Two Tricks for the Price of One: Linear Filters and Their Transposes

Abstract

Many image processing operations exist in pairs: a forward-mapping version and a reverse-mapping version [Wolberg 94, Wolberg et al. 00]. In this paper, we show that the forward and reverse versions of spatially-varying convolution filters are transposes of each other. We will then show how an implementation of a function that operates linearly on a set of variables may be transformed into an implementation of its transpose function. This gives a mechanical procedure to convert a reverse spatially-varying convolution into a forward one and vice versa. Although this approach is general-purpose, we focus on one particular type of application: applying this transformation to fast algorithms for reverse convolution based on running sums or summed-area tables [Lewis 95, Crow 84] yielding novel fast algorithms for forward convolution. For many practical applications, such as simulating depth of field and motion blur, the forward convolution can often yield more visually appealing results while the reverse-mapping algorithm has traditionally been more straightforward to implement.