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An efficient image denoising method for various images under various resolutions using non-related dictionary learning approach.

INTRODUCTION

Background Study:

One of the most important processes in image processing is imaging Denoising. Denoising is an effective process where it leads to increase the quality of an image. The image quality determines image processing results like image segmentation, feature extraction. Denoising is one of the main processes in image processing as well as a component in other processes. Using various ways an image can be denoised. Performance of a Denoising method is measured in terms of noise removed and edge preservation. Satellite images, underwater images, remote sensor images, medical images and other images are essentially applied for image denoising in order to improve the image processing results. Image denoising plays an indispensable role in medical image. Most of the Medical images have low distinction objects, tarnished by unsystematic noise in the input process. During communication and retrieval of an image there is more possibility of corruption. Due to the rapid growth of media transmission and the very high demand of image transmission with highest quality it is essential to improve the quality of images using various effective techniques. In the development of image achievement and communication, noise always exists unavoidably. Image processing is a significant technology in image denoising as a process and component in various other processes. So it is essential to have image denoising process to progress the excellence of image.

The unwanted signal in the noise is termed as Image noise. It is a random variation of color information and brightness in images, and is usually an aspect of electronic noise. Noise can be occurred in an image by adding spurious and extraneous noise. It is an undesirable by-product of image capture that adds spurious and extraneous information. Image denoising is the main important task and it is very much essential for medical image processing, satellite image processing, industrial vision system and etc. Availability of the noise in the image is in the form of a pattern integrated with the probabilistic characteristics. Denoising is basic task required by medical analysis. Noise elimination causes blurring of images, the excellence of image is also lowered. Nonlinear models can hold edges in a better mode than linear models. Best image denoising method removes the noise without affecting the edge information of the image. Basically conventional linear models are used for denoising, whereas one of the conventional models is using Filters. Filters are assigned by a threshold value for removing the noise by comparing the noise level occurred in the image. Also filter used various parameters under various conditions for removing the noise.

An extensive number of direct and nonlinear sifting calculations [1] have been produced to diminish commotion from spoiled pictures to enhance picture quality. Pictures might be discolored by clamor. Clamor is there as an aftereffect of the electronic hardware of cameras or in the picture transmission period. The most widely recognized kind of clamor is the white or Gaussian commotion where its energy is consistently scattered over the ghostly and spatial spaces and its mean is zero. There is a broad assortment of commotion sorts yet we concentrate on the most vital sorts, they are; Gaussian clamor, dot commotion, poison commotion, motivation commotion, salt and pepper clamor. Diverse systems are utilized to denoise the picture. They are spatial channels which comprise of Gaussian channel, Median filer, Adaptive channel. Be that as it may, the uproarious picture rebuilding strategies are for the most part as the accompanying three:

> Mean Filter Principle

> The Median Filter

> Wavelet Transform

The picture data passed on as picture is actually spoiled by Gaussian commotion which is a standard issue in picture handling. This added substance irregular commotion can be evacuated utilizing wavelet denoising system because of the ability to catch the vitality of a sign in few vitality evolving values. Here in this examination, an assessment has been made on appropriateness strategies for picture denoising to expel commotion utilizing unique procedures. The execution of the picture denoising is appeared as far as PSNR and visual execution. The outcome indicated curvelets change gave preferable PSNR and visual execution over wavelet change and different techniques. There are numerous ways is accessible to denoise a picture. They are: Simple straight smoothing channel, for example, Gauss channel [2-4], will bring about the subtle element data loss of picture. As of late, countless denoising calculations mean of nonlinear channel has showed up. Basic calculations incorporate an assortment of versatile middle channel calculations: the wavelet edge [5-9] (additionally called wavelet shrinkage) calculation, the anisotropic dispersion condition calculation [10-13], the aggregate variety minimization calculation [14-17], non-nearby mean channel calculation [18-21], and so on. For some purposes this kind of denoising is acceptable. One big advantage of linear noise removal models is the speed. But drawback of the linear models is that they are not able to preserve edges in a well manner. Edges, which are recognized as discontinuities in the image, are tarnished out. Nonlinear models on the other hand can handle edges in a better way than linear models. Nonlinear filter is a signal-processing device where its output is not a linear function of its input. A common method addressed as classical image denoising problem in most of the research work is described below.

An input image I is measured including an additive zero-mean white and homogeneous Gaussian noise (n) including standard deviation [sigma]. The measured in J can be represented as:

J = I + n (1)

Filters used for spatial representation are direct and high speed processing tools of images. Spatial filters are capable filters can able to obtain the spectral structure of the images to do image processing efficiently. For example a 2D Gaussian filter is represented in the form of mathematical representation is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Here it is motivated to desire an approach for removing various kinds of noise exists in J. The result should be very close to the original image I. But the median filter based noise reduction is calculated for fetching the mean values of the current pixel values (x, y) as the gray pixelsg(x, y) on the original image I(x, y). It means:

g(x, y) = 1/M [[summation].sub.f [member of] s]f(x, y) (3)

Where, s is the mean template, and M is the total number of pixels comprises of current pixels and the template. Another spatial filter which can do edge-preserving based image denoising is adaptive filter, where it can be represented in mathematical form is:

[??](x, y) = g(x, y) - [[sigma].sup.2.sub.n]/[[sigma].sup.2.sub.L] [g(x, y) - [[mu].sub.L]] (4)

Sparse Representation:

One of the modeling tools which is a powerful and promising method applied for signal processing and image processing is sparse representation [22, 23]. Initially the signal or image is converted into sparse form over a redundant dictionary. At the same time it is also described as an optimized problem as:

[min.sub.[alpha]] [[parallel][alpha][parallel].sub.0] subject to [[parallel][PSI][alpha] - x[parallel].sup.2.sub.2] [less than or equal to] [epsilon] (5)

Where, x [member of] [[Real part].sup.n] denotes the signal, [PSI] [member of] [[Real part].sup.n x k] (k > n) denotes the completed dictionary, [epsilon] denotes the error boundary value and [alpha] denotes the co-efficient. [[parallel][alpha][parallel].sub.0] is the non-zero entries in the coefficients. This sparse representation based image denoising is mainly used for medical images like MRI and CT-scan during image segmentation, image reconstruction and classification of image disease [24, 25].

Sparse Representation based Image denoising for Diffusion Weighted Images:

DWI images are always taken as sequence of images acquired for quantify the water diffusion information in each voxel. Denoising on DWI applied on each 2D image in the sequence. The denoising model can be written in the form of formulae is:

[??] = [min.sub.[alpha]] [[parallel][alpha][parallel].sub.0] subject to [[parallel]y - [PSI][alpha][parallel].sup.2.sub.2] [less than or equal to] C[n.sup.2][[sigma].sup.2] (6)

Where, y denotes the noisy image, C is a constant value and the std of Rician noise is represented as [sigma].

arg [min.sub.[alpha]]([[parallel]y - [PSI][alpha][parallel].sup.2.sub.2] + [mu][[parallel][alpha][parallel].sub.0]) (7)

Where, [mu] is the penalty factor denotes the image value loss. And the approximation of the noise-removed image can be computed by solving [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

NRDL Approach:

The input image is read from various resources and experimented. The entire functionality of the NRDL approach followed in this paper is shown in Figure-1. Initially the input mage read and divided into blocks. According to the size of the image the number blocks is obtained. All the blocks are under same size.

[FIGURE 1 OMITTED]

In the experiment blkproc is a predefined function used to determine the blocks and it will be specified as overlapping each other. It can be visualized by an additional rows and columns given in the outside of the figure. These blocks values are taken into account during the process of blocks. If there is an overlapped block then the blkproc transfers the extended block to the specific function. Figure-2 shows the overlapped areas for few of the blocks in a 15 x 30 matrix where 1 x 2 overlaps. The entire 4 x 8 block has one row overlapped in above and below. Similarly two columns overlapped on both sides. The outer shadow box denotes the overlapping.

[FIGURE 2 OMITTED]

Dividing into blocks can help to apply the image processing procedure efficiently and speedily. More number of blocks may have the same content, so called redundancy. Instead applying the same process (e.g. noise removal) in all the redundant blocks, it is easy and fast to take one common block and process it. Then it will be used to replace all the other blocks at the same time in the entire image. This functionality reduces the time complexity, comparison complexity and computational time.

1.1. Simple Dictionary learning Model:

Let us consider a group of samples Y = ([y.sub.1], [y.sub.2], ..., [y.sub.K]) [member of] [R.sup.n x K] for learning purpose which can provide a redundant dictionary D = ([d.sub.1], [d.sub.2], ..., [d.sub.m]) [member of] [R.sup.n x m] from the learning. Since each sample [y.sub.k] (k =1, 2, ... K) can be expressed as a sparse vector as [x.sub.k](k=1, 2, ..., K) via sparse representation.

From the above discussion the dictionary learning problem can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

From the above formula, X = ([x.sub.1], [x.sub.2], ..., [x.sub.K]) [member of] [R.sup.m x K] is coefficient matrix and a [T.sub.0] denotes the sparsity.

1.2. Non Related Dictionary Learning Algorithm:

A small portion of relevancy in the redundant dictionary is the most important index for measuring the expression ability of the redundant dictionary. The ability of the weaker small portion of relevancy is indicating stronger dictionary expression. The small portion of relevancy of the dictionary is written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

From the above definition, R([d.sub.i], [d.sub.j]) [member of] [0, 1];

At the time of [d.sub.i] and [d.sub.j] are orthorhombic to each other, R([d.sub.i], [d.sub.j]) = 0;

When [d.sub.i] = [rho][d.sub.j] is a non-zero constant), R([d.sub.i], [d.sub.j]) = 1; From the equation (uy) the relevant dictionary D can be obtained as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The standard vector [parallel][d.sub.i][parallel]=1 represents the small portion, the relevancy of the dictionary D can be simplified as:

R(D) = [summation over (i,j)] [R.sup.2] ([d.sub.i], [d.sub.j]) = [[parallel][D.sup.T]D[parallel].sup.2.sub.F] (11) In the real dictionary learning model (8), due to more relevancy constraints increasing, the number of irrelevant is also increased. On the basis of increasing the relevancy constraint of the dictionary D, non-related dictionary learning model can be expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

From the above formula, [lambda] is a positive constant, [[parallel][D.sup.T]D[parallel].sup.2.sub.F] is the second item of the objective function where it can make D as a relatively strong irrelevance.

1.3. Selecting the model:

There are two steps are followed to fetch the solution of the model are:

> Setting up the redundant dictionary D then solve sparse co-efficient matrix X.

> Setting up the sparse coefficient matrix X and update the dictionary D.

If D is fixed then X is written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

For series of sparse coding problems, the model can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

If X is fixed then D can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

After standardizing the problem, the dictionary problems should satisfy the condition H H 2 ^and the above model can be rewritten as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

The above models are iteratively used for a number of convergence times in order to fetch the numerical solution ^which will be updated in the redundant dictionary D.

4.4. Algorithm NonRelatedDictionaryLearning ():
{

Input: Sample Y = ([y.sub.1], [y.sub.2], ..., [y.sub.K])
[member of] [R.sup.n x K]

Initialization: random dictionary D, iteration time L; integer l=1;

1. create spare representation coefficient matrix X

2. for 1 = 1 to L

3.    solve D to fetch [??], update D = [??]

4.    small portion weaker data [d.sup.l.sub.i] =
      [d.sup.l.sub.i]/[[parallel][d.sup.l.sub.i][parallel].sub.2];

5. end 1

6. output = [D.sup.L]


4.5. Image Denoising Using Non-Related Dictionary Learning Approach

The image denoising with additive noise is represented as

Y = X + n (18)

Where Y is the noisy image of X (original input Image) and n is the noise. According to our NRDL approach the noisy image Y is divided into number of blocks K, whereas each block is overlapped. The size of the block is [square root of n] x [square root of n] where block is written in the form of a column vector as [y.sub.k] [member of] [R.sup.n]. Then from K number of image blocks, L number of blocks are chosen randomly and created a set {[p.sub.1]}(1 = 1, 2, 3, ..., L); [P.sub.1] is trained to fetch the non-related redundant dictionary D [member of] [R.sup.n x m]. Where this problem can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

From the above expression, P = {[p.sub.1], [p.sub.2], ..., [p.sub.L]}, Q = {[q.sub.1], [q.sub.2], ..., [q.sub.L]}. Finally the NRDL algorithm fetches the dictionary D.

4.6.Image Denoising:

It is assumed that, D is obtained by learning from the original image I. Also the sparse representation coefficient of the entire image block [y.sub.k] [member of] [R.sup.n] in dictionary D and the sparse representation coefficient [[alpha].sub.k] can be expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

From the above formula D = [[d.sub.1], [d.sub.2], ..., [d.sub.m]] and [d.sub.j] is the jth column of the vector D. various algorithms are adopted to obtain the sparse representation coefficient [[alpha].sub.k] corresponding to the denoised image block [x.sub.k] = D[[alpha].sub.k] is obtained. The image block is joined according to position and the overlapped parts of the image blocks which are averaged to obtain the denoised integral image can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Where [R.sub.k] is used to extract the matrix of the kth image block. For example, [R.sub.h], Y denotes the kth image block of image Y.

Experiment and Results:

Our NRDL approach is programmed in MATLAB software, running in Intel- Pentium, [Core-i.sup.5], 2.64 GHz processor with 8 GB RAM, and 1TB secondary memory. The execution time for 512 x 512 images is about 90 s. An efficient implementation based experiment on a computer language is expected to improve the execution time and reduce the complexity. The main objective of this paper is to improve the quality of the image in terms of noise reduction for various kinds of images like satellite images, medical images, underwater images and natural images. To do that from various repositories like UCI, NCBI-databases, Biomedical Informatics Research Network Data Repository [26] and other web-resources different kinds of images are taken for experiment. The experiment is carried out in MATLAB software. The resolution of the images is different. White, Gaussian and Rician noise is added to the images for in order to produce cleaned images. The images are patched in a range from 8 x 8 to 2 x 2. The following figure shows the image divided into block segregation and sparse representation. Here natural and artificial noises are applied with the NRDL techniques and the results are compared to evaluate the performance.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

From the collected input images all the images are experimented and results are produced. The obtained results for some images under each category like medical, still-images, agriculture are given in Figure-1. The first column shows the input images and the second column shows the noise removed in first level and the third column shows the output image which is the cleaned image (Noise removed). The final column shows the name of the image, type of the image, and the obtained value of the parameters defining the quality of the image like MSE and PSNR are given.

[FIGURE 5 OMITTED]

The input images are experimented in an order where the images are applied with normal median filter for removing the overall noise on the texture. Then the images are divided into blocks. Finally the image blocks are applied non-related redundant learning method and the noise are removed. The NRDL approach is applied for image denoising with a scaling factor [sigma] where it is helps to assign noise level manually. In this experiment the image representation includes with scaling factor [sigma] = 3. Since most of the existing approaches like NEDI, DFDF, SAI and SME are designed the image interpolation with [sigma] = [2.sup.n] where n is an integer. The following table shows the comparison results of the proposed approach with some of the existing approach results.

The obtained MSE and PSNR results from the NRDL approach are compared with the existing results using NARM and ScSR method [27].

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

The above Table-1, Fig.6 and Fig.7 shows the comparison of PSNR and MSE values obtained from NRDL and the existing ScSR and NARM methods respectively. From the figures and table, it is clear and identified that the proposed approach is better than the existing approach due to the obtained PSNR and MSE values. The quality of the output images obtained from NRDL is better than the ScSR and NARM method results.

Conclusion:

The main objective of this paper is to improve the quality of the image by denoising method. Non-related redundant learning based noise removal method is applied for denoising. The denoising process leads to improve the quality of the images where it increases the accuracy in image processing results like segmentation, extraction and feature extraction. This paper concentrates on learning the image by verifying the blocks and eliminates the redundant blocks in order to reduce the computational complexity and time. The quality of the image is measured by PSNR and MSE values computed on the images. From the experiment the PSNR and MSE results are measured and it proves that the proposed approach is efficient than the existing approaches. Hence NRDL approach is proved as an efficient approach for denoising.

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(1) Subha. S and (2) Thanushkodi. K

(1) Assistant Professor, KLN College of Engineering, Sivagangai District, Pottapalayam, Tamil Nadu, India--630612.

(2) Director, Akshaya College of Engineering, Kinathukadavu, Coimbatore, Tamil Nadu, India- 642109.

Address For Correspondence:

Subha. S, Assistant Professor, KLN College of Engineering, Sivagangai District, Pottapalayam, Tamil Nadu, India-- 630612.

E-mail: subavarshini1308@gmail.com Phone:+91-9578848832.

Received 22 June 2016; Accepted 28 August 2016; Available online 31 August 2016
Table 1: Performance Evaluation of NRDL, Comparing PSNR, MSE with
ScSR and NARM

Images      Lena     Retinal   Mammogram   Face     Apple

NRDL        32.14    27.13     26.23       32.43    35.11
            0.8967   0.8423    0.9123      0.7987   0.9453
ScSR [27]   30       26.08     23.84       31.1     32.29
            0.8472   0.8138    0.8461      0.7653   0.9073
NARM        31.16    26.86     25.57       31.9     34.8
            0.8693   0.8293    0.8993      0.7847   0.9284
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Publication:Advances in Environmental Biology
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Date:Aug 1, 2016
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