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).
The nonlinearity of @math makes the bilateral filter computationally expensive in its standard form. However, they remain attractive as a number of works have been dedicated to accelerate them. Paris and Durand derived criteria for downsampling in space and intensity to come up with a fast approximation of the bilateral filter @cite_20 . A constant-time algorithm for fast bilateral filtering has been proposed in @cite_0 @cite_8 . achieved substantial acceleration at the cost of quantization @cite_3 .
@cite_0 @cite_3 @cite_20 @cite_8
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The bilateral filter is a nonlinear filter that smoothes a signal while preserving strong edges. It has demonstrated great effectiveness for a variety of problems in computer vision and computer graphics, and fast versions have been proposed. Unfortunately, little is known about the accuracy of such accelerations. In this paper, we propose a new signal-processing analysis of the bilateral filter which complements the recent studies that analyzed it as a PDE or as a robust statistical estimator. The key to our analysis is to express the filter in a higher-dimensional space where the signal intensity is added to the original domain dimensions. Importantly, this signal-processing perspective allows us to develop a novel bilateral filtering acceleration using downsampling in space and intensity. This affords a principled expression of accuracy in terms of bandwidth and sampling. The bilateral filter can be expressed as linear convolutions in this augmented space followed by two simple nonlinearities. This allows us to derive criteria for downsampling the key operations and achieving important acceleration of the bilateral filter. We show that, for the same running time, our method is more accurate than previous acceleration techniques. Typically, we are able to process a 2 megapixel image using our acceleration technique in less than a second, and have the result be visually similar to the exact computation that takes several tens of minutes. The acceleration is most effective with large spatial kernels. Furthermore, this approach extends naturally to color images and cross bilateral filtering. This paper presents three novel methods that enable bilateral filtering in constant time O(1) without sampling. Constant time means that the computation time of the filtering remains same even if the filter size becomes very large. Our first method takes advantage of the integral histograms to avoid the redundant operations for bilateral filters with box spatial and arbitrary range kernels. For bilateral filters constructed by polynomial range and arbitrary spatial filters, our second method provides a direct formulation by using linear filters of image powers without any approximation. Lastly, we show that Gaussian range and arbitrary spatial bilateral filters can be expressed by Taylor series as linear filter decompositions without any noticeable degradation of filter response. All these methods drastically decrease the computation time by cutting it down constant times (e.g. to 0.06 seconds per 1MB image) while achieving very high PSNRpsilas over 45 dB. In addition to the computational advantages, our methods are straightforward to implement. 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. We propose a new bilateral filtering algorithm with computational complexity invariant to filter kernel size, so-called O(1) or constant time in the literature. By showing that a bilateral filter can be decomposed into a number of constant time spatial filters, our method yields a new class of constant time bilateral filters that can have arbitrary spatial and arbitrary range kernels. In contrast, the current available constant time algorithm requires the use of specific spatial or specific range kernels. Also, our algorithm lends itself to a parallel implementation leading to the first real-time O(1) algorithm that we know of. Meanwhile, our algorithm yields higher quality results since we are effectively quantizing the range function instead of quantizing both the range function and the input image. Empirical experiments show that our algorithm not only gives higher PSNR, but is about 10× faster than the state-of-the-art. It also has a small memory footprint, needed only 2 of the memory required by the state-of-the-art for obtaining the same quality as exact using 8-bit images. We also show that our algorithm can be easily extended for O(1) median filtering. Our bilateral filtering algorithm was tested in a number of applications, including HD video conferencing, video abstraction, highlight removal, and multi-focus imaging.