Fast image transforms using diophantine methods

Abstract

Many image transformations in computer vision and graphics involve a pipeline when an initial integer image is processed with floating point computations for purposes of symbolic information. Traditionally, in the interests of time, the floating point computation is approximated by integer computations where the integerization process requires a guess of an integer. Examples of this phenomenon include the discretization interval of rho and theta in the accumulator array in classical Hough transform, and in geometric manipulation of images (e.g., rotation, where a new grid is overlaid on the image). The result of incorrect discretization is a poor quality visual image, or worse, hampers measurements of critical parameters such as density or length in high fidelity machine vision. Correction techniques include, at best, anti-aliasing methods, or more commonly, a "kludge" to cleanup. In this paper, we present a method that uses the theory of basis reduction in Diophantine approximations; the method outperforms prior integer based computation without sacrificing accuracy (subject to machine epsilon).