Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 5245948, 13 pages

http://dx.doi.org/10.1155/2016/5245948

## A Residual-Based Kernel Regression Method for Image Denoising

^{1}College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China^{2}Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA

Received 7 October 2015; Revised 4 January 2016; Accepted 18 January 2016

Academic Editor: Kacem Chehdi

Copyright © 2016 Jiefei Wang 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

We propose a residual-based method for denoising images corrupted by Gaussian noise. In the method, by combining bilateral filter and structure adaptive kernel filter together with the use of the image residuals, the noise is suppressed efficiently while the fine features, such as edges, of the images are well preserved. Our experimental results show that, in comparison with several traditional filters and state-of-the-art denoising methods, the proposed method can improve the quality of the restored images significantly.

#### 1. Introduction

In image processing, noise reduction with preserving fine features is a long-standing research problem. Many applications such as image segmentation and medical image analysis require effective noise suppression to obtain reliable results. In recent years, many efficient mathematical methods such as kernel regression based methods and locally adaptive based methods for the purpose of image denoising have been developed in the literature (see [1–10] and the references cited therein). For example, Tomasi and Manduchi proposed the well-known bilateral filtering for image filtering in [1], which exploits the local structural patterns to regularize the image filtering procedure and assume that images are locally smooth except at edges. The bilateral filter can preserve certain details of images, while more information inherent in the image still remains to be discovered and utilized. Tschumperlé [2] proposed a common framework for image filtering which is based on the iterative local diffusion in the image guided by the local structure tensor. The method makes use of the local gradients of the image to control the shape of the kernel and hence to keep texture details of the image efficiently. However, for a very noisy image, the gradients of the image are likely to be corrupted by the noise and thus might raise huge errors in terms of gradient estimator in smooth areas, which might generate negatively “ripple” phenomena. To overcome the drawbacks mentioned above, Takeda et al. [3] further proposed an iterative steering kernel regression filter, of which an iterative filtering procedure is used to exploit the output (less noisy) image of each iteration to improve the local orientation estimates used in the kernel in the next iteration. The implementation of the method is very time-consuming, even exceeding that of nonlocal methods [11, 12]; and it shows instability in estimating the local gradients of the image under high-level noisy case. López-Rubio and Florentín-Núñez [4] proposed to adopt a general kernel regression framework to 3D MR image denoising. The method is rooted on a zeroth-order 3D kernel regression, which calculates a weighted average of the pixels over a regression window; and the weights are obtained from the similarities among small sized feature vectors associated to each pixel. One common feature for these kernel regression methods is the exploration of the local structural regularity properties in natural images/videos. Based on this observation, the kernel regression based methods (or combined with other methods) are also successfully applied to image and video deblurring, upscaling, interpolation, fusion [3], superresolution [6, 13–16], registration [17], JPEG image deblocking [18], and so forth. On the other hand, the residual-based image denoising methods have also been studied. For instance, in [19], the corrupted image was first denoised with the total variation minimization algorithm; then the residual image was denoised with the adaptive Wiener filter; and finally, based on a statistical test, the filtered residuals were added back to the denoised image. Similarly, in [20], the authors first adopted nonlocal means filter to denoise the degraded image and added the final filtered residual image to the denoised image to recover the lost image details. In [21], the image denoising was implemented in the wavelet transform domain. In the method, based on the kurtosis statistic of the residual image, a criterion was proposed for determining wavelet coefficient thresholds.

In this paper, different from the residual-based methods in [19–21], we propose a new residual-based denoising method, which combines the bilateral filter and the local structure adaptive kernel filter by making use of their respective advantages. The proposed method can prevent the fine details loss caused by the bilateral kernel regression method and avoid image ripple phenomena produced from the local structure adaptive kernel method. Numerical experiments demonstrate that, for images contaminated with Gaussian noise, the performance of our combined approach is generally superior to those of some traditional and state-of-the-art restoration methods in terms of the image quality measured by PSNR and SSIM [22].

#### 2. Classic Kernel Regression Methods

In this section, we briefly review bilateral kernel and structural adaptive kernel methods which will be used in the development of our new method later.

Classical regression methods for image processing rely on a specific model of the signal of interest and calculate the parameters of the model in the presence of noise. Compared to the parametric methods, kernel regression methods [23, 24] rely on the data itself to dominate the structure of the model. Generally, the traditional kernel regression methods are based on solving the following optimization problem: where is a set that consists of the index numbers of the neighbors of ; is the pixel observation at ; denotes the estimate of the pixel at location of the image. is a generic kernel (e.g., Gaussian, exponential, or other forms) which is used as the penalty for distance between and , so that the influence of on decreases as the distance between these two points increases. Considering as the first coefficient of some regression function on the coordinates , (1) is essentially a zero-order estimation. In order to approximate the image local structure better, higher order estimation was used in [3, 25]. Specifically, assuming that the image is locally smooth to some order, we can relay on the local expansion of the function using the Taylor series where denotes the operator that extracts the lower triangular part of a matrix and stacks it to a column vector. Then, the optimal regression coefficients can be calculated from the following optimization problem: Although all regression coefficients can be estimated effectively, the first element is the desired pixel value estimation at . For the details of computing , one can refer to [3, 23, 24].

As mentioned above, the traditional kernel methods simply take the distance of the pixels into consideration. That is, large weight is assigned to nearby pixels, while smaller weights are assigned to farther pixels. This may lead to obscurity of the image, especially on the edges. To relieve the problem, Tomasi and Manduchi [1] proposed the bilateral filter as solving the following optimization problem: where . Here, the kernel functions and are used for penalizing the spatial distance between the pixel of interest and its neighbors and the radiometric “distance” between the corresponding pixels and . The main idea of bilateral kernel is that, when calculating weights, it considers both the pixel distance and space distance between the points at and . Then, due to considerable variation of the pixel values around the image edges, the values of the associated weights are relatively small, which contributes to preserving the edges of the image. In addition, the generalizations of the bilateral filter, including the choices of various kernels and higher order estimation, were reported in [10, 26, 27].

Although the bilateral filter can solve the indistinction problem of image margins to some extent, only distance information of the pixels is used, which might make a loss of details in the textures of image margins. Tschumperlé [2] further proposed to utilize correlation information of pixels to estimate the local directions of the image. And he constructed a local structure adaptive kernel as follows: where is the diffusion tensor controlling the spatial structure of the kernel and is the parameter that controls the bandwidth of the structural kernel. In order to obtain , we construct a structure tensor, which can be defined as follows: where is a Gaussian kernel and and are the -gradient and -gradient at coordinates , respectively. Here we use Prewitt operator [23, 24] to estimate the gradient. After performing eigenvalue decomposition on , we get where are the eigenvalues of with , reflecting the strength of the gradient along each eigenvector direction and and are the eigenvectors of . According to methods for finding diffusion tensor described in [2, 24], we set where Here, is a sensible parameter so that the shape of the local structure adaptive kernel will respond to the local gradient of the image.

#### 3. A Residual-Based Joint Denoising Method

Bilateral filter and structure adaptive denoising methods described above can restore the noisy images efficiently. For example, the original noise-free image “Lena” shown in Figure 1(a) is degraded by an additive white Gaussian noise with noise variances , as shown in Figure 2(a). We test the two methods on the Noisy “Lena" image, respectively. As we can see, the bilateral filter with PSNR = 27.56 dB, SSIM = 0.86 and the structure adaptive denoising method with PSNR = 30.20 dB, SSIM = 0.84 exhibit acceptable image quality as shown in Figures 2(b) and 2(c), respectively. On the other hand, as shown in the figure, the bilateral filter can smooth the Noisy “Lena” efficiently but erase certain details of the image in the meanwhile; and the local self-adaptive method can deliver the fine features, such edges, of the image, whereas it might generate image ripples caused by noise in the smooth area. Based on this observation, we here propose a new method, that is, a residual feedback-based joint image denoising method using bilateral filter and structure adaptive methods.