A two dimensional field theory for motion computation. First order approximation; translatory motion of rigid patterns.

Abstract

The local extraction of motion information from brightness patterns by individual movement detectors of the correlation-type is considered in the first part of the paper. A two-dimensional field theory of movement detection is developed by treating the distance between two adjacent photoreceptors as a differential. In the first approximation of the theory we only consider linear terms of the time interval between the reception of a contrast element and its delayed representation by the detector and linear terms of the spatial distances between adjacent photoreceptors. As a result we may neglect terms of higher order than quadratic in a Taylor series development of the brightness pattern. The responses of pairs of individual movement detectors are combined to a local response vector. In the first approximation of the detector field theory the response vector is proportional to the instantaneous pattern velocity vector and linearly dependent on local properties of the moving pattern. The linear dependence on pattern properties is represented by a two by two tensor consisting of elements which are nonlinear, local functionals of the moving pattern. Some of the properties of the tensor elements are treated in detail. So, for instance, it is shown that the off-diagonal elements of the tensor disappear when the moving pattern consists of x- and y-dependent separable components. In the second part of the paper the tensor relation leading to the output of a movement detector pair is spatially integrated. The result of the integration is an approximation to a summation of the outputs of an array of detector pairs. The spatially integrated detector tensor relates the translatory motion vector to the resultant output vector. It is shown that the angle between the motion vector and the resultant output vector is always smaller than +/- 90 degrees whereas the angle between the motion vector and local response vectors, elicited by detector pairs, may cover the entire angular range. In the discussion of the paper the limits of the field theory for motion computation as well as its higher approximations are pointed out in some detail. In a special chapter the dependence of the detector response on the pattern properties is treated and in another chapter questions connected with the so called aperture problem are discussed. Furthermore, properties for compensation of the pattern dependent deviation angle by spatial physiological integration are mentioned in the discussion.