Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 340182, 9 pages

http://dx.doi.org/10.1155/2015/340182

## Image Denoising via Asymptotic Nonlocal Filtering

^{1}School of Mathematics and Statistics, Xidian University, Xi’an 710071, China^{2}School of Science, Xi’an Shiyou University, 18 Second Dianzi Road, Yanta District, Xi’an, Shaanxi 710065, China^{3}School of Mathematics and Computer Science, Ningxia University, Yinchuan 750021, China

Received 6 February 2014; Revised 5 October 2014; Accepted 7 October 2014

Academic Editor: Dan Simon

Copyright © 2015 Xiaoyan Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The nonlocal means algorithm is widely used in image denoising, but this algorithm does not work well for high-intensity noise. To overcome this shortcoming, we establish a coupled iterative nonlocal means model in this paper. Considering the computation complexity of the new model, we realize it by using multiscale wavelet transform and propose an asymptotic nonlocal filtering algorithm which can reduce the influence of noise on similarity estimation and computation complexity. Moreover, we build a new nonlocal weight function based on the structure similarity index. Simulation results indicate that the proposed approach cannot only remove the noise but also preserve the structure of image and has good visual effects, especially for highly degenerated images.

#### 1. Introduction

Noise in images introduced by formation and transmission is unavoidable, which makes postprocessing difficult. Denoising is always an important research in image processing and computer vision. Numerous denoising techniques have been proposed in the image processing literature including variation regularization [1–6], Partial Differential Equation [7, 8], wavelet shrinkage [9], and neighborhood filtering [10, 11]. In particular, the nonlocal means (NLM) algorithm, which is first presented in [12] by Buades et al., has drawn much research attention lately due to its excellent performance.

How to measure the similarity of original clean patches through noisy patches is the heart of the NLM algorithm. The Gaussian weighted distance is used to measure the similarity on different image patches in NLM, which is easily influenced by noise and does not use the structure information in the patch. In order to improve the accuracy of similarity estimation, multiple weight functions have been developed [13–16]. Tasdizen [13] proposed a principle neighborhood dictionary that measures the patch similarity in the domain of principle component analysis. Other transform domain based NLM algorithms are also proposed such as [14, 15]. Kervrann et al. [16] proposed a Bayesian NLM framework that measures the patch similarities based on the statistical distribution of noise. Rehman and Wang [17] directly used the structure similarity (SSIM) index to estimate the nonlocal weights (SSIM-based nonlocal means, SSIM-NLM). All these weights functions can generate good denoising results for low-intensity noise, while not working well for high-intensity noise.

In order to reduce the influence of noise, iterative NLM algorithm has been proposed in [18], and it has been extended in [19, 20]. They update the filtered image in iterative process, while estimating the nonlocal weights over original noisy image which will reintroduce noise. Another iterative NLM algorithm has been proposed in [21] from variation formulation. Though the similarity of patches may be measured more accurately from the denoised image rather than the noisy image, the residual noise is still in the results because the weighted averaging is implemented on noisy image. From statistics, Deledalle et al. proposed a similar iterative approach in [22]. The above two kinds of iterative NLM algorithms have better denoising effect; however, these approaches only update one variable during iteration. This noncorresponding could bring deviation. In order to avoid it, [23] established a new regularization functional by using the maximum entropy to estimate the nonlocal weights. This model can obtain better results because the nonlocal weights and filtered image both are updated in iterative process. However, its computational complexity is high.

In fact, the iterative NLM algorithm switches strong filtering into multiple relatively weak filtering. As we know, multiscale transform is an effective tool to realize asymptotic approximation of the signal. Based on this, multiscale NLM algorithms have been proposed in [24, 25]. Compared to [25] (multiscale NLM, MSNLM), our algorithm proposed in [24] can better preserve the details because subimage has orientation information with wavelet transform. Thus, in this paper, we use multiscale wavelet transform to realize the new coupled iterative NLM model and call it asymptotic nonlocal filtering (ANLF). In ANLF algorithm, the noisy image is transformed into a series of spatial subimages; then the nonlocal weights are estimated by relatively clean low-frequency part and weighted averaging is applied at the same layer to remove the noise. ANLF has the same way as [24] but out of different considerations. Moreover, we give a new nonlocal weight function based on SSIM index [26].

To summarize, in this paper, we propose a novel asymptotic nonlocal filtering for image denoising. Extensive experimental results show our method is valid. The contributions of this paper are threefold.(i)We establish a new NL regularization functional and deduce the corresponding coupled iterative NLM model.(ii)Using multiscale wavelet transform to realize the coupled iterative NLM model, the new method reduces the computational complexity.(iii)A new nonlocal weight function is given based on SSIM index which can provide more accurate similarity estimation.

#### 2. Coupled Iterative Nonlocal Model

In this section, we review the NLM algorithm and previous related iterative NLM algorithms. Then, we establish a coupled iterative nonlocal model which makes a good tradeoff between noise suppression and edge preservation.

##### 2.1. NLM Algorithm

We assume that the input noisy image is where : () is composed of the original clean image and the independent additive white Gaussian noise with zero mean and variance . NLM algorithm estimates the gray value of each pixel as the weighted average of all pixels whose Gaussian neighborhood looks like the neighborhood at pixel : where , is Gaussian function with standard deviation , is a normalizing factor, and is the decay parameter of weight function.

The similarity function represents the affinity between different pixels of the image, and it is defined as a similarity measure based on the -norm between intensity values. The discrete version of the NLM algorithm can be represented as where and is a small neighborhood centered at the pixel , is the weighted -norm and is the standard deviation of the Gaussian kernel, and is a normalizing factor.

##### 2.2. Previous Iterative NLM Algorithm

In 2005, Kindermann et al. [27] first proposed nonlocal regularization functional based on the calculus of variation and diffusion processes. And the following weighted nonlocal functional is proposed by Gilboa and Osher [18]: where , , , .

By using fixed point method, the corresponding iterative scheme is obtained:

The weights in (5) depend on original noisy image and are not involved in the iterative process. This will lead to reintroduced noise.

Different from the method shown in (4), Brox et al. [21] proposed another variation formulation based on a trivial variation principle, which can be written as

And they obtained the corresponding iterative scheme by using fixed point method:

Deledalle et al. [22] presented further iterative weighted maximum likelihood denoising with probabilistic patch-based weights (iPPB), which updates the weights in a data-driven way using both the noisy patch and iteratively filtered results. Same as (7), iPPB can give more accurate similarity weights, but its weighted averaging is always implemented over original noisy image .

##### 2.3. Coupled Iterative Nonlocal Model

The averaging procedure of the above iterative algorithms does not correspond to the weight function because there is only variable ( or ) in updating. This strategy is not reasonable from the perspective of game theory. Inspired by the nonlocal regularization for inverse problems [23], we directly use the maximum entropy approach to estimate and consider the following joint optimization: where is the regularization parameter.

Using the alternating minimum and fixed point method for (8), we obtain

It is obvious that (9) is a neighborhood filtering (Gaussian filtering), and the regularization parameter of (8) is just the filtering parameter which controls smoothing degree. This coupled system avoids the disadvantages of the methods mentioned in Section 2.2 through the mutual influence between and .

Assume that the total number of image pixels is , that the patch size is , and that the searching window is of size . The complexity of NLM is . The complexity of iterative NLM algorithm is , where is the number of iterations. It will not achieve the intention edge-preserving if is too small. Such as, literature [22] used 25 iterations to ensure iPPB algorithm to reach convergence. That is to say, the complexity of proposed coupled iterative NLM is high. Moreover, the number of iterations and the selection criterion of smoothing parameter are hard to be determined. In order to solve these problems, we propose an asymptotic nonlocal filtering algorithm in the next section.

#### 3. Asymptotic Nonlocal Filtering

For high-intensity noise, the coupled iterative NLM algorithms perform better than NLM in edge-preserving because they switch a strong filtering into multiple weaker filtering. It is well known that multiscale transform is the effective tool to realize asymptotic approximation of the signal and when the signal is decomposed into different subbands, the noise is decomposed and its variance is reduced layer by layer. According to this, we realize the coupled iterative NLM system by using multiscale wavelet transform.

##### 3.1. The Framework of ANLF

Wavelet transform is used to realize image multiscale decomposition because it is simple and can maintain the structure of image better than others (such as LP [25]).

Let the noisy image belong to zero layer. Using the Mallat algorithm [28], is decomposed into layers. At layer , the low-frequency part and the high-frequency part are, respectively, reconstructed from scale coefficient and wavelet coefficients , , , and then , ().

After wavelet transform, the noise of obeys or approximately obeys Gaussian distribution, so NLM can be applied to directly and we can obtain the denoised image by and the weights are computed by : where is the decay parameter of the weights at layer , depending on the noise variance of .

Then, we replace by and repeat the process above at the layer . This process will not stop until the layer . We call this algorithm ANLF or -ANLF and show whole process in Algorithm 1.