This document summarizes an improved fixed point method for image restoration. It begins by introducing the problem of image restoration and describing common image degradation models. It then presents the traditional fixed point method using Tikhonov regularization. The improved method proposes changing the regularization parameter from larger to smaller values across iterations. Experimental results show the improved method performs better than other popular algorithms at solving motion and Gaussian degradation, producing reconstructed images with less noise and more detail.
Robust Image Denoising in RKHS via Orthogonal Matching PursuitPantelis Bouboulis
We present a robust method for the image denoising task based on kernel ridge regression and sparse modeling. Added noise is assumed to consist of two parts. One part is impulse noise assumed to be sparse (outliers), while the other part is bounded noise. The noisy image is divided into small regions of interest, whose pixels are regarded as points of a two-dimensional surface. A kernel based ridge regression method, whose parameters are selected adaptively, is employed to fit the data, whereas the outliers are detected via the use of the increasingly popular orthogonal matching pursuit (OMP) algorithm. To this end, a new variant of the OMP rationale is employed that has the additional advantage to automatically terminate, when all outliers have been selected.
Image Restitution Using Non-Locally Centralized Sparse Representation ModelIJERA Editor
Sparse representation models uses a linear combination of a few atoms selected from an over-completed
dictionary to code an image patch which have given good results in different image restitution applications. The
reconstruction of the original image is not so accurate using traditional models of sparse representation to solve
degradation problems which are blurring, noisy, and down-sampled. The goal of image restitution is to suppress
the sparse coding noise and to improve the image quality by using the concept of sparse representation. To
obtain a good sparse coding coefficients of the original image we exploit the image non-local self similarity and
then by centralizing the sparse coding coefficients of the observation image to those estimates. This non-locally
centralized sparse representation model outperforms standard sparse representation models in all aspects of
image restitution problems including de-noising, de-blurring, and super-resolution.
In topological inference, the goal is to extract information about a shape, given only a sample of points from it. There are many approaches to this problem, but the one we focus on is persistent homology. We get a view of the data at different scales by imagining the points are balls and consider different radii. The shape information we want comes in the form of a persistence diagram, which describes the components, cycles, bubbles, etc in the space that persist over a range of different scales.
To actually compute a persistence diagram in the geometric setting, previous work required complexes of size n^O(d). We reduce this complexity to O(n) (hiding some large constants depending on d) by using ideas from mesh generation.
This talk will not assume any knowledge of topology. This is joint work with Gary Miller, Benoit Hudson, and Steve Oudot.
Robust Image Denoising in RKHS via Orthogonal Matching PursuitPantelis Bouboulis
We present a robust method for the image denoising task based on kernel ridge regression and sparse modeling. Added noise is assumed to consist of two parts. One part is impulse noise assumed to be sparse (outliers), while the other part is bounded noise. The noisy image is divided into small regions of interest, whose pixels are regarded as points of a two-dimensional surface. A kernel based ridge regression method, whose parameters are selected adaptively, is employed to fit the data, whereas the outliers are detected via the use of the increasingly popular orthogonal matching pursuit (OMP) algorithm. To this end, a new variant of the OMP rationale is employed that has the additional advantage to automatically terminate, when all outliers have been selected.
Image Restitution Using Non-Locally Centralized Sparse Representation ModelIJERA Editor
Sparse representation models uses a linear combination of a few atoms selected from an over-completed
dictionary to code an image patch which have given good results in different image restitution applications. The
reconstruction of the original image is not so accurate using traditional models of sparse representation to solve
degradation problems which are blurring, noisy, and down-sampled. The goal of image restitution is to suppress
the sparse coding noise and to improve the image quality by using the concept of sparse representation. To
obtain a good sparse coding coefficients of the original image we exploit the image non-local self similarity and
then by centralizing the sparse coding coefficients of the observation image to those estimates. This non-locally
centralized sparse representation model outperforms standard sparse representation models in all aspects of
image restitution problems including de-noising, de-blurring, and super-resolution.
In topological inference, the goal is to extract information about a shape, given only a sample of points from it. There are many approaches to this problem, but the one we focus on is persistent homology. We get a view of the data at different scales by imagining the points are balls and consider different radii. The shape information we want comes in the form of a persistence diagram, which describes the components, cycles, bubbles, etc in the space that persist over a range of different scales.
To actually compute a persistence diagram in the geometric setting, previous work required complexes of size n^O(d). We reduce this complexity to O(n) (hiding some large constants depending on d) by using ideas from mesh generation.
This talk will not assume any knowledge of topology. This is joint work with Gary Miller, Benoit Hudson, and Steve Oudot.
Lec-07: Feature Aggregation and Image Retrieval System [notes]
Image retrieval system performance metrics, precision, recall, true positive rate, false positive rate; Bag of Words (BoW) and VLAD aggregation.
Optimized linear spatial filters implemented in FPGAIOSRJVSP
Linear spatial filters (LSF) are used for filtering of digital images with the purpose of blurring, noise reduction, detail enhancement etc. The realization of LSF confronts the capital problem of a lot of operations needed for their computation. In this paper, described is an approach for optimizing of LSF by utilizing parallel algorithms and their hardware implementation on FPGA. A model and an algorithm based on partial sums and aimed at calculating the filtered pixels are presented. Defined are criteria for comparing of the different types of linear filters. A schematic diagram of an FPGA-based DSP operational block is shown. VHDL is utilized for the hardware design. Conducted are studies focused on comparing the partial sums and the non-partial sums based methods of filtering. It is ascertained that the methods employing partial sums reduce the number of operations to the size of the window (3, 5, 7,...) . These FPGA-based LSF are suitable for applications using threshold detecting, edge detection or image detail enhancement.
Image Retrieval with Fisher Vectors of Binary Features (MIRU'14)Yusuke Uchida
Recently, the Fisher vector representation of local features has attracted much attention because of its effectiveness in both image classification and image retrieval. Another trend in the area of image retrieval is the use of binary feature such as ORB, FREAK, and BRISK. Considering the significant performance improvement in terms of accuracy in both image classification and retrieval by the Fisher vector of continuous feature descriptors, if the Fisher vector were also to be applied to binary features, we would receive the same benefits in binary feature based image retrieval and classification. In this paper, we derive the closed-form approximation of the Fisher vector of binary features which are modeled by the Bernoulli mixture model. In experiments, it is shown that the Fisher vector representation improves the accuracy of image retrieval by 25% compared with a bag of binary words approach.
Different Approach of VIDEO Compression Technique: A StudyEditor IJCATR
The main objective of video compression is to achieve video compression with less possible losses to reduce the
transmission bandwidth and storage memory. This paper discusses different approach of video compression for better transmission of
video frames for multimedia application. Video compression methods such as frame difference approach, PCA based method,
accordion function, fuzzy concept, and EZW and FSBM were analyzed in this paper. Those methods were compared for performance,
speed and accuracy and which method produces better visual quality.
ALEXANDER FRACTIONAL INTEGRAL FILTERING OF WAVELET COEFFICIENTS FOR IMAGE DEN...sipij
The present paper, proposes an efficient denoising algorithm which works well for images corrupted with
Gaussian and speckle noise. The denoising algorithm utilizes the alexander fractional integral filter which
works by the construction of fractional masks window computed using alexander polynomial. Prior to the
application of the designed filter, the corrupted image is decomposed using symlet wavelet from which only
the horizontal, vertical and diagonal components are denoised using the alexander integral filter.
Significant increase in the reconstruction quality was noticed when the approach was applied on the
wavelet decomposed image rather than applying it directly on the noisy image. Quantitatively the results
are evaluated using the peak signal to noise ratio (PSNR) which was 30.8059 on an average for images
corrupted with Gaussian noise and 36.52 for images corrupted with speckle noise, which clearly
outperforms the existing methods.
GENERIC APPROACH FOR VISIBLE WATERMARKINGEditor IJCATR
In this paper generic image watermarking technique is used for the copyright protection of color images. Watermarking
with monochrome and translucent images based on One-to-One compound mapping of the values of the image pixels, which provide us
the recovered image without any loss. Both the translucent full color and Opaque monochrome images are used in this paper. Two-fold
monotonically increasing compound mapping is used to get more typical visible watermarks in the image. Measures have been taken to
protect it from hackers.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Digital image processing using matlab: basic transformations, filters and ope...thanh nguyen
How to use Matlab to deal with basic image manipulations.
Negative transformation
Log transformation
Power-law transformation
Piecewise-linear transformation
Histogram equalization
Subtraction
Smoothing Linear Filters
Order-Statistics Filters
The Laplacian
The Gradient
Lec-07: Feature Aggregation and Image Retrieval System [notes]
Image retrieval system performance metrics, precision, recall, true positive rate, false positive rate; Bag of Words (BoW) and VLAD aggregation.
Optimized linear spatial filters implemented in FPGAIOSRJVSP
Linear spatial filters (LSF) are used for filtering of digital images with the purpose of blurring, noise reduction, detail enhancement etc. The realization of LSF confronts the capital problem of a lot of operations needed for their computation. In this paper, described is an approach for optimizing of LSF by utilizing parallel algorithms and their hardware implementation on FPGA. A model and an algorithm based on partial sums and aimed at calculating the filtered pixels are presented. Defined are criteria for comparing of the different types of linear filters. A schematic diagram of an FPGA-based DSP operational block is shown. VHDL is utilized for the hardware design. Conducted are studies focused on comparing the partial sums and the non-partial sums based methods of filtering. It is ascertained that the methods employing partial sums reduce the number of operations to the size of the window (3, 5, 7,...) . These FPGA-based LSF are suitable for applications using threshold detecting, edge detection or image detail enhancement.
Image Retrieval with Fisher Vectors of Binary Features (MIRU'14)Yusuke Uchida
Recently, the Fisher vector representation of local features has attracted much attention because of its effectiveness in both image classification and image retrieval. Another trend in the area of image retrieval is the use of binary feature such as ORB, FREAK, and BRISK. Considering the significant performance improvement in terms of accuracy in both image classification and retrieval by the Fisher vector of continuous feature descriptors, if the Fisher vector were also to be applied to binary features, we would receive the same benefits in binary feature based image retrieval and classification. In this paper, we derive the closed-form approximation of the Fisher vector of binary features which are modeled by the Bernoulli mixture model. In experiments, it is shown that the Fisher vector representation improves the accuracy of image retrieval by 25% compared with a bag of binary words approach.
Different Approach of VIDEO Compression Technique: A StudyEditor IJCATR
The main objective of video compression is to achieve video compression with less possible losses to reduce the
transmission bandwidth and storage memory. This paper discusses different approach of video compression for better transmission of
video frames for multimedia application. Video compression methods such as frame difference approach, PCA based method,
accordion function, fuzzy concept, and EZW and FSBM were analyzed in this paper. Those methods were compared for performance,
speed and accuracy and which method produces better visual quality.
ALEXANDER FRACTIONAL INTEGRAL FILTERING OF WAVELET COEFFICIENTS FOR IMAGE DEN...sipij
The present paper, proposes an efficient denoising algorithm which works well for images corrupted with
Gaussian and speckle noise. The denoising algorithm utilizes the alexander fractional integral filter which
works by the construction of fractional masks window computed using alexander polynomial. Prior to the
application of the designed filter, the corrupted image is decomposed using symlet wavelet from which only
the horizontal, vertical and diagonal components are denoised using the alexander integral filter.
Significant increase in the reconstruction quality was noticed when the approach was applied on the
wavelet decomposed image rather than applying it directly on the noisy image. Quantitatively the results
are evaluated using the peak signal to noise ratio (PSNR) which was 30.8059 on an average for images
corrupted with Gaussian noise and 36.52 for images corrupted with speckle noise, which clearly
outperforms the existing methods.
GENERIC APPROACH FOR VISIBLE WATERMARKINGEditor IJCATR
In this paper generic image watermarking technique is used for the copyright protection of color images. Watermarking
with monochrome and translucent images based on One-to-One compound mapping of the values of the image pixels, which provide us
the recovered image without any loss. Both the translucent full color and Opaque monochrome images are used in this paper. Two-fold
monotonically increasing compound mapping is used to get more typical visible watermarks in the image. Measures have been taken to
protect it from hackers.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Digital image processing using matlab: basic transformations, filters and ope...thanh nguyen
How to use Matlab to deal with basic image manipulations.
Negative transformation
Log transformation
Power-law transformation
Piecewise-linear transformation
Histogram equalization
Subtraction
Smoothing Linear Filters
Order-Statistics Filters
The Laplacian
The Gradient
A Comparative Study of Histogram Equalization Based Image Enhancement Techniq...Shahbaz Alam
Four widely used histogram equalization techniques for image enhancement namely GHE, BBHE, DSIHE, RMSHE are discussed. Some basic definitions and notations are also attached. All analysis are done by using MATLAB . Pictures are taken from the book "Digital Image Processing" by Rafael C. Gonzalez and Richard E. Woods. The presentation slide was made for my B.Sc project purpose.
Kernel Estimation of Videodeblurringalgorithm and Motion Compensation of Resi...IJERA Editor
This paper presents a videodeblurring algorithm utilizing the high resolution information of adjacent unblurredframes.First, two motion-compensated predictors of a blurred frame are derived from its neighboring unblurred frames via bidirectional motion compensation. Then, an accurate blur kernel, which is difficult to directly obtain from the blurred frame itself, is computed between the predictors and the blurred frame. Next, a residual deconvolution is employed to reduce the ringing artifacts inherently caused by conventional deconvolution. The blur kernel estimation and deconvolution processes are iteratively performed for the deblurred frame. Experimental results show that the proposed algorithm provides sharper details and smaller artifacts than the state-of-the-art algorithms.
Performance Analysis of Image Enhancement Using Dual-Tree Complex Wavelet Tra...IJERD Editor
Resolution enhancement (RE) schemes which are not based on wavelets has one of the major
drawbacks of losing high frequency contents which results in blurring. The discrete wavelet- transform-based
(DWT) Resolution Enhancement scheme generates artifacts (due to a DWT shift-variant property). A wavelet-
Domain approach based on dual-tree complex wavelet transform (DT-CWT) & nonlocal means (NLM) is
proposed for RE of the satellite images. A satellite input image is decomposed by DT-CWT (which is nearly
shift invariant) to obtain high-frequency sub bands. Here the Lanczos interpolator is used to interpolate the highfrequency
sub bands & the low-resolution (LR) input image. The high frequency sub bands are passed through
an NLM filter to cater for the artifacts generated by DT-CWT (despite of it’s nearly shift invariance). The
filtered high-frequency sub bands and the LR input image are combined by using inverse DTCWT to obtain a
resolution-enhanced image. Objective and subjective analyses show superiority of the new proposed technique
over the conventional and state-of-the-art RE techniques.
Surveillance refers to the task of observing a scene, often for lengthy periods in search of particular objects or particular behaviour. This task has many applications, foremost among them is security (monitoring for undesirable behaviour such as theft or vandalism), but increasing numbers of others in areas such as agriculture also exist. Historically, closed circuit TV (CCTV) surveillance has been mundane and labour Intensive, involving personnel scanning multiple screens, but the advent of reasonably priced fast hardware means that automatic surveillance is becoming a realistic task to attempt in real time. Several attempts at this are underway.
WAVELET DECOMPOSITION AND ALPHA STABLE FUSIONsipij
This article gives a new method of fusing multifocal images combining the Laplacian pyramid and the wavelet decomposition using the stable distance alpha as a selection rule. We start by decomposing multifocal images into several pyramid levels, then applying the wavelet decomposition to each level. the originality of this work is to use the stable distance alpha to fuse the wavelet images at each level of the Pyramid. To obtain the final fused image, we reconstructed the combined image at each level of the pyramid. We compare our method to other existing methods in the literature and we deduce that it is almost better.
1. 第 6 卷 第 3 期
2013 年 6 月
中国光学
Chinese Optics
Vol. 6 No. 3
June 2013
收稿日期: 2013-02-11; 修订日期: 2013-04-13
基金项目: Major State Basic Research Development Program of China( 973 Program,No. 2009CB72400603) ; The National
Natural Science: Scientific Instrumentation Special Project( No. 61027002) ; The National Natural Science Foun-
dation of China( No. 60972100)
文章编号 1674-2915( 2013) 03-0318-07
Improved fixed point method for image restoration
YAN Xue-fei,XU Ting-fa*
,BAI Ting-zhu
( Key Laboratory of Photoelectric Imaging Technology and System of the Ministry of Education,
College of Optical Engineering,Beijing Institute of Technology,Beijing 100081,China)
* Corrponding author,E-mail: xutingfa@ 163. com
Abstract: We analyze the fixed point method with Tikhonov regularization under the periodic boundary condi-
tions,and propose a changable regularization parameter method. Firstly,we choose a bigger one to restrain
the noise in the reconstructed image,and get a convergent result to modify the initial gradient. Secondly,we
choose a smaller one to increase the details in the image. Experimental results show that compared with other
popular algorithms which solve the L1 norm regularization function and Total Variation ( TV) regularization
function,the improved fixed point method performs favorably in solving the problem of the motion degradation
and Gaussian degradation.
Key words: image restoration; periodic boundary condition; Tikhonov regularization; change regularization pa-
rameter
改进的固定点图像复原算法
阎雪飞,许廷发
*
,白廷柱
( 北京理工大学 光电学院 光电成像技术与系统教育部重点实验室,北京 100081)
摘要: 研究了周期边界条件下,Tikhonov 正则化的固定点算法,提出了变化正则化参数的方法。首先对正则化参数取较
大值,抑制复原图像中的噪声,通过得出的收敛结果来修正初始梯度; 然后对正则化参数取较小值,以增强复原图像中的
细节。实验结果表明,与当前求解 L1 范数正则化函数和全变分正则化函数的流行算法比较,本文算法对于运动模糊与高
斯模糊图像的复原效果更佳。
关 键 词: 图像复原; 周期边界条件; Tikhonov 正则化; 变正则化参数
中图分类号: TP391. 4 文献标识码: A doi: 10. 3788 /CO. 20130603. 0318
2. 1 Introduction
In modern society,images are important media for
human to obtain the information. However, the
images are destroyed inevitably during imaging,
copying,and transmission,such as image informa-
tion is polluted by the noise. While,the high-
quality images are necessary in many areas. So the
image restoration is a very important task. It is one
of the earliest and most classical nonlinear inverse
problems in imaging processing,dating back to the
1960's.
2 Image degradation model
In this class of problems,the image blurring is mod-
eled as:
g = Hf + n , ( 1)
Where f( f∈RAB × 1
) is a AB × 1 vector and repre-
sents the unknown A × B original image. g ( g ∈
RAB × 1
) is a AB × 1 vector and represents the A × B
noisy blurred image. n( n∈RAB × 1
) is a AB × 1 vec-
tor and represents the white Gaussian noise. H( H∈
RAB × AB
) represents the blur operator. In the case of
spatially invariant blurs,it is the matrix representa-
tion of the convolution operation. The structure of H
depends on the choice of boundary conditions,that
is,the underlying assumptions on the image outside
the field of view.
We consider the original image:
f =
1 2 3
4 5 6
7 8 9
. ( 2)
Zero boundary condition means that all pixels
outside the borders are assumed to be zero,which
can be pictured as embedding the image f in a large
image:
fext =
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 1 2 3 0 0 0
0 0 0 4 5 6 0 0 0
0 0 0 7 8 9 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
,( 3)
In this case,H is a Block Toeplitz with Toeplitz
Blocks( BTTB) matrix.
Periodic boundary condition means that the im-
age repeats itself in all directions,which can be pic-
tured as a large image embedded with image f:
fext =
1 2 3 1 2 3 1 2 3
4 5 6 4 5 6 4 5 6
7 8 9 7 8 9 7 8 9
1 2 3 1 2 3 1 2 3
4 5 6 4 5 6 4 5 6
7 8 9 7 8 9 7 8 9
1 2 3 1 2 3 1 2 3
4 5 6 4 5 6 4 5 6
7 8 9 7 8 9 7 8 9
,( 4)
In this case,H is a Block Circulant with Circulant
Blocks( BCCB) matrix.
Reflexive ( Neumann ) boundary condition
means that the scene outside the boundaries is a mir-
ror image of the scene inside the image boundaries,
which can be pictured as a large image embedded
with image f:
fext =
9 8 7 7 8 9 9 8 7
6 5 4 4 5 6 6 5 4
3 2 1 1 2 3 3 2 1
3 2 1 1 2 3 3 2 1
6 5 4 4 5 6 6 5 4
9 8 7 7 8 9 9 8 7
9 8 7 7 8 9 9 8 7
6 5 4 4 5 6 6 5 4
3 2 1 1 2 3 3 2 1
. ( 5)
In this case,H is the sum of BTTB,Block Toeplitz
with Hankel Blocks( BTHB) ,Block Hankel with To-
eplitz Blocks( BHTB) and Block Hankel with Hankel
913第 3 期 徐国权,等: 光纤光栅传感技术在工程中的应用
3. Blocks( BHHB) matrices[1]
.
In this paper we propose an improved fixed point
method with Tikhonov regularization that we claim
performs better than other popular algorithms in both
subjective and objective judgements of the image res-
toration ability. It changes regularization parameters
from bigger ones to small ones with iterations,and
modifies the initial gradient at every iteration to in-
crease the details in the reconstructed image.
3 Tikhonov regularization function
As we all know,H is often singular or very ill-con-
ditioned. So the problem of estimating f from g is an
Ill-posed Linear Inverse Problem( ILIP) . A classical
approach to ILIP is the Tikhonov regularization
which can be expressed as:
f = argmin
1
2
‖Hf - g‖
2
+
λ
2
‖Df‖
2
. ( 6)
where λ( λ > 0) is the regularization parameter,D
is the penalty matrix. ‖Df‖
2
is the regularizer or
regularization functional,which has many choices
such as: L1 norm regularization functional[2-6]
,To-
tal-Variation ( TV ) regularization functional[7-10]
.
There are algorithms for solving L1 norm regulariza-
tion functional such as: Gradient Projection for Space
Resonstruction ( GPSR ) algorithm[11]
, which in-
cludes GPSR-basic algorithm,GPSR-bb algorithm
and SpaRSA-monotone algorithm[12]
. There are al-
gorithms for solving TV regularization functional such
as: TwIST algorithm[13-14]
,SALSA algorithm[15-16]
,
FTVd algorithm[17-20]
and fixed point method[23]
.
The fixed point method can be expressed as:
k = 0 , ( 7)
f0 = 0 . ( 8)
Begin the fixed point iterations:
Dk = D( fk ) , ( 9)
rk = HT
( Hfk - g) + αDk fk , ( 10)
M = HT
H + αDk , ( 11)
sk = M-1
rk , ( 12)
fk+1 = fk + sk , ( 13)
k = k + 1 . ( 14)
Generally,when rk < 0. 001,we stop the itera-
tions.
4 Improved fixed point method
When the boundary condition of image blurring is
the periodic boundary condition,we can choose pen-
alty matrix as the negative Laplacian with periodic
boundary condition L,which is a B × B partitioned
matrix:
L =
L0 - I Θ … - I
- I L0 - I … Θ
Θ - I L0 … Θ
- I Θ Θ … L
0
, ( 15)
where I is an A × A identity matrix,Θ is a A × A ze-
ro matrix.
L0 =
4 - 1 0 … - 1
- 1 4 - 1 … 0
0 - 1 4 … 0
- 1 0 0 …
4
. ( 16)
We use the matlab function psf = fspecial( 'mo-
tion',71,56) to create the motion blurring kernel
point spread function,and use g = imfilter( f,psf,'
conv','same',' circular') to get the motion blurred
image; g = imnoise( g,'Gaussian',0,1 × 10 - 3
) to
add white Gaussian noise of zero mean and standard
deviation 1 × 10 - 3
. We run the fixed point method
and compare the reconstructed images when λ is dif-
ferent.
In the Fig. 1,there are 256 pixel × 256 pixel o-
riginal image Lena( a) ,motion blurring kernel noisy
blurred image ( b) ,and image reconstructed when
λ =10( c) ,λ =1( d) ,λ =10 - 1
( e) ,λ =10 - 2
( f) .
We can find that when we choose a bigger λ,
restored image has less noise and also less details;
however,when we choose a smaller λ,reconstruc-
ted image has more noise and also more details.
Therefore,we propose the change regularization pa-
023 中国光学 第 6 卷
4. Fig. 1 Image restoration comparison of different regu-
larization parameters
rameter method. Firstly,we choose a bigger λ to re-
strain the noise in the reconstructed image. Running
the fixed point method to get the convergent result f,
and convolving with H to modify the initial gradient
r0 to the real value. We use the new f,r0 as the ini-
tial ones to run the fixed point method again. Sec-
ondly,we choose a smaller λ to increase the details
in the reconstructed image. Repeating the above
process until the suitable value is found:
1. Give the initial value: λ = λ0 ,k < 1,f0 = 0,
r0 = - HT
g;
2. Run the fixed point method to get the conver-
gent result f;
3. f0 = f,λ = λ·k,r0 = Af0 - HT
g,back to 2.
4 Experimental results
Simulations are carried out to test the proposed algo-
rithm. The Mean Structural Similarity Index Method
( MSSIM) [24]
is selected to evaluate the restored im-
age quality. The SSIM index is defined as follows:
SSIM( X,Y) =
( 2μx μy + C1 ) ( 2σxy + C2 )
( μ
2
x + μ
2
y + C1 ) ( σ
2
x + σ
2
y + C2 )
, ( 17)
Where X and Y are the original and the reconstruc-
ted images,respectively; μx and μy are the mean in-
tensity,σx and σy are the standard deviation; and
σxy is the covariance.
C1 = ( K1 L) 2
, ( 18)
C2 = ( K2 L) 2
, ( 19)
Where L is the dynamic range of the pixel values
( 255 for 8-bit grayscale images) ,and K1 and K2 are
small constants.
MSSIM( X,Y) =
1
MΣ
M
j = 1
SSIM( xj ,yj ) ,( 20)
Where xj and yj are the image contents at the jth local
window; and M is the number of local windows of
the image.
We choose the image Lena as the test target and
compare the proposed algorithm with several state-of-
the-art algorithms. We choose the GPSR-bb algo-
rithm as GPSR algorithm.
The parameters for the proposed algorithm are
λ0 = 10,k = 0. 8. The parameters for the others are
defaults in the papers by the authors. The parame-
ters for the GPSR-bb algorithm are αmin = 10 - 30
,
αmax = 1030
,λ = 0. 35. The parameters for the SpaR-
SA-monotone algorithm are λ = 0. 035,M = 0,σ =
10 - 5
,η = 2. The parameters for the TwIST algo-
rithm are α = 1. 960 8,β = 3. 921 2. The parameter
for the SALSA algorithm is λ = 0. 025. The parame-
ter for the FTVd algorithm is λ = 50 000.
For all the algorithms,we choose the same
maximum iteration number as 10000; the same stop
criterion as the relative error difference between two
iterations is less than 1 × 10 - 7
. All computations
were performed under windows 7 and matlab V 7. 10
( R2012a) running on a desktop computer with an
Intel Core 530 duo CPU i3 and 2GB of memory.
123第 3 期 YAN Xue-fei,et al. : Improved fixed point method for image restoration
5. 5. 1 Motion degradation
We use the matlab function psf = fspecial( 'mo-
tion',71,56) to create the motion blurring kernel
point spread function,and use g = imfilter( f,psf,
'conv','same','circular') to get the motion blurred
image,and use g = imnoise ( g,' Gaussian',0,
1 × - 3
) to add white Gaussian noise of zero mean
and standard deviation 1 × 10 - 3
.
In the Fig. 2,there are 256 pixel × 256 pixel o-
riginal image Lena( a) ,motion blurring kernel noisy
blurred image( b) ,image restored by the improved
fixed point method( c) ,GPSR-bb( d) ,SpaRSA-mon-
otone( e) ,TwIST( f) ,SALSA( g) and FTVd( h) .
Fig. 2 Restoration comparison of motion degradations
Tab. 1 contains the MSSIM and time compari-
son between the proposed algorithm and several
state-of-the-art algorithms with motion degradation.
Tab. 1 Restoration comparison of motion degradations
Several algorithms MSSIM Time/s
Improved fixed point method 0. 537 2 4. 196
GPSR bb 0. 386 9 1. 451
SpaRSA-monotone 0. 372 1 1. 108
TwIST 0. 342 7 1. 232
SALSA 0. 097 3 0. 577 2
FTVd 0. 000 8 3. 931
5. 2 Gaussian degradation
We use the matlab function psf = fspecial
( 'Gaussian',[21,28],10) to create the Gaussian
blurring kernel point spread function,and use g =
imfilter( f,psf,'conv','same','circular') to get the
Gaussian blurred image,and use g = imnoise ( g,
'Gaussian',0,1 × 10 - 3
) to add white Gaussian
noise of zero mean and standard deviation 1 × 10 - 3
.
Fig. 3 Restoration comparison of Gaussian degradations
223 中国光学 第 6 卷
6. In the Fig. 3,there are 256 pixel × 256 pixel o-
riginal image Lena ( a) ,Gaussian blurring kernel
noisy blurred image( b) ,image restored by the im-
proved fixed point method ( c ) ,GPSR-bb ( d ) ,
SpaRSA-monotone( e) ,TwIST( f) ,SALSA( g) and
FTVd( h) .
Tab. 2 contains the MSSIM and the time com-
Tab. 2 Restoration comparison of
Gaussian degradations
Several algorithms MSSIM Time/s
Improved fixed point method 0. 492 8 4. 321
GPSR bb 0. 434 4 2. 246
SpaRSA-monotone 0. 418 7 1. 981
TwIST 0. 423 9 2. 246
SALSA 0. 217 5 0. 670 8
FTVd 0. 001 0 5. 476
parison between the proposed algorithm and several
state-of-the-art algorithms,with the Gaussian degra-
dation. From the above comparisons,we can see
that under the high noise level,the improved fixed
point method performs better than the others,and
the FTVd algorithm is totally unfit.
6 Conclusion
We propose,analyze,and test the improved fixed
point method for solving the Tikhonov regularization
under the periodic boundary conditions in image res-
toration problem. In experimental comparisons with
state-of-the-art algorithms,the proposed algorithm a-
chieves remarkable performance in both subjective
and objective judgments of the image restoration a-
bility.
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Authors' biographies:
YAN Xue-fei ( 1979—) , male, PhD
student. He received his B. S. degree in
electronics engineering from Shanxi Uni-
versity in 2002,and received his M. S.
degree in Optical Engineering from Bei-
jing Institute of Technology in 2007. His
research interests include image deblur-
ring. E-mail: yxfamyself@ sina. com
BAI Ting-zhu( 1955—) ,male,M. PhD
supervisor,Professor,PhD. He received
his B. S. ,M. S. and Ph. D. degrees in
Optical Engineering from Beijing Institu-
te of Technology in 1982,1987 and
2001,respectively. His research inter-
ests include electro-optical imaging tech-
nology,photoelectric detection technolo-
gy, and image information processing
technology. E-mail: tzhbai@ bit. edu. cn
XU Ting-fa( 1968—) ,male,M. PhD supervisor,Professor,post doctorate. He received his B. S. and M. S.
degrees in physics from Northeast Normal University in 1992 and 2000,respectively,and received his Ph. D.
degree in Optical Engineering from Changchun Institute of Optics,Fine Mechanics and Physics,Chinese A-
cademy of Sciences in 2004. He finished studies in mobile postdoctoral station of South China University of
Technology in 2006. His research interests include photoelectric imaging detection and recognition,target
tracking,photoelectric measurement. E-mail: xutingfa@ 163. com
423 中国光学 第 6 卷