Distance Transformation on Two-Dimensional Irregular Isothetic Grids

Abstract

In this article, we propose to investigate the extension of the E 2 DT (squared Euclidean Distance Transformation) on irregular isothetic grids. We give two algorithms to handle different structurations of grids. We first describe a simple approach based on the complete Voronoi diagram of the background irregular cells. Naturally, this is a fast approach on sparse and chaotic grids. Then, we extend the separable algorithm defined on square regular grids proposed in [22], more convenient for dense grids. Those two methodologies permit to process efficiently E 2 DT on every irregular isothetic grids. The authors would like to thank Pr. Annick Montanvert for her constructive suggestions during the preparation of this article.

A Novel Algorithm for Distance Transformation on Irregular Isothetic Grids

“…We can first consider that the distance between two cells R and R is the distance between their centers. If we denote p = (x R , y R ) and p = (x R , y R ) these points, and d 2 e (p, p ) the squared Euclidean distance between them, the I-CDT (Center-based DT on I-grids) of a cell R is defined as follows [16]:…”

“…More precisely, as in the regular case, the I-CDT can be linked with the VD of the background cells centers (i.e. Voronoi sites or VD sites) [16]. The VD of a set of points P = {p i } is a tiling of the plane into Voronoi cells (or VD cells) {C pi } [3].…”

“…If we now consider the background/foreground frontier to compute the I-BDT, our definition implies that we compute a VD of segments, and not a classical VD of points (see Figure 2 for an example of these diagrams computed on a simple I-grid). Hence, a simple approach to compute the I-CDT is to pre-compute the complete VD of the background points [16], and to locate foreground points in the VD. In this case, the I-CDT computation has a O(n log n B ) time complexity, where n is the total number of cells, and n B is the number of background cells.…”

“…Many of those methodologies can be linked to the computation of a discrete Voronoi diagram of the background pixels [6,2]. In a previous work [16], we introduced a definition of E 2 DT on I-grids based on the background cells centers. After showing the link between this process and the computation of the Voronoi diagram of those background points, we proposed two completing algorithms to handle various configurations of I-grids.…”

“…This paper is organized as follows: (1) we present a new way to compute the E 2 DT on I-grids, based on the border of the background cells (Section 2). We also illustrate its relation with the computation of a Voronoi diagram of segments; (2) in Section 3, we describe an algorithm based on the work of R. Maurer et al [10] to efficiently compute this new transformation in 2-D, but extensible to higher dimensions; (3) we finally show in Section 4 experimental results to compare our new contribution with the techniques we developed in [16] in terms of speed and time complexity.…”