Date Published: May 10, 2019
Publisher: Public Library of Science
Author(s): Khuram Naveed, Bisma Shaukat, Shoaib Ehsan, Klaus D. Mcdonald-Maier, Naveed ur Rehman, Ahmadreza Baghaie.
Two novel image denoising algorithms are proposed which employ goodness of fit (GoF) test at multiple image scales. Proposed methods operate by employing the GoF tests locally on the wavelet coefficients of a noisy image obtained via discrete wavelet transform (DWT) and the dual tree complex wavelet transform (DT-CWT) respectively. We next formulate image denoising as a binary hypothesis testing problem with the null hypothesis indicating the presence of noise and the alternate hypothesis representing the presence of desired signal only. The decision that a given wavelet coefficient corresponds to the null hypothesis or the alternate hypothesis involves the GoF testing based on empirical distribution function (EDF), applied locally on the noisy wavelet coefficients. The performance of the proposed methods is validated by comparing them against the state of the art image denoising methods.
The acquisition and transmission normally corrupt an image by introducing an additive noise. In this regard, image denoising algorithms are utilized to suppress noise while preserving the desired image features. Let xp,q denote a pixel of a noisy N × N sized image X at location (p, q), acquired from an acquisition device, a transmission medium or a reconstruction process as
where sp,q denotes the pixels of the true image S while ηp,q denotes noise at pixel location (p, q). In matrix form, the above equation can be written as
The goal of denoising is to estimate the true signal S from its noisy observation X. Here, η is considered an independent Gaussian noise N(0,σ2) with zero mean and arbitrary variance σ2.
Two novel image denoising methods are proposed which employ GoF test on the wavelet coefficients of the noisy image obtained by using DWT and DTCWT respectively. The DT-CWT exhibits approximate translation invariance and directional selectivity which helps it to suppress the artifacts otherwise present in the DWT based denoising results. We denote the proposed denoising methods as the GoFShrink based on the DWT and the DT-CWT.
In this section we present the computational cost of the GoFShrink based on DWT. The computational cost of the GoFShrink based on DT-CWT will be four times to that of GoFShrink based on DWT, provided the length of filters used by both transforms is exactly the same.
This section presents the performance comparison of the proposed algorithms against the state of the art in image denoising. The peak signal to noise ratio (PSNR) has been employed as the measure of quantitative performance, given as
The mean squared error (MSE) is calculated as
where sp,q denotes pixels of the true image S of size N × N and s^p,q represents the pixels of the denoised image S^. Note that MSE of noisy image is equal to the variance of the noise σ2.
A class of multiscale image denoising algorithms have been proposed which employ the goodness of fit test on multiple image scales obtained from discrete wavelet transform (DWT) and dual tree complex wavelet transform (DT-CWT). The Anderson Darling (AD) statistics have been employed, within the framework of GoF test, on the wavelet coefficients of the noisy image to compute the distance between the empirical distribution function (EDF) of local coefficients and the CDF of reference Gaussian noise. A local thresholding function is then used to classify the wavelet coefficients as belonging to signal or noise depending on the given probability of false alarm (Pfa) and the estimated AD statistic. The signal coefficients are retained while the noise coefficients are discarded to yield the denoised image. While the current work only deals with the case of Gaussian noise, the proposed scheme has potential to remove any type of noise with prior knowledge of the noise distribution. The proposed methods have been shown to outperform the state-of-the-art image denoising methods on a variety of input images ranging from standard test datasets to medical and diffusion images. The results have revealed that from the two proposed methods, the GoFShrink-DT (based on DT-CWT) has outperformed the GoFShrink-TI (based on DWT) which was expected given directional selectivity and translation invariance of the DT-CWT transform.