Theoretical Foundations of Gaussian Convolution by Extended Box Filtering

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.