Some Solutions to Problems in Depth Vision from Differential Geometry and Deconvolution Methods.

Abstract

Two distinct parts are presented. The first part is an application of manifold theory and geometry to mathematical modeling problems in depth vision. The second part is an application of deconvolution methods to the problem of constructing converging sequences of approximating functions from sampled values of convolutions. The connection between the two parts is this: the first part depends on differential methods; the second part provides converging algorithms for such methods. The first part begins with a model for objects. A measure is introduced and smoothness is assumed almost everywhere. The radiometric notion of sterance is modeled using differential forms on the sphere bundle (of three dimensional Euclidean space). It is shown that sterance, even if known on a neighborhood in the sphere bundle, does not uniquely determine the objects. However, sterance on a neighborhood can be used to construct a submersion, hence to determine codimension on submanifolds. A similar construction is carried out for sterance given on the sphere bundle over a curve (motion stereo with the path known). The properties of this construction are used to motivate the constructions for modeling the problem of depth vision based on time varying sterance on the sphere bundle over a point. We introduce bundles of bases, integral manifolds, and vector fields and one forms with vanishing Lie derivative with respect to the position vector field. We study the flows of vector fields that are isometrics of the integral manifolds. The second part begins with an analysis of deconvolution methods for convolution operators that are characteristic functions of n- dimensional cubes. Such operators (for squares) are approximations of the impulse response of photo-detectors in vision systems. A complete description for the the implementation of the method for sampled data is given. The primary accomplishment is that the error analysis is presented with explicit error bounds throughout. The final chapter is an analysis of the properties of the deconvolution methods in the presence of additive noise. For the class of methods and noise studied it is shown that there is no penalty for the use of deconvolution methods with photo-detectors.