Parallel Banding Algorithm to Compute Exact Distance Transform with the GPU

1. Parallel Banding Algorithm to Compute Exact Distance Transform with the GPU Thanh-Tung Cao Ke Tang Anis Mohamed Tiow-Seng Tan

2. Euclidean Distance • if p = (p1, p2,..., pn) and  q = (q1, q2,..., qn) are two points in Euclidean n-space, then the distance from p to q, or from q to p is given by:

3. Voronoi diagram • A set of points (called seeds, sites, or generators) is specified beforehand and for each seed there will be a corresponding region consisting of all points closer to that seed than to any other.

4. Euclidean Distance Transform 100 100

5. Exact Euclidean Distance Transform • Let Si,jbe the nearest site, among all sites in row j,of the pixel (i, j), and let Si = {Si,j| Si,j NULL, j = 0, 1, 2, . . . , n−1} be the collection of such closest sites for all pixels in column i. Note that Si,jisNULLwhen there is no site in row j. • Let Pi be the set of sites whose Voronoi regions intersect the pixels in column i. These sites are termed the proximate sites of column i.

6. Fact 1 • Consider column i and let b(i1, j) and b′(i2, j) be two sites in row j. If |i1 − i| < |i2 − i|, then the Voronoi region of b′ cannot intersect column i. • This fact means that for each column i, there can be at most one site along a row that can potentially be a proximate site, or basically, Pi ⊆ Si. As a result, |Pi| ≤ n.

7. Fact 2 • Consider columni and let a(i1, j1), b(i2, j2), c(i3, j3) be any three sites with j1 < j2 < j3. Let the intersection of the perpendicular bisector of a and b and column i be p(i, u), and that of b and c be q(i, v). If u > v then the Voronoi region of b cannot intersect column i. • When the mentioned situation happens, we say that a and cdominateb on column i. In this case, bPi.

8. Fact 3 • Let q(i, v) and p(i, u) be two pixels in column i such that u > v, and let a(i1, j1) and c(i3, j3) be the closest sites to q and p respectively. Then we have j1 ≤ j3. • This fact means that the proximate sites of column ihave their Voronoi regions appear in exactly the same order as when they are sorted by their y-coordinates.

9. Exact Euclidean Distance Transform • Phase 1: For each pixel (i, j), compute Si,j. • Phase 2: Compute the set Pi for each column i using Si. • Phase 3: Compute the closest site for each pixel (i, j) using Pi

10. Parallel Banding Algorithm Phase 1 - Band Sweeping Band 1 Band 2 Band 3 Band 4

11. Parallel Banding Algorithm si i Phase 1 - Band Sweeping Band 1 Band 2 Band 3 Band 4 si,j si,j (i,j)

12. Parallel Banding Algorithm Phase 1 - Band Sweeping Band 1 Band 2 Band 3 Band 4

13. Parallel Banding Algorithm Phase 1 - Band Sweeping Band 1 Band 2 Band 3 Band 4

14. Parallel Banding Algorithm Phase 1 - Band Sweeping Band 1 Band 2 Band 3 Band 4

15. Parallel Banding Algorithm Phase 2 - Hierarchical Merging Band 1 Band 2 Band 3 Band 4

16. Parallel Banding Algorithm • Phase 2 - Hierarchical Merging si i Si,1 Si,2 Si,3 Si,4 Si,5 Si,6 Si,7 Band 1

17. Parallel Banding Algorithm • Phase 2 - Hierarchical Merging si i Si,1 Si,2 Si,3 Si,4 Si,5 Si,6 Si,7 Band 1 Si,2 Si,1 stack

18. Parallel Banding Algorithm • Phase 2 - Hierarchical Merging si i Si,1 Si,2 Si,3 Si,4 Si,5 Si,6 Si,7 (i,v) Si,3 Band 1 v  u Si,2 Si,1 (i,u) stack

19. Parallel Banding Algorithm • Phase 2 - Hierarchical Merging si i Si,1 Si,2 Si,3 Si,4 Si,5 Si,6 Si,7 (i,v) Band 1 Si,3 Si,2 Si,1 (i,u) stack

20. Parallel Banding Algorithm • Phase 2 - Hierarchical Merging si i (i,u) Si,1 Si,2 Si,3 Si,4 Si,5 Si,6 Si,7 Si,4 Band 1 v u Si,5 Si,6 Si,3 Si,7 Si,2 Si,1 (i,v) stack

21. Parallel Banding Algorithm • Phase 2 - Hierarchical Merging si (i,v) i Si,1 Si,2 Si,3 Si,4 Si,5 Si,6 Si,7 Si,4 Band 1 (i,u) v  u Si,2 Si,1 stack

22. Parallel Banding Algorithm • Phase 2 - Hierarchical Merging si (i,v) i Si,1 Si,2 Si,3 Si,4 Si,5 Si,6 Si,7 Si,4 (i,u) Band 1 v  u Si,2 Si,1 stack

23. Parallel Banding Algorithm • Phase 2 - Hierarchical Merging si i Si,1 Si,2 Si,3 Si,4 Si,5 Si,6 Si,7 (i,v) Si,5 Band 1 (i,u) v  u Si,4 Si,2 Si,1 stack

24. Parallel Banding Algorithm • Phase 2 - Hierarchical Merging si i Si,1 Si,2 Si,3 Si,4 Si,5 Si,6 Si,7 Si,6 Band 1 Si,5 Si,4 Si,7 Si,2 Si,1 stack

25. Parallel Banding Algorithm • Phase 2 - Hierarchical Merging si i (i,u) Si,1 Si,2 Si,3 Si,4 Si,5 Si,6 Si,7 Si,7 Band 1 v u Si,6 Si,5 Si,4 Si,2 Si,1 stack (i,v)

26. Parallel Banding Algorithm • Phase 2 - Hierarchical Merging si i Si,1 Si,2 Si,3 Si,4 Si,5 Si,6 Si,7 (i,u) Si,7 Band 1 v u Si,5 Si,4 (i,v) Si,2 Si,1 stack

27. Parallel Banding Algorithm • Phase 2 - Hierarchical Merging si i (i,v) Si,1 Si,2 Si,3 Si,4 Si,5 Si,6 Si,7 Si,7 Band 1 v  u (i,u) Si,4 Si,2 Si,1 stack

28. Parallel Banding Algorithm • Phase 2 - Hierarchical Merging si i Si,1 Si,2 Si,3 Si,4 Si,5 Si,6 Si,7 Band 1 Si,7 Si,4 Si,2 Si,1 stack

29. Parallel Banding Algorithm i • Phase 2 - Hierarchical Merging Si,7 Si,4 Band 1 Si,2 Si,1 stack i Si,8 Si,9 Band 2 Si,12 Si,13

30. Parallel Banding Algorithm i • Phase 2 - Hierarchical Merging Band 1 Si,7 Band 2 Si,8 Si,4 Si,9 Si,2 Si,1 Si,12 Si,13 stack

31. Parallel Banding Algorithm i • Phase 2 - Hierarchical Merging Band 1 Si,12 Si,13 Si,9 Si,8 Si,7 Band 2 Si,4 Si,2 Si,1 stack

32. Parallel Banding Algorithm Phase 2 - Hierarchical Merging i proximate texture Band 1

33. Parallel Banding Algorithm Phase 3 - Block Coloring Case 1 pi i a a b m3 b

34. Parallel Banding Algorithm Phase 3 - Block Coloring Case 2 pi i a a b m3 b

35. Parallel Banding Algorithm Phase 3 - Block Coloring Case 2 pi i a a b m3 a a b b b

36. Parallel Banding Algorithm

37. Parallel Banding Algorithm

38. Parallel Banding Algorithm

39. Parallel Banding Algorithm