Optical flow cannot be computed locally, since only one independent measurement is available from the image sequence at a point, while the flow velocity has two components. A second constraint is needed. A method for finding the optical flow pattern is presented which assumes that the apparent velocity of the brightness pattern varies smoothly almost everywhere in the image. An iterative implementation is shown which successfully computes the optical flow for a number of synthetic image sequences. The algorithm is robust in that it can handle image sequences that are quantized rather coarsely in space and time. It is also insensitive to quantization of brightness levels and additive noise. Examples are included where the assumption of smoothness is violated at singular points or along lines in the image.

The problem of oscillatory polynomial interpolants arising from equally spaced mesh points is considered. It is shown that by making use of adaptive approaches that the oscillations may be contained and that the resulting polynomials are data bounded and monotone on each interval. This is achieved at the cost of using a different polynomial on each sub-interval. Computational results for a number of challenging functions including a number of problems similar to Runge’s function with as many as 511 points per interval are shown. keywords Adaptive polynomial interpolation, data bounded polynomials, Runge’s function

The theoretical study of micron-scale quantum-mechanical systems generally begins with two assumptions about the potential: that there is no background potential, and that any confining potential is hard-walled. In this thesis, we will look at a phenomenon that is seen when these assumptions are not made, in the context of electron conductance through two-dimensional electron gasses (2DEGs).

This thesis is concerned primarily with the development of algorithms for estimating and segmenting image motion fields that contain discontinuities. An error-weighted regularization algorithm for image motion field estimation is proposed as a computationally attractive alternative to stochastic optimization based schemes. Block matching errors in the local motion measurement process are used in the regularization functional in order to avoid oversmoothing across motion boundaries. A second algorithm, anisotropic regularization, improves on the local measurement process, by employing alternative matching criteria and matching window organization. A selective confidence measure derived from anisotropic local measurements is used to further improve the error-weighted regularization.