A Novel Algorithm for Distance Transformation on Irregular Isothetic Grids

Abstract

In this article, we propose a new definition of the E 2 DT (Squared Euclidean Distance Transformation) on irregular isothetic grids such as quadtree/octree or run-length encoded d-dimensional images. We describe a new separable algorithm to compute this transformation on every grids, which is independent of the background representation. We show that our proposal is able to efficiently handle various kind of classical irregular two-dimensional grids in imagery, and that it can be easily extended to higher dimensions. 1 for j = 0 to n2 − 1 do {First stage along X} 2 if A(0, j) = 0 then B(0, j) ← 0; 3 else B(0, j) ← ∞; 4 for i = 1 to n1 − 1 do 5 if A(i, j) = 0 then B(i, j) ← 0; 6 else 7 if B(i − 1, j) = 0 then B(i, j) ← TX (i) − TX(i − 1) − HR(i − 1, j); 8 else B(i, j) ← TX (i) − TX (i − 1) + B(i − 1, j); 9 for i = n1 − 2 to 0 do 10 if B(i + 1, j) < B(i, j) then 11 if B(i + 1, j) = 0 then B(i, j) ← TX (i + 1) − TX (i) − HL(i, j);

Separable algorithms for distance transformations on irregular grids

“…We recall that the VD of a set of points P ¼ fp i g is a tiling of the plane into Voronoi cells (or VD cells) fC p i g (de Berg et al, 2000;Voronoi, 1908). The I-CDT may be computed with such a structure, as we discussed in (Vacavant et al, 2008). If we now consider the background/foreground frontier to compute the I-BDT, Eq.…”

“…(2)) (Vacavant et al, 2009a) employs cell borders, and computes the shortest distance between a foreground cell centre and the background/foreground boundary. We sum up in Table 1 the advantages and the drawbacks of each model.…”

“…1 for an example of these diagrams computed on a simple I-grid). Hence, a simple approach to compute the I-CDT (presented in Algorithm 1 s to pre-compute the complete VD of the background points (Vacavant et al, 2008), and to locate foreground points in the VD. In this case, the I-CDT computation has a Oðn log n B Þ time complexity, where n is the total number of cells, and n B is the number of background cells.…”

“…The first extension, the Centre-based DT (Eq. (1)) (Vacavant et al, 2008) is based on the cell centres of the grid, where d 2 e ðp; p 0 Þ the squared Euclidean distance between the centres p = (x R , y R ) and p 0 ¼ ðx R 0 ; y R 0 Þ. The Border-based DT (Eq.…”

“…In (Vacavant et al, 2008;Vacavant et al, 2009a), we introduced two kinds of extensions of the E 2 DT on I-grids. The first one is based on the distance between cell centres, while the second one computes the distance between a cell centre and the nearest foreground/background frontier.…”