Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps January 2006 Theo Schouten Harco Kuppens Egon van den Broek

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps Distance Maps D(p) = min { dist(p,q), q  O} Euclidean distance City-block distance easy to calculate, 4 line program, slow

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps Calculation forward + backward raster scan, local distances –1966 Rosenfeld and Pfaltz: city-block distance –1986 Borgefors: chamfer distances region growing methods euclidean distances can not be done these ways semi-exact ED’s ordered propagation+ corrections at tile boundaries intersection Voronoi diagram with rows + dimensionality reduction: O(N) algorithms recently one with low time constant C. Maurer, R. Qi, V. Raghaven

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps Fast Exact Euclidean Distance (FEED) Maps D(p) = if (p  O) then 0 else  each q  O feeds its ED to each p: D(p) = min ( D(p), ED(q,p)) 1.restrict q : only border pixels of O –having one of its neighbors  O 2.restrict feed distance: only p’s which are closer to B than to another q  O 3.do something smart with the ED calculation: –use ED 2, pre-calculation

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps Bisection hyperplane’s time: –search for q’s –administrate bisection planes, area to feed should be smaller than the time –saved by doing less updates

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps Search and administration process keep bounding box local around B radial search lines, only first q up to a maximum stop when small special cases many parameters rather easy to adjust

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps 3D-2D implementation comparison more (type) neighbors: 6-plane, 12-line, 8-point effectively non-uniform memory access time m(l); x+1: m(l+1); y+1: m(l+width); z+1: m(l+width*height) 2D: pre-calculated ED’s stored in a matrix 3D: recalculated ED 2 30% faster 2D: bounding box + filling per quadrant around B 3D: single bounding box, 1 loop over Z for filling 3D: special searches to reduce bb in Z

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps Images generated 64x64x64 128x128x128 “or” and “xor” roughening surfaces 1024 images

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps Timing and accuracy comparison 3D- FEED City Block CH 3,4,5 i CH 3,4,5 f SemiEx EDT Linear Time AMD ms Intel ms wrong voxels (%) av abs error max abs error av rel error (%) max rel error (%) AMD rel time Intel rel time

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps Time vs % object pixels

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps Conclusion principles Fast Exact Euclidean Distance (FEED) 3-D implementation, 5 other DT, 1024 images fast ; less easy to implement 2D video with stationary and moving objects adding influence of moving objects per frame FEED faster than CH3,4 faster than City-Block future: –more dependent on image content -> faster –adaptable to anisotropic voxels –dimension independent: 2D, 3D… 4D, 5D

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps The End

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps slices through the shown images

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps Disjunct Voronoi tiles

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps Random dot images

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps OLDTiming and accuracy comparison 3D- FEED City Block CH 3,4,5 i CH 3,4,5 f SemiEx EDT Linear Time AMD ms Intel ms wrong voxels (%) av abs error max abs error av rel error (%) max rel error (%) AMD rel time Intel rel time

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps OLDConclusion principles Fast Exact Euclidean Distance (FEED) 3-D implementation, 4 other DT, 1024 images up to 2x faster semi-exact EDT, 3x slower City-Block same speed as CH3,4,5 ; less easy to implement future: –more dependent on image content -> faster –adaptable to non-square voxels –dimension independent: 2D, 3D… 4D, 5D –2D video with stationary and moving objects adding influence of moving objects per frame FEED faster than CH3,4 faster than City-Block