 in all of our
experiment.
We also show how the number of bands of HSI influences
the denoising capacity of our model. From Figure 15, we can
observe that the denoising results become gradually better with
larger number of bands. This suggests the information con-
tained in one band could be utilized to recover other bands, such
that the global correlation along the spectrum can be effectively
exploited by our model.
Figure 16 displays the effects of numbers of iteration on de-
noising performance with respect to different noise levels. Gen-
erally, more stronger noise intensity will require more iteration
times to achieve better performance, while at a expense of com-
puting efficiency. we can see when noise intensity is relatively
small (e.g. σ=15), running algorithm in more than 2 iterations
would successively degenerate the performance. Though the
degradation issue is not observed during 5 iterations in strong
corruption cases (e.g. σ=50,75,100), the performance in-
crement through iterations becomes limited while significantly
increasing the computational cost. Therefore, we suggest the
use of {1, 2, 3, 4, 5}for σ={15,25,50,75,100}in the simu-
lated data experiments respectively.
5. Conclusion
In this paper, we presented an effective Low-rank Bayesian
Tensor Factorization based HSI denoising method, which con-
sidered two intrinsic characteristics of HSIs: the nonlocal self-
similarity across space and the global correlation across spec-
trum. In order to sufficiently embed these useful priors into our
model, the LBTF is utilized to describe the spatial-spectral cor-
relation of each FBP formed by block matching. This model
was effectively solved by our deterministic algorithm derived
under the variational Bayesian framework. Besides, an iterative
denoising framework was introduced for the purpose of further
enhancing the denoising capability of our method. The experi-
mental results on simulated and real HSI denoising showed that
12
the proposed method outperformed many state-of-the-art meth-
ods and demonstrated the effectiveness of the proposed method.
We encode the noise structure as Gaussian distribution in
our hierarchical probabilistic model. Since in real case, the sta-
tistical distribution of noise structure may be hard to be deter-
mined, it is worth investigating more effective noise model to
model the noise from the real world in future.
6. Acknowledgements
We thank the anonymous reviewers for their helpful com-
ments and suggestions to improve this paper. This work was
supported by the National Science Foundation of China under
Grants no. 61672096.
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