A MAP-MRF Approach for Wavelet-Based Image Denoising

Abstract

Image denoising is a required pre-processing step in several applications in image processing and pattern recognition, from simple image segmentation tasks to higher-level computer vision ones, as tracking and object detection for example. Therefore, estimating a signal that is degraded by noise has been of interest to a wide community of researchers. Basically, the goal of image denoising is to remove the noise as much as possible, while retaining important features, such as edges and fine details. Traditional denoising methods have been based on linear filtering, where the most usual choices were Wiener, convolutional finite impulse response (FIR) or infinitie impulse response (IIR) filters. Lately, a vast literature on non-linear filtering has emerged Barash (2002); Dong & Acton (2007); Elad (2002); Tomasi & Manduchi (1998); Zhang & Allebach (2008); Zhang & Gunturk (2008), especially those based on wavelets Chang et al. (2000); H. et al. (2009); Ji & Fermuller (2009); Nasri & Nezamabadi-pour (2009); Yoon & Vaidyanathan (2004) inspired by the remarkable works of Mallat (1989) and after Donoho (1995). The basic wavelet denoising problem consists in, given an input noisy image, dividing all its wavelet coefficients into relevant (if greater than a critical value) or irrelevant (if less than a critical value) and then process the coefficients from each one of these groups by certain specific rules. Usually, in most denoising applications soft and hard thresholding are considered, in a way that filtering is performed by comparing each wavelet coefficient to a given threshold and supressing it if its magnitude is less than the threshold; otherwise, it is kept untouched (hard) or shrinked (soft). Soft-thresholding rule is generally preferred over hard-thresholding for several reasons. First, it has been shown that soft-thresholding has several interesting and desirablemathematical propertiesDonoho (1995), Donoho & Johnstone (1994). Second, in practice, the soft-thresholding method yields more visually pleasant images over hard-thresholding because the latter is discontinuous and generates abrupt artifacts in the recovered images, especially when the noise energy is significant. Last but not least, some results found in the literature Chang et al. (2000) conclude that the optimal soft-thresholding estimator yields a smaller estimation error than the optimal hard-thresholding estimator. However, for some classes of signals and images, hard-thresholding results in superior estimates to that of soft-thresholding, despite some of its disadvantages Yoon & Vaidyanathan (2004). To tackle this problem, several hybrid thresholding functions have been proposed in the literature. 0