We propose a bilateral filter with a locally controlled domain kernel for directional edge-preserving smoothing. Traditional bilateral filters use a range kernel, which is responsible for edge preservation, and a fixed domain kernel that performs smoothing. Our intuition is that orientation and anisotropy of image structures should be incorporated into the domain kernel while smoothing. For this purpose, we employ an oriented Gaussian domain kernel locally controlled by a structure tensor. The oriented domain kernel combined with a range kernel forms the directional bilateral filter. The two kernels assist each other in effectively suppressing the influence of the outliers while smoothing. To find the optimal parameters of the directional bilateral filter, we propose the use of Stein's unbiased risk estimate (SURE). We test the capabilities of the kernels separately as well as together, first on synthetic images, and then on real endoscopic images. The directional bilateral filter has better denoising performance than the Gaussian bilateral filter at various noise levels in terms of peak signal-to-noise ratio (PSNR).
Considerable work has also been done on optimizing the parameters of the bilateral filter for improving denoising performance. Peng and Rao used Stein's unbiased risk estimate (SURE) to find the optimal parameters of the Gaussian bilateral filter @cite_17 @cite_12 . Kishan and Seelamantula achieved this goal for a bilateral filter with a raised cosine range kernel @cite_10 . Chen and Shu used Chi-square unbiased risk estimate (CURE) for optimizing bilateral filter parameters in squared magnitude MR images @cite_14 @cite_21 .
@cite_14 @cite_21 @cite_10 @cite_12 @cite_17
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Edge-preserving smoothing is widely used in image processing and bilateral filtering is one way to achieve it. Bilateral filter is a nonlinear combination of domain and range filters. Implementing the classical bilateral filter is computationally intensive, owing to the nonlinearity of the range filter. In the standard form, the domain and range filters are Gaussian functions and the performance depends on the choice of the filter parameters. Recently, a constant time implementation of the bilateral filter has been proposed based on raised-cosine approximation to the Gaussian to facilitate fast implementation of the bilateral filter. We address the problem of determining the optimal parameters for raised-cosine-based constant time implementation of the bilateral filter. To determine the optimal parameters, we propose the use of Stein's unbiased risk estimator (SURE). The fast bilateral filter accelerates the search for optimal parameters by faster optimization of the SURE cost. Experimental results show that the SURE-optimal raised-cosine-based bilateral filter has nearly the same performance as the SURE-optimal standard Gaussian bilateral filter and the Oracle mean squared error (MSE)-based optimal bilateral filter. A statistical approach is presented for determining the optimal parameters of a bilateral filter for suppressing noise in an image. A closed-form solution to obtaining Stein's unbiased estimate of the bilateral filter-based image estimation risk is derived. The values of the parameters are then determined by minimization of the estimated risk, which is equivalent to maximization of the estimated signal to noise ratio. Experimental results show that the method provides estimates of noise intensity that are close to the actual, and the performance of the bilateral filter, in terms of providing good tradeoff between noise suppression and edge preservation, is improved significantly by using the optimal parameters estimate. Bilateral filter is an edge-preserving diffusion filter and is widely used for image denoising. The denoising quality is sensitive to the choice of the bilateral filter parameters. Here, we propose to use the Chi-Square Unbiased Risk Estimate (CURE) for optimizing bilateral filter parameters in squared magnitude MR images, which follow a non-central chi-square distribution with two degrees of freedom. CURE is an unbiased estimate of the Mean Squared Error (MSE) and it does not require any knowledge of the noise-free data. We derive explicit analytical expressions for CURE in the setting of bilateral filters and present experimental results that confirm the optimality of CURE. We propose to employ bilateral filters to solve the problem of edge detection. The proposed methodology presents an efficient and noise robust method for detecting edges. Classical bilateral filters smooth images without distorting edges. In this paper, we modify the bilateral filter to perform edge detection, which is the opposite of bilateral smoothing. The Gaussian domain kernel of the bilateral filter is replaced with an edge detection mask, and Gaussian range kernel is replaced with an inverted Gaussian kernel. The modified range kernel serves to emphasize dissimilar regions. The resulting approach effectively adapts the detection mask according as the pixel intensity differences. The results of the proposed algorithm are compared with those of standard edge detection masks. Comparisons of the bilateral edge detector with Canny edge detection algorithm, both after non-maximal suppression, are also provided. The results of our technique are observed to be better and noise-robust than those offered by methods employing masks alone, and are also comparable to the results from Canny edge detector, outperforming it in certain cases.