Restoration of Images With High-Density Impulsive Noise Based on Sparse Approximation and Ant-Colony Optimization

In this work, we propose an image denoising approach, specifically for “salt-and-pepper noise,” based on the optimized sparse approximation for restoring images contaminated by high-density impulse noise. The proposed method first uses the inverse-distance weighting-based prediction to estimate noise-recovered pixels. It then utilizes DCT-based sparse approximation to further refine the denoised results with the ant colony optimization. Experiments on an image benchmark dataset demonstrate that the proposed method yields better results compared to the state-of-the-art image noise removal methods.


I. INTRODUCTION
Image communication and acquisition under unfavorable conditions often cause captured images to corrupt with highdensity impulse noise [1]. Such image noise can not only degrade visual quality but negatively affect the performance of various computer-vision applications, such as peoplecounting, crowd analysis, action recognition, human tracking, and so on [1]. Therefore, developing a robust and effective denoising method is essential. In general, conventional denoising methods for removing impulse noise often use median filtering [2] or average filtering. Toh and Isa [3] developed a noise adaptive fuzzy switching median filter to remove impulse noise from corrupted images, obtaining a denoised image by using fuzzy computation via a weighted smoothing for the original and filtered pixels.
Esakkirajan et al. proposed to use the mean and median filtering conjunctively deal with the high-density noise of an image. An adaptive weighted mean filter [4]

recovers noisy
The associate editor coordinating the review of this manuscript and approving it for publication was Yi Zhang . images with adaptively-sized kernels. Hsieh et al. [5] developed a fast median filter using two forms of the filtering window to restore images corrupted with high-density impulse noise. The denoising approaches based on the median filtering can generate decent results as long as acquired images are not heavily contaminated by impulse noise. In other words, these approaches cannot deal with images having high-density impulse noise, especially with the noise level over 50%. Erkan et al. [6] proposed to use two-pass median filtering with the selective window size for image noise removal. The approach chooses the window size in the first pass, where it contains at least one non-noise pixel, and then applies median filtering. If there are still noise pixels in the window, median filtering is applied again in the second pass to remove the rest of the noise pixels. The main disadvantage of it is that fake edges often exist in its denoised results.
There has been research done using learning-based techniques for image noise removal recently. In particular, sparse approximation approaches with a dictionary learned to restore images with high-density impulse noise for better results than those of median-filter-based methods. Chen et al. [7] first detected noise candidates via the conjunctive utilization of the adaptive median filter and the adaptive center-weighted median filter, producing denoised images through sparse representation learning from noisefree images. Stanković et al. [8], [9] proposed to deal with images taken in impulsive disturbance environments using a gradient-based iterative algorithm to locate and then remove noise pixels. Peng et al. [10] proposed an overlapped and adaptive Gaussian smoothing method with convolutional refinement networks for denoising. Liu et al. [11] proposed a feature extraction algorithm based on sparse and low-rank representation. Ma et al. [12] extended it to get better denoising results by utilizing the total variation regularization based on the sparse representation prior. Jiang et al. [13] developed an approach that uses weighted encoding with sparse nonlocal regularization for high-density impulse noise removal, in which soft impulse pixel detection is applied through weighted encoding to deal with impulse noise. Aggarwal and Majumdar [14] regarded image denoising as a data fidelity minimization problem with l 1 -norm regularization. They then used a split Bregman-based algorithm to solve the problem via the general analysis prior.
Zhang et al. [15] proposed feed-forward denoising convolutional neural networks (DnCNNs) that use residual connections and batch normalization to try to deal with more general image denoising tasks at different scales. However, it is often too general to work well on more specific image noise. One common disadvantage these learning-based methods have is that they often fail to recover images having a high percentage of noise pixels. For example, they usually cannot handle images with a noise level exceeding 70% (see Fig. 1 (a)). It could introduce excessive artifacts in denoised results. In such cases, there are too few noise-free pixels available in the image to sufficiently facilitate the sparse representation based on learning-based techniques.
To address this issue, we propose an effective sparse approximation method based on a DCT dictionary with the ant colony optimization for high-density impulse noise removal. The proposed method consists of two primary modules: a Sparse Representation (SR) module and an Ant-colony Optimization (AO) module.
In order to have enough non-noise information for sparse approximation, the proposed method first adopts an inversedistance weighting-based prediction to recover noise-tainted pixels in the proposed SR module. Next, it seeks a better prediction of noise-recovered pixels based on ant-colony optimization in the proposed AO module. Fig. 1 (b) shows the abstract flowchart for the proposed method. To sum up, the primary contributions of this paper are: 1) We propose a novel denoising method through an effective combination of the SR and AO modules to better reconstruct the corrupted images. 2) As far as we know, we are the first to adopt ant-colony optimization for further improving the visual quality of denoised results. 3) We evaluate our method with numerous experiments and demonstrate that our method outperforms other state-ofthe-art methods. The remainder of this paper is organized as follows. The proposed method is described in detail in Section II. Section III discusses the comparisons between the proposed method and other state-of-the-art learning-based methods. Section IV concludes the paper.

II. PROPOSED DENOISING METHOD
This paper proposes a novel noise removal approach based on sparse approximation using ant-colony optimization to remove high-density impulse noise from a corrupted image and then recover the image. As illustrated in Fig. 2, our approach consists of two major modules: a sparse representation module and an ant-colony optimization module.
Since noise-free pixels in the input image with high-density noise are not enough for general learning-based denoising methods to learn sparse representations, these methods often fail to reconstruct the de-noised image effectively because of insufficient training patterns. To overcome this problem, the proposed SR module employs the inverse-distance weighting based prediction model to estimate noise-fixed pixels, which serve as non-noise information for learning sparse approximation. Note that the sparse representation module was primarily presented in our previous work [16].
Based on the inverse-distance weighting-based prediction model [17], different sampled noise-free pixels may have different weights for estimating recovered noise pixels. Thus, to VOLUME 8, 2020  seek a better reconstruction for a de-noised image, ant-colony optimization is used in the proposed AO module.
Initially, the impulse noise detector [18]- [20] is adopted to discriminate noise pixels and noise-free pixels in an image. The impulse noise detector can be expressed as follows: where I (x) denotes the intensity value of a pixel at position x, and B (x) denotes the binary noise mask. If the mask is labeled '1', it indicates that the corresponding pixel is either the maximal or minimal intensity values (i.e., 0 or 255 for an 8-bit image). The label '0' represents that the pixel is a noisefree pixel. The detailed description of these two modules is given in the following subsections. Illustration of the distance between the noise pixel (indicated by *) and its surrounding noise-free pixels (marked in red circles) in the inverse-distance weighting-based prediction model.

A. SPARSE REPRESENTATION MODULE
First of all, we decompose the input image I into overlapped patches with size √ n × √ n, where n is the width of the input square image I . As demonstrated in Fig. 3, we introduce a noise-free pixel counter NF i for the i th patch, where we use the counter to record the number of noise-free pixels. It can be expressed as follows: where (i) denotes the coordinate set for the pixels in the i th patch. The minimal value of NF min over overlapped image patches is used as the number of sampled pixels to predict potential noise-fixed pixels in the proposed inverse-distance weighting-based (IDW) prediction model [17]. Note that the number of noise-free pixels might be greater than or equal to the number of sampled pixels. Thus, the inverse-distance weighting based prediction model can be expressed as follows: where and V (x) represents the predicted intensity of a potential noise-fixed pixel at position x within the ith overlapped image patch which is interpolated with a set of given intensity of noise-free pixels I (x), s.t. ∀B (x) = 0; d is the Euclidean distance between the noise pixel (i.e., I (x), s.t. ∀B (x) = 1) and the noise-free pixel (i.e., is the sum of weights within the ith overlapped image patch; p is a predefined power factor. Here, we empirically set p = 11 in our experiment.
According to the sparsity-based regularization principle, each overlapped image patch can be sparsely represented as I i ≈ φα I ,i by solving the following l 1 -minimization problem using the DCT-based dictionary φ ∈ n×M ,s.t.n ≤ M [7]: where I i − φα i 2 2 and α i 1 are the data-fidelity term and the sparse regularization term, respectively. Next, we can reconstruct a de-noised image from the set of sparse codes {α I ,i } by using the least-square solution [21] as follows: where M is the total number of overlapped patches with size √ n × √ n in the image I . Here, in order to reconstruct a de-noised patch, the predicted noise-fixed pixels within the ith overlapped image patch are regarded as the measured patch by which to provide a sufficient training pattern for sparse approximation (See Fig. 5). Therefore, we reformulate the minimization task in Eq. (5) into: where the term I i − φα i 2 2 denotes each patch-error with a predefined tolerance . Note that the ith overlapped image patch I i is composed of both noise-free pixels and predicted noise-fixed pixels. The error-constrained orthogonal matching pursuit is employed in the proposed SR module for solving the above minimization task [22].

B. ANT-COLONY OPTIMIZATION MODULE
Ant-colony optimization was first introduced in [23] and is regarded as an adaptive meta-heuristic optimization method inspired by nature for solving the combinatorial optimization problems [24]. The principles of ant colony optimization include: 1) constructing ant solutions that can balance pheromone trails (i.e., characteristics of past solutions, with a problem-specific heuristic); VOLUME 8, 2020 2) reinforcing and evaporating pheromone; 3) searching locally for improved solutions. The key steps of the proposed ant-colony optimization are described below.
Step 1) Constructing ant solutions: In general, there is a set of solutions S = {S 1 , . . . , S l , . . . , S k } (also called trails) that can satisfy all the constraints in the set, in which each decision variable has a series of values assigned by s 1 1 , . . . , s i l , . . . , s M k . Moreover, the solution is feasible for the given optimization problem. The set of solutions is randomly initialized while the series of values in s l i is restricted to 0 ≤ s i l < n ∧ ∀B i (x) = 0 for each solution. Therefore, a complete ant solution is composed of an integer vector of size k × M .
Step 2) Reinforcing and evaporating pheromones: After the set of solution is generated, each series of values in s l i can be used to reconstruct an overlapped image patch in the proposed SR module. The quality of each solution (i.e., the de-noised image) is evaluated by using the no-reference Q metric. This no-reference metric can be expressed as follows: where z 1 and z 2 denote the singular values representing the energy observed in the dominant direction V 1 and the perpendicular direction V 2 , respectively. These directions can be obtained as follows: where G denotes the gradient matrix over a local window of size √ n × √ n. Hence, the dominant orientation of the local window can be obtained by computing the SVD of G [25]. Scores attained by the aforementioned Q metric with higher values indicate superior noise removal effects.
Next, each pheromone is reinforced and evaporated in order to facilitate the construction of a better trail (i.e., solution) that is likely to be feasible and observe the weight and cost constraints. The better trail S 1 possesses a higher weight ω ranked by Q and labeled by l = {1, . . . , k}. The weight ω can be calculated by where k is the number of solution in a set, and q is a tolerance factor that is set to 0.5 in our experiments. Therefore, the new trail is represented by where δ is the mutation factor. If the random value of rand (0, 1) exceeds the mutation factor δ, then the trail is randomly updated by previous trail s i cdf(rand(0,1)) , where the cdf (·) denotes the cumulative probability density of weight ω. Otherwise, we create a new solution via Step 1. The lower mutation factor δ possesses higher Q score climbing in order to overcome the local optimum problem [26], as indicated in Fig. 6.
Step 3) Searching locally for improved solutions: Finally, a selection process is performed from the solution set (see Fig. 7) for the next generation. In the solution set S, the solutions assigned with the top ten lowest labels l = {1, . . . , k} are maintained into the next generation. By executing the above steps iteratively, the best solutions can be preserved from generation to generation. Therefore, the proposed method is able to effectively remove impulse noise from corrupted images for each level of high-density noise, as shown in Fig. 8.

III. EXPERIMENTAL RESULTS
In this section, the experimental results for high-density impulse noise removal using the proposed approach and three other state-of-the-art approaches are conducted for

A. QUALITATIVE EVALUATION
As can be seen in Figs. 9-14, we visually demonstrate the reconstruction efficacy of each compared method through eight different images corrupted with high-density impulse noise levels ranging from 70% to 90%. Additionally, we include the corresponding peak signal-to-noise ratio (PSNR) of each reconstruction result.
As can be seen in the third to fifth columns of Figs. 9-14, the reconstructed images obtained through the approaches of Aggarwal and Majumdar [14], Ma et al. [12], and Jiang et al. [13] appear blurrier and with more serious artifacts compared to the results produced through the proposed method. Aggarwal et al.'s approach [14] considered the impulse noise removal problem as an l 1 -norm regularized VOLUME 8, 2020 FIGURE 11. Illustration of ''Lake'' image reconstructed via each compared method as impulse noise increases from 70% to 90%. l 1 -norm data fidelity minimization problem and thereby a noise-free image was produced by solving this problem via a general analysis prior using a split Bregman-based algorithm. Ma et al.'s approach [12] conjunctly utilized the sparse representation prior and total variation regularization to seek better recovery results. Jiang et al.'s approach [13] employed weighted encoding with sparse nonlocal regularization for high-density impulse noise removal, in which soft impulse pixel detection is employed via weighted encoding to deal with the impulse noise. Additionally, both the image sparsity prior and nonlocal self-similarity prior are used to represent the noise-free image. However, these methods employ insufficient noise-free information with which to predict the unknown corrupted patches for learning sparse approximation when the impulse noise in the corrupted image   [15] does not work at all for images with impulsive noise.
As can be observed in the sixth column of Figs. 9-14, our approach is capable of yielding clearer results than the other state-of-the-art approaches, and those results possess higher PSNR scores. This is because the proposed SR module employs an inverse-distance weighting-based prediction model to produce potential noise-fixed pixels by which to sufficiently provide non-noise pixels for sparse approximation. Additionally, ant-colony optimization is used with the no-reference Q metric in the proposed AO module to seek an optimized prediction of a de-noised image. As such, the PSNR scores in Figs. 9-14 demonstrate that the corrupted 99186 VOLUME 8, 2020 FIGURE 15. Comparison of PSNR results between each compared method for fifteen high-density noise images, where the percentage of noise ranges from 70% to 90%. images are recovered more effectively through our approach than through the others.

B. QUANTITATIVE EVALUATION
Next, the quantitative results obtained by the proposed approach are compared with those obtained by the approaches of Aggarwal and Majumdar [14], Ma et al. [12], Jiang et al. [13], Erkan et al. [6], and Zhang et al. [15], in terms of the average PSNR [27] over 70 gray scale image sets. Note that higher rates produced by the PSNR metric indicate superior noise removal effects.
As shown in Fig. 15, the PSNR scores of images reconstructed via the approaches of Aggarwal and Majumdar [14], Jiang et al. [13], and Erkan et al. [6] indicate those reconstructions degrade as the impulse noises increased. Regarding the approach of Ma et al. [12], while the PSNR score increases for its reconstructed images, they inevitably suffered from artifacts, and the PSNR is still lower than that of the other compared approaches. The PSNR score of Zhang et al.'s approach [15] shows that it cannot deal with images with such noise at all. The PSNR scores for the images reconstructed by our method did not degrade as much as noise increased, and thus outperformed the results from the approaches of Aggarwal and Majumdar [14], Ma et al. [12], Jiang et al. [13], Erkan et al. [6], and Zhang et al. [15]. The results also show that the proposed method is capable of performing superior noise removal while recovering texture information effectively.

IV. CONCLUSIONS
This paper has presented a new de-noising method based on optimized sparse approximation using ant-colony optimization for high-density impulse noise removal from a corrupted image. The proposed method allows high-density impulse noise removal from corrupted images, which is strongly required in several computer-vision systems. Unlike existing methods that use the corrupted texture information or remnant noise-free information to reconstruct a de-noised image, our method employs the inverse-distance weighting based prediction model to produce potential noise-fixed pixels for sparse approximation learning. This gives the proposed method the ability to remove high-density impulse noise from the corrupted image. Additionally, the proposed method adopt the ant-colony optimization to seek an optimized non-noise image reconstruction, in which the no-reference Q metric is used to evaluate the reconstructed image at each generation for improving the quality of the reconstruction result in accordance with its original value. A comprehensive evaluation of the results produced by the different compared methods via qualitative and quantitative assessments for image noise removal is conducted in this paper. Our experimental results, using various test images with varying levels of highdensity noise and comparing the results through different evaluations, demonstrate the effectiveness of the proposed method, its ability to outperform the other state-of-the-art sparse approximation approaches, and its robustness for highdensity impulse noise removal.   Engineering, in 2014. Since 2020, he has been a Guest Editor of MDPI Mathematics. VOLUME 8, 2020