Toward a mathematical theory of perception

Abstract

A new technique for the modelling of perceptual systems called formal modelling is developed. This technique begins with qualitative observations about the perceptual system, the so-called perceptual symmetries, to obtain through mathematical analysis certain model structures which may then be calibrated by experiment. The analysis proceeds in two different ways depending upon the choice of linear or nonlinear models. For the linear case, the analysis proceeds through the methods of unitary representation theory. It begins with a unitary group representation on the image space and produces what we have called the fundamental structure theorem. For the nonlinear case, the analysis makes essential use of infinite-dimensional manifold theory. It begins with a Lie group action on an image manifold and produces the fundamental structure formula. These techniques will be used to study the brightness perception mechanism of the human visual system. Several visual groups are defined and their corresponding structures for visual system models are obtained. A new transform called the Mandala transform will be deduced from a certain visual group and its implications for image processing will be discussed. Several new phenomena of brightness perception will be presented. New facts about the Mach band illusion along with new adaptation phenomena will be presented. Also a new visual illusion will be presented. A visual model based on the above techniques will be presented. It will also be shown how use of statistical estimation theory can be made in the study of contrast adaptation. Furthermore, a mathematical interpretation of unconscious inference and a simple explanation of the Tolhurst effect without mutual channel inhibition will be given. Finally, image processing algorithms suggested by the model will be used to process a real-world image for enhancement and for "form" and texture extraction.