Lower and upper bounds for scaling factors used for integer approximation of 3D anisotropic chamfer distance operator

Abstract

For 3D images composed of successive scanner slices (e.g. medical imaging, confocal microscopy or computed tomography), the sampling step may vary according to the axes, and specially according to the depth which can take values lower or higher than 1. Hence, the sampling grid turns out to be parallelepipedic. In this paper, 3D anisotropic local distance operators are introduced. The problem of coefficient optimization is addressed for arbitrary mask size. Lower and upper bounds of scaling factors used for integer approximation are given. This allows, first, to derive analytically the maximal normalized error with respect to Euclidean distance, in any 3D anisotropic lattice, and second, to compute optimal chamfer coefficients. As far as large images or volumes are concerned, 3D anisotropic operators are adapted to the measurement of distances between objects sampled on non-cubic grids as well as for quantitative comparison between grey level images. © 2009 Springer Berlin Heidelberg.