Moment and Hypergeometric Filters for High Precision Computation of Focus, Stereo, and Optical Flow

Abstract

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. Because of the non-stationary nature, 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 results of algorithms, there has been no quantitative way to measure or eliminate the negative effects of finite-width windows. To address this problem, we introduce two sets of filters, "moment" filters and "hypergeometric" filters. Due to their recursive properties, these filters allow the effects of finite-width windows and foreshortening to be explicitly analyzed and eliminated. We develop one paradigm to solve general one-parameter extraction problems using hypergeomtric filters. We apply these paradigms to problems of focus and stereo, in which one parameter is extracted at every pixel location, and optical flow, in which two parameters are extracted. We demonstrate that our algorithms based on moment filters and hypergeometric filters achieve much higher precision than other techniques.