08-10-2016, 12:09 PM
[attachment=72736]
Abstract—Nonlocal self-similarity of images has attracted
considerable interest in the field of image processing and led
to several state-of-the-art image denoising algorithms, such as
BM3D, LPG-PCA, PLOW and SAIST. In this paper, we propose a
computationally simple denoising algorithm by using the nonlocal
self-similarity and the low-rank approximation. The proposed
method consists of three basic steps. Firstly, our method classifies
similar image patches by the block matching technique to form
the similar patch groups, which results in the similar patch
groups to be low-rank. Next, each group of similar patches is
factorized by singular value decomposition (SVD) and estimated
by taking only a few largest singular values and corresponding
singular vectors. Lastly, an initial denoised image is generated
by aggregating all processed patches. For low-rank matrices,
SVD can provide the optimal energy compaction in the least
square sense. The proposed method exploits the optimal energy
compaction property of SVD to lead a low-rank approximation
of similar patch groups. Unlike other SVD-based methods,
the low-rank approximation in SVD domain avoids learning
the local basis for representing image patches which usually
is computationally expensive. Experimental results demonstrate
that the proposed method can effectively reduce noise and be
competitive with the current state-of-the-art denoising algorithms
in terms of both quantitative metrics and subjective visual quality
INTRODUCTION
D
URING acquisition and transmission, images are inevitably
contaminated by noise. As an essential and important
step to improve the accuracy of the possible subsequent
processing, image denoising is highly desirable for numerous
applications, such as visual enhancement, feature extraction
and object recognition
The purpose of denoising is to reconstruct the original
image from its noisy observation as accurately as possible,
while preserving important detail features such as edges and
textures in the denoised image. To achieve this goal, over the
past several decades, image denoising has been extensively
studied in the signal processing community, and numerous
denoising techniques have been proposed in the literature.
Generally, denoising algorithms can be roughly classified into
three categories: spatial domain methods, transform domain
methods and hybrid methods [3], [4]. The first class utilizes
the spatial correlation of pixels to smooth the noisy image, the
second one exploits the sparsity of representation coefficients
of the signal to distinguish the signal and noise, and the
third one takes advantage of spatial correlation and sparse
representation to suppress noise.
Spatial domain methods, also called spatial filters, estimate
each pixel of the image by performing a weighted average
of its local/nonlocal neighbors, in which the weights can be
determined by their similarities and higher weights are given to
similar pixels. Therefore, spatial filters can be further divided
into local filters and nonlocal filters. In [5], Smith et al.
proposed a structure preserving local filter called SUSAN,
which uses the intensity distance as a quantitative measure
of the similarity between pixels. Tomasi et al. [6] proposed
the bilateral filtering by generalizing the SUSAN filter, in
which both the intensity and spatial distances are used to
measure the similarity between pixels. Although these local
filters are effective for preserving edges, they cannot perform
very well when the noise level is high. The reason is that
the severe noise destroys the correlations of pixels within
local regions [7]. To overcome this disadvantage of local
filters, Buades et al. [8] proposed the nonlocal mean (NLM)
filter, which estimates each pixel by a nonlocal averaging
of all the pixels in the image. The amount of weighting
for a pixel is based on the Euclidean distance between the
patch centered around the pixel being denoised and the one
centered around a given neighboring pixel. In essence, NLM
uses the structural redundancy, namely self-similarity which
is inherent in natural images, to estimate each pixel. NLM
can be considered as an extension of the bilateral filter by
the means of replacing point-wise photometric distances with
patch distances. Several variants of NLM have been proposed
to improve the adaptivity of the nonlocal filter [9], [10]. Talebi
et al. [3] proposed a spatially adaptive iterative filtering (SAIF)
to improve the performance of NLM. Recently, there has been
a growing interest in exploiting the self-similarity of images to
suppress noise. Chatterjee et al. [11] proposed a patch-based
locally optimal wiener (PLOW) filter , which also exploits
the structural redundancy for image denoising and achieves
the near optimal performance in the minimum mean-squared
error (MMSE) sense [12]. In [13], Zhang et al. proposed a
two-direction nonlocal (TDNL) variational model for image
denoising by using the horizontal and vertical similarities in
the matrix formed by similar image patches. SAIF, PLOW and
TDNL are currently considered to be state of the art in spatial
domain denoising methods.
Transform domain methods assume that the image can be
sparsely represented by some representation basis, such as
wavelet basis and its directional extensions. Due to the sparsity
of representation coefficients, noise is uniformly spread
throughout the coefficients in the transform domain, while
most of image information is concentrated on the few largest
ones. Therefore, noise can be effectively distinguished by different
coefficient shrinkage strategies, including BayesShrink
[14], ProbShrink [15], BiShrink [16], MultiShrink [17], and
SUREShrink [18], [19]. Despite its remarkable success in
dealing with point and line singularities, the fixed wavelet
transform fails to provide an adaptive sparse representation
for the image containing complex singularities. In order to
overcome the problems caused by using the fixed transforms,
Aharon et al. [20] proposed an adaptive representation
technique called K-SVD, which uses a greedy algorithm to
learn an overcomplete dictionary for image representation and
denoising. Under the assumption that each image patch can be
represented by the learned dictionary, Elad et al. [21] proposed
a K-SVD based denoising algorithm, in which each image
patch can be expressed as a linear combination of few atoms
of the dictionary. Although the dictionary-based methods are
more robust to noise, they are computationally expensive.
Spatial-based filters and transform-based filters have
achieved great success in image denoising. Their overall
performance, however, does not generally surpass the hybrid
methods. Due to its impressive performance, the most wellknown
hybrid method for image denoising is the blockmatching
and 3-D filtering (BM3D) reported by Dabov et
al. in [22], which groups similar patches into 3D arrays and
deals with these arrays by a sparse collaborative filtering. To
the best of our knowledge, it is the first one that utilizes
both nonlocal self-similarity and sparsity for image denoising.
However, the fixed 3-D transform is not able to deliver a
sparse representation for image patches containing edges,
singularities or textures. Thus, BM3D may introduce visual
artifacts. In [23], Dabov et al. proposed an improved BM3D
filter (called BM3D-SAPCA) which exploits adaptive-shape
patches and principle component analysis (PCA). Although
BM3D-SAPCA achieves state-of-the-art denoising results, its
computational cost is very high (see Table III). Zhang et
al. [24] proposed an adaptive image denoising scheme by
using PCA with local pixel grouping (LPG-PCA). This method
uses block matching to group the pixels with similar local
structures, transforms each group of pixels by using locally
learned principle component analysis basis, and shrinks PCA
transformation coefficients by using the linear minimum mean
square-error estimation (LMMSE) technique. Both LPG-PCA
and BM3D-SAPCA use the PCA basis to represent image
patches. A key difference between them is that LPG-PCA
applies PCA on 2-D groups of fixed-size image patches, while
BM3D-SAPCA applies PCA on 3-D groups of adaptive-shape
image patches. In [25], He et al. presented an adaptive hybrid
method called ASVD, which uses SVD to learn the local basis
for representing image patches. Another SVD-based denoising
method is called spatially adaptive iterative singular-value
thresholding (SAIST) [26]. This method uses SVD as a sparse
representation of image patches and reduces noise in images
by iteratively shrinking the singular values with BayesShrink.
BM3D-SAPCA and SAIST are considered to be the current
state of the art in image denoising.
In this paper, we propose a simple and efficient denoising
method by combining patch grouping with SVD. The proposed
method first groups image patches by a classification algorithm
to achieve many groups of similar patches. Then each group
of similar patches is estimated by the low-rank approximation
in SVD domain. The denoised image is finally obtained by
aggregating all processed patches. The SVD is a very suitable
tool for estimating each group because it provides the optimal
energy compaction in the least square sense [27]. This implies
that we can achieve a good estimation of the group by taking
only a few largest singular values and corresponding singular
vectors. While ASVD uses SVD to learn a set of local
basis for representing image patches and SAIST uses SVD
as a sparse representation of image patches, the proposed
method exploits the optimal energy compaction property of
SVD to lead a low-rank approximation of image patches.
Experiments indicate that the proposed method achieves highly
competitive performance in visual quality, and it also has a
lower computational cost than most of existing state-of-theart
denoising algorithms.
The rest of this paper is organized as follows. In Section
II, we briefly review image representation tools for the sake
of completeness. We present the proposed algorithm in detail
in Section III, which fuses the nonlocal self-similarity and the
low-rank approximation by using patch clustering and SVD. In
Section IV, we report the experimental results of our method
to validate its efficacy and compare it with the state-of-the-art
methods. In Section V, we discuss the differences between
our method and other state-of-the-art methods. Finally, we
conclude this paper with some possible future work in Section
VI.
I
Abstract—Nonlocal self-similarity of images has attracted
considerable interest in the field of image processing and led
to several state-of-the-art image denoising algorithms, such as
BM3D, LPG-PCA, PLOW and SAIST. In this paper, we propose a
computationally simple denoising algorithm by using the nonlocal
self-similarity and the low-rank approximation. The proposed
method consists of three basic steps. Firstly, our method classifies
similar image patches by the block matching technique to form
the similar patch groups, which results in the similar patch
groups to be low-rank. Next, each group of similar patches is
factorized by singular value decomposition (SVD) and estimated
by taking only a few largest singular values and corresponding
singular vectors. Lastly, an initial denoised image is generated
by aggregating all processed patches. For low-rank matrices,
SVD can provide the optimal energy compaction in the least
square sense. The proposed method exploits the optimal energy
compaction property of SVD to lead a low-rank approximation
of similar patch groups. Unlike other SVD-based methods,
the low-rank approximation in SVD domain avoids learning
the local basis for representing image patches which usually
is computationally expensive. Experimental results demonstrate
that the proposed method can effectively reduce noise and be
competitive with the current state-of-the-art denoising algorithms
in terms of both quantitative metrics and subjective visual quality
INTRODUCTION
D
URING acquisition and transmission, images are inevitably
contaminated by noise. As an essential and important
step to improve the accuracy of the possible subsequent
processing, image denoising is highly desirable for numerous
applications, such as visual enhancement, feature extraction
and object recognition
The purpose of denoising is to reconstruct the original
image from its noisy observation as accurately as possible,
while preserving important detail features such as edges and
textures in the denoised image. To achieve this goal, over the
past several decades, image denoising has been extensively
studied in the signal processing community, and numerous
denoising techniques have been proposed in the literature.
Generally, denoising algorithms can be roughly classified into
three categories: spatial domain methods, transform domain
methods and hybrid methods [3], [4]. The first class utilizes
the spatial correlation of pixels to smooth the noisy image, the
second one exploits the sparsity of representation coefficients
of the signal to distinguish the signal and noise, and the
third one takes advantage of spatial correlation and sparse
representation to suppress noise.
Spatial domain methods, also called spatial filters, estimate
each pixel of the image by performing a weighted average
of its local/nonlocal neighbors, in which the weights can be
determined by their similarities and higher weights are given to
similar pixels. Therefore, spatial filters can be further divided
into local filters and nonlocal filters. In [5], Smith et al.
proposed a structure preserving local filter called SUSAN,
which uses the intensity distance as a quantitative measure
of the similarity between pixels. Tomasi et al. [6] proposed
the bilateral filtering by generalizing the SUSAN filter, in
which both the intensity and spatial distances are used to
measure the similarity between pixels. Although these local
filters are effective for preserving edges, they cannot perform
very well when the noise level is high. The reason is that
the severe noise destroys the correlations of pixels within
local regions [7]. To overcome this disadvantage of local
filters, Buades et al. [8] proposed the nonlocal mean (NLM)
filter, which estimates each pixel by a nonlocal averaging
of all the pixels in the image. The amount of weighting
for a pixel is based on the Euclidean distance between the
patch centered around the pixel being denoised and the one
centered around a given neighboring pixel. In essence, NLM
uses the structural redundancy, namely self-similarity which
is inherent in natural images, to estimate each pixel. NLM
can be considered as an extension of the bilateral filter by
the means of replacing point-wise photometric distances with
patch distances. Several variants of NLM have been proposed
to improve the adaptivity of the nonlocal filter [9], [10]. Talebi
et al. [3] proposed a spatially adaptive iterative filtering (SAIF)
to improve the performance of NLM. Recently, there has been
a growing interest in exploiting the self-similarity of images to
suppress noise. Chatterjee et al. [11] proposed a patch-based
locally optimal wiener (PLOW) filter , which also exploits
the structural redundancy for image denoising and achieves
the near optimal performance in the minimum mean-squared
error (MMSE) sense [12]. In [13], Zhang et al. proposed a
two-direction nonlocal (TDNL) variational model for image
denoising by using the horizontal and vertical similarities in
the matrix formed by similar image patches. SAIF, PLOW and
TDNL are currently considered to be state of the art in spatial
domain denoising methods.
Transform domain methods assume that the image can be
sparsely represented by some representation basis, such as
wavelet basis and its directional extensions. Due to the sparsity
of representation coefficients, noise is uniformly spread
throughout the coefficients in the transform domain, while
most of image information is concentrated on the few largest
ones. Therefore, noise can be effectively distinguished by different
coefficient shrinkage strategies, including BayesShrink
[14], ProbShrink [15], BiShrink [16], MultiShrink [17], and
SUREShrink [18], [19]. Despite its remarkable success in
dealing with point and line singularities, the fixed wavelet
transform fails to provide an adaptive sparse representation
for the image containing complex singularities. In order to
overcome the problems caused by using the fixed transforms,
Aharon et al. [20] proposed an adaptive representation
technique called K-SVD, which uses a greedy algorithm to
learn an overcomplete dictionary for image representation and
denoising. Under the assumption that each image patch can be
represented by the learned dictionary, Elad et al. [21] proposed
a K-SVD based denoising algorithm, in which each image
patch can be expressed as a linear combination of few atoms
of the dictionary. Although the dictionary-based methods are
more robust to noise, they are computationally expensive.
Spatial-based filters and transform-based filters have
achieved great success in image denoising. Their overall
performance, however, does not generally surpass the hybrid
methods. Due to its impressive performance, the most wellknown
hybrid method for image denoising is the blockmatching
and 3-D filtering (BM3D) reported by Dabov et
al. in [22], which groups similar patches into 3D arrays and
deals with these arrays by a sparse collaborative filtering. To
the best of our knowledge, it is the first one that utilizes
both nonlocal self-similarity and sparsity for image denoising.
However, the fixed 3-D transform is not able to deliver a
sparse representation for image patches containing edges,
singularities or textures. Thus, BM3D may introduce visual
artifacts. In [23], Dabov et al. proposed an improved BM3D
filter (called BM3D-SAPCA) which exploits adaptive-shape
patches and principle component analysis (PCA). Although
BM3D-SAPCA achieves state-of-the-art denoising results, its
computational cost is very high (see Table III). Zhang et
al. [24] proposed an adaptive image denoising scheme by
using PCA with local pixel grouping (LPG-PCA). This method
uses block matching to group the pixels with similar local
structures, transforms each group of pixels by using locally
learned principle component analysis basis, and shrinks PCA
transformation coefficients by using the linear minimum mean
square-error estimation (LMMSE) technique. Both LPG-PCA
and BM3D-SAPCA use the PCA basis to represent image
patches. A key difference between them is that LPG-PCA
applies PCA on 2-D groups of fixed-size image patches, while
BM3D-SAPCA applies PCA on 3-D groups of adaptive-shape
image patches. In [25], He et al. presented an adaptive hybrid
method called ASVD, which uses SVD to learn the local basis
for representing image patches. Another SVD-based denoising
method is called spatially adaptive iterative singular-value
thresholding (SAIST) [26]. This method uses SVD as a sparse
representation of image patches and reduces noise in images
by iteratively shrinking the singular values with BayesShrink.
BM3D-SAPCA and SAIST are considered to be the current
state of the art in image denoising.
In this paper, we propose a simple and efficient denoising
method by combining patch grouping with SVD. The proposed
method first groups image patches by a classification algorithm
to achieve many groups of similar patches. Then each group
of similar patches is estimated by the low-rank approximation
in SVD domain. The denoised image is finally obtained by
aggregating all processed patches. The SVD is a very suitable
tool for estimating each group because it provides the optimal
energy compaction in the least square sense [27]. This implies
that we can achieve a good estimation of the group by taking
only a few largest singular values and corresponding singular
vectors. While ASVD uses SVD to learn a set of local
basis for representing image patches and SAIST uses SVD
as a sparse representation of image patches, the proposed
method exploits the optimal energy compaction property of
SVD to lead a low-rank approximation of image patches.
Experiments indicate that the proposed method achieves highly
competitive performance in visual quality, and it also has a
lower computational cost than most of existing state-of-theart
denoising algorithms.
The rest of this paper is organized as follows. In Section
II, we briefly review image representation tools for the sake
of completeness. We present the proposed algorithm in detail
in Section III, which fuses the nonlocal self-similarity and the
low-rank approximation by using patch clustering and SVD. In
Section IV, we report the experimental results of our method
to validate its efficacy and compare it with the state-of-the-art
methods. In Section V, we discuss the differences between
our method and other state-of-the-art methods. Finally, we
conclude this paper with some possible future work in Section
VI.
I