Image Restoration UsingNonlocally Centralized Sparse Representation and histogram Clipping

Abdussayeed khanInt. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622,Vol. 5, Issue 6, ( Part -2) June 2015, pp.112-115
www.ijera.com 112|P a g e
Image Restoration UsingNonlocally Centralized Sparse
Representation and histogram Clipping
Abdussayeed khan, M.Sc. Electronics,Khalifa UniversityAbuDhabi
Abstract-
Due to the degradation of observed image the noisy, blurred, distorted image can be occurred .To restore the
image informationby conventional modelsmay not be accurate enough for faithful reconstruction of the original
image. I propose the sparse representations to improve the performance of based image restoration. In this
method the sparse coding noise is added for image restoration, due to this image restoration the sparse
coefficients of original image can be detected. The so-called nonlocally centralized sparse representation
(NCSR) model is as simple as the standard sparse representation model, fordenoising the image here we use the
histogram clipping method by using histogram based sparse representation to effectively reduce the noise and
also implement the TMR filter for Quality image. Various types of image restoration problems, including
denoising, deblurring and super-resolution, validate the generality and state-of-the-art performance of the
proposed algorithm.
Keywords —Image restoration, nonlocal similarity, sparserepresentation,histogram clipping.
I. Introduction
Image classification is one of the most important
topics in computer vision. Recently, sparse coding
technique attractsmore and more attention because of its
effectiveness in extracting global properties from signals. It
recovers a sparse linear representation of a query datum
with respect to a set of non-parametric basis set, known as
dictionary [3, 24]. In image classification problem, methods
based on sparse coding or its variants mainly collect a set
of image patches to learn the dictionariesby representing an
image with a histogram of local features, Bag of Words
(BoW) models [25] have shown excellent performance,
especially its robustness to spatial variations. Considering
spatial information with BoW, Lazebniket al. [10] built a
spatial pyramid and extended the BoW model by
partitioning the image into sub-regions and
computinghistograms of local features. Yang et al. [29]
further extended Spatial Pyramid Matching (SPM) by using
sparse coding. They provided a generalized vector
quantization for sparse coding followed by multi-scale
spatial max pooling.
Recent years have seen a great deal of renewed
interests, enthusiasm and progress in sparsity-based image
processing, particularly in image restoration. However,
quite surprisingly, most published algorithms for image
processing and analysis based themselves on the sparsity of
luminance component of the image signal and overlooked
the sparsities induced by spectral correlations. This leaves a
slack in the performance of these algorithms. Mairalet.
alextended the K-SVD algorithm to color images in the
searching of a dictionary based sparse representation of
color images [2]. In this paper, to pick up the performance
slack we investigate ways to formulate spectral correlations
into inherent and computationally amenable sparse
representations of multispectral images. Our investigation
begins with an image formation model of digital color
Cameras. This image model and mild assumptions on
illumination conditions and imaged objects reveal intrinsic
sparsity properties of natural images. It turns out that
theseallows the newly revealed sparsities of color images to
be readily exploited by a `1 minimization process, or by
linear programming algorithmically. Upon the conclusion
of our technical development, it will become self-evident
how the new results of this paper can be integrated into the
general framework of image restoration and used as strong
domainknowledge toimprove the solution of the
corresponding inverse problem.
In this paper we improve the sparse representation
performanceby proposing a nonlocallycentralized sparse
representation (NCSR) model. To faithfully reconstruct the
original image, the sparse code αy [refer to Eq. (3)] should
be as close as possible to the sparse codes αx [refer to Eq.
(2)] of the original image. In other words, the difference
υα= αy –αx(called as sparse coding noise, SCN in short, in
this work) should be reduced and hence the quality of
reconstructed image ˆx = _αy can be improved because ˆx
− x ≈ _αy −_αx = _υα. To reduce the SCN, we centralize
the sparse codes to some good estimation of αx. In practice,
a good estimation of αx can be obtained by exploiting the
rich amount of nonlocal redundancies in the observed
image
The proposed NCSR model can be solved effectively
byconventional iterative shrinkage algorithm [9], which
allows the remainder of the paper has the following flow of
presentation. The image formation model is reviewed in
Section II, which leads to the sparse representation that is
detailed in Section III. Typical applications of color image
denoisingand deconvolution are investigated in Section IV.
And finally, Section V concludes the paper.
RESEARCH ARTICLE OPEN ACCESS

II. NONLOCALLY CENTRALIZED
SPARSEREPRESENTATION (NCSR)
Following the notation used in [19], for an image x ∈
_N, let xi = Rix denote an image patch of size√n×√n
extracted at location i, where Riis the matrix extracting
patch xi from x at location i. Given an dictionary _ ∈
_n×M, n ≤ M, each patch can be sparsely represented as xi
≈ _αx,iby solving an l1-minimization problem αx,i=
argminαi{||xi − _αi||22+λ||αi||1}. Then the entire image x
can be represented by the set of sparse codes {αx,i}. The
patches can be overlapped to suppress the boundary
artifacts, and we obtain a redundant patch-based
representation. Reconstructing x from {αx,i} is an over-
determined system, and a straightforward least-square
solution is [19]: x ≈ (_Ni=1 RTiRi )−1_Ni=1 (RTi _αx,i ).
For the convenience of expression, we let
(1)
Whereαx denotes the concatenation of all αx,i. The above
equation is nothing but telling that the overall image is
reconstructed by averaging each reconstructed patch of xi.
In the scenario of image restoration (IR), the observed
image is modeled as y = Hx+ υ. The sparsity-based IR
method recovers x from y by solving the following
minimization problem:
(2)
The image x is then reconstructed as ˆx = αy.
2.1. Sparse Coding Noise:
In order for an effective IR, the sparse codes αy obtained
by solving the objective function in Eq. (5) are expected to
be as close as possible to the true sparse codes αx of the
original image x. However, due to the degradation of the
observed image y (e.g., noisy and blurred), the sparse code
αy will deviate from αx, and the IR quality depends on the
level of the sparse coding noise (SCN), which is defined as
the difference between αy and αx
(3)
ImageLena as an example. In the first experiment, we add
Gaussian white noise to the original image x to get the
noisy image y (the noise level σn= 15). Then we compute
αx and αy by solving Eq. (2) and Eq. (5), respectively. The
DCT bases are adopted in the experiment. Then the SCN
υα is computed. In Fig. 2(a-1), we plot the distribution of
υα corresponding to the 4th atom in the dictionary. We also
plot these distributions in log domain in Fig. 2(b-1) ∼ (b-3).
This observation motivates us to model υα with a Laplacian
prior, as will be further discussed in Section III-A.
2.2. Modeling of NCSR:
The definition of SCN υα indicates that by suppressing the
SCN υα we could improve the IR output ˆx. However, the
difficulty lies in that the sparse coding vector αx is
unknown so that υα cannot be directly measured.
Nonetheless, if we could have some reasonably good
estimation of αx, denoted by β, available, then αy − β can
be a good estimation of the SCN υα. To suppress υα and
improve the accuracy of αy and thus further improve the
objective function of Eq., we can propose the following
centralized sparse representation (CSR)
(4)
Whereβiis some good estimation of αi, γ is the
regularization parameter and p can be 1 or 2. In the above
CSR model, while enforcing the sparsity of coding
coefficients αi, the sparse codes are also centralized to
some estimate of αx (i.e., β) so that SCN υα can be
suppressed. One important issue of sparsity-based IR is the
selection of dictionary _.
. Hence we propose the following sparse coding model:
(5)
2.3. Nonlocal Estimate of Unknown Sparse Code:
Generally, there can be various ways to make an estimate
of αx, depending on how much the prior knowledge of αx
wehave. If we have many training images that are similar to
the original image x, we could learn the estimate β of αx
from the training set. However, in many practical situations
the training images are simply not available. On the other
hand, the strong nonlocal correlation between the sparse
coding coefficients, as shown in Fig. 1, allows us to learn
the estimate β from the input data. Based on the fact that
natural images often contain repetitive structures, i.e., the
rich amount of nonlocal redundancies [30], we search the
nonlocal similar patches to the given patch iin a large
window centered at pixel i. For higher performance, the
search of similar patches can also be carried out across
different scales at the expense of higher computational
complexity, as shown in [31]. Then a good estimation of αi,
i.e., βi, can be computed as the weighted average of those
sparse codes associated with the nonlocal similar patches
(including patch i)to patch i. For each patch xi , we have a
set of its similar patches, denoted by i. Finally βican be
computed from the sparse codes of the patches within
i.Denote by αi,qthe sparse codes of patch xi,qwithin set i.
Then βican be computed as the weighted average of αi,q
(6)
Whereωi,qis the weight. Similar to the nonlocal means
approach [30], we set the weights to be inversely
proportional to the distance between patches xi and xi,q
(7)
III. ALGORITHM OF NCSR
3.1. Parameters Determination:

In Eq. (8) or Eq. (11) the parameter λ that balances the
fidelity term and the centralized sparsity term should be
adaptively determined for better IR performance. In this
subsection we provide a Bayesian interpretation of the
NCSR model, which also provides us an explicit way to set
the regularization parameter λ. In the literature of wavelet
denoising, the connection between Maximum a Posterior
(MAP) estimator and sparse representation has been
established [28], and here we extend the connection from
the local sparsity to nonlocally centralized sparsity. For the
convenience of expression, let’s define θ = α − β. For a
given β, the MAP estimation of θ can be formulated as
(8)
The likelihood term is characterized by the Gaussian
distribution where θ and β are assumed to be independent.
In the prior probability P(θ ), θ reflects the variation of α
from its estimation β. If we take β as a very good
estimation of the sparse coding coefficient of unknown true
signal, then θy= αx – β is basically the SCN associated with
αy, and we have seen in Fig. 2 that the SCN signal can be
well characterized by the Laplacian distribution. Thus, we
can assume that θ follows i.i.d. Laplacian distribution, and
the joint prior distribution P(θ ) can be modeled as
(9)
Hence, for a given β the sparse codes α can then be
obtained by minimizing the following objective function
Compared with Eq. (8), we can see that the l1-norm (i.e.,p=
1) should be chosen to characterize the SCN term αi−βi.
Comparing Eq. (16) with Eq. (8), we have
(10)
3.3. Histogram-Based SparseRepresentation:
Histogram-based representations have been widely used
with the feature descriptors, e.g., HOG [4], BoW [25], and
GLOH [17]. It provides very compact representationand
captures global frequency of low-level features. In this
section, we present a framework that determines the
component-level importance of histogram information and
Combines it with a sparse representation, which is referred
to as Histogram-Based Component-Level Sparse
Representation (HCLSP) for the rest of this paper.
3.3.1. Component Level Importance:
Suppose we have a training set of image groups.
Eachimage group is defined as a class. For the training set,
we have C classes, for each class index p = 1; : : : ;C.
DenoteX(p) = [x(p)1 ; : : : ; x(p) n ] in Rm_nto be a set of
training samples from class p, with each individual sample
x(p)iin Rm. The dictionaries trained from class p are
represented by D (p). Given the training data and the
dictionaries, the sparse coefficient vector _ can be obtained
by solving equation (2) using LARS-Lasso or other
standard algorithms. Denote the reconstruction error for the
training set X (p) by using the dictionaries D (p) as:
(11)3.4.
Histogram Equalization:
Histogram equalization is a technique for adjusting image
intensities to enhance contrast. Let f be a given image
represented by mc matrix of integer pixel intensities
ranging from 0 to L − 1. L is the number of possible
intensity values.
IV. EXPERIMENTAL RESULTS
To verify the IR performance of the proposed NCSR
algorithm we conduct extensive experiments on image
denoising, deblurring and super-resolution. The basic
parameter setting of NCSR is as follows: the patch size is 7
× 7 and K = 70. For image denoising, δ = 0.02, L = 3, and J
= 3; for image deblurring and super-resolution, δ = 2.4, L =
5, andJ = 160. To evaluate the quality of the restored
images, the PSNR and the recently proposed powerful
perceptual quality metric FSIM [32] are calculated. Due to
the limited page space, we only show part of the results in
this paper, and all the experimental results can be
downloaded on the clearer and much more details are
recovered. Considering that the estimated kernel will have
bias from the true unknown blurring kernel, these
experiments validate that NCSR is robust to the kernel
estimation errors.
Original image Output image
Fig .1 conversion of image to gray scale image
We also test the proposed NCSR deblurring method on real
motion blurred images. Since the blur kernel estimation is a
non-trivial task, we borrowed the kernel estimation method
from [34] to estimate the blur kernel and apply the
estimated blur kernel in NCSR to restore the original
images. In Fig. 8 we present the deblurring results by the
blind deblurringmethod of [34] and the proposed NCSR
method. We can seethat the images restored by our
approach are much clearer and much more details are
recovered. Considering that the estimated kernel will have
bias from the true unknown blurring kernel, these
experiments validate that NCSR is robust to the kernel
estimation errors

Original ImageHistogramSpacialadaptive
equalizationhistogram
equalization
SpacialpreservingequalizationHistogram clipping
Fig.2 image denoisingoutoutimage with the quality
Zoom before Iterated TMR Zoom after Iterated
TMR
Fig.3TMR filtered Image
Iterated TMR Filter of Image (b) Iterated TMR
Filter of(c)
Iterated TMR Filter of ImageIterated TMR filter
of (d) of image (e)
Fig.4Denoising image
V. CONCLUSION
In this paper we presented a novel nonlocally
centralized sparse representation (NCSR) model for image
restoration. The sparse coding noise (SCN), which is
defined as the difference between the sparse code of the
degraded image and the sparse code of the unknown
original image, should be minimized to improve the
performance of sparsity-based image restoration. To this
end, we proposed a centralized sparse constraint, which
exploits the image nonlocal redundancy, to reduce the
SCN. The Bayesian interpretation of the NCSR model was
provided and this endows the NCSR model an iteratively
reweighted implementation. An efficient iterative shrinkage
function was presented for solving the l1-regularized
NCSR minimization problem. Experimental results on
image denoising, deblurring and super-resolution
demonstrated that the NCSR approach can achieve highly
competitive performance to other leading denoising
methods, and outperform much other leading image
deblurring and super-resolution methods. And also
implement the histogram clipping and TMR filter for
Quality of picture.
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