This paper addresses mathematical model for signal restoration based on fractional order total variation (FOTV) for multiplicative noise. In alternating minimization algorithm the Newton method is coupled with time-marching scheme for the solutions of the corresponding PDEs related to the minimization of the denoising model. Results obtained from experiments show that our model can not only reduce the staircase effect of the restored images but also better improve the PSNR as compare to other existed methods.
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.
Investigation on the Pattern Synthesis of Subarray Weights for Low EMI Applic...IOSRJECE
In modern radar applications, it is frequently required to produce sum and difference patterns sequentially. The sum pattern amplitude coefficients are obtained by using Dolph-Chebyshev synthesis method where as the difference pattern excitation coefficients will be optimized in this present work. For this purpose optimal group weights will be introduced to the different array elements to obtain any type of beam depending on the application. Optimization of excitation to the array elements is the main objective so in this process a subarray configuration is adopted. However, Differential Evolution Algorithm is applied for optimization method. The proposed method is reliable and accurate. It is superior to other methods in terms of convergence speed and robustness. Numerical and simulation results are presented.
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.
Investigation on the Pattern Synthesis of Subarray Weights for Low EMI Applic...IOSRJECE
In modern radar applications, it is frequently required to produce sum and difference patterns sequentially. The sum pattern amplitude coefficients are obtained by using Dolph-Chebyshev synthesis method where as the difference pattern excitation coefficients will be optimized in this present work. For this purpose optimal group weights will be introduced to the different array elements to obtain any type of beam depending on the application. Optimization of excitation to the array elements is the main objective so in this process a subarray configuration is adopted. However, Differential Evolution Algorithm is applied for optimization method. The proposed method is reliable and accurate. It is superior to other methods in terms of convergence speed and robustness. Numerical and simulation results are presented.
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Sparse representation models uses a linear combination of a few atoms selected from an over-completed
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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.
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been proposed, which is based on three different sized cascaded filtering windows. The
differences between the current pixel and its neighbors aligned with four main directions. A
direction index is used for each edge aligned with a given direction. Then, the minimum of these
four direction indexes is used for impulse detection for each and every masking window.
Depending on the minimum direction indexes among the three windows one window is selected.
The filtering is done on this selected window. Extensive simulations showed that the MDWCMM
filter provides good performances of suppressing impulse with low noise level as well as for highly corrupted images from both gray level and colored benchmarked images.
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2017-12, Keio University, Projection-based Regularized Dual Averaging for Sto...asahiushio1
This talk is about `Projection-based Regularized Dual Averaging (PDA)`, which is a stochastic optimizer we proposed in 2017 ICASSP paper [1]. This is also the presentation for my master thesis defense at Keio University.
[1] Asahi Ushio, Masahiro Yukawa “Projection-based Dual Averaging for Stochastic Sparse Optimization” Proceedings of ICASSP 2017
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.
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.
Bayesian modelling and computation for Raman spectroscopyMatt Moores
Raman spectroscopy can be used to identify molecules by the characteristic scattering of light from a laser. Each Raman-active dye label has a unique spectral signature, comprised by the locations and amplitudes of the peaks. The Raman spectrum is discretised into a multivariate observation that is highly collinear, hence it lends itself to a reduced-rank representation. We introduce a sequential Monte Carlo (SMC) algorithm to separate this signal into a series of peaks plus a smoothly-varying baseline, corrupted by additive white noise. By incorporating this representation into a Bayesian functional regression, we can quantify the relationship between dye concentration and peak intensity. We also estimate the model evidence using SMC to investigate long-range dependence between peaks. These methods have been implemented as an R package, using RcppEigen and OpenMP.
Image Acquisition and Representation
A Simple Image Formation Model
Image Sampling and Quantization
Image Interpolation
Image quantization
Nearest Neighbor Interpolation
A MODIFIED DIRECTIONAL WEIGHTED CASCADED-MASK MEDIAN FILTER FOR REMOVAL OF RA...cscpconf
In this paper a Modified Directional Weighted Cascaded-Mask Median (MDWCMM) filter has
been proposed, which is based on three different sized cascaded filtering windows. The
differences between the current pixel and its neighbors aligned with four main directions. A
direction index is used for each edge aligned with a given direction. Then, the minimum of these
four direction indexes is used for impulse detection for each and every masking window.
Depending on the minimum direction indexes among the three windows one window is selected.
The filtering is done on this selected window. Extensive simulations showed that the MDWCMM
filter provides good performances of suppressing impulse with low noise level as well as for highly corrupted images from both gray level and colored benchmarked images.
2017-03, ICASSP, Projection-based Dual Averaging for Stochastic Sparse Optimi...asahiushio1
We present a variant of the regularized dual averaging (RDA) algorithm for stochastic sparse optimization. Our approach differs from the previous studies of RDA in two respects. First, a sparsity-promoting metric is employed, originated from the proportionate-type adaptive filtering algorithms. Second, the squared-distance function to a closed convex set is employed as a part of the objective functions. In the particular application of online regression, the squared-distance function is reduced to a normalized version of the typical squared-error (least square) function. The two differences yield a better sparsity-seeking capability, leading to improved convergence properties. Numerical examples show the advantages of the proposed algorithm over the existing methods including ADAGRAD and adaptive proximal forward-backward splitting (APFBS).
2017-12, Keio University, Projection-based Regularized Dual Averaging for Sto...asahiushio1
This talk is about `Projection-based Regularized Dual Averaging (PDA)`, which is a stochastic optimizer we proposed in 2017 ICASSP paper [1]. This is also the presentation for my master thesis defense at Keio University.
[1] Asahi Ushio, Masahiro Yukawa “Projection-based Dual Averaging for Stochastic Sparse Optimization” Proceedings of ICASSP 2017
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.
OPTIMIZATION OF COST 231 MODEL FOR 3G WIRELESS COMMUNICATION SIGNAL IN SUBURB...Onyebuchi nosiri
ABSTRACT Radio propagation models are important tools adopted for the characterization and optimization of wireless communication signals in a propagation environment. In this paper, the Cost 231 model was optimized for 3G wireless communication signal in Aggrey Road (04˚ 45̀ .06˝N, 07˚ 02̀ .24˝E), a suburban area in Port Harcourt, Nigeria. Phone-Based Drive Test was adopted for field measurement using TEMS 11 network simulator. The optimization process was implemented through the use of Least Square Algorithm taking into account the initial offset parameters and the slope of the model curve in Cost 231 model for the process. The performance of the optimized model using the statistical tools in Minitab-14 software was evaluated for the Mean Absolute Deviation (MAD) and the Mean Squared Deviation (MSD) respectively. The results obtained demonstrated the following parametric values: 1.196, 2.01 and 1.179, 1.94 for Cost 231 and Optimized models respectively. The optimized model is recommended for deployment for a better and accurate path loss prediction on the environment of study.
OPTIMIZATION OF COST 231 MODEL FOR 3G WIRELESS COMMUNICATION SIGNAL IN SUBURB...Onyebuchi nosiri
Radio propagation models are important tools adopted for the characterization and optimization of wireless communication signals in a propagation environment. In this paper, the Cost 231 model was optimized for 3G wireless communication signal in Aggrey Road (04˚ 45̀ .06˝N, 07˚ 02̀ .24˝E), a suburban area in Port Harcourt, Nigeria. Phone-Based Drive Test was adopted for field measurement using TEMS 11 network simulator. The optimization process was implemented through the use of Least Square Algorithm taking into account the initial offset parameters and the slope of the model curve in Cost 231 model for the process. The performance of the optimized model using the statistical tools in Minitab-14 software was evaluated for the Mean Absolute Deviation (MAD) and the Mean Squared Deviation (MSD) respectively. The results obtained demonstrated the following parametric values: 1.196, 2.01 and 1.179, 1.94 for Cost 231 and Optimized models respectively. The optimized model is recommended for deployment for a better and accurate path loss prediction on the environment of study
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.
An enhanced fletcher-reeves-like conjugate gradient methods for image restora...IJECEIAES
Noise is an unavoidable aspect of modern camera technology, causing a decline in the overall visual quality of the images. Efforts are underway to diminish noise without compromising essential image features like edges, corners, and other intricate structures. Numerous techniques have already been suggested by many researchers for noise reduction, each with its unique set of benefits and drawbacks. Denoising images is a basic challenge in image processing. We describe a two-phase approach for removing impulse noise in this study. The adaptive median filter (AMF) for salt-and-pepper noise identifies noise candidates in the first phase. The second step minimizes an edge-preserving regularization function using a novel hybrid conjugate gradient approach. To generate the new improved search direction, the new algorithm takes advantage of two well-known successful conjugate gradient techniques. The descent property and global convergence are proven for the new methods. The obtained numerical results reveal that, when applied to image restoration, the new algorithms are superior to the classical fletcher reeves (FR) method in the same domain in terms of maintaining image quality and efficiency.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Performance of MMSE Denoise Signal Using LS-MMSE TechniqueIJMER
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The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
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International journal of engineering and mathematical modelling vol2 no1_2015_1IJEMM
Our efforts are mostly concentrated on improving the convergence rate of the numerical procedures both from the viewpoint of cost-efficiency and accuracy by handling the parametrization of the shape to be optimized. We employ nested parameterization supports of either shape, or shape deformation, and the classical process of degree elevation resulting in exact geometrical data transfer from coarse to fine representations. The algorithms mimick classical multigrid strategies and are found very effective in terms of convergence acceleration. In this paper, we analyse and demonstrate the efficiency of the two-level correction algorithm which is the basic block of a more general miltilevel strategy.
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When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
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In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
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Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
mathematical model for image restoration based on fractional order total variation
1. International Journal of Advanced Engineering, Management and Science (IJAEMS) [Vol-3, Issue-2, Feb- 2017]
https://dx.doi.org/10.24001/ijaems.3.2.19 ISSN: 2454-1311
www.ijaems.com Page | 111
Mathematical Model for Image Restoration Based
on Fractional Order Total Variation
Akhtar Rasool1
, Noor Badshah2
, Asmat Ullah3
2, 1
Basic Sciences Department, University of Engineering and Technology KP Peshawar (Pak).
3
Department of Engineering Mechanics, Hohai University, No.1 Xikang Road, Nanjing, Jiangsu 210098, PR China.
Abstract—This paper addresses mathematical model for
signal restoration based on fractional order total variation
(FOTV) for multiplicative noise. In alternating minimization
algorithm the Newton method is coupled with time-marching
scheme for the solutions of the corresponding PDEs related to
the minimization of the denoising model. Results obtained
from experiments show that our model can not only reduce the
staircase effect of the restored images but also better improve
the PSNR as compare to other existed methods.
Keywords— Total Variation (TV), Fractional Order
Derivative, Speckle Noise, Image Restoration.
I. INTRODUCTION
Image denoising has various applications in different fields
including pattern recognition, remote sensing, preprocessing
for computer vision and medical images. It is still a valid
challenge for researchers. Two important types of image
denoising are additive noise and multiplicative noise [1, 2]. In
additive noise removal model, the original image 𝑢 is
considered to be contaminated by Gaussian additive noise 𝑛.
Our goal is to restore 𝑢 from g = u + n. The total variation
(TV) has been developed by Rudin, Osher and Fatemi. TV
based method is edge preserving [1]. We discuss
multiplicative noise in this paper. Consider the noisy image
𝑔: Ω → R, the original image 𝑢 is contaminated by some
noise 𝜈 satisfying the Gamma law with mean 1. This type of
noise mostly exists in ultrasound imaging, sonar and laser
imaging and synthetic aperture radar images. Multiplicative
noise are more complex as compared to additive noise. Rudin
et al early approached a variational model [3]. Second
approach for multiplicative noise has been proposed by
Aubert and Aujol, see [4] for more details. Keeping in mind
the statistical properties of multiplicative noise 𝑛, we
approximate 𝑔 by treating the following constrained
minimization problem
min
𝑢
∫ |𝐷𝑢|Ω
𝑑𝑥𝑑𝑦 (1)
Subject to: ∫
𝑔
𝑢
𝑑𝑥𝑑𝑦 = 1,Ω
∫ (
𝑔
𝑢
− 1)
2
𝑑𝑥𝑑𝑦 = 𝜎2
.Ω
The mean of noise is 1 with standard deviation 𝜎2
as listed
above. TV regularization methods remove noise and
simultaneously preserve sharp intensity boundaries but
sometimes textures are smoothed with noise in noise removing
process [1-3]. The staircase effects can be reduced by
generalizing the regularization term which can be performed in
two ways. First approach is based on high order derivative; see
[5-7].
Secondly, the regularization term can be generalized by using
fractional-order derivatives. In recent past, some general
fractional-order additive noise models have been treated in
[8,9]. Chambolle used the fitting term ||𝑔 − 𝑢||2
2
in ROF
model and hence introduced an efficient projection method
for additive noise [10]. Obtained results indicate that the
performance of fractional-order TV are better than integer high
order TV. Fractional order models have competitive
performance with other integer order based models in noise
removing as well as staircase reduction. Moreover Aubert and
Aujol developed a multiplicative noise model “AA-model”,
which is given by
min
𝑢
∫ (𝑙𝑜𝑔𝑢 +
𝑔
𝑢
) 𝑑𝑥𝑑𝑦 + 𝜆 ∫ |𝐷𝑢|𝑑𝑥𝑑𝑦ΩΩ
(2)
Recently Zhang Jun and Wei Zhihui introduced fractional
order AA Model by replacing TV term in AA model with
fractional order TV.
As the function (logu +
g
u
) is not convex everywhere,
therefore Haung et al [11] introduced the auxiliary variable
w = logu and introduced a strictly convex function
𝑚𝑖𝑛 𝑤,𝑢 ∫ (𝑤 + 𝑔𝑒−𝑤)Ω
𝑑𝑥𝑑𝑦 + 𝛼1|𝑤 − 𝑢| 𝐿2
2
+
𝛼2 ∫ |𝐷𝑢|𝑑𝑥𝑑𝑦Ω
, (3)
with two regularization parameters 𝛼1 and 𝛼2. As 𝑢 = 𝑒−𝑤
is
strictly convex for all value of 𝑤 which ensures that the
solution of the variational problem is unique. TV based
models have better performance in preserve sharp edges,
reducing the noise but often these models cause the staircase
effect. We develop fractional-order TV based method for
multiplicative noise in Sect. 2. In Sect. 3, we use alternating
minimization algorithm to find minimizer of our proposed
problem .In Sect. 4, we give some numerical results to validate
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the performance of the proposed methodology. Finally, the
paper is concluded in the last Section.
II. THE PROPOSED APPROACH
We introduce a convex function for restoring images
contaminated by multiplicative noise in this section. Our new
model is based on fractional Order AA model and HNW
algorithm. Replacing TV term in model (3) with Fractional
order TV, we introduce the following model.
𝑚𝑖𝑛 𝑤,𝑢 ∫ (𝑤 + 𝑔𝑒−𝑤
)𝑑𝑥𝑑𝑦
Ω
+𝛼1||𝑤 − 𝑢|| 𝐿2
2
+ (𝛼2 + 𝛼3 𝑢2
)|𝐷 𝛼
𝑢|, (4)
with three positive regularization parameters 𝛼1, 𝛼2 and 𝛼3.
The fractional-order total variation norm provides a better
performance in preserving image edges. It also reduces
staircase effect. Moreover 𝑢 and 𝑤 preserve edges at the same
point; see [9]. The use of fractional-order TV to 𝑤 is quite
better for preserving sharp edges and reducing staircase effect.
The main advantage is that u = ew
is positive for all 𝑤, even
though 𝑢 in the objective function (2) required to be positive.
In our proposed model if we assign 𝑤 a large negative number,
𝑢 is still a small non zero positive number.
III. NUMERICAL SCHEME
We use an alternating minimization algorithm to solve the
FOTV problem. We split eqn. (4) in the following two
subproblems.
(i) First fix 𝑢 and find the solution of
min
w
∫ (w + ge−w
)dxdyΩ
+ α1||𝑤 − 𝑢||L2
2
. (5)
(ii) Then fix 𝑤, find the solution of
min
u
α1||w − u||L2
2
+(α2 + α3u2) ∫ |Dα
u|dxdy.Ω
(6)
Applying discrete setting, the first problem gets the form:
min
w
∑ (𝑤𝑖 + 𝑔𝑒−𝑤 𝑖) + 𝛼1||𝑤 − 𝑢||2
2
.𝑛2
1 (7)
The minimizer of the above problem can be obtained by
solving the following nonlinear system
1 − 𝑔𝑒−𝑤 𝑖 + 2𝛼1(𝑤𝑖 − 𝑢𝑖) = 0, 𝑖 = 1,2, . . . 𝑛2
. (8)
The 2nd derivative w.r.t 𝑤 of the first term in (5) is 𝑔𝑒−𝑤
,
which is positive for positive value of 𝑔. Thus the strictly
convex function (5) has a unique minimum which we compute
with Newton method as
wi+1 = wi −
1 − ge−wi + 2α1(wi − u)
ge−wi + 2α1
, i = 0,1,2, . ..
The Euler-Lagrange equation of FOTV model (6) is
𝜕𝑢
𝜕𝑡
=
2𝛼1
𝛼2+𝛼3 𝑢2 (𝑤𝑖,𝑗
𝑛
− 𝑢𝑖,𝑗) + (−1) 𝛼
𝑑𝑖𝑣 𝛼 ∇ 𝛼 𝑢 𝑖,𝑗
|∇ 𝛼 𝑢 𝑖,𝑗|
(9)
𝜕𝑢
𝜕𝑛
= 0 𝑜𝑛 𝜕Ω, u(0) = u0.
The numerical scheme for the steady-state equation (9) is
given by
𝑢𝑖,𝑗
𝑛+1
= 𝑢𝑖,𝑗
𝑛
+ 𝜂𝜏|𝑤𝑖,𝑗 − 𝑢𝑖,𝑗| + (−1) 𝛼
𝑑𝑖𝑣 𝛼
∇ 𝛼
𝑢𝑖,𝑗
|∇ 𝛼 𝑢𝑖,𝑗|
,
𝜂 =
2𝛼1
𝛼2+𝛼3 𝑢2 with B. Conditions
𝑢0,𝑗
𝑛
= 𝑢1,𝑗
𝑛
, 𝑢 𝑁,𝑗
𝑛
= 𝑢 𝑁−1,𝑗
𝑛
,
𝑢𝑖,0
𝑛
= 𝑢𝑖,1
𝑛
, 𝑢𝑖,𝑁
𝑛
= 𝑢𝑖,𝑁−1
𝑛
.
Algorithm
We take initial guess 𝑢(0)
to be the noisy image.
(i) First fix𝑢(𝑘−1)
, compute by Newton's algorithm 𝑤(𝑘)
=
arg min
w
∑ (𝑤𝑖 + 𝑔𝑛2
𝑘=0 𝑒−𝑤 𝑖) + 𝛼1||𝑤 − 𝑢 𝑘−1||2
2
(ii) Then fix 𝑤(𝑘)
, compute 𝑢(𝑘)
by using
𝜕𝑢
𝜕𝑡
=
2𝛼1
𝛼2 + 𝛼3 𝑢2
||𝑤(𝑘)
− 𝑢|| + (−1) 𝛼
𝑑𝑖𝑣 𝛼
(
∇ 𝛼
𝑢
|∇ 𝛼 𝑢|
)
(iii) Exit with an approximate denoised image 𝑢 𝑘
= 𝑒 𝑤(𝑘)
if
the stopping criteria
||𝑢 𝑘+1−𝑢 𝑘||
||𝑢 𝑘+1||
2
< 10−4
is satisfied, otherwise
we continue with 𝑘 = 𝑘 + 1 for next iteration.
IV. NUMERICAL RESULTS
This section describes some experimental results of image
denoising to claim the superiority of our method. We consider
some images contaminated by multiplicative noise. Figure 1
shows clean images. Results obtained from experiments
indicate the superiority of our proposed method. We use
PSNR to compare the capabilities of the restored images as
𝑃𝑆𝑁𝑅 = 20 log10
255
||𝑢 − 𝑢 𝑡𝑟𝑢𝑒||
Where 𝑢 𝑡𝑟𝑢𝑒 and 𝑢 denote the noise-free signal and recovered
signal respectively.
5. International Journal of Advanced Engineering, Management and Science (IJAEMS) [Vol-3, Issue-2, Feb- 2017]
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Test 3.
(a)Barbara, 0.03 (b) AA Method (c) Proposed Method
Fig.10: (a) Noisy Signal, 𝜎2
= 0.03 (b) Restored Signal, AA-Method, 𝜆 = 35 (c) Restored Signal, Proposed Method with 𝛼1 =
450, 𝛼2 = 0.0001, 𝛼3 = 650.
(a) Barbara, 0.1 (b) AA Method (c) Proposed Method
Fig.11: (a) Noisy Signal, 𝜎2
= 0.1 (b) Restored Signal, AA Method, 𝜆 = 33 (c) Restored Signal, Proposed Method with 𝛼1 =
450, 𝛼2 = 0.00001, 𝛼3 = 650.
Table 1: Restoration results for face images
Image
Variance
"𝜎2
"
Fractional Order
𝛼
AA-Model
PSNR
Proposed Model
PSNR
Face
Size(256 × 256)
0.3 1.9 26.035 27.193
0.1 1.1 28.525 29.009
0.03 1.9 31.934 33.340
0.01 1.9 35.037 36.420
Table 2: Restoration results for Peppers images
Image
Variance
"𝜎2
"
Fractional Order
𝛼
AA-Model
PSNR
Proposed Model
PSNR
Peppers
Size(256 × 256)
0.1 1.1 26.152 26.301
0.03 1.1 29.228 29.355
0.01 1.1 32.142 32.201
Table 3: Restoration Results for Barbara Images.
Image
Variance
"𝜎2
"
Fractional Order
𝛼
AA-Model
PSNR
Proposed Model
PSNR
Barbara
Size(256 × 256)
0.03 1.1 28.162 28.211
0.1 1.1 25.821 25.910
In first experiment we test two dimensional signal “face”
256 × 256. We have compared the denoising capabilities of
the proposed algorithm with fractional order AA method.
Numerous values of 𝜆 have been tested for fractional order AA
algorithm and the best amongst them are mentioned in this
paper. As the regularization parameters perform well in
denoising so we have chosen the three regularization
parameters for increasing the PSNR. In Table 1 we have
mentioned the highest PSNR of fractional order AA method
and the proposed method. Peppers image of size 256 × 256 is
6. International Journal of Advanced Engineering, Management and Science (IJAEMS) [Vol-3, Issue-2, Feb- 2017]
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used to make the second experiment. Best results are chosen
among the obtained results of the performed experiements
through fractional order AA method. As the quality of the
restored image depends upon the regularization parameters so
we have chosen the suitable values for regularization
parameters. In table 2 we have shown the comparison of our
results with those obtained from fractional order AA
Algorithm. The PSNR values of the recovered peppers images
are listed in table 2. The values mentioned in table 2 clearly
shows that the PSNR values of the proposed method are
bigger than fractional order AA-Method. In third test we
consider the barbara image. The PSNR values shown in table
3 showing the better performance of the proposed algorithm.
V. CONCLUSION
This paper describes a new mathematical approach for image
denoising with fractional-order TV. Coupling the Newton
algorithmwith time- marching scheme for the solutions of
PDEs related to the minmization of denoising model based on
fractional order total variation, gives good results.The
proposed method can suppress noise very well while it can
preserve details of the recovered image. Results obtained from
experiements show that the efficiency of the proposed
approach are better than what obtained from other existed total
variation restoration methods.
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