Moment and hypergeometric filters for high precision computation of focus, stereo and optical flow

Many low level visual computation problems such as focus, stereo, optical flow, etc., can be formulated as problems of extracting one or more parameters of a non-stationary transformation between two images. Finite-width windows are widely used in various algorithms to extract spatially local information from images. While the choice of window width has a very profound impact on the quality of algorithmic results, there has been no quantitative way to measure or eliminate the negative effects of finite-width windows. To address this problem and the foreshortening problem caused by non-stationarity, we introduce two novel sets of filters: ''moment'' filters and ''hypergeometric'' filters. The recursive properties of these filters allow the effects of finite-width windows and foreshortening to be explicitly analyzed and eliminated. We apply the moment filter approach to the focus and stereo problems, in which one parameter is extracted at every pixel location. We apply the hypergeometric approach to the optical flow problem, in which two parameters are extracted. We demonstrate that algorithms based on moment filters and hypergeometric filters achieve much higher precision than other state-of-art techniques.