The Fourier-Argand Representation: An Optimal Basis of Steerable Patterns

Computing the convolution between a 2D signal and a corresponding filter with variable orientations is a basic problem that arises in various tasks ranging from low level image processing (e.g. ridge/edge detection) to high level computer vision (e.g. pattern recognition). Through decades of research, there still lacks an efficient method for solving this problem. In this paper, we investigate this problem from the perspective of approximation by considering the following problem: what is the optimal basis for approximating all rotated versions of a given bivariate function? Surprisingly, solely minimising the L
²
-approximation-error leads to a rotation-covariant linear expansion, which we name Fourier-Argand representation. This representation presents two major advantages: 1) rotation-covariance of the basis, which implies a “strong steerability” - rotating by an angle α corresponds to multiplying each basis function by a complex scalar e
^-ikα
; 2) optimality of the Fourier-Argand basis, which ensures a few number of basis functions suffice to accurately approximate complicated patterns and highly direction-selective filters. We show the relation between the Fourier-Argand representation and the Radon transform, leading to an efficient implementation of the decomposition for digital filters. We also show how to retrieve accurate orientation of local structures/patterns using a fast frequency estimation algorithm.