A Fast Constant-Time Approximation for Locally Adaptive Bilateral Filtering

Abstract

Smoothing is a fundamental task in low-level image processing that is used to suppress irrelevant details while preserving salient image structures. The simplest smoothing mechanism is to average neighboring pixels using a spatial kernel. While this works well when the kernel is narrow, it inevitably results in blurring of edges when the kernel is wide. This problem can be alleviated using a range kernel along with the spatial kernel. The range kernel automatically damps out the smoothing action near an edge and is turned o in homogeneous regions where greater smoothing is required. A canonical prototype in this regard is the bilateral filter in which both kernels are Gaussian. A flip side of the range kernel is that it makes the bilateral filter non-linear and computationally expensive. However, several fast algorithms have been proposed in the literature that allow the filter to be implemented in real-time. The focus of this thesis is on a generalization of the classical bilateral filter in which the center and width of the range kernel are allowed to change from pixel to pixel. This so-called adaptive bilateral filter was originally proposed for image sharpening and noise removal, but it can also be used for other applications. Similar to the classical bilateral filter, its brute-force implementation requires intense computations. However, most fast algorithms for classical bilateral filtering require the range kernel to be fixed, and hence cannot be extended for the adaptive counterpart. For the first time, we propose a fast algorithm for adaptive bilateral filtering. The algorithm is constant-time in that the computational complexity does not scale with the the width of the spatial kernel. At the core of the algorithm is the observation that the filtering can be performed purely in range space using an appropriately defined local histogram. By replacing the histogram with a polynomial and the finite sum in range-space with an integral, we can approximate the filter using a series of definite integrals. We derive an efficient algorithm from this analytic approximation using the following innovations: the polynomial is fitted by matching its moments to those of the target histogram (this is done using fast convolutions), and the integrals are recursively computed using integration-by-parts. The proposed algorithm can achieve at least 20 acceleration over the brute-force computation, without perceptible distortions in visual quality. We demonstrate the effectiveness of our algorithm for sharpening, removal of compression artifacts, texture filtering, and saliency-driven detail enhancement