Chamfer Distances Research Articles

Exact medial axis with euclidean distance

Medial Axis (MA), also known as Centres of Maximal Disks, is a useful representation of a shape for image description and analysis. MA can be computed on a distance transform, where each point is labelled to its distance to the background. Recent algorithms allow one to compute Squared Euclidean Distance Transform (SEDT) in linear time in any dimension. While these algorithms provide exact measures, the only known method to characterise MA on SEDT, using local tests and Look-Up Tables (LUT), is limited to 2D and small distance values [Borgefors, et al., Seventh Scandinavian Conference on Image Analysis, 1991]. We have proposed [Remy, et al., Pat. Rec. Lett. 23 (2002) 649] an algorithm which computes the LUT and the neighbourhood to be tested in the case of chamfer distances. In this article, we adapt our algorithm for SEDT in arbitrary dimension and show that results have completely different properties.

Medial axis for chamfer distances: computing look-up tables and neighbourhoods in 2D or 3D

Medial axis, also known as centres of maximal disks, is a representation of a shape, which is useful for image description and analysis. Chamfer or weighted distances are discrete distances which allow to approximate the Euclidean distance with integers. Medial axis extraction for chamfer distances is discussed in the literature, but only for simple cases. The principle is to use local tests and look-up tables. In this paper, we give an algorithm which computes for any chamfer distance in 2D or 3D, the look-up table and, very important, the neighbourhood to be tested.

A cost optimal parallel algorithm for weighted distance transforms

The distance transform and the nearest feature transform are useful operations in image processing. These transforms are based on various kinds of distance functions because the distance functions have different efficiency or usefulness. In this paper, we consider these transforms based on the weighted distance, which is a generalization of many distances, such as city block, chessboard and chamfer distances. This paper presents a parallel algorithm for these transforms of an n× n binary image. The algorithm runs in O ( log n) time using n 2/ log n processors on the EREW PRAM and in O ( log log n) time using n 2/ log log n processors on the common CRCW PRAM. The algorithm also runs in O( n 2/ p 2+ n) time on a p× p mesh and in O (n 2/p 2+(n log p)/p) time on a p 2 processor hypercube (for 1⩽ p⩽ n). From these complexities, the algorithm is cost optimal on all models. Also we obtained an Ω( log n) lower bound for the transform on the CREW PRAM. This implies that the algorithm is time optimal on the EREW PRAM.

Chamfer metrics, the medial axis and mathematical morphology

This paper describes a number of efficient algorithms for morphological operations which use discs defined by chamfer distances as structuring elements. It presents an extension to previous work on extending metrics (such as the p-q-metrics). Theoretical results and algorithms are presented for p-q-r-metrics, which are not extending. These metrics can approximate the Euclidean metric close enough for most practical situations. The algorithms are based on an analysis of the structure of shortest paths in the p-q-r-metric and of the set of values this metric can assume. Efficient algorithms are presented for the medial axis and the opening transform. The opening transform algorithm is two orders of magnitude faster than a more straightforward algorithm.

Metric subgraphs of the chamfer metrics and the Melter-Tomescu path generated metrics

Chamfer metrics are determined by local distances which are chosen to ensure that each geodesic lies within one of the cones determined by the mask and contains only edges in the directions of the bounding rays of the cone. It is shown that the chamfer distances calculated within a set are the same as those calculated in the whole space if and only if the set is convex in each of the local distance directions. The result does not hold when the local distances allow more general geodesics. The results for chamfer metrics are related to corresponding results for the metrics generated by the two-, three- and four-direction graphs studied by Melter and Tomescu.