Biomedical images enhancement based on swarm optimization and differential evolution technique

In this paper, we have introduced an effective technique to remove the noise in the MRI and CT medical images during the process of acquisition, transmission, storage or compression. Removing these noise from medical images must be done without affecting relevant features of the image. Many techniques, such as genetic algorithm, Particle Swarm Optimization, Dynamic Multi-Swarm Particle Swarm Optimization and matching pursuit algorithm are used for denoising MRI and CT images. These techniques need more time to remove noise from medical images and find local point optimal. The proposed Differential Evolution based on Matching Pursuit (DE-MP) is used to detect best atom dictionary. The initial dictionary is created from an anisotropic atom. To evaluate the performance of the proposed techniques, the results of the proposed algorithm were compared with the other algorithm. The numerical results show that the performance of the proposed algorithm is more efficient and faster than other algorithms for medical images denoising.


Introduction
*Medical images play an essential role in analysis duties with applications in various knowledge domains, such as diagnoses illnesses and medicine. There are many problems facing medical specialists in the medical field such as: noise suppression and blurring. Although several image denoising techniques have been suggested in the literature, noise suppression in images still remains a challenging problem for researchers since the process of removing the noise from images can cause the removal of pertinent image features, such as edges and corners (Singh and Wadhwani, 2015). Therefore, one of the biggest problems is removing noise from images. Noise suppression can introduce artifacts or cause image blurring, which makes image denoising a complicated task. There are different methods have been proposed to remove noise from images; however, each one examines specific facts of the problem (Farouk et al., 2016).
Image denoising algorithms can be divided into three types: image denoising algorithm based on transform domain filtering, image denoising algorithm based on spatial filtering and image denoising based on learning. The denoising algorithm based on transform domain filtering includes wavelet transform, Fourier transform, Block-matching and 3D filtering (BM3D) algorithm, etc. (Geng et al., 2016, He et al., 2014. The second type is the denoising algorithm based on spatial filtering includes bilateral filtering, Gaussian filtering, guide filtering, nonlocal mean filtering, etc. (Geng et al., 2016, He et al., 2014. The third type is the denoising algorithm based on learning involves K-singular value decomposition (K-SVD) algorithm (He et al., 2014) large-scale sparse clustering (LSSC) algorithm  and clustering based sparse representation (CSR) algorithm (Yang et al., 2015). K-SVD is a dictionary learning algorithm. It is used to create a dictionary for a sparse representation (SR), via singular value decomposition (SVD) method. It works by iteratively alternating between sparse coding the input data based on the current dictionary, and updating the atoms in the dictionary to better fit the data. The K-SVD improved with different type of dictionary to form atoms of it such as: DCT (Elad and Aharon, 2006), Gabor Wavelet (Khedr et al., 2012), Log-Gabor  and Log Gabor Wavelet (Farouk et al., 2016). In Ruiz-Reyes et al. (2005), the space-alternating generalized expectationmaximization (SAGE) algorithm is used to estimate the values of parameter vector and minimum description length. In Wax and Ziskind (1989), the minimum description length (MDL) basic principle was incorporated in the algorithm to estimate the model order (i.e., the determination of the number of echoes). The matching pursuit (MP) is made by estimating and adding. MP algorithm has been applied for medical image denoising. The main problem of MP formula is to estimate the basic parameters of the original signal. First, MP matches a function to the original signal. Then, this function is subtracted from the signal to obtain the signal residue. In Toledo et al. (2013), GA algorithm is used to remove noise from the digital image and the numerical results proved that GA algorithm is better than traditional (state-of-the-art) denoising methods. In Davis (1991), GA based thresholding techniques are used in image denoising. It is used wavelet transform to remove Gaussian noise from medical images. GA algorithm finds the best threshold value and the level of decomposition for the Bayesian thresholding by the use of stochastic and randomized search algorithm compared with Visu shrink, sure shrink and Bayes shrink. The numerical results show that GA based thresholding techniques are better than Visu shrink, sure shrink and Bayes shrink. In Liu (2015), GA algorithm is used to derive the denoising results. In Korurek et al. (2010), GA algorithm is used to evaluate parameter values in a model for a near field effect of X-ray source. Also, GA-MP algorithm is used to reduce the computational cost of computing the projection of the signal. Genetic algorithm (GA) is still local optimization algorithm. So, it is applied to obtain the best solution does not give an optimal result and compromise between speed and accuracy, GA algorithm try to maximize this relation (Jansi and Subashini, 2013;Cui et al., 2014). Because of the randomness of the GA algorithm, the final solution will be suboptimal. PSO algorithm is presented by Eberhart and Kennedy (1995). In Ashour et al. (2015), cuckoo search (CS) is used to find the optimal parameter settings for log transform and log transform is used in medical image enhancement algorithm. The numerical results proved that CS-log transform is faster than PSO algorithm. In Jansi and Subashini (2013), PSO is used to find the optimal value of the regularization parameter of total variation method that helps total variation method to remove noise in magnetic resonance imaging (MRI) images. In Hu et al. (2015), PSO is used to find the optimal weighted factor that was inserted into weighted sub wavelets then classical Shearlet transform is used to decompose noised image into these sub wavelets under multiscale and multi-orientation. PSO is simple algorithm but it has been slow so that, PSO improved with MP algorithm to find the best atom searching problem (Liang and Suganthan, 2006).
DMS-PSO algorithm is appeared to avoid the disadvantages of PSO algorithm. In the DMS-PSO, the population is divided into several small groups. In Chen et al. (2013), the DMS-PSO-MP was developed by Discrete Coefficient Mutation (DCM) strategy to improve the local searching ability of DMS-PSO in the MP approach over the anisotropic atom dictionary.
Experimental results show that DMS-PSO with DCM strategy is better than other popular versions of PSO.
This paper aims to combine DE and MP algorithms for denoising medical images, the anisotropic atom is used to generate over-complete dictionary and DE is applied to detect best atom of dictionary based on MP algorithm. Generally speaking, each algorithm has some filtering and threshold parameters. Taking variety kinds of images into account, it is a key problem of how to set these parameters in denoising algorithms under different conditions to achieve better performance and short execution. The values of the parameter vector and optimal model can be estimated by merging DE algorithm with MP algorithm. The rest of this paper is organized as follows: Section 2 sparse representation of images, matching pursuit of images and differential evolution algorithm are discussed. Section 3 the proposed algorithm DE-MP is discussed. The numerical results of DE algorithm in image denoising is discussed and compared with other methods in section 4. Finally, conclusions are shown in Section 5.

Sparse representation of images
Sparse representation of images is used to generate a set of atoms ∈ that form the dictionary D∈R m×k and it is defined mathematical equation as:

=
(1) Eq. 1 can be rewritten as Eq. 2: where ||.|| 0 is the L 0 semi-norm, which counting the non-zero entries of a vector, ||.|| 2 is the L 0 -norm, which represents the Euclidean length of a vector. l is a threshold that control the sparseness of the vector . The over-complete dictionary is generated by anisotropic atom. DE algorithm is used to select the best atom from the dictionary and matching pursuit algorithm (MP) is used to find sparse representation x i of .

Matching pursuit of images
MP is a greedy algorithm that is used to utilize signal decomposition based on a redundant dictionary called the over-complete dictionary with each element called an atom (Davis, 1991). Best atom can be selected from atoms of the dictionary when MP applied. MP depends on over-complete dictionary, in each iteration, the best matching atom can be available in over-complete dictionary (Hu et al., 2015). Let = {g γ } γ∈Γ is the atoms of the dictionary, for arbitrary image of size × , is the set of all indexes and ||g γ || = 1 (Jansi and Subasini, 2013). Let is an arbitrary signal. The initial step of a Matching Pursuit is always to approximate y by the projecting it on a vector 0 ∈ =< , 0 > 0 + where < , 0 > 0 is projection of onto atom 0 and is the residual of the original signal. Since the residual is orthogonal to 0 (Liang and Suganthan, 2006).
where || 0 || 2 = 1. In term || || 2 = || || 2 − | < , 0 > | 2 must be minimized, so 0 ∈ must be chosen to maximize| < , 0 > | . After iterations, the original signal can be reconstructed via the selected atom : The process of generating the is the one of the main steps in denoising processes. So, the Gaussian function is used as in Eq. 7. The essential function is a Gaussian in one axis and the second derivative of the Gaussian in the various other axis (Liang and Suganthan, 2006).
Eq. 7 is used to generate the dictionary with translation, rotation, translation and scaling factor in x and y directions, i.e., , , , , respectively. Anisotropic atom was used to find ℎ, .

Differential evolution algorithm
In Storn and Price (1997) DE algorithm developed by Storn and Price. It really is inspired by natural selection and natural genetics. The main difference between DE and GA algorithms is that GA algorithm based on the crossover (recombination) while DE algorithm based on mutation operation. DE algorithm is same as genetic algorithms (i.e., DE algorithm have the same steps of GA algorithm).
The DE algorithm is heuristic algorithm that has three advantages; finding the best global minimum of a multi-modal search space regardless the values of the initial parameter, it using a little number of control parameters and its convergence faster than GA, PSO and DMS-PSO algorithms (Karaboga and Cetinkaya, 2004). Fig. 1 shows the main steps of DE algorithm. In the DE algorithm, the solution has the form: where , are the size of population and number of parameters respectively The DE algorithm can be presented as the following steps. The first step of DE algorithm is called initialization, lower (L) and upper (U) bound of each solution are defined and then randomly generate initial populations P and should cover all entire solutions (i.e. , , , , ) (Bhandari et al., 2016).
Then the fitness function is measured. Then, randomly select three solutions 1 , , 2 , 3 , from population P (mutation step). DE algorithm generates new vector called donor vector by adding the weight difference of two of the vectors to the third as equation where the mutation factor is constant ∈ [0, 2], , +1 is called the donor vector. The next step is called crossover (recombination). Here, new vector called trial vector , +1 is developed from the elements of both target vector , and donor vector , +1 . Elements of the donor vector enter the trial vector with probability Crossover Rate (CR).
CR is constant ∈ [0, 1]. Where = 1,2, … , ; = 1,2, … , is a random integer from [1, 2, 3. . . n]. The final step of DE algorithm called Selection step. Which a new vector called trail vector is created. The target vector , is compared with the trial vector , +1 and the one with the lowest function value is admitted to the next generation.

The proposed algorithm
In this section the proposed algorithm is introduced (i.e., Denoising image based on DE-MP algorithm). It can be presented as the following: Noisy image y and error tolerance are considered inputs to the proposed algorithm. Anisotropic atom is used Eq. 7 to form the over complete dictionary. The vector which is defined as ,0 , ,1 , ,2 , … , , −1 ∈ could be gotten = ( ,0 , ,1 , ,2 , … , , −1 ) then randomly initialized the population of = 300 to be represented in one column of over complete dictionary D. The problem = + must be solved where sparse representation of image can be defined as = ( 1 , 2 , 3 , … , ) .

Fig. 1: Shows the main steps of DE algorithm
The previous problem can be solved as min , || − || 2 . || || < ε, ε ∈ R If the error tolerance ε <= 0.01 now, the image is really cleaned. Then, compute the performance of clean image. If this condition is not satisfied (i.e., ε <= 0.01) then, use Eq. 11 to find mutation operators on populations. The donor vector and target vector are used to find trial vector in crossover step. Use Eq. 13 to select best atom of over complete dictionary. Then, use MP algorithm to find sparse representation of image and then go to solve the previous problem. Simply DE-MP denoising algorithm can be presented in Fig. 2. The denoised image based on DE-MP can be given as following steps:

DE-MP denoising algorithm Input: noisy image and initialize error tolerance .
Step 1: generate atoms of dictionary (use Eq. 5).
Step 2: construct a random initial population of and dimension (Dim=300).
Step 7: use MP to find sparse representation of image.
Step 8: go to step 3, to calculate fitness function.
All experiments are performed on a PC running MATLAB(R) R2012b 64-bit in Windows(R) 10 with 4 GB RAM and Intel(R) CORE(R) i5 M-430 processor is used to conduct the whole experiment. In this paper, the performance is measured by using two methods shown as the following: the performance was measured at different noise levels. Where the MSE (Mean Squared Error) is: where 0 and are the original and denoised images respectively. is represents the numbers of rows of pixels of the images 0 and . is represents the number of columns of pixels of the images and .

Fig. 2: Algorithm flowchart of MP optimized by improved DE
where shows maximum generation. In this paper, are set to 0.9, 0.2 respectively. In Table 1 and Table 2, the PSO-MP, DMS-PSOMP, GA-MP and the proposed algorithm have been developed using anisotropic atom. The MP algorithm improved with DMS-PSO, PSO, GA and DE algorithm, in which the best results are highlighted (in bold red). In Table Fig. 3-8. The average time comparison of different algorithms with image resolution 128 × 128 and 64 × 64 is shown in Fig. 9, it is clear that the proposed algorithm takes shorter execution time than other algorithms. In Table 2, the SSIM is used to measure the performance of medical images. The best results are highlighted (in bold red). It is clear that proposed algorithm which improved with anisotropic atom is better than PSO, DMS-PSO and GA algorithms. Table  3 shows that execution time of DMS-PSO, Standard PSO, GA-MP and the proposed algorithm that developed by anisotropic atom with different values of sigma σ = 10, 30, 50, 70, and 90. The best results are highlighted (in bold red). It is clear that proposed algorithm is faster than other algorithms.

Conclusion
This paper presented image denoising method using a Differential Evolution algorithm to suppress noise from medical images. A representation of populations is proposed based on the pixel matrix in such way that tailor-made crossover and mutation operators are designed. Preliminary experiments conducted on a set of medical images indicate the proposed algorithm, producing superior results compared to PSO-MP, DMS-PSO-MP and GA-MP denoising methods. The reported computational results indicated that several final solutions (images) can be returned by the proposed method within a short execution. The image quality that obtained by the proposed algorithm increases along increase in population size. The solution quality measured using PSNR and SSIM for these final solutions. In fact, the proposed method was able to outperform some of these methods in all results, taking into account the average and the best solutions found.