Fast space-variant elliptical filtering using box splines

TL;DR

This paper introduces a fast, efficient algorithm for space-variant elliptical filtering in image processing, approximating Gaussian-like windows with controllable size, shape, and orientation using box splines, achieving constant computation per pixel.

Contribution

The paper presents a novel method utilizing radially-uniform box splines for efficient space-variant elliptical filtering with fixed computations per pixel, regardless of filter shape or size.

Findings

Achieves O(1) computations per pixel for space-variant elliptical filtering.

Uses box splines to approximate anisotropic Gaussians with controllable parameters.

Demonstrates efficient filtering with varying size, elongation, and orientation.

Abstract

The efficient realization of linear space-variant (non-convolution) filters is a challenging computational problem in image processing. In this paper, we demonstrate that it is possible to filter an image with a Gaussian-like elliptic window of varying size, elongation and orientation using a fixed number of computations per pixel. The associated algorithm, which is based on a family of smooth compactly supported piecewise polynomials, the radially-uniform box splines, is realized using pre-integration and local finite-differences. The radially-uniform box splines are constructed through the repeated convolution of a fixed number of box distributions, which have been suitably scaled and distributed radially in an uniform fashion. The attractive features of these box splines are their asymptotic behavior, their simple covariance structure, and their quasi-separability. They converge to Gaussians with the increase of their order, and are used to approximate anisotropic Gaussians of varying covariance simply by controlling the scales of the constituent box distributions. Based on the second feature, we develop a technique for continuously controlling the size, elongation and orientation of these Gaussian-like functions. Finally, the quasi-separable structure, along with a certain scaling property of box distributions, is used to efficiently realize the associated space-variant elliptical filtering, which requires O(1) computations per pixel irrespective of the shape and size of the filter.