Modeling Space-variant Visual Acuity And Image Compression As Eigenvalue Problems

Abstract

We have suggested that sine-wave based acuity gratings (such as Gabor functions) should be replaced by a new set of test functions - functions which act as the eigenfunctions of an appropriate operator (Stewart & Pinkham, in press). The Hermite functions emerge as an important special case, and have recently been applied to problems in image analysis (Martens, 1990). The Hermite functions are generated as the eigenfunctions of a differential operator which produces a simple and appealing mathematical model of space-variant acuity. The operator’s mathematical properties are in better agreement with the spatial inhomogeneity of the visual field than is the Fourier kernel. Surprisingly, these functions are their own Fourier transformations, and so are easily related to the Fourier transforms which are used in image analysis. If we are correct, human factors work on visual acuity can be placed on a coherent mathematical foundation, and new approaches to evaluating the quality of visual images will emerge. Just as important, a space-variant model of human visual acuity includes as a special case, the space-invariant models of image analysis used in engineering studies of image processing. The Hermite decomposition of the stimulus is directly related to the eigenfunctions of our space-variant acuity operator. The theory of Markov chains can be tied directly to the components of such a Hermite decomposition. The Hermite breakdown of two-dimensional and one-dimensional fractional Brownian motion produces components having corresponding spectra. Consequently, the information inherent in two- dimensional fractional Brownian motion is preserved under the Hermite decomposition, and finds a natural expression as a Markov process. Markov models are widely used for automatic speech recognition. Similar methods can be used to capture the inherent structure of visual images. Some natural scenes can be represented by fractals. Many images can be treated as