Theoretical Foundations of Gaussian Convolution by Extended Box Filtering

Abstract

Abstract. Gaussian convolution is of fundamental importance in linear scale-space theory and in numerous applications. We introduce iterated extended box filtering as an efficient and highly accurate way to compute Gaussian convolution. Extended box filtering approximates a continuous box filter of arbitrary non-integer standard deviation. It provides a much better approximation to Gaussian convolution than conventional iterated box filtering. Moreover, it retains the efficiency benefits of iterated box filtering where the runtime is a linear function of the image size and does not depend on the standard deviation of the Gaussian. In a detailed mathematical analysis, we establish the fundamental properties of our approach and deduce its error bounds. An experimental evaluation shows the advantages of our method over classical implementations of Gaussian convolution in the spatial and the Fourier domain.

A Survey of Gaussian Convolution Algorithms

“…Extended box filter (algorithm 8). Gwosdek, Grewenig, Bruhn, and Weickert [30] extend the box filter to allow a fractional radius,…”

An Analysis of the SURF Method

“…Second moment comparison One may for instance choose to match the second order moment σ 2 of the 1D Gaussian g σ and the variance of the corresponding box filter, as suggested by [8]. This leads to the relation…”

“…SURF scale parameter analogy Note that box filters are only used to approximate the first and second order of Gaussian derivatives in the SURF algorithm, and not to approximate Gaussian filtering like in [8]. However, when considering the approximation of the second order Gaussian derivative…”

Computing an Exact Gaussian Scale-Space

“…These methods can be described either as approximations of the continuous Gaussian convolution or as linear filters designed to satisfy some of the previously introduced properties expressed in the discrete framework. Other methods, not discussed in this work, include the use of recursive filters [15,3] or the iteration of extended box filters [7]. They provide fast and accurate approximations of the Gaussian convolution for large σ values but they crudely approximate the Gaussian function for low values of σ (typically σ ≤ 1), making them unsuitable for an accurate computation of the Gaussian scale-space.…”