Journal of Electrical and Computer Engineering

Volume 2015, Article ID 459285, 12 pages

http://dx.doi.org/10.1155/2015/459285

## A Novel Directionlet-Based Image Denoising Method Using MMSE Estimator and Laplacian Mixture Distribution

School of Electrical Engineering and Automation, Anhui University, Hefei 230601, China

Received 1 October 2014; Revised 29 December 2014; Accepted 23 February 2015

Academic Editor: Igor Djurović

Copyright © 2015 Yixiang Lu 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

A novel method based on directionlet transform is proposed for image denoising under Bayesian framework. In order to achieve noise removal, the directionlet coefficients of the uncorrupted image are modeled independently and identically by a two-state Laplacian mixture model with zero mean. The expectation-maximization algorithm is used to estimate the parameters that characterize the assumed prior model. Within the framework of Bayesian theory, the directionlet coefficients of noise-free image are estimated by a nonlinear shrinkage function based on weighted average of the minimum mean square error estimator. We demonstrate through simulations with images contaminated by additive white Gaussian noise that the proposed method is very competitive when compared with other methods in terms of both peak signal-to-noise ratio and visual quality.

#### 1. Introduction

The recent advancement in multiscale geometric analysis has promoted an enormous amount of research in the field of image processing, including image denoising. The distortion of images by additive white Gaussian noise is common during the images acquisition, processing, compression, and transmission. The goal of image restoration is to reconstruct a plausible estimate of the original image from the distorted image or observation and keep the image features as much as possible at the same time.

As a representative of multiscale geometric analysis tools, wavelet transform (WT) [1] played an important role in the application of image denoising and has achieved great success over the last decade. There are plenty of works on image denoising using wavelet thresholding since Donoho [2, 3] has done the pioneering work for signal denoising using WT. The method described in [3] has near-optimal properties in the minimax sense and performs well in simulation studies of one-dimensional curve estimation. The image denoising methods using threshold mainly consist of three steps: (1) representing the image sparsely using WT; (2) determining an appropriate threshold and comparing the coefficients with it, preserving or modifying the few high-magnitude transform coefficients that convey mostly the true-signal energy and discarding the rest which are mainly due to noise; (3) reconstructing the denoised image using inverse WT based on the thresholded coefficients. Due to their effectiveness and simplicity that operate on one wavelet coefficient at a time, various wavelet-based thresholding methods were proposed for image denoising [4–6]. Although the denoising performance of the wavelet thresholding is better than that of spatial filters (such as the filters proposed in [7–9]), the problem of how to develop an optimal threshold used for distinguishing between the coefficients that are mainly due to signal and those mainly due to noise has not been solved well. Thus, for the issue of image denoising, the wavelet-based thresholding algorithms with threshold are not an optimal strategy.

To improve the denoised results, the prior probabilistic knowledge of the noise-free image and noise image in wavelet domain is exploited in the process of image denoising. In the statistical wavelet-based denoising approaches, instead of using an arbitrary threshold to shrink the noisy wavelet coefficients, the shrinkage function is designed by minimizing a Bayesian risk, typically under the minimum mean square error (MMSE) criterion [10], minimum mean absolute error (MMAE) criterion [11], or maximum a posteriori (MAP) criterion [12]. In general, modeling the statistics of natural images is a challenging task, partly because of the high dimension of the signal. However, the power of statistical image models can be substantially improved by representing the spatial image using WT, since the WT nearly decorrelates the dependencies between image pixels. By exploiting the nature of the histogram of the wavelet coefficients, different noise reduction algorithms try to incorporate the prior model that best matches the true image coefficients. In [13], Mallat proposed that the wavelet coefficients show non-Gaussian behavior and sharp peak centering around zero with heavy-tailed distribution and used the generalized Gaussian distribution (GGD) with the shape parameter to fit the coefficients. Since then, the GGD has been commonly used to model the image wavelet coefficients and accordingly derived different image denoising algorithms based on some criteria [14, 15]. For example, Moulin and Liu [15] first assumed that the wavelet coefficients follow a simple independent and identically distributed GGD prior, then estimated the shape parameter using Mallat’s algorithm proposed in [13], and finally developed a shrinkage estimator using MAP criterion based on the assumed prior. In addition to the GGD prior, several other PDFs have also been proposed to model the wavelet coefficients of the image, such as Gaussian scale mixture (GSM) distribution [16, 17], -stable distribution [18], and Bessel -form distribution [19]. Among these proposed models used for the purpose of denoising, the GSM model accounts for both marginal and pairwise joint distributions of the wavelet coefficients, and thus the corresponding denoising methods outperform the methods using other prior models. Although the -stable model provides a more accurate fitness to the histogram of the WT, this probability density function (PDF) has no closed-form expression even though its characteristic function has one, and its parameter estimation is poor, specially in the presence of noise [20]. This disadvantage hampers its wide application for image denoising in practice. It has been shown that the application of prior knowledge in the noise reduction algorithms results in a performance better than that of the wavelet-based denoising methods using thresholds. However, due to the inherent drawbacks of the wavelet bases which lack anisotropy and have poor directional selectivity, wavelets fail to capture the geometric regularity along the singularities of surfaces (e.g., the edges and contours), thus seriously affecting the performances of the wavelet-based methods.

To exploit the anisotropic regularity of a surface along edges, several image representations with elongated basis functions have been proposed to capture the geometric regularity of a given image, including curvelets [21], bandelets [22], and contourlets [23]. They were also introduced into the field of image denoising because of their ability of efficiently representing the anisotropic regularity [24, 25]. For example, in [25], Eslami and Radha modeled the joint PDF of parents and children of the coefficients in semi-translation-invariant contourlet transform and implemented image denoising by constructing a bivariate shrinkage function under MAP criterion. However, since the directional filter banks used in these representation tools are inseparable, the construction of directional filters is very complex, which is the major disadvantage of these transforms. Moreover, the process of image decomposition using directional filters is more time consuming than decomposition using wavelets.

Apart from these image representation tools mentioned above, another related image representation scheme called sparse representation [26, 27] was proposed and extended to the image denoising community. Sparse representation accounts for most or all information of a signal with a linear combination of a small number of elementary signals called atoms which are chosen from a so-called overcomplete dictionary. The results reported in [27] demonstrated that the denoising performance based on learned dictionaries surpasses that of wavelet-based methods. However, the algorithm imposes a very high computational burden, since it utilizes highly overcomplete dictionaries obtained via a preliminary training procedure.

In this paper, a new denoising method based on directionlet transform is proposed. Because of anisotropy and multidirectionality of directionlets, they allow us to capture more image features which can enhance the denoising performance. To recover the noise-free image using the Bayesian technique, we model the directionlet coefficients to be a two-state Laplacian mixture PDF with zero mean and use the expectation-maximization (EM) algorithm which is a specialization to the mixture density estimation problem to estimate the parameters involved in the assumed PDF. The noise-free coefficients are estimated by using average MMSE estimator.

The rest of this paper is organized as follows. The directionlet transform is briefly reviewed in Section 2. The statistical model of the directionlet coefficients of the noise-free image is studied in Section 3 as well as the discussion on the goodness of fit. Moreover, the design of the MMSE estimator is also presented in this section. The performance of the proposed algorithm is evaluated by using experimental results and compared with other existing denoising methods in Section 4. Finally, some conclusions are drawn in Section 5.

#### 2. Directionlet Transform

Discontinuity curves (i.e., the edges and contours) presented in the images are highly anisotropic and they are characterized by a geometrical coherence. These features are not properly captured by the standard two-dimension (2D) WT that uses isotropic basis functions. Because of this, when an image is decomposed by the standard 2D WT, many wavelet bases intersect the discontinuity curves and result in many large magnitude coefficients. Obviously, in such cases, WT is not an optimal sparse representation for images. To efficiently capture these elongated structures characterized by geometric regularity along different directions (not only the horizontal and vertical), the directionlet transform [28] based on lattice is proposed recently. This transform is constructed by using two concepts: directionality and anisotropy, and they are realized through two steps, respectively. First, use an integer lattice which is represented by a nonunique generator matrixto partition the discrete space, where the two vectors (with slope ) and (with slope ) determine two directions which are called transform direction and alignment direction, respectively. By sampling the image with generator matrix , one can get cosets. Second, achieve the anisotropy by using an unequal number of 1D wavelet transform steps along the two directions; that is, the 1D transform is applied more along one direction than the other direction. It should be noticed that if , then the iterated 1D transform should be applied in each coset separately. For notational simplicity, the directionlet transform is denoted as SAWT() in the rest of this paper, where and are the numbers of 1D WT along transform direction and alignment direction in each iteration, respectively. The basis functions of SAWT are called directionlets, which include elongated functions that are very suitable for image representation. The frequency decomposition of SAWT() along 0° for images is shown in Figure 1.