A fast image denoising algorithm based on TLS model and sparse representation Example for Paper MS Word Template for the Journal Top Margin Left1

A fast image denoising algorithm based on TLS model and sparse representation Example for Paper MS Word Template for the Journal Top Margin Left1.5, Font size 16 pt., Font Face Arial Bold, Align Center, Line Spacesingle Eman K. Elsayed A. Y. J. Akossou1, and Dina R Salem R. Palm2 and Mohammed Aly 3 1Mathematical and Computer science Dept., Faculty of Science, Alazhar University (girls branch), Cairo, Egypt 1Facult dAgronomie, Universit de Parakou, BP 123, Parakou (Bnin) EmailEmail HYPERLINK [email protected] [email protected]@example.com 2Business Technology Dept., School of the Business, Canadian International College, CIC, Cairo, Egypt 2Facult Universitaire des Sciences Agronomiques de Gembloux, Avenue de la Facult dAgronomie, 8, B-5030 Gembloux (Belgique) EmailEmail HYPERLINK file///[email protected] [email protected] 3Mathematical and Computer science Dept., Faculty of Science, Zagazig University, Zagazig, Egypt Email HYPERLINK [email protected] [email protected] [email protected] ABSTRACT The sparse representation algorithms (SRs) plays important role to remove the noise from noisy images. The Quantum Particle Swarm Optimization based on the matching pursuit algorithm (QPSO-MP) denoising algorithm that takes advantage of the SRs and the meta-heuristic algorithms has shown hopeful results with respect to noise reduction. Despite effective combination between the meta-heuristic and SRs, the learning dictionary and the fixed populations make these algorithms computational exacting, which largely locates its applicability in many applications. To solve this problem, the fast version of the QPSO-MP denoising algorithm, called FQPSO-MP denoising algorithm, based on a prelearned dictionary and Dynamic-Multi-Swarm (DMS), were proposed in this paper. The pre-learned dictionary produces the best average performance of the denoised images. So, the learned dictionary is no longer required to provide the best average performance of the denoising images. While the denoising approach is applied, the learned dictionary is obtained. So, the execution time of the pre-learned dictionary is shorter than the learned dictionary. The proposed algorithm is compared to the original QPSO-MP and the state of the art algorithms after modifications. These comparisons proof that the PSNR results and the execution time of the proposed algorithm are better than other algorithms. Research papers should be accompanied by an abstract, which will appear in front of the main body of the text. It should be written in complete sentences and should summarize the aims, methods, results and conclusions in less than 250 words. The abstract should be comprehensible to readers before they read the paper and abbreviations, citations and mathematical equations/notations should be avoided. Margin Left1.5 inch, Right1.5 inch, Font size9 pt. Font Face Arial italic Align justify Line Spacesingle Keywords matching pursuit, denoising image, sparse representation, Dynamic-Multi-Swarm, Quantum Particle Swarm Optimization. Regression, data structure, prediction, simulation. Mathematics Subject Classification 94A08, 68U10.62J12, 62G99 Computing Classification System Computing methodologies Image processing.I.4 1. INTRODUCTION Paragraph Margin Left1 inch, Right1 inch, Font size10 pt. Font Face Arial, Align justify, Line Space 1.3 pt. The contamination of image with noise is more introduced during image acquisition, transmission, storage, or compression. Images can be infected by different types of noise such as Gaussian, impulsive, speckle and combined noise. Image processing has many preprocessing steps. That is why image denoising is an important preprocessing step of image processing 1-10, 31. The goal of image denoising is removing the noise from the corrupted images to estimate its original Images while maintaining as much as possible the relevant features and fine details of the images (edges and textures). In the last few years, different image denoising algorithms have been improved to enhance the quality of the noisy images that corrupted with various kind of noise. In 9, 10 shows a more overall overview of the denoising algorithms. Although there are various types of images denoising algorithms, their performance in terms of removing noise and execution time are remain not ideal. There are three types of the image denoising algorithms, they are based on the transform domain filtering, the spatial filtering or the learning. The example of the image denoising algorithms based on the transform domain filtering is a wavelet, Fourier transforms, Block-Matching and 3D filtering (BM3D) algorithms, etc. 11, 12. The bilateral, Gaussian, guide, and the nonlocal mean filters are examples of the second type of the image denoising algorithms that are based on the spatial filtering 13, 14. The examples of the third type, based on learning, are the K-Singular Value Decomposition (K-SVD) 15, 16, Large-Scale Sparse Clustering (LSSC) 17, 18 and the Clustering based Sparse Representation (CSR) algorithms 19. These algorithms are developed to enhance the quality of corrupted images. The key of the SR-based denoising algorithm is to learn the suitable dictionary, which can minutely fit the local structure of the images. When the dictionary is related to the performance of the sparse coding, and if it matches the image features, then the SR-based denoising algorithm is successful 20. The dictionary is divided into two types. The first type is the fixed dictionary such as wavelet, Gabor 23, Gabor wavelet 21, 22, log Gabor 24, log Gabor wavelet 25 and discrete cosine transform (DCT) 16. The second type is the learning dictionary such as K-mean Singular Value Decomposition (K-SVD) 15, 16, Large Scale Sparse Clustering (LSSC) 17, 18, principal component analysis (PCA) 26 and locally learned dictionary (KLLD) 27. The atoms of the learning dictionary algorithms are developed iteratively. Utilization of the learning dictionary in the denoising algorithm is still a challenging problem for the researchers. In 16 the K-SVD algorithm is an iterative procedure and takes a long time SVD. The CPU need to almost some minutes or more for the dictionary learning for a normal problem. So, the application of K-SVD algorithm is encumbered in large-scale images. Utilization of the learning dictionary in the denoising algorithm is a remained problem. The researchers face a challenging problem to find an efficient image denoising method. SR algorithm is the most common type of algorithm used for denoising images affected by the Gaussian noise. In SR, the atoms of an over-complete dictionary have some characteristics, such as bandpass, oriented and localized. The method that is used to generate the atoms of an over-complete dictionary has to check the previous characteristics. In denoising algorithms, setting the filtering and the threshold parameters is a key problem, taking into account the different types of images and under different conditions, to achieve better performance and minimum execution time. In 28, the Particle Swarm Optimization (PSO) is used to reach the optimal value of the regularization parameter of the total variation method, to remove the noise in the magnetic resonance imaging (MRI) images. In 41 the over-complete dictionary was trained by using the K-SVD algorithm, followed by the Batch-Orthogonal Matching Pursuit (Batch-OMP) instead of the Orthogonal Matching Pursuit (OMP) algorithm to enhance the denoising algorithm. In 26 the Non-locally Centralized Sparse Representation (NCSR) algorithm combines the ideas of the SR (learning dictionary) and the non-local self-similarity of image patches within a given image. In NCSR algorithm, the image is denoised by using the PCA and the K-means or even the image patches extracted from the corrupted image itself. The patches are clustered into K clusters, and a PCA sub-dictionary is learned for each cluster. Then, only one of the compact PCA sub-dictionary is selected to be coded, which leads to better image denoising results. In general, the noise reduction and computational efficiency are applied to evaluate the denoising performance of the images. In fact, they are clashing precondition for an overall analysis of the denoising algorithm. So, the denoising application needs to a trade-off between the computational cost and noise reduction. The goal of this paper is reducing the computational complexity of the learning dictionary and enhancing the denoising performance of the test images. This paper proposes a new fast DMS-QPSO algorithm based on the Matching Pursuit (MP) and the pre-learned dictionary generated by the Translation, Libration, and Screw (TLS) model. There are two improvements over the QPSO algorithm that achieve a decrease in the computational complexity and a decreasing in the execution time. The rest of this paper is organized as follows, in section 2, the MP algorithm is discussed. In section 3, the original algorithm QPSO is discussed. The TLS model is discussed in section 4. The proposed method with two modifications over the original QPSO algorithm is discussed in section 5. The experiment results and efficiency of the proposed algorithm and other denoising algorithms are tested and evaluated in section 6. The paper is concluded in section 7. In the establishment of the prediction model, three stages are fundamental possible selection of the variables, the estimation of the coefficients of the variables selected and the validation of the model. Ideally, this validation should be done on different observations. But in most practical situations, the selection of the variables, the estimation of the coefficients and the validation are done using the same sample. Indeed, it is often difficult to have separate samples for the various stages of modeling, because the dataset available to the researcher is frequently too small to use part of it to establish the regression model and the remaining for its validation. Sometimes, the number of predictors is higher than the number of observations. The objective of this work is to bring some useful information for the users, especially those who do not have the possibility to validate the models from external data. In a more concrete way, we propose to examine the predictive value of a regression model by calculating a coefficient, similar to the multiple coefficient of determination, which we call coefficient of determination of prediction. It is denoted EMBED Equation.DSMT4 and is defined, for EMBED Equation.DSMT4 new observations, as follows EMBED Equation.DSMT4. In this relation, EMBED Equation.DSMT4 indicates the actual value of the dependent variable for the new individual EMBED Equation.DSMT4 (EMBED Equation.DSMT4). EMBED Equation.DSMT4, is the predicted value for this individual given by the regression model, EMBED Equation.DSMT4 is the arithmetic mean of EMBED Equation.DSMT4 observations of the dependent variable in the sample which was used to establish the model. 2. MATCHING PURSUITGENERATION OF THE DATA MP is a greedy algorithm that utilizes the signal decomposition based on a redundant dictionary with each element called an atom 23, 35. The best atom can be selected from the atoms of the dictionary when MP is applied, they can be available in each iteration. Let QUOTE is the atoms of the dictionary, for an arbitrary image of size QUOTE . QUOTE is the set of all indexes QUOTE and QUOTE 29. Let QUOTE is an arbitrary signal. The first step of an MP is always to approximate QUOTE by projecting it on a vector QUOTE QUOTE (1) Where QUOTE is the projection of QUOTE into the atom QUOTE , and QUOTE is the residual of the original signal, since the residual QUOTE is orthogonal to QUOTE 30 QUOTE (2) QUOTE (3) In term, QUOTE QUOTE must be minimized, so QUOTE must be chosen to maximize QUOTE QUOTE . After QUOTE iterations, the original signal can be reconstructed via the selected atom QUOTE QUOTE (4) The process of generating QUOTE is one of the main steps in the denoising processes. In equation (5) the Gaussian function is used. The essential function is a Gaussian in one axis and the second derivative of the Gaussian in the various another axis. The TLS model is used a Gaussian function to form the pre- learned dictionary. The Gaussian function can be shown by the following formula. QUOTE (5) Equation (5) is used to generate the dictionary with the TLS rotation in x and y directions, i.e. QUOTE respectively. TLS model is used to find QUOTE . QUOTE (6) QUOTE (7) Where QUOTE are a translation in QUOTE directions, screw rotation angle, arbitrary columns of the population respectively, number of parameters of TLS model is 5, QUOTE . 3. MATCHING PURSUIT QUANTUM-BEHAVED PARTICLE SWARM OPTIMIZATION (QPSO) QPSO algorithm is motivated by the awareness of quantum mechanics with the original PSO algorithm 32. QPSO is the update version of PSO algorithm and even better, not only in its search ability but also in its accuracy. The particles of QPSO algorithm can appear on any search space and these particles depend on the delta potential. The QPSO algorithm can ride the fault of the standard PSO algorithm, but the global convergence with probability QUOTE cannot be guaranteed. The positions and velocities of the particles cannot be measured at the same time in the quantum space. The state of the particle is measured by the wave function QUOTE . The wave function QUOTE is the probability appearance of the particles in a certain position and can be measured using the formula QUOTE , then the probability distribution function can be obtained 33, 34. In QPSO algorithm, the position of the particles can be updated using the following equation (8) QUOTE (8) Where QUOTE is the local attractor, and QUOTE is the random number between 0 and 1. The local attractor QUOTE is defined as the following formula (9). QUOTE (9) Where QUOTE is the best position of the ith particle, QUOTE is the global position of all particles and QUOTE is the random number distributed between 0 and 1.The parameter QUOTE can be measured by equation (10). QUOTE QUOTE (10) Where QUOTE is defined as the average optimal position of all the particles. The QUOTE can be determined by equation (11). QUOTE (11) The parameter QUOTE is defined as the contraction-expansion coefficient, it can be determined by the following equation (12). QUOTE (12) Where QUOTE is the total number of iterations and QUOTE the current number of iteration. Table 1. Shows pseudo-code of the original QPSO algorithm. The original QPSO algorithmStep 1 initialize the particles QUOTE and let each particle equal to the position (i.e. QUOTE ) Step 2 update the best position of each particles QUOTE according to the fitness function and also update the global best QUOTE of each particle Step 3 use equation (11) to update the average optimal position of all the particles QUOTE Step 4 use equation (9) to update the local attractor of the particles QUOTE Step 5 use equation (8) to find the new position QUOTE Step 6 Repeat steps 2 to 5 until the algorithm satisfies the end condition. 4. TLS MODEL Cruickshank is the first one who used the Translation Libration – Screw rotation (TLS) model where it can be used to describe the movements of the atomic groups. Three matrices are defined in the TLS model, the T, L and S, that can define the combination of the atomic vibrations and librations. The TLS model can be utilized in the molecular systems involving correlated motions 36. The elements of the matrices should satisfy several conditions to be converted into essential molecular movement. When these conditions are not satisfied, the matrices may be no describe concerted molecular movements. The atoms can be given in a few bases denoted M, and the matrices of the TLS in the L basis can be measured by the following formulas QUOTE . (13) Where QUOTE is the transition matrix from basis QUOTE into basis QUOTE . In addition, the matrices of the TLS in the M basis can be measured by the following formulas QUOTE . (14) Libration ( QUOTE ) to locally orient the atom along the contours. Translation ( QUOTE ) to move the atom all over the image. Screw rotation ( QUOTE ) to adopt the contour smoothness. The equations (13) or (14) are used to find QUOTE (i.e. equation (6) and (7)). In this paper, the equation (13) is used to find QUOTE . 5. THE PROPOSED ALGORITHMThe realization of this work supposes the availability of a great number of repetitions of samples responding to the same known theoretical model. In practice, as the theoretical model is unknown, we use the Monte-Carlo method based on the generation of the data by computer according to a fixed theoretical model. 52.1. Theoretical Pre-learned dictionary To generate the dictionary that is used to characterize the image content effectively, some considerations must be satisfied. The first one is the training of the dictionary on the corrupted image, and the second one is directly the training of the dictionary on a high-quality image instead of the noisy image. The dictionary must be generated from another clean image where its contents are different from the contents of the noisy image. Therefore, the reconstruction error is probably cannot be determined by the change of the contents of the image during the denoising approach. Table 2 shows the pre-learned dictionary that is trained on some high-quality images to find the quality of the reconstructed images. In Table 2, it is clear that all PSNR results are higher than 138 decibels, this means that the quality of the reconstructed image is convincing. In this paper, the TLS model uses the pre-learned dictionary rather than the dictionary structured during runtime. Thats why the execution time is reduced significantly. The best reconstruction results dont always be obtained by the adaptive dictionary. In the two kinds of the dictionaries, the reconstruction error of the test images is an insignificant difference. It is clear that the contents of the test images in Fig.1 are different from the contents of the test images in Fig.2 and Fig.3, which are appropriate to measure the robustness of the pre-learned dictionary approach. The denoising results of the QPSO-MP algorithm are not sensitive to the contents of the test images used for constructing the dictionary. Thus, the TLS model is used as a pre-learned dictionary to speed up the QPSO-MP algorithm. model Figure 1. (a) Image1, (b) Image2, and (c) Image3. The size of all images is 512×512. We consider the traditional theoretical model of multiple linear regressions as EMBED Equation.DSMT4 Figure 2. (d) Lena, (e) Cameraman, and (f) Couple. The size of all images is 512×512. Figure 3. (g) Meloid. Kralli, (h) Meloid. Duyts, and (i) Meloid. Pore. The size of all images is 490×330. Table 2. Summary of the reconstruction error results on the test images in terms of PNSR. The images to be reconstructed are listed in the tables rows, whereas each column names an image used for constructing the dictionary. The bold values denote the dictionary trained on itself. ImagesImg1Img2Img3LenaLiving roomCameramanMeloid. kralliMeloid. duytsMeloid. poreImg1145.54142.02147.82140.25145.02140.59145.97145.84139.96Img2140.13144.92150.3143.64142.97147.83143.44140.97141.56Img3143.32142.54153.49141.99140.85142.95149.37146.09147.67Lena138.09141.03144.19146.51147.95144.87145.14149.01149.28Living room139.63141.31141.77144.39149.09148.6142.62152.13150.22Cameraman142.31142.54143.82143.98145.53152.61143.65138.45147.82Meloid. kralli138.69143.98151.87138.94144.98139.84154.87149.33140.59Meloid. duyts142.07140.76148.69141.47148.52145.26149.56153.76148.87Meloid. pore144.89140.78146.67145.79147.64150.74147.94150.5150.98 where EMBED Equation.DSMT4 is an EMBED Equation.DSMT4 vector observations of the dependent variables, EMBED Equation.DSMT4 is the matrix EMBED Equation.DSMT4 of EMBED Equation.DSMT4 explanatory variable, EMBED Equation.DSMT4 the vector of EMBED Equation.DSMT4 theoretical residuals and EMBED Equation.DSMT4 the vector of the theoretical regression coefficients. It is supposed that the residuals are independent random variable of the same normal distribution of null mean and constant variance EMBED Equation.DSMT4. The parameters to be simulated are EMBED Equation.DSMT4, EMBED Equation.DSMT4 and EMBED Equation.DSMT4, while the vector EMBED Equation.DSMT4 is calculated by the model. 52.2. Controlled factorsDMS of population DMS-QPSO algorithm is based on the standard version of the QPSO algorithm. The difference between the local version of the QPSO and DMS-QPSO algorithms, the swarms are dynamic and swarms are small sizes 37. The neighborhood topology of the DMS-QPSO algorithm has two important properties the DMS divide the whole population of QPSO algorithm into small swarms. Repeatedly, these swarms are re-clustered by using various regrouping schedules, and the information of these swarms is exchanged between them. The search of the DMS-QPSO algorithm is shown in the following figure. Figure 4. DMS-QPSOs search. 5.2. FQPSO-MP algorithm The first modification is using the TLS model to generate the pre-learned dictionary from high quality and texture images instead of the over-complete dictionary obtained from the given noisy image itself or the denoised intermediate image. The second modification is using the DMS over the QPSO (DMS-QPSO) algorithm. The population is divided into small groups by using DMS. Each group will use its own members to search the space. In this approach, the diversity of entire population will not be decreased because the information produced by every swam will be exchanged. The members of each swarm group will randomly distribute in QUOTE generation (i.e. QUOTE is the Regroup period). Table 3. Shows pseudo-code of the proposed FQPSO-MP algorithm based on the MP algorithm with two major improvements to the original QPSO algorithm. The proposed (FQPSO-MP) algorithm. Input Noisy image and the iteration number QUOTE . R Regroup Period (R3) m Population size of each group (m5) n Number of groups (n5) Initialization Stage Step 1 Set the initial estimate as QUOTE . Step 2 Use TLS model to generate pre-learned dictionary. Step 3 divide a random initial population into groups (DMS). Step 4 QUOTE evaluate the fitness function if QUOTE then Output Denoised image QUOTE . Else Step 5 Use DMS-QPSO to select the best atom of a dictionary. Step 6 Use MP to find the sparse representation of an image. Step 7 Go to step 4, to calculate fitness function. 6. EXPERIMENT RESULT The factors controlled for the theoretical models are the number of explanatory variables EMBED Equation.DSMT4, the number of observations (EMBED Equation.DSMT4), the index of collinearity of the explanatory variables EMBED Equation.DSMT4, the index of decrease of the regression coefficients EMBED Equation.DSMT4 and the theoretical coefficient of determination EMBED Equation.DSMT4. EMBED Equation.DSMT4 where EMBED Equation.DSMT4 is the value of coefficient EMBED Equation.DSMT4, EMBED Equation.DSMT4 the index of decrease of the regression coefficients and EMBED Equation.DSMT4 a constant. 62.13. Methods of regression studiedComparison between the proposed and the original algorithms To guarantee a fair comparison between the proposed algorithm and the local version of the QPSO algorithm, all conditions are the same, the patch size is 8×8, and the test images are the same having the same size. The PSNR is used to evaluate the quality of the denoised images. It is necessary to measure the performance and the execution time of the denoised images. The computational complexity is evaluated to the proposed and QPSO-MP algorithms. The experiment is conducted on a DELL PC, running Windows 7 Enterprise with 2.4 GHz Intel Core (TM) i5-M520 CPU, (R), MATLAB R2016 64-bit and 4 GB RAM. 6.1.1. Computational complexity The role of the pre-learned dictionary generated by the TLS model is effective in the decrease of the execution time of the proposed algorithm. Table 4 shows the execution time of the proposed and QPSO-MP algorithms on the test images. It is clear that the execution time of the proposed algorithm is smaller than the execution time of the QPSO-MP algorithm. Replacing the adaptive dictionary (Anisotropic atom) with the pre-learned dictionary is the reason behind this difference in the execution time. The pre-learned dictionary can be used directly in the proposed algorithm (it doesnt require any time to be executed), while the execution time needed for generating the dictionary from an image is more than 70 seconds. This means that the computational complexity of the proposed algorithm is less than the local version of the QPSO-MP algorithm. Fig.6 shows the average performance of the proposed algorithm that is better than the performance of the QPSO-MP algorithm. Fig.5 presents the average execution time of the proposed algorithm, and its found that it is shorter than its correspondent value for the original QPSO-MP algorithm. This shows the potential benefits of the proposed algorithm. Figure 5. Average execution time on all the test images using the proposed and original QPSO-MP denoising algorithm. Figure 6. Average PSNR results on all test images using the proposed and original QPSO-MP denoising algorithm. 6.1.2. Noise reduction Table. 4 summarizes the PSNR values of the denoised images for the proposed and QPSO-MP algorithms. The proposed algorithm can be applied to images with large size, because of its minimum execution time and it produces the best performance of the denoised images. The average PSNR values of the proposed algorithm are greater than the QPSO-MP algorithm. The main reason behind this is the division of the population in the proposed algorithm into groups. Fig.7 and Fig.8 show the difference of the denoised results between the proposed and QPSO-MP algorithms at the same sigma QUOTE and using the same image. This means that the proposed algorithm implements as good as the QPSO-MP algorithm while the running time is quite reduced. The conclusion of this experiment is that the best-denoised images can be achieved by the pre-learned dictionary instead of the adaptive dictionary obtained on the noisy image during runtime. The TLS model can be used to form the pre-learned dictionary that fits most of the medical, biological and the natural images. In summary, the final results prove that the proposed algorithm is excellent for removing noise (noise reduction) and execution time (computational complexity). Figure 7. Denoised results on Cameraman image with moderate noise corruption ( QUOTE 30) using the proposed and QPSO-MP algorithms, respectively. (a) Original image (b) corrupted image (c) denoised image using the proposed algorithm, PSNR 30.77 (d) denoised image using the QPSO-MP algorithm, PSNR29.16. Figure 8. Denoised results on image 3 with moderate noise corruption ( QUOTE 50) using the proposed and QPSO-MP algorithms, respectively. (a) Original image (b) corrupted image (c) denoised image using proposed algorithm, PSNR 28.38 (d) denoised image using the QPSO-MP algorithm, PSNR25.68. Image Noise ( QUOTE )MethodsTimeProposed algorithmQPSO-MP algorithmThe Execution time of the proposed algorithmThe Execution time of the QPSO-MP algorithmImg11037.934.992731052.23032.3129.59221991.65029.6826.162241096.17027.8823.71225940.89026.3721.88231917.2Img21036.2635.2229452.33031.2229.33229423.45028.5926.04223387.17026.6323.58228393.69025.2421.75253468.7Img31036.5534.38229402.13031.0228.95210387.85028.3825.68212379.77026.4123.34195425.49024.8621.57147374.1Lena1035.432.7196.4403.43030.6128.68181.9377.95028.0125.69182.15387.17026.423.42183.43779025.3221.72184.3376.2Living room1033.2628.82193.93955.13027.7826.25190.839555025.224.13186.5224.137023.8122.37200.79906.169022.8520.87197.13400.55Cameraman1036.7234.42189.83977.33030.7729.16185.559735028.525.95181.2211467026.7723.46183.411120.89025.3121.62184.319761030.3321.64128.9246.93023.7721.23121.7238Meloidkralli5022.2020.57114.5278.77021.5419.73124.8279.89020.8818.88113.4299.51029.7821.92114.1374.73023.4221.48114.4297.5Meloid duyts5022.0120.77120312.27021.4119.94138.82339020.9319.08111.4309.91029.5622.86110.72913023.5022.26110555.8Meloid. pore5022.2921.41111.7325.37021.4520.40117.9237.19020.7319.43116.3241.5Table 4. PSNR and execution time comparison of the proposed and the QPSO-MP algorithms under different noise levels from 10 to 90. The bold values denote the highest PSNR values and the shortest execution time. 6.2. Comparison between the proposed and the state of the art The denoising performance of the proposed algorithm is compared with three other denoising algorithms, 1) K-SVD 16, 2) Bilateral filter 31, and 3) Wiener filter 38. In this paper, the software package of the previous algorithms is used to obtain the denoising performance results. For a fair comparison between the proposed algorithm and the state of the art algorithms, the tested images 39, 40, 42 are the same, having the same size and corrupted with the additive Gaussian noise (GN) QUOTE where QUOTE is the estimated noise deviation with noise levels QUOTE Table 5, summarizes the denoising results (PSNR) using four different algorithms under the different Gaussian noise levels. Table 5 shows that the proposed algorithm provides the best performance results. Fig.9 presents the average overall performance of the tested algorithms. Fig.10 and Fig.11 show the difference of the denoised results between the proposed and the state of the art algorithms at the same sigma QUOTE and at the same image. The experimental results of this paper show that the proposed algorithm has the lowest execution time. In addition, it shows that the Bilateral and Wiener filters are faster than the proposed algorithm. The average performance of the proposed algorithm is higher than the other algorithms. The proposed algorithm realizes a certain trade-off between the quality of denoised images and the execution time. The proposed algorithm is applied to medical, biological and natural images, to satisfy and test different types of images. After all, it is demonstrated that the proposed algorithm is a competitive algorithm. Figure 9. The average PSNR results on the tested images using the proposed and the state of the art algorithms. Figure 10. Denoised results on the Living Room image with moderate noise corruption ( QUOTE 30) using the proposed, KSVD algorithms, Bilateral and Wiener filters respectively. (a) Original image (b) corrupted image (c) denoised image using the proposed algorithm, PSNR 27.78 (d) denoised image using the K-SVD algorithm, PSNR26.58 (e) denoised image using the Bilateral filter, PSNR 18.73 (f) denoised image using the Wiener filter, PSNR24.61. Figure 11. Denoised results on the Img1 image with moderate noise corruption ( QUOTE 50) using the proposed, K-SVD algorithms, Bilateral and Wiener filters respectively. (a) Original image (b) corrupted image (c) denoised image using the proposed algorithm, PSNR 29.68 (d) denoised image using the K-SVD algorithm, PSNR28.4 (e) denoised image using the Bilateral filter, PSNR15.92 (f) denoised image using the Wiener filter, PSNR20.19. ImageNoise ( QUOTE )MethodsK-SVDBilateralWienerProposed algorithmImg11037.6829.0835.1737.93031.3320.0829.6432.315028.415.9220.1929.687026.4713.3117.2627.889024.8611.4815.0626.37Img21036.2728.834.5136.293029.8719.7624.6331.225026.4115.6520.2128.597024.2713.1217.2526.639022.9611.3515.0925.24Img31036.8129.2334.3836.953029.7719.9424.6231.025026.315.8320.1728.387024.1413.2617.2926.419022.3311.4514.2124.86Lena1035.0328.1334.2635.43029.2418.7024.5930.615026.3814.6220.2128.017024.5912.2917.2526.49023.3110.7114.0825.32Living room1032.9428.1633.1733.263026.5818.7324.6127.785023.7514.5620.1925.27022.3912.1517.2523.819021.6510.6414.0622.85Cameraman1036.0328.2932.1736.723029.9719.0224.6330.775027.0914.8720.1928.57025.2312.4317.2626.779023.7510.8513.9125.311030.2428.0629.1630.333023.2818.7423.6123.77Meloid kralli5021.2314.7320.1622.207020.5612.3617.2521.549019.9910.8215.0820.881028.5328.1529.1329.783022.8118.7122.9823.42Meloid duyts5021.6914.6720.1822.017021.1412.2917.2721.419020.7710.7615.0720.931028.3228.1929.1529.563023.2118.7223.2323.50Meloid pore5022.0114.5820.1822.297021.2612.1817.2721.45Table 5. PSNR comparison of the proposed and the K-SVD algorithms, the Bilateral and the Wiener filters under different noise levels from 10 to 90. The bold values denote the higher PSNR values. On the one hand, we considered the classical method of least squares without variables selection and on the other hand, the stepwise selection method of variables is used. These methods were adopted, because they are among the most used methods, and are available in almost all statistical software. The selection of variables is based on the t test of Student or F test of Snedecor for significance of the regression coefficients. We used the same level of significance for the introduction and the exclusion of a variable in the model. Two theoretical levels were retained 0.15 and 0.05. 3. RESULTS 3.1. Effects of the various factors on the coefficient EMBED Equation.DSMT4 The analysis of table 1 shows that coefficient EMBED Equation.DSMT4 is more often lower than the theoretical coefficient of determination. The ratio increases as the sample size increases, for a given value of EMBED Equation.DSMT4 and EMBED Equation.DSMT4. Table 1 Average observed values of EMBED Equation.DSMT4, expressed in proportion of EMBED Equation.DSMT4, according to EMBED Equation.DSMT4, EMBED Equation.DSMT4 and EMBED Equation.DSMT4. EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4Complete model58-14.39-4.76-1.540.06102000.820.930.970.993050-0.150.520.770.90306000.910.960.980.99Model selected (EMBED Equation.DSMT4)58-1.66-0.310.260.6551000.790.920.960.981017-0.990.150.560.81102000.890.960.980.993050-0.400.440.750.90306000.930.981.001.00Model selected (EMBED Equation.DSMT4)306000.931.001.001.00 For known values of EMBED Equation.DSMT4 and EMBED Equation.DSMT4, the ratio EMBED Equation.DSMT4/EMBED Equation.DSMT4 depends little on the values of k and n. We also note that the ratio is weaker for the low values of EMBED Equation.DSMT4. Finally, the use of variables selection tends to increase the ratio. 3.2. Determination of the levels of factors combinations leading to a null predictive value In order to obtain results easily usable in practice, we determined the validity limits of the equations for the purpose of prediction by being unaware of the effect of factors EMBED Equation.DSMT4 and EMBED Equation.DSMT4 on the prediction. These limits are obtained by determining the levels of the ratio EMBED Equation.DSMT4 leading to a zero value of EMBED Equation.DSMT4. These levels give on average the thresholds of combinations of factors from which the model led to predictions of quality lower than the prediction given by the arithmetic mean of the dependent variable of the sample. Figure 1. Evolution of the ratio EMBED Equation.DSMT4/EMBED Equation.DSMT4 according to the sample size on logarithmic scale in X-coordinate, for EMBED Equation.DSMT4, EMBED Equation.DSMT40.40. From this table, we note that this size varies according to the method used to establish the model. It is higher for the complete models and decreases gradually with the intensity of the selection. It also decreases as the theoretical value EMBED Equation.DSMT4 increases. 4. DISCUSSION AND CONCLUSION Several authors documented criteria that assess the quality of a model. These criteria are based on the difference between the estimated model and the presumed known theoretical model. In the present study, the criterion used compares to new observations resulting from the same population as individuals of the sample, the variability of the errors of prediction, when the predictions are carried out by a regression equation and on the other hand when these predictions are equal to the arithmetic mean EMBED Equation.DSMT4 of the dependent variable in the sample. It thus gives an idea of the improvement of the quality of prediction by taking into account the explanatory variables. It also informs about the validity limits of a prediction model. The plan of simulation considers data of varied structures. In particular, we considered the case where all the explanatory variables available are indeed present in the theoretical model (EMBED Equation.DSMT4) and the case where certain explanatory variables available are not present in the theoretical model. This approach makes it possible to be close to the situations often encountered in practice. 75. REFERENCES Paragraph Margin Left1 inch, Right1 inch, Font size10 pt. Font Face Arial, Align justify, Line Space single pt. Akossou, A.Y.J., 2005, Impact de la structure des donnes sur les prdictions en rgression linaire multiple. PhD Thesis, Fac. Univ. Sci. Agron., Gembloux, Belgium, 215p. Bendel, R.B., Afifi, A.A., 1977, Comparison of stopping rules in forward stepwise regression. J. Amer. Stat. Assoc. 72, 46-53. Copas, R.D., 1983, Regression, prediction and shrinkage. J. R. Stat. Soc. B 45,311-354. Dempster, A.P., Schatzoff, M., Wermuth N., 1977, A simulation study of alternatives to ordinary least squares. J. Amer. Stat. Assoc. 72, 77-106. Meg, B. C., 1988, Determining the optimum number of predictors for linear prediction equation. Amer. Meteo. Soc. 116, 1623-1640. Miller, A.J., 1990, Subset selection in regression. Monographs on statistics and applied probability 40. Chapman and Hall. Palm, R., De Bast, A., Lahlou, M., 1991, Comparaison des modles agromtorologiques de type statistique empirique construits partir de diffrents ensembles de variables mtorologiques. Bull. Rech. Agron. Gembloux 26, 71-89. Roecker, E.B., 1991, Prediction error and its estimation for subset-selected models. Technometrics 33, 459-468. rtise-Services/Chair-groups/Plant-Sciences/Laboratory-of-Nematology/Nematode-in-the-picture/Pictures.html https//www.wur.nl/en/Expertise-Services/Chair-groups/Plant-Sciences/Laboratory-of-Nematology/Nematode-in-the-picture/Pictures.html, accessed 23 February 2018.
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