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Manuscript Draft
Manuscript Number: INS-D-13-547R1
Title: Joint Image Denoising Using Adaptive Principal Component Analysis and Self-similarity
Article Type: Full length article
Keywords: Image denoising; Dimensionality reduction; Principal component analysis; Singular value
decomposition; Parallel analysis
Corresponding Author: Dr. Yong-Qin Zhang, Ph.D.
Corresponding Author's Institution: Peking University
First Author: Yong-Qin Zhang, Ph.D.
Order of Authors: Yong-Qin Zhang, Ph.D.; Jiaying Liu, Ph.D.; Mading Li, M.E.; Zongming Guo, Ph.D.

Cover Letter
Dear Editors,
We would like to submit the enclosed manuscript entitled "Joint Image Denoising Using Adaptive
Principal Component Analysis and Self-similarity", which we wish to be considered for publication in
"Information Sciences". No conflict of interest exits in the submission of this manuscript, and
manuscript is approved by all authors for publication. I would like to declare on behalf of my
co-authors that the work described was original research that has not been published previously, and
not under consideration for publication elsewhere, in whole or in part. All the authors listed have
approved the manuscript that is enclosed. In this paper, we propose an efficient joint denoising
algorithm based on adaptive principal component analysis and self-similarity (APCAS) that improves
the predictability of pixel intensities in reconstructed images. The proposed algorithm consists of two
successive steps: the joint denoising strategy without iteration, the self-similarity based image patch
clustering and parallel analysis based adaptive principal component analysis for the low-rank
approximation. The experimental results validate its generality and effectiveness in a wide range of
the noisy images. The proposed algorithm not only produces very promising denoising results that
outperforms the state-of-the-art methods, but also adapts to a variety of noise levels. The medical
images, and the images or videos in surveillance, traffic and remote sensing systems will benefit from
our proposed methods.
I hope this paper is suitable for "Information Sciences". The following is a list of possible reviewers
for your consideration:
1) Wenxuan Shi, E-mail: shiwx@163.com ;
2) Hai-Bo Huang, E-mail: huang7855@163.com ;
3) Yong-An Liu, E-mail: liuan86@126.com.
We deeply appreciate your consideration of our manuscript, and we look forward to receiving
comments from the reviewers. If you have any queries, please don’t hesitate to contact me at the
address below.
Thank you and best regards.
Yours sincerely,
Yong-Qin Zhang
Corresponding author: Yong-Qin Zhang
Institute of Computer Science & Technology, Peking University
Address: No.128 Zhongguancun North Road, Haidian District, Beijing, P.R. China
Tel: +86-10-82529641 Postcode: 100080
E-mail: zhangyongqin@pku.edu.cn ; zyqwhu@gmail.com

Revision Note
Authors' Response to Reviewers’ Comments
Dear Editors and Reviewers,
Thanks a lot for the editor's and the reviewers' suggestions concerning our manuscript
entitled "Joint Image Denoising Using Adaptive Principal Component Analysis and
Self-similarity" (Manuscript ID: INS-D-13-547). We would like to express our sincere thanks to
the editor and the reviewers for their careful reading and valuable suggestions on our manuscript.
These suggestions are not only very useful and helpful for improving the quality of our paper, but
also have great guiding significance to our researches. We have considered those suggestions
carefully and revised the manuscript accordingly. The main corrections in the paper and the
responses with detailed explanations to the reviewers’ comments are given below for the editors
and the reviewers’ perusal. We hope that this revision is satisfactory for publication in the journal.
Thank you again and best regards.
Yours sincerely,
Yong-Qin Zhang
Institute of Computer Science and Technology, Peking University,
No.128 Zhongguancun North Road, Haidian District, Beijing 100871, P.R. China.
Email: zhangyongqin@pku.edu.cn

The Editor's and the reviewers' comments are as follows:
***********************************************************
The Editor’s comments:
Dear Dr. Zhang,
We have received the decision on your paper entitled "Joint Image Denoising Using Adaptive Principal
Component Analysis and Self-similarity". In view of these comments the Editor-in-Chief, Professor
Witold Pedrycz, has decided that the paper can be reconsidered for publication after major revisions.
We look forward to receiving the revised version of your paper together with a reply to the reports and
a summary of the revisions made.
Kind regards,
Witold Pedrycz
Editor-in-Chief
Information Sciences
Response:
Thank the editor for his valuable suggestions. After further studying the theories and applications
on image denoising, we have considered these suggestions carefully and revised the manuscript
accordingly. The main corrections in the paper and the responses to the editor’s and the reviewers’
comments are given below. If you have any other suggestion or anything else we should do, please
do not hesitate to let us know.
***************************************************
Reviewer #1: Contribution:
Auithors describe a joint denoising strategy based on adaptive principal component analysis (PCA) and
self-similarity (APCAS).
Their proposal computes two successive steps: dimensionality reduction of similar patch groups, and
the collaborative filtering.
At the end, they use collaborative Wiener filtering for residual noise removal.
Paper presents interesting and promising results. However, in my opinion it should be sent for review to
a more specialized journal regarding image processing strategies.
Things to do to improve paper quality:
1. At the begining of figures 4, 5 and 6 add the original subímages.
Response to Reviewer #1:
Response:
We are very grateful to the reviewer for pointing this out. According the reviewer’s
suggestions, the original subimages have been separately added at the beginning of Figure 4,
5, and 6 in this revised manuscript (see Figure 5, 6, and 7 in Subsection 3.1).

2. Revise paper. Writing could be improved. For example, in pp. 17, intead of: Besides this set of
experiments on the simulated noisy images, we also validate our denoising algorithm...
Through experiments authors have shown that their proposal performs better than some reported
methdos.
A better writing would be:
Besides this set of experiments on the simulated noisy images, we also validated our denoising
algorithm...
Other examples:
In pp. 5, you wrote:
The relationships between the central pixel, the target patch, the adjacent patch and the local search
window are described in Figure 1. The difference metric can be calculated based on Euclidean distance
in this formula:
A better writing cvould be:
Relationships between the central pixel, the target patch, the adjacent patch and the local search
window can be appreciated in Figure 1.
Prase: The difference metric can be calculated based on Euclidean distance in this formula:...
is difficult to underestand...
Response:
Thank the reviewer for his constructive suggestions and comments. We have considered those
suggestions carefully and revised the manuscript accordingly. Moreover, we have asked
several experienced writing colleagues for advice to correct grammar and spelling errors, and
also have checked the whole manuscript carefully to improve the quality of this English paper.
The English writing of this manuscript has been improved in this revision. The corrections of
the mistakes mentioned by the reviewer are also given below.
In Page 17, Change “we also validate” to “we also validated”;
In Page 6, Change “are described in Figure 1” to “can be appreciated in Figure 2”;
In Page 6, Change “The difference metric can be calculated based on Euclidean distance in this
formula:” to “The similarity metric between the candidate patch and the target patch can be calculated
using the Euclidean distance in this formula:”.
Although the reviewer suggested that this manuscript should be sent for review to a more
specialized journal regarding image processing strategies in his or her opinion, we found that
the prestigious international journal “Information Sciences” has published numerous papers in
the fields of image processing and computer vision in recent years. For example,
• D. Alassi, and R. Alhajj, Effectiveness of template detection on noise reduction and
websites summarization, Information Sciences, 219 (2013) 41-72.
• XiaoHong Han, XiaoMing Chang, An intelligent noise reduction method for chaotic
signals based on genetic algorithms and lifting wavelet transforms, Information Sciences,
218(1) (2013) 103-118.
• T. C. Lin, A new adaptive center weighted median filter for suppressing impulsive noise
in images, Information Sciences, 177(4) (2007) 1073-1087.
Thank the reviewer again for his or her careful reading and valuable suggestions, which
helped to significantly improve the quality of our paper.

Reviewer #2:
This paper proposes a de-noising method using PCA and self-similarity. However This paper should
be accepted subject to the following conditions:
(1) In this work, it said "The proposed algorithm not only pro-duces very promising denoising results
that outperform the state-of-the-art methods", then experimental results are so so from table 1.
(2)The algorithm has not provide quantitive evaluation index of edge preservation in this paper.
(3)For MRI, I can not find out outperform the best state-of-the-art methods, such as "image regions
containing the abundant edges".
(4) Please explain how to improve the predictability of pixel intensities.
(5)what is really goal to dimensional reduction in another words, how to understand dimensional
reduction and denoising
Response to Reviewer #2:
Response:
(1) We are very sorry that there is a logical error for these sentences. This mistake has been
corrected in the revised manuscript. The redescription of these sentences is also given as
follows:
The experimental results validate its generality and effectiveness in a wide range of the noisy
images. The proposed algorithm not only produces very promising denoising results that
outperform the state-of-the-art methods, but also adapts to a variety of noise levels.
(2) We are very grateful to the reviewer for pointing this out. Besides noise suppression
assessment (See SSIM results in Table 1, and Figure 5-7), we have also provided the
quantitive evaluation of edge preservation index (Figure of Merit, FOM \cite{Yu02,Yue06})
for the proposed algorithm in comparison with BM3D \cite{Dabov07}, PLPCA
\cite{Deledalle11}, and LPG-PCA \cite{LeiZhang10} in the revision (See Table 2).
References:
\bibitem[]{Yu02}
Y. Yu, and S. T. Acton, Speckle reducing anisotropic diffusion, IEEE Transactions on Image
Processing, 11(11) (2002) 1260-1270.
\bibitem[]{Yue06}
Y. Yue, M.M. Croitoru, A. Bidani, J.B. Zwischenberger, and J.W. Clark. Nonlinear Multiscale
Wavelet Diffusion for Speckle Suppression and Edge Enhancement in Ultrasound Images.
IEEE Transactions Medical Imaging, 25(3) (2006) 297-311.
\bibitem[]{Dabov07}
K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, Image Denoising by Sparse 3D
Transform-Domain Collaborative Filtering, IEEE Transactions on Image Processing, 16(8)
(2007) 2080-2095.
\bibitem[]{Deledalle11}
C.~A. Deledalle, J. Salmon, and A.~S. Dalalyan, Image denoising with patch based PCA:
local versus global, Proceedings of the British Machine Vision Conference 2011 (BMVC),
63(3) (2011) 782-789.

\bibitem[]{LeiZhang10}
L. Zhang, W. Dong, D. Zhang, and G. Shi, Two-stage image denoising by principal
component analysis with local pixel grouping, Pattern Recognition, 43(4) (2010) 1531-1549.
(3) We are very grateful to the reviewer for pointing this out. Since the MR images were
zoomed out too much so that the reviewer can’t distinguish the image details in the previous
version of this paper, we have displayed the MR images with the appropriate size in the
revised manuscript. As seen from the visual comparisons in Figure 8-11 and the SNR results
\cite{Dietrich07,Gilbert07,YuJing11} in Table 4 in the revision, it is found that the proposed
algorithm outperforms the best state-of-the-art methods (e.g., BM3D \cite{Dabov07}),
especially for the edge preservation and the structure recovery (See Subsection 3.2).
References:
\bibitem[]{Dietrich07}
O. Dietrich, J.~G. Raya, S.~B. Reeder, M.~F. Reiser, and S.~O. Schoenberg. Measurement of
signal-to-noise ratios in MR images: influence of multichannel coils, parallel imaging, and
reconstruction filters, Journal of Magnetic Resonance Imaging, 26(2) (2007) 375-85.
\bibitem[]{Gilbert07}
G. Gilbert. Measurement of signal-to-noise ratios in sum-of-squares MR images. Journal of
Magnetic Resonance Imaging, 26(6) (2007) 1678.
\bibitem[]{YuJing11}
J. Yu, H. Agarwal, M. Stuber, and M. Schar, Practical Signal-to-Noise Ratio Quantification for
Sensitivity Encoding: Application to Coronary MRA, Journal of Magnetic Resonance
Imaging, 33(6) (2011) 1330-1340.
(4) Thank the reviewer for his or her constructive suggestions and comments. To improve the
predictability of pixel intensities, we proposed the joint denoising algorithm using adaptive
principal component analysis and self-similarity. Without using iteration, it consists of two
successive steps: the low-rank approximation based on parallel analysis, and the empirical
M N
Wiener filtering. Let Y denote an observed noisy image with its coordinate domain XR .
First, for each pixel
y x and its target patch Y , the patch group
x
Z
x
is formed by finding
similar patches in a local search window. Next, after singular value decomposition (SVD) of
the patch group Z x
, the signal dimension K is determined by parallel analysis applied to
the singular values for separating the signal and the noise in the SVD domain. Then, after
the inverse SVD transform, the denoised patch group Z ˆ
x
is obtained by the low-rank
approximation, where most of the noise is eliminated. After that, the whole denoised image
S ˆ t is acquired by the weighted averaging of all the estimates of each pixel for further noise
reduction. Finally, after the empirical Wiener shrinkage coefficients G
are reconstructed
x
from the power spectrum coefficients of 3D transform of the denoised image S ˆ t , the final
whole noiseless image S ˆ f is obtained by the collaborative Wiener filtering for further noise
removal. Thus, the predictability of pixel intensities is improved by our proposed algorithm.
(5) Thank the reviewer for his or her constructive suggestions and comments. In the SVD
domain of the similar patch group, the signal components almost concentrate on the first

several largest singular values, whereas the small singular values are almost noise and
redundancy. Thus, it’s easier to distinguish the signal from the noise by the singular values in
the SVD domain. As one of the successful dimensionality reduction techniques, parallel
analysis of the proposed algorithm can adaptively determine the number of signal components
almost without loss of signal information. So the goal of dimensionality reduction is to
remove noise and redundancy by discarding the small singular values. The denoised similar
patch group is obtained by the low-rank approximation after the inverse SVD transform.
We are very grateful to the reviewer for his or her valuable suggestions. We have tried our
best to revise the English of the whole manuscript carefully to avoid typo mistakes, grammar
errors, spelling errors and vague descriptions. After reorganization of the language description,
the revised manuscript is clear and concise. Moreover, the qualitative and quantitative
evaluations of both edge preservation and noise suppression have been provided in the
revision. We hope that the revised paper is now acceptable for the reviewer and the readers.
Reviewer #3:
This paper proposes a new image denoising strategy consisting of two successive steps, which are
dimensionality reduction of similar patch groups, and the collaborative filtering. The authors use the
self-similarity to cluster similar patch groups and then they use a parallel analysis method to adaptively
select the signal components.
They experimented and compared the proposed approach with other state-of-the-art methods for image
denoising and obtained promising results.
The paper is generally well-written and well-organized. My first concern is the pseudo-code algorithm
is given as is without any comments and follow-text describing the details of the algorithm. It would be
good if the authors insert comments at appropriate places in the algorithm to make it easily
understantable. My another concern is the images in Figure 7 are too dark and difficult to interpret. It
would be nice if the authors make these images lighter in color.
Please also mention about the following related papers in your related work by comparing these
approaches with yours.
Kivanc Kose, Volkan Cevher, A. Enis Cetin, Filtered Variation method for denoising and sparse signal
processing, Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing
(ICASSP), 2012.
Sinan Gezici, Ismail Yilmaz, Omer N. Gerek, A. Enis Cetin, Image denoising using adaptive subband
decomposition, Proceedings of the International Conference on Image Processing, 2001.
Response to Reviewer #3:
Response:
(1)We are very grateful to the reviewer for his valuable suggestions. The comments have been

inserted at appropriate places in the pseudo-code of the proposed algorithm. The following
concise descriptions also explain the procedures of the proposed algorithm. Now it’s easy for
the reviewers and the readers to understand the details of the proposed algorithm (See
Subsection 2.6).
(2) We are very grateful to the reviewer for pointing this out. In the previous version of this
paper, MR images in Figure 8 were too dark and zoomed out too much so that they were
difficult to be interpreted correctly. Considering that MR images are grayscale, we scaled
these images to the right size instead of making them lighter in color. They have been
separated into four figures and displayed with the appropriate size in Figure 8 to Figure 11 in
the revised manuscript. Now these MR images are easy to be interpreted, whose edges and
details can be distinguishable from the visual comparisons in Figure 8 to Figure 11.
(3)We are very grateful to the reviewer for pointing this out. All the recent related literatures
including \cite{Gezici01} and \cite{Kose12} have been added in this revision. However, since
there are no source codes of these approaches \cite{Gezici01} and \cite{Kose12} at hand, the
comparison between them and our proposed algorithm hasn’t been given in the revised
manuscript. In addition, the proposed algorithm has been compared with the state-of-the art
methods, e.g., BM3D \cite{Dabov07}, PLPCA \cite{Deledalle11}, and LPG-PCA
\cite{LeiZhang10} for the quantitative and qualitative evaluations in this revision.
References:
\bibitem[]{Kose12}
K. Kose, V. Cevher, and A.~E. Cetin, Filtered Variation method for denoising and sparse
signal processing, Proceedings of IEEE International Conference on Acoustics, Speech and
Signal Processing (ICASSP), Kyoto, (Mar. 25-30, 2012) 3329-3332.
\bibitem[]{Gezici01}
S. Gezici, I. Yilmaz, O.~N. Gerek, and A.~E. Cetin, Image denoising using adaptive subband
decomposition, Proceedings of International Conference on Image Processing (ICIP),
Thessaloniki, 1 (Oct. 7-10, 2001) 261-264.
Reviewer #4:
The manuscript presents a novel algorithm for image denoising (APCAS).
My main concerns fall into two categories: (1) evaluation and (2) presentation. However, I've decided
to vote for a major revision.
(1) EVALUATION issues
The problem of the evaluation section is partly rooted in the lack of context. Since the authors did not
specify in which applications their approach is useful, they chose the first evaluation metrics at hand.
Which is not good (it's too generic)
The synthetic images chosen are supposed to be for a general viewer. Hence, the evaluation lacks
ground truth data from subjective assessments (Mean Opinion Score). I do not request to perform the
MOS (although it is advisable) but the lack of it is a drawback in the evaluation process. The PSNR,

although used sometimes, is too generic and non-informative. As far as I am concerned (i don't know
the opinion of other reviewers) it can be skipped. The SSIM is a better choice since it is grounded on
the human visual system. However, a table with SSIM results is not present. Please, add the table with
SSIM and a discussion section on the results.
Furthermore, although I do not request it (although it is advisable), the paper would benefit from a
larger set of test images and a statistical test (ANOVA or paired t-test) between the SSIM results of the
proposed method and the baseline methods.
With regards to the MR images, which are a good choice (the context is specific), the authors ran into
the problem of the lack of the baseline image (= image without the noise). Here, the best thing to do
would be to have an expert (phisician) provide the scores for the image quality. Again, i do not request
this, but it is a drawback in the evaluation and gives little credibility to the work presented. Since I've
seen different ways in which the CNR and SNR have been implemented, the authors should provide
more details on their implementation of the metrics.
The final issue related to the evaluation is the results. Both in the synthetic and MR images the
proposed method does not outperform any of the baseline algorithms, according to the results. In some
cases it is better in some cases not. The authors must provide a more thorough discussion on when their
approach is better than the compared ones. Furthermore, they should remove bold and generic
statements about the quality of their algorithm (e.g. "It is evident that the proposed algorithm achieves a
better edge and structure preservation than other state-of-the-art methods.") and replace them with more
objective observations.
(2) PRESENTATION issues
The introduction with related work is relatively well structured. The authors list a number of related
techniques and their drawbacks. However, when they introduce their approach it is not clear how it
differs from the related work, i.e. what is the contribution of the presented work. If the proposed work
is just another implementation of image denoising with no clear advantages over the existing
approaches, then it is not OK. Please state very clearly how your contribution differs from the others.
When you state "... both the combined denoising strategy and adaptive dimensionality reduction
approach of similar patch groups" it is not clear what the contribution is.
In Sec. 2 the authors jump too quikly to the description of the algorithm's details. Please provide a
subsection (2.1) with an overview of the procedure along with a figure showing various steps of the
workflow.
Another very important issue is the lack of domain. The authors simply state that the noise is a gaussian
noise. This is too generic. Please specify which applications (acquisition devices, processing devices ...)
add that kind of noise. Provide strong arguments for the choice of the noise model. Put your work in
context.
Last but not least, the manuscript would benefit from an improvement of english. Although the prose is

generally good there are some oddities that can be classified into two groups: (i) language (e.g. a/the
should be placed correctly) and (ii) style (e.g. you cannot say " ... as is known to all ..." in a scientific
publication. Please be objective and base your statements on facts. It would be better something like "...
it has been shown in [1,2,3,4,5] that ..."). A skilled scientific writer should be able to remove these
oddities.
Table 2 caption is not clear. What are the values in the table
Response to Reviewer #4:
Response:
(1)Thank the reviewer for his constructive suggestions and comments. The task of the
proposed algorithm is to remove noise in MR images. In fact, for the proposed algorithm, we
did simulation first and then validated it with real MR images in the experiments. In this
subsection 3.1, the subjective and objective evaluations of our proposed algorithm for
synthetic images were carried out for noise removal and edge preservation. As the reviewer
pointed out that the PSNR is too generic and non-informative, we have provided SSIM and
FOM results instead of PSNR results in Table 1 and Table 2, respectively in this revision. Due
to the lack of experimental conditions and professionals, we are very sorry that we are not
able to provide subjective assessments (Mean Opinion Score). For the subjective assessment,
Figure 5 to Figure 7 also show visual comparisons of the denoising results of the proposed
algorithm and BM3D, PLPCA, and LPG-PCA in this revision. Although the reviewer pointed
out that the paper would benefit from a larger set of test images and a statistical test, we have
found that the SSIM may not be accurate enough to distinguish image quality from the
perspective of subjective evaluation. For example, the visual comparison of the “Lena”
images in Figure 5 shows that Figure 5(b) the proposed algorithm looks better than Figure 5(c)
BM3D, but the SSIM of Figure 5(b) the proposed algorithm (PSNR=31.33dB, SSIM=.8424)
is lower than that of Figure 5(c) BM3D (PSNR=31.20dB, SSIM=.8440). Therefore, a larger
set of test images and a statistical test have not been added in this revised manuscript.
For the MR images, there is no baseline image for quantitative objective assessment. Due to
lack of an expert (physician), we are very sorry that we are unable to provide the scores for
the image quality. However, it’s easy for the people to distinguish MR image quality from
Figure 8 to Figure 11 in this revision. To validate the performance of the proposed algorithm,
we have adopted a no-reference image quality metric (signal-to-noise ratio, SNR)
\cite{Dietrich07, Gilbert07, YuJing11} to evaluate noise suppression of the proposed method
and the baseline methods. The details on their implementation of the SNR metric have been
provided in this revision. The SNR results have been given in Table 4 in the revised
manuscript.
From simulation results on synthetic images, the proposed algorithm outperforms the baseline
methods in most cases. Although the SSIM/PSNR results of the proposed algorithm are not
better than those of the baseline methods in some cases, the visual comparisons in Figure 5-7

show that the proposed algorithm has better effect of edge retention and texture recovery than
the baseline methods, even if its SSIM or PSNR is a little lower than those of them in some
cases. For experimental results on real MR images, the SNR results show that the proposed
algorithm is better than BM3D and PLPCA except for LPG-PCA. However, in this revision,
the visual comparisons in Figure 8-11 show that the proposed algorithm is better than all of
them for noise reduction and edge preservation, where LPG-PCA is worst due to the missing
edges and details. Moreover, the bold and generic statements have been removed, and more
objective observations have been added in this revision. In general, the proposed algorithm is
better than the BM3D, PLPCA, and LPG-PCA in most cases, especially for the recovery of
edges and fine structures.
References:
\bibitem[]{Dietrich07}
O. Dietrich, J.~G. Raya, S.~B. Reeder, M.~F. Reiser, and S.~O. Schoenberg. Measurement of
signal-to-noise ratios in MR images: influence of multichannel coils, parallel imaging, and
reconstruction filters, Journal of Magnetic Resonance Imaging, 26(2) (2007) 375-85.
\bibitem[]{Gilbert07}
G. Gilbert. Measurement of signal-to-noise ratios in sum-of-squares MR images. Journal of
Magnetic Resonance Imaging, 26(6) (2007) 1678.
\bibitem[]{YuJing11}
J. Yu, H. Agarwal, M. Stuber, and M. Schar, Practical Signal-to-Noise Ratio Quantification for
Sensitivity Encoding: Application to Coronary MRA, Journal of Magnetic Resonance
Imaging, 33(6) (2011) 1330-1340.
(2) Thank the reviewer for his constructive suggestions and comments. After the content
refinement and the language reorganization, we have expatiated on our highlights that are
different from the related work. The main contribution has been given in this revision, e.g.,
“self-similarity-based image patch clustering”, “the low-rank approximation based on parallel
analysis”, and “joint image denoising framework without iteration”.
An overview of the procedures along with a figure has been provided to show various steps of
the workflow in Subsection 2.1 in this revision (See Figure 1 in Subsection 2.1).
The goal of this study is to remove noise in MR images, which is an important concern that
impairs their diagnostic accuracy. Depending on the used reconstruction method, noise in MR
images can be Gaussian or Rician distributed with uniform or nonuniform variance across the
image \cite{Manjon10}. Most of denoising methods in the literatures have been developed
assuming a Gaussian noise distribution with a spatially independent variance. In fact, the
Gaussian assumption could be valid on MR images when SNR is larger than two
\cite{Gudbjartsson95}. For simplicity, the additive Gaussian noise model is adopted to
describe the noisy MR images for simulation \cite{Dabov07, LeiZhang10, Deledalle11}.
References:

\bibitem[]{Gudbjartsson95}
H. Gudbjartsson, and S. Patz, The Rician Distribution of Noisy MRI Data, Magnetic
Resonance in Medicine, 34(6) (1995) 910-914.
\bibitem[]{Manjon10}
J.~V. Manjon, P. Coupe, L. Marti-Bonmati, D.~L. Collins, and M. Robles, Adaptive non-local
means denoising of MR images with spatially varying noise levels, Journal of Magnetic
Resonance Imaging, 31(1) (2010) 192-203.
We are very sorry that there are some oddities and language errors in the original paper. All
the mistakes have been corrected in this revised manuscript. We have tried our best to revise
the English of the whole manuscript carefully to avoid typo mistakes, grammar errors,
spelling errors and vague descriptions. Moreover, we have asked several colleagues who are
skilled authors to proofread the manuscript. We believe that the English is now acceptable for
the reviewers and the readers.
Thanks again for all the valuable comments and suggestions. We hope the revised manuscript
has properly addressed all the concerns and is worthy of publication.

INS_Highlights.docx
Highlights:
(1) We propose the novel joint denoising strategy consisting of two successive steps.
(2) The self-similarity is used to cluster similar patch groups.
(3) The parallel analysis method adaptively selects signal components.
(4) The low rank approximation exploits the local and nonlocal signal information.
(5) The results prove its validity for the synthetic images and the real MR images.

*Manuscript (including abstract)
Click here to view linked References
Joint Image Denoising Using Adaptive Principal
Component Analysis and Self-similarity
Yong-Qin Zhang, Jiaying Liu, Mading Li, Zongming Guo
Institute of Computer Science and Technology, Peking University, Beijing 100871, China
Abstract
The nonlocal means (NLM) has attracted enormous interest in image denoising
problem in recent years. In this paper, we propose an efficient joint
denoising algorithm based on adaptive principal component analysis (PCA)
and self-similarity that improves the predictability of pixel intensities in reconstructed
images. The proposed algorithm consists of two successive steps
without iteration: the low-rank approximation based on parallel analysis,
and the collaborative filtering. First, for a pixel and its nearest neighbors,
the training samples in a local search window are selected to form the similar
patch group by the block matching method. Next, it is factorized by
singular value decomposition (SVD), whose left and right orthogonal basis
denote local and nonlocal image features, respectively. The adaptive PCA
automatically chooses the local signal subspace dimensionality of the noisy
similar patch group in the SVD domain by the refined parallel analysis with
Monte Carlo simulation. Thus, image features can be well preserved after
dimensionality reduction, and simultaneously the noise is almost eliminated.
Then, after the inverse SVD transform, the denoised image is reconstructed
from the aggregate filtered patches by the weighted average method. Finally,
the collaborative Wiener filtering is used to further remove the noise. The
experimental results validate its generality and effectiveness in a wide range
of the noisy images. The proposed algorithm not only produces very promising
denoising results that outperforms the state-of-the-art methods in most
cases, but also adapts to a variety of noise levels.
0 † Corresponding Author. Tel: +86-10-82529641; fax: +86-10-82529207. E-mail address:
{guozongming,zhangyongqin}@pku.edu.cn
Preprint submitted to Information Sciences July 1, 2013

Keywords: Image denoising, Dimensionality reduction, Principal
component analysis, Singular value decomposition, Parallel analysis
1. Introduction
Image denoising is still a challenging problem in the fields of image processing
and computer vision. It refers to the recovery of a digital image
that has been contaminated by some types of noise, e.g., Gaussian noise,
or Rician noise, while preserving image features such as the edges and the
textures. The problem of image denoising [34, 29] was first studied in 1970s.
After the development of wavelet transform in late 1980s, many denoising
methods based on wavelet transform and its variants have appeared in the
literatures [13, 39, 6, 17, 40, 35]. However, they often blur the sharp edges
and smooth out the fine structures.
Since the nonlocal means (NLM) algorithm [5] was published by Buades
et al., many more powerful denoising techniques have been proposed in the
past several years [1, 14, 30, 15, 9, 42, 53, 12, 27, 56, 2, 55, 21, 31]. After a
brief review, there are two basic categories for image denoising approaches.
One of them is the spatial filters, which can be further classified into linear
filters and non-linear filters. Some of the recent popular linear spatial filters
are bilateral filtering [38], Wiener filtering [18], NLM [5] and Total Least
Squares(TLS) [23]. Furthermore, manyvariantsoftheNLMmethod[5]were
also developed to improve its weight calculation, e.g., Stein’s unbiased risk
estimate (SURE) [43, 36], the principle neighborhood dictionary (PND) [42]
and the MMSE approach [28]. Similarly, the typical non-linear spatial filters
are total variation regularization (TV) [37], Kernel Regression (KR) [41] and
the diffusion filter [4]. The other category is transforming domain filtering
methods, which can also be further divided into the non-data adaptive transformsincluding
wavelet-based variants[40, 35] anddataadaptive transforms,
such as K-SVD [1], BM3D [9], principal component analysis (PCA) [54, 10]
and independent component analysis (ICA) [8].
One main direction of these works is to find sparse representations of
signals built on the globally or locally adaptive basis. Assuming that each
imagepatch canbesparsely represented, K-SVDalgorithm[1]anditsvariant
[51] learn a sparse and redundant basis of image neighborhoods to remove
noise. But they ignore the characteristics of the human visual perception
2

that the edges and textures of the image contribute greatly to the subjective
assessment of image quality. The KR method with recursive iterations
proposed by Takeda et al. [41] has expensive computation, which is difficult
to achieve real-time processing. He et al. [22] proposed an image denoising
method using the adaptive thresholding scheme by singular value decomposition.
However, its denoising performance depends on three free parameters,
which are quite tricky and difficult to tune for optimal values. The patchbased
PLPCA method [10] adopts the hard thresholding technique directly
for the elements of the eigenvectors, whereas it also damages the sharp edges
and the fine structures. Furthermore, its thresholding value is based on the
known noise deviation. However, in fact, the noise deviation is unknown
in most cases. To the best of our knowledge, the best existing state-ofthe-art
filtering methods are mostly based on the optimal Wiener filter [9] or
equivalently Linear-Minimum MeanSquared Error (LMMSE) [54]. Although
they have very good performance for reducing additive white Gaussian noise
(AWGN) from the noisy image, they have not yet reached the limit of noise
removal [7].
These denoising methods mentioned above have the drawback that while
removing noise, they may also smooth the edges and the fine structures in
the image. To mitigate this drawback, different from the existing denoising
methods, we propose an advanced denoising algorithm based on adaptive
principal component analysis and self-similarity (APCAS). The proposed algorithm
consists of two successive steps without iteration: the low-rank approximation
based on parallel analysis, and the collaborative filtering. The
image self-similarity is exploited to construct similar patch groups. Parallel
analysis is used to choose the signal dimensionality of the coefficients in the
SVD domain of the similar patch group. This dimensionality reduction technique
can adaptively determine the number of signal components in noisy
environments. The low-rank approximation of the true image is employed
to perform the empirical Wiener filtering to further reduce the noise. Our
main contributions of the proposed algorithm include the joint denoising
strategy without iteration, the self-similarity based image patch clustering
and parallel analysis based adaptive principal component analysis for the
low-rank approximation. Experimental results show that the proposed algorithm
achieves highly competitive performance with the best state-of-the-art
methods from subjective and objective measures of image quality, and can
outperform them in most cases.
3

The rest of this paper is organized as follows. In Section 2, the proposed
APCAS algorithm for image denoising is described. Section 3 shows
the simulation and experimental results of the developed algorithm, and the
comparison to the state-of-the-art methods. Finally, the discussions and conclusions
are drawn in Section 4.
2. Proposed Denoising Algorithm
2.1. Method Preview
In real-world digital-imaging devices, the acquired images are often contaminated
by device-specific noise. Due to the existence of random noise in
the acquisition process, magnetic resonance (MR) images are generally the
most noisy. In the MR literatures [20, 32], the noise in MR images is assumed
to be Rician distributed with uniform or nonuniform variance across
the image. However, Most of denoising methods in the literatures [9, 54, 10]
have been developed assuming a Gaussian noise distribution with a spatially
independent variance. In fact, the Gaussian assumption could be valid on
MR images when SNR is larger than two [20]. For simplicity, the additive
Gaussian noise model is adopted to simulate the noisy MR images for validation
and evaluation of the denoising methods. As in the previous literatures
[9, 54, 10], a simple noise model of independent additive type is generally
used to describe the noisy image for simulation in the following formula:
y(x) = s(x)+υ(x) (1)
where x is a two-dimensional spatial coordinate; y is the observed image, s
is the ideal noise-free image, and υ represents additive white Gaussian noise
(AWGN) with zero mean and standard deviation σ n .
Besides giving an image an undesirable appearance, the noise can cover
and reduce the visibility of certain features within the image, which weakens
the clinical diagnostic accuracy. Thus, it is necessary to remove the noise
from MR images. Because of its advantage of not increasing the acquisition
time and motion artifacts, postprocessing filtering techniques have been traditionally
andextensively used in MR imagedenoising. In brief, theoverview
of procedures of our proposed method is described as follows (See Figure 1):
(1) Image pacth clustering. For each target patch, its corresponding patch
group is formed by finding similar patches in the observed noisy image;
4

(2) Signal dimension estimation. Parallel analysis is used to estimate the
signal dimension of each similar patch group;
(3) SVD-based low-rank approximation. The denoised image is obtained
with the weighted averaging of the aggregate filtered patches that are
constructed with the low-rank approximation;
(4) Empirical Wiener filtering. With shrinkage coefficients from the denoised
result in the first step, the collaborative Wiener filtering further
removes the noise.
Figure 1: The workflow of the proposed APCAS algorithm.
2.2. Image Patch Clustering
Since most ofthehumanvisual perceptionandunderstanding ofanimage
is conveyed by its edge structures and texture patterns. To preserve the edge
and the texture, the patch-based image representation is modeled instead
of the pixel-based image for noise reduction, where each patch contains a
pixel and its nearest neighbors. For an observed noisy image Y with its
coordinate domain X ⊂ R M×N , let y(x) be a pixel at a position x in the
image Y. Assuming that P is the number of pixels in the patch, the patchbased
image Y x denotes a patch of fixed size √ P × √ P extracted from Y,
where x is the coordinate of the central pixel of the patch. That is, Y x is a
reshaped vector of size P ×1, which contains the √ P × √ P pixels consisting
of a central pixel y(x) and its nearest neighbors in the observed image Y.
5

In order to remove the noise from the input noisy image Y, the dataadaptive
SVD transform is used to separate the image signal and the noise.
The high degree of self-similarity and redundancy widespreadly exists within
any natural image. Through effective analysis of a signal or image, the varioussub-dictionaries
withatomsshould bepicked upthataremorphologically
similar to the features in the signal or image, respectively. For each target
pixel y(x) located in the central position of the target patch Y x , there are
totallyLpossible training patches of the same size √ P× √ P in the √ L× √ L
local search window. However, there may be very different adjacent patches
from the given target patch so that taking all the √ L× √ L patches as the
training samples will cause inaccurate estimation of the target patch vector
Y x . Thus, it is necessary to choose and cluster the training samples that are
similar to the target patch for the full use of both local and nonlocal information
before applying the SVD transform. The problem of patch classification
has several different solutions, e.g., block matching [9], K-means clustering
[3] and affinity propagation (AP) [16]. For simplicity, we employ the block
matching method for image patch clustering.
LetY x = [y x (1),y x (2),··· ,y x (P)] T denotethecentraltargetpatch. For
typographical convenience, Y l ,l = 1,2,··· ,L − 1 represents the candidate
adjacent patches of the same size √ P × √ P in the √ L× √ L search window
Ψ x . The block matching method is used to construct the patch group based
on the similarity measurement between the adjacent patch and the target
patch. The relationships between the central pixel, the target patch, the
adjacent patch and the local search window can be appreciated in Figure 2.
The similarity metric between the candidate patch and the target patch can
be calculated using the Euclidean distance in this formula:
e l = 1 P∑
(y l (p)−y x (p)) 2 (2)
P
p=1
Then, through the addition of the scalar error 0 between the target patch
and itself, we build the error vector e = [0,e 1 ,··· ,e L−1 ] T . After the error
vector e is sorted in the ascending order, the top most similar patches are
chosen to construct the patch group. Assume that we select L s similar patch
vectors to reconstruct the patch group Z υ x for the target patch Y x, where L s
is a preset number. For each target patch Y x , its corresponding patch group
Z υ x consisting of the training patches can be expressed in this form:
Z υ x = [Y x ,Y 1 ,··· ,Y Ls−1] T (3)
6

To separate the image signal and the noise effectively, the number L s of
most similar patches should be large enough. The patch clustering matrix
Z υ x or its transposition (Z υ x) T will be used to avoid the problem of rank deficiency
in computing the SVD of the covariance matrix of Z υ x . For each noisy
measurement Z υ x , the next procedure is to estimate its underlying noiseless
counterpart dataset Z x = [S x ,S 1 ,··· ,S Ls−1] T .
Figure 2: The relationships between the central pixel, the target patch, the
adjacent patch and the local search window.
2.3. Signal Dimension Estimation
The PCA operator transfers a set of correlated variables into a new set
of uncorrelated variables, whose eigenfunctions form the basis for a signal
decomposition. We find that most of the energy of a function defined on
the graph is mainly concentrated on the top several principal components.
Moreover, the principal components of the original image signal can be captured
by analyzing the eigenfunctions generated from noisy data, whereas
the eigenvectors of the small singular values are almost all noises. It can
also be implemented by eigenvalue decomposition of a data covariance (or
correlation) matrix or singular value decomposition (SVD) of a data matrix.
Note that the data matrix is usually pre-treated for each attribute by the
mean centering method.
The denoising problem of the noisy patch group Z υ x is indeed how to
select the optimal number of the principal components in the SVD transform
domain. In this paper, we employ dimensionality reduction technique to
analyze the eigenvalues of the covariance matrix of the patch group Z υ x . By
subtracting the sample mean value from each column, we have computed the
7

zero-centered matrix from the patch group Z υ x in this formula:
¯Z υ x (l,p) = Zυ x (l,p)−M x(l,p) (4)
whereM x (l,p) = [1,··· ,1] T ∑
× 1 L s
L s
Zx υ(l,p); [1,··· ,1]T isthecolumn vector
l=1
of size L s ×1; l = 1,··· ,L s ; and p = 1,··· ,P.
To reduce calculation time in an efficient way on the condition of L s ≥ P
(or L s < P), the covariance matrix (¯Z υ x)T¯Z υ x (or ¯Z υ x(¯Z x) υ T
) instead of the
zero-centered patch group ¯Z υ x is used to be factorized in this form:
(¯Z υ x
)T¯Z υ x = VΣ2 V T (5)
where the symbol T denotes the transpose operator, and V is the unitary
matrix of eigenvectors derived from (¯Z υ x)T¯Z υ x . Σ is a P ×P diagonal matrix
with its singular values λ 1 ≥ λ 2 ≥ ··· ≥ λ r ≥ 0 and r = rank (¯Zυ )
x .
In fact, the eigenvalues of the covariance matrix of the noise υ are not
the same. Therefore, we can not take a preset fixed threshold to select the
signal components. It has been shown in [56] that if the eigenvalues are
small enough, the discard of less significant components does not lose much
information. From the diagonal singular values, only the first K primary
eigenvectors are retained by dimensionality reduction based on parallel analysis.
However, the parameter K should be not only large enough to allow
fitting the characteristics of the data, but also small enough to filter out the
non-relevant noise and redundancy.
There are variousdimensionality reduction methods proposed inthe literatures
for determining the number of components to retain in data analysis.
The parallel analysis (PA) method firstly introduced by Horn [24] compares
theobserved eigenvalues tobeanalyzed withthoseofanartificial dataset obtained
from uncorrelated normal variables. Subsequently, the improvements
to the original parallel analysis method have been proposed using Monte-
Carlo simulations instead of the normal distribution assumption[26]. It is
proved that PA is one of the most successful methods for determining the
number of true principal components. In this paper, we adopt the proposed
parallel analysis with Monte Carlo simulation to choose the top K largest
values.
Let λ p for p = 1,··· ,P denote the singular values of the zero-centered
patch group ¯Z υ x sorted in the descending order. Similarly, let α p denote the
8

sorted singular values of the artificial data. Therefore, the proposed parallel
analysis estimates signal subspace dimensionality of noisy data as follows:
K = max{p = 1,··· ,P|λ p ≥ α p } (6)
The intuition is that α p is a threshold value for the singular value λ p below
which the p ′ th component is judged to have occurred because of chance.
Currently, it is recommended to use the singular value that corresponds to a
given percentile, such as the95 th ofthe distribution of singular values derived
from the random data set.
In our algorithm, without any assumption of a given random distribution,
we generate the artificial data by randomly permuting each element
of all the patch vectors located in a search window. Let y l (p) denote the
p ′ th element of the training patch vector Y l (or Y x ) in the patch group Z υ x .
For each P elements of the patch vector Y l (or Y x ), multiple random permutations
of the coordinates p = 1,··· ,P of the noisy data matrix Z υ x are
generated by the uniform distribution. Thus, the mean value, maximum,
minimum and random distribution of the artificial data is satisfied for each
image patch-based vector. Then singular values of the random artificial data
are computed by the SVD transform which keeps the marginal distributions
intact while breaking any interdependency between them. For the L s by P
synthetic matrix C s , after multiple times (e.g. 10) of Monte Carlo simulations,
summary statistics (e.g. 95 th percentile) can be used to extract the
P singular values and order them from the largest to the smallest. Then
parallel analysis is applied and the two lines denote singular values of the
simulated data C s and the zero-centered patch group ¯Z υ x, respectively. The
intersection of the two lines is the cutoff for determining the number of the
signal subspace dimension presented in the noisy image.
While the traditional PCA [44] adopts the thresholding technique to estimate
data dimensionality, the proposed adaptive PCA can automatically
determine signal subspace dimensions in noisy data by the parallel analysis
technique. Our developed refined parallel analysis using Monte Carlo
simulation were compared with the traditional PCA [44] for dimensionality
reduction in our experiments. Suppose that a 25×25 fragment of lena image
corrupted by additive zero-mean Gaussian noise with standard deviation 20.
In this case, the signal dimensionality of the given noisy patch group Z υ x with
patch size 5 × 5 pixels was separately estimated by the proposed adaptive
PCA and the traditional PCA [44]. Figure 3 shows the numbers of its signal
9

subspace dimension estimated by the proposed adaptive PCA technique and
the traditional PCA [44] are separately 5 and 19. We can see that the proposed
adaptive PCA approach is better than the traditional PCA method
for separating the signal and the noise from the noisy image.
2.4. SVD-Based Low-Rank Approximation
For each target pixel y(x), the similar candidate patches are selected by
the block matching method to form the corresponding patch group Z υ x =
[Y x ,Y 1 ,··· ,Y Ls−1] T . The patch-based correlated patch groups Z υ x (x ∈ X)
[see Equation (3)] can be used for component analysis of the noisy image Y.
The zero-centered patch group ¯Z υ x is factorized by the SVD transform in this
formula:
¯Z υ x = UΣV T (7)
where Σ is the diagonal matrix with the singular values λ 1 ≥ ··· ≥ λ r .
V = [V 1 ,V 2 ,··· ,V P ] and U = [U 1 ,U 2 ,··· ,U Ls ] are the unitary matrices
of eigenvectors, which represent the orthogonal dictionaries of nonlocal bases
and local bases, respectively. The zero-centered noisy patch group ¯Z υ x is decomposed
into a sum of components from the largest to the smallest singular
values.
In fact, most of the energy of the true image is concentrated on few highmagnitude
transform coefficients, whereas the corresponding eigenimages of
the small singular values are almost all noises. After automatically determining
signal dimension of the noisy zero-centered patch groups ¯Z υ x, the low
rank approximation [47, 33] is used to estimate the original noiseless image
S by reducing noise in the observed image Y. Through our adaptive signal
dimension estimation [see Equation (6)], the noiseless image can be reconstructed
by the SVD-based low rank approach. The straightforward way to
restore a noiseless image is to directly apply the inverse SVD transform to
the noisy similar patch groups Z υ x in a reduced dimensionality representation.
That is, the inverse SVD transform is implemented to approximate
the true noise-free image with the matrices Σ K = diag{λ 1 ,λ 2 ,··· ,λ K },
U K = [U 1 ,U 2 ,··· ,U K ], and V K = [V 1 ,V 2 ,··· ,V K ]. For each target
pixel, the denoised patch group Ẑ x and its weight matrix W x are separately
estimated as follows:
Ẑ x = U K Σ K V T K +M x (8)
10

Figure 3: The eigenimages, the singular values, and the reconstructed images
generated from a 25×25 image block with patch size 5×5 pixels of the test
lena image corrupted by additive zero-mean Gaussian noise with standard
deviation 20. (a) the original image block; (b) the noisy image block; (c) the
eigenimages; (d) the numbers of its signal subspace dimensionality estimated
by the proposed adaptive PCA approach, and the traditional PCA [44] are
5, and 19,respectively; (e) the reconstructed image block (K = 19); (f) the
reconstructed image block (K = 5).
W x (l,p) =
{ 1−K/P,K < P;
1/P, K = P.
where M x is the mean value [see Equation (4)] of the patch group Z υ x.
(9)
11

After applying such procedures to each pixel y(x), the relevant denoised
patch group is estimated with the low rank approach by adaptive PCA technique,
and its weight is empirically determined. Since these denoised patches
are overlapping, multiple estimates of each pixel in the image are combined
to reconstruct the whole image. The weighted averaging procedure is carried
out to suppress the noise further. The whole filtered image Ŝ t is obtained by
aggregating all the estimates of each pixel in this formula:
Ŝ t = 1 M×N
∑
W t
x=1
where the total weight matrix W t = M×N ∑
x=1
Ẑ x W x (10)
W x .
In addition, the proposed efficient algorithm is implemented by reducing
the number of the image patch groups. A step of J s ∈ X pixels is used
for the noisy image in both horizontal and vertical directions. Thus, the
number of similar patch groups is approximately MN/Js 2 rather than MN.
Hence most ofthenoisewill beremoved by using theadaptive PCAapproach
and the weighted averaging scheme in Subsections 2.2-2.4. However, there
is still some unpleasant residual noise in the denoised image, especially for
terribly noisy images. As the observed image contains the strong noise,
the image patches are seriously corrupted by noise, which leads to image
patch clustering errors and the biased estimation of the SVD transform.
Consequently, it is necessary to further suppress the noise residual of the
denoised output Ŝ t .
2.5. Empirical Wiener Filtering
After the first phase of noise removal by the low rank approximation, we
can employ the empirical Wiener filtering [9] for the second phase to further
improve the denoising performance of the output Ŝ t . Because there is less
noise in the output Ŝ t , the close approximation of the true patch distance
is calculated with the denoised output Ŝ t instead of the noisy image Y [see
Equation (2)]. Thus, the coordinates of the similar patches for each pixel are
grouped into the index set:
{
}
∥
Ω x = x ∈ X : ∥Ŝt l −Ŝt x∥ 2 x < τ (11)
2/P
12

where Ŝ t x is the √ P x × √ P x target patch with its central pixel ŝ t (x) in the
denoised output Ŝ t ; Ŝ t l is the √ P x × √ P x candidate patch in the local search
window; ‖·‖ 2 2 is the squared l2 −norm; and τ is a threshold value.
Then the index set Ω x is used to construct two patch groups Ŝ t Ω x
and Y Ωx
from the denoised output Ŝ t and the observed noisy image Y, respectively.
From the power spectrum coefficients of 3D transform of the denoised patch
groupŜ t Ω x
, theempirical Wiener shrinkagecoefficients forthecase ofadditive
white noise are found as:
)∣
∣ ∣∣
2 (Ŝt Ωx
G Ωx =
where σ 2 is the noise variance.
∣
∣T3D
wie
∣T3D
wie
(Ŝt Ωx
)∣ ∣∣
2
+σ
2
(12)
Subsequently, the empirical Wiener filtering of the noisy patch groupY Ωx
is executed though multiplying the 3D transform coefficients T3D wie(Y
Ω x
) by
the Wiener shrinkage coefficients G Ωx . Whereafter, the estimated noiseless
patch group is acquired by the inverse 3D transform as follows:
(
Ŝ wie
Ω x
= T3D
wie−1 GΩx T3D wie (Y Ωx ) ) (13)
Note that such an empirical Wiener filter is adaptive because its weight coefficients
depend on the spectrum of the output image Ŝ t . However, the
patch-wise estimation for each pixel is generally biased, correlated, and have
different variances. Moreover, the total sample variance in the corresponding
estimates of the patch groups is σ 2 ‖G Ωx ‖ 2 2
on the assumption that the
additivenoiseissignal-independent. Tosuppress these distortions, theaggregation
weights are empirically assigned for the estimates of the patch groups
Ŝ wie
Ω x
as follows:
W Ωx = σ −2 ‖G Ωx ‖ −2
2
(14)
Therefore, after a weighted average of the patch-wise estimates, the final
noiseless image Ŝ f is obtained as:
Ŝ f =
M×N ∑
x=1
M×N
∑
x=1
W Ωx Ŝ wie
Ω x
W Ωx
, ∀x ∈ X (15)
13

2.6. Algorithm Summary
The following pseudo codes give further clarification of the specific implementation
of the proposed algorithm for an input noisy image:
Algorithm 1 Pseudocode of the Proposed Algorithm
Input: Y M×N , P,L and L s .
Output: Ŝ f
for each x = 1 to M ×N do
/∗Convert the pixel-based image to the patch-based image ∗/
H(x,1 : P) ← Y
(
m+1 : √ P,n+1 : √ P
end for
/∗The SVD-based low-rank approximation using parallel analysis ∗/
Ŝ t = zeros(M ×N,P); W t = zeros(M ×N,P);
for each x = 1 to M ×N do
Ψ x ← L; Y x = H(x,:); Y Ψx = H(Ψ x ,:);
Ψ s x = BlockMatching(Y x,Y Ψx ,L s );
Zx υ = H(Ψs x ,:); M x = mean(Zx υ);
¯Z x υ = Zυ x −M x; (¯Zυ )T
x
¯Zυ x = VΣ 2 V T ; C T C = VΛ 2 V T ;
λ = diag(Σ); α = diag(Λ);
K = max{p = 1,··· ,P|λ p ≥ α p }; /∗Parallel analysis ∗/
Ẑ x = U K Σ K VK T +M x;
Ŝ t (Ψ s x ,:) = Ŝt (Ψ s x ,:)+W xẐx;
W t (Ψ s x ,:) = Wt (Ψ s x ,:)+W x;
end for
I = zeros
(
M + √ P −1,N + √ P −1
)
;
)
; Q = I;
/∗The weighted averaging of the aggregate estimates of each pixel ∗/
for each a,b = 1 to √ P do
id = (b−1)× √ P +a;
Π a = a : M +a−1; Π b = b : N +b−1; )
I(Π a ,Π b ) = I (Π a ,Π b )+reshape(Ŝt (:,id),[M,N] ;
Q(Π a ,Π b ) = Q(Π a ,Π b )+reshape(W t (:,id),[M,N]);
end for
Ŝ t = I/Q; )
Ŝ f = WienerFiltering(Ŝt ; /∗The empirical Wiener filtering ∗/
14

The proposed algorithm can be summarized as follows. Let Y denote
an observed noisy image with its coordinate domain X ∈ R M×N . First, for
each pixel y(x) and its target patch Y x , the corresponding patch group Z υ x
is formed by finding similar patches in a local search window. Next, after
performing singular value decomposition (SVD) of the patch group ¯Z υ x , the
signal dimension K is determined by parallel analysis applied to the singular
values λ for separating the signal and the noise in the SVD domain. Then,
after the inverse SVD transform, the denoised patch group Ẑx is obtained
by the low-rank approximation, where most of the noise is eliminated. After
that, the whole denoised image Ŝt is acquired by the weighted averaging of
all the estimates of each pixel for further noise removal. Finally, after the
shrinkage coefficients G Ωx are reconstructed from the power spectrum of 3D
transform of the denoised image Ŝt , the final noiseless image Ŝf is obtained
by the collaborative Wiener filtering for further noise reduction.
3. Results and Analysis
In this work, numerical simulation and experimental studies were carried
out to verify the performance of the proposed algorithm subjectively
and objectively. In the objective evaluation, the full-reference image quality
assessment (FR-IQA) was adopted for synthetic images in numerical simulation,
whereas the no-reference image quality assessment (NR-IQA) was used
for real MR images in the experiments.
3.1. Simulation Results on Synthetic Images
To test the performance of the proposed algorithm comprehensively, we
have implemented the qualitative and quantitative evaluation on multiple
test images from standard image databases [46]. The corrupted images synthetically
damaged by white Gaussian noise using the noise model (1). For
variant noise levels, the benchmark for image denoising evaluation includes
the noise-free test images shown in Figure 4.
A quantitative comparison was performed between the performances of
the proposed denoising algorithm and the state-of-the-art methods published
recently [9, 10, 54]. The well-known full-reference quality metrics were considered
for measuring the similarity between the filtered image and the original
noise-free image in terms of noise suppression and edge preservation,
respectively. Both Peak Signal-to-Noise Ratio (PSNR) [25] and Structural
SIMilarity (SSIM) [45] were adopted to evaluate noise suppression perfor-
15

Figure 4: The test images in our experiments, from left to right, top to
bottom: Lena, Baboon, Barbara, Boat, Bridge, Monarch, House, and Brain.
Theseimagespresentawiderangeofedges, textures, details, andfrequencies.
mance of different denoising methods, respectively. The edge preservation
performance of different denoising methods was also quantified by the edge
preservation index called Figure of Merit (FOM) [48, 50], respectively. The
FOM [48, 50] is defined as
FOM =
1
max(n d ,n r )
∑n d
i=1
1
1+γd 2 i
(16)
where n d is the number of detected edge pixels in the test noisy image, n r is
thenumber ofreferenceedgepixelsinthenoise-freeimage, d i istheEuclidean
distance between the ith detected edge pixel and the nearest reference edge
pixel, and γ is a constant typically set to 1/9. The Laplacian of Gaussian
method was used to detect the edges.
In this simulation, the test images were degraded by Gaussian noise with
zero means and different deviation 10, 20, 30 and 50, respectively. To verify
the performances of noise suppression and edge preservation, our developed
algorithm was compared with the current state-of-the-art methods, such as
BM3D [9], PLPCA [10] and LPG-PCA [54]. The SSIM results for these test
images are presented in Table 1. And the FOM results for these test images
are also given in Table 2. To further inspect the effectiveness of our proposed
algorithm, the detailed denoised results of the proposed algorithm, BM3D
16

[9], PLPCA [10], and LPG-PCA [54] for the fragments of the Lena, Baboon
and Barbara images are shown in Figure 5 to Figure 7, respectively.
As is shown in Table 1, the SSIM results of our proposed algorithm outperform
those of PLPCA [10], and LPG-PCA [54] in most cases, and can
be competitive with those of BM3D [9]. However, the FOM results in Table
2 shows that our proposed algorithm are generally superior to all of them
in the edge preservation. The visual comparisons in Figure 5 to Figure 7
demonstrate that the proposed algorithm have better recovery of textures
and edges than the baseline methods. Therefore, our proposed algorithm
outperforms the best existing state-of-the-art methods, e.g., BM3D [9] not
only in visual comparisons but also in the edge preservation in most cases.
As can be seen from the simulation results, it works well for a wide variety
of noisy images and can effectively produce sharp edges and rich textures.
(a) (b) (c) (d) (e)
Figure 5: Visual comparisons of the denoising results of the proposed
algorithm and other state-of-the-art methods for the Lena image corrupted
with Gaussian noise with standard deviation 30. From left to
right: (a) Original subimage; (b) the proposed algorithm (PSNR=31.33dB,
SSIM=.8424); (c) BM3D [9] (PSNR=31.20dB, SSIM=.8440); (d)
PLPCA [10] (PSNR=30.26dB, SSIM=.7956); and (e) LPG-PCA [54]
(PSNR=30.66dB, SSIM=.8380).
3.2. Experimental Results on Real MR Images
Besides the simulations on noisy synthetic images, we also validated our
denoising algorithm on real MR images because the characteristics of the
fine structures (especially the edges and the textures) in MR images is very
importantformedicaldiagnosis. Inthisexperiment, therealMRimageswere
acquired from the MR scanner at 3T (MAGNETOM Tim Trio, Siemens,
Germany). As objective no-reference image quality metric, the signal-tonoise
ratio (SNR) [11, 19, 49] was used for measuring the noise in an input
17

(a) (b) (c) (d) (e)
Figure 6: Visual comparisons of the denoising results of the proposed
algorithm and other state-of-the-art methods for the Baboon
image corrupted with Gaussian noise with standard deviation 50.
From left to right: (a) Original subimage; (b) the proposed algorithm
(PSNR=23.46dB, SSIM=.5762); (c) BM3D [9] (PSNR=23.13dB,
SSIM=.5320); (d) PLPCA [10] (PSNR=23.01dB, SSIM=.5199); and (e)
LPG-PCA [54] (PSNR=22.83dB, SSIM=.5116).
(a) (b) (c) (d) (e)
Figure 7: Visual comparisons of the denoising results of the proposed
algorithm and other state-of-the-art methods for the Barbara
image corrupted with Gaussian noise with standard deviation 50.
From left to right: (a) Original subimage; (b) the proposed algorithm
(PSNR=27.54dB, SSIM=.8014); (c) BM3D [9] (PSNR=27.21dB,
SSIM=.7928); (d) PLPCA [10] (PSNR=25.98dB, SSIM=.6792); and (e)
LPG-PCA [54] (PSNR=26.20dB, SSIM=.7667).
18

Table 1: The SSIM results of BM3D [9], PLPCA [10], LPG-PCA [54], and
the proposed algorithm separately applied on subset of test images, e.g.,
Lena, Baboon, Barbara, Boat, Bridge, Monarch, House, and Brain. These
noise-free images are corrupted by Gaussian noise with variant noise levels
σ n = 10, 20, 30, and50, respectively. Top left: BM3D [9], Top right: PLPCA
[10], Bottom left: LPG-PCA [54], Bottom right: Proposed. The best result
among them is highlighted in each cell.
σ n 10 20 30 50
Lena .9169 .9078 .8767 .8538 .8440 .7956 .7987 .6286
512×512 .9145 .9162 .8728 .8664 .8380 .8424 .7774 .7884
Baboon .8900 .8950 .7801 .7714 .6830 .6676 .5320 .5199
512×512 .8844 .8770 .7642 .7909 .6562 .7049 .5116 .5762
Barbara .9418 .9337 .9046 .8802 .8665 .8233 .7928 .6792
512×512 .9407 .9421 .8991 .8990 .8543 .8717 .7667 .8014
Boat .8874 .8875 .8265 .8065 .7789 .7375 .6996 .5878
512×512 .8822 .8920 .8111 .8261 .7540 .7774 .6698 .6962
Bridge .9068 .9059 .7904 .7789 .6963 .6818 .5695 .5447
512×512 .9004 .8926 .7742 .7930 .6701 .7065 .5390 .5905
Monarch .9560 .9423 .9210 .8915 .8865 .8360 .8239 .6910
256×256 .9551 .9533 .9154 .9120 .8782 .8802 .7994 .8118
House .9199 .9032 .8726 .8469 .8489 .7940 .8154 .6283
256×256 .9122 .9201 .8684 .8640 .8388 .8391 .7867 .7996
Brain .9010 .8921 .8325 .7987 .7796 .7214 .6846 .5501
256×256 .8963 .8992 .8189 .8329 .7557 .7797 .6569 .6954
.9150 .9084 .8506 .8285 .7980 .7571 .7146 .6037
Average
.9107 .9116 .8405 .8480 .7807 .8002 .6884 .7199
noisy MR image and the denoised results of the proposed algorithm and the
current popular methods [9, 10, 54], respectively. The measurement of SNR
in MR images is commonly based on the signal statistics in two separate
regions of interest (ROIs) from a single image: one in the tissue of interest,
and one in background air. The SNR [11, 19, 49] was defined as
SNR stdv = S mean
σ stdv
(17)
where S mean is the mean value of pixel intensities in a region of interest (ROI)
within the object, and σ stdv is the standard deviation of noise in a chosen
background ROI outside the object, free from signals or ghosting artifacts.
19

Table2: TheFOMresultsofBM3D[9], PLPCA[10], LPG-PCA[54], andthe
proposed algorithm separately applied on subset of test images, e.g., Lena,
Baboon, Barbara, Boat, Bridge, Monarch, House, and Brain. These noisefree
images are corrupted by Gaussian noise with variant noise levels σ n =
10, 20, 30, and 50, respectively. Top left: BM3D [9], Top right: PLPCA
[10], Bottom left: LPG-PCA [54], Bottom right: Proposed. The best result
among them is highlighted in each cell.
σ n 10 20 30 50
Lena .9168 .9210 .8463 .8207 .7965 .7713 .6776 .6984
512×512 .8913 .9195 .7932 .8619 .7353 .8133 .6283 .7128
Baboon .9470 .9416 .8712 .8464 .7861 .7524 .6231 .6890
512×512 .9312 .9368 .8458 .8767 .7725 .8071 .6644 .6949
Barbara .9480 .9394 .8686 .8304 .8162 .7649 .6878 .6907
512×512 .9241 .9406 .8042 .8758 .7329 .8291 .6336 .7407
Boat .9318 .9254 .8354 .7970 .7672 .7168 .6245 .6521
512×512 .8919 .9313 .7573 .8499 .6650 .7854 .5514 .6718
Bridge .9426 .9397 .8504 .8186 .7427 .7067 .5724 .6121
256×256 .9210 .9300 .8007 .8510 .6865 .7607 .5400 .5724
Monarch .9728 .9739 .9135 .8936 .8945 .8709 .8093 .7978
256×256 .9658 .9711 .8795 .9220 .8410 .8937 .7587 .8245
House .9394 .9522 .8994 .9001 .9078 .8912 .8180 .8321
256×256 .9263 .9403 .8679 .9093 .8209 .9347 .7284 .8434
Brain .9297 .9159 .8046 .7529 .7588 .6875 .5698 .5676
256×256 .8858 .9286 .7219 .8206 .6486 .7661 .5045 .5960
.9410 .9386 .8612 .8325 .8087 .7702 .6728 .6925
Average
.9172 .9373 .8088 .8709 .7378 .8238 .6262 .7071
The four volunteers with informed consent in accordance with our institution’s
human subject policies participated in the study. They were scanned
with the four pulse sequences, i.e., spin echo (SE), turbo spin echo (TSE),
and gradient echo (GR), whose imaging parameter values are given in Table
3. These four images, i.e., T1 SE, TOF, TSE T1, and TSE T2, provide a
testing set which presents a good diversity: different anatomical structures,
and different noise intensities. To show the robustness of the proposed algorithm
with real MR images, we used the same parameters in all examples.
Table 4 shows the SNR results of the comparison between the proposed algorithm
and other state-of-the-art methods for the noisy MR images. Figure
20

Table 3: The four pulse sequences used in the human experiments with these
imaging parameters, i.e., Repetition Time (TR) (msec), Echo Time (TE)
(msec), Flip Angle (FA) ( ◦ ), Echo Train Length (ETL), Slice Thickness (ST)
(mm), and Pixel Bandwidth (PB) (Hz).
Images Sequence TR TE FA ETL ST PB
T1 SE SE 2000 131 120 37 0.36 449
TOF SE 327 14 90 1 3 130
TSE T1 GR 19 3.09 25 1 0.70 250
TSE T2 SE 1120 26 174 7 2 130
Table 4: The SNR results of the comparison between the proposed algorithm
and BM3D [9], PLPCA [10], and LPG-PCA [54] for the four noisy MR
images, i.e., T1 SE, TOF, TSE T1, and TSE T2.
Images Noisy Proposed BM3D[9] PLPCA[10] LPG-PCA[54]
T1 SE 24.77 190.85 183.40 124.37 458.53
TOF 27.40 77.90 71.98 39.59 116.30
TSE T1 12.21 116.47 102.72 102.21 168.96
TSE T2 15.51 293.25 221.83 83.86 502.28
8 to Figure 11 present the visual comparison results of the proposed algorithm,
BM3D [9], PLPCA [10], and LPG-PCA [54] for the different noisy
MR images, respectively.
As can be seen from Table 4, the SNR results of the proposed algorithm
outperformsthose of BM3D [9], andPLPCA [10], andarelower thanthose of
LPG-PCA[54]. However, the visual comparisons of these denoising methods
in Figure 8 to Figure 11 show that LPG-PCA[54] smooths out the edges and
anatomical structures of MR images. It is observed that the results of our
proposed algorithm have sharper edges and clearer anatomical structures
than those of the state-of-the-art methods. Strictly speaking, in fact the
AWGN model (1) is not very accurate to describe the noise in MR images.
TheGaussianassumptioninthesimulationonsyntheticimagesmaybecauses
that the proposed algorithm does not outperform the existing best state-ofthe-art
methods in some cases, e.g., BM3D [9]. These experiments verify
that the proposed algorithm can be effectively applied to noisy MR images
with different noise levels for noise removal, and achieves a better edge and
structure preservation than other state-of-the-art methods.
21

(a) (b) (c)
(d)
Figure 8: Visual comparisons of the denoising results of the proposed algorithm
and other state-of-the-art methods for the T1 SE MR images. From
left to right: (a) Noisy MR images; (b) the proposed algorithm; (c) BM3D
[9]; (d) PLPCA [10]; and (e) LPG-PCA [54].
(e)
4. Conclusions and Future Work
In this paper, we addressed the problem of automatically determining
the signal dimensionality for the similar patch groups from the given noisy
image. For a pixel in the search window, the parallel analysis with Monte
Carlo simulation is employed to estimate the signal component dimensionality
from its related high-dimensional noisy patch groups. After the inverse
SVD transform, the approximation of the true noiseless patch groups is obtained
by the low-rank approach. Thus, our proposed adaptive PCA can discard
a considerable part of noises almost without loss of signal information.
The weighted averaging aggregation operator is used to reduce noise further.
22

(a) (b) (c)
(d)
Figure 9: Visual comparisons of the denoising results of the proposed algorithm
and other state-of-the-art methods for the TOF MR images. From
left to right: (a) Noisy MR images; (b) the proposed algorithm; (c) BM3D
[9]; (d) PLPCA [10]; and (e) LPG-PCA [54].
(e)
Then, the reconstructed noise-free image with better denoising performance
is achieved by the empirical Wiener filtering applied to the first-stage denoised
image. The experimental results of real MR images demonstrate that
the proposed denoising algorithm can outperform the best state-of-the-art
methods both visually and quantitatively, especially for image regions containing
the abundant edges, and the fine structures. Moreover, our proposed
adaptive PCA approach can be extended to many applications, particularly
multivariate analysis and machine learning. Finally, combined with sparse
representation, theproposed algorithm will be further studied on theoptimal
parameters of the search window, the patch size and its shape in the future.
Acknowledgements
This work was supported by National Basic Research Program (973 Program)
of China under Contract No.2009CB320907, National Natural Science
23

(a) (b) (c)
(d)
Figure 10: Visual comparisons of the denoising results of the proposed algorithm
and other state-of-the-art methods for the TSE T1 MR images. From
left to right: (a) Noisy MR images; (b) the proposed algorithm; (c) BM3D
[9]; (d) PLPCA [10]; and (e) LPG-PCA [54].
(e)
Foundation of China under contract No.61201442, and China Postdoctoral
Science Foundationunder contract No.2013M530481. Theauthorswouldlike
to thank Yu Ding for his helpful discussions and precious advices from Davis
Heart and Lung Research Institute, The Ohio State University.
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