Recursive 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 extracting parameter(s) of a nonstationary transformation between two images. Because of the nonstationary nature, finite width filters have to be used to extract information from images. In this paper, we present a theory of manipulating information extracted from finite width filters. We will propose two novel filters, ``moment filters'' and ``hypergeometric filters''. Unlike the infinite width filters which are independent as long as they have different frequency, the moment filters we propose are interdependent even they have different peak frequency, and the interdependency can be well modeled. We demonstrate that the moment filters drastically improve the precision of focus and stereo, and also provide a new approach to the long perplexing problem of foreshortening. Furthermore, the hypergeometric filters are able to provide a complete and nonredundant decomposition of the local signal. And a general problem of extracting parameter(s) of a nonstationary transformation can be formulated as a multidimensional minimivzation problem. We apply this technique to the optical flow problem, and its performance is very encouraging comparing to other techniques.