Parallel Banding Algorithm for Distance Transform

1. Thanh-Tung Cao Ke Tang Anis Mohamed Tiow-Seng Tan

2. Outline • Distance transform and applications • Related works • Exact Euclidean distance transform • Parallel Banding Algorithm (PBA) • Experiment results i3D 2010 - Washington DC, USA

3. Distance Transform • Given a grid N = ndand a set of sites S, compute for every grid point the distance to the nearest site. n n Distance transform Voronoi diagram i3D 2010 - Washington DC, USA

4. Applications • Image processing, graphics, computational geometry... Morphological operation Integer medial axis Stipple drawing i3D 2010 - Washington DC, USA

5. Related Work • Approximate: Vector propagation approach • Danielsson [1980], O(n). • GPU based: • Jump flooding [Rong and Tan, 2006, 2007] • Schneider et al. [2009] • Exact: Voronoi diagram-based • Maurer et al. [2003], O(n), parallelizable. • PRAM: Lee et al [2003], Wang et al [2001] • GPU: PBA is the first. i3D 2010 - Washington DC, USA

6. Euclidean Distance Transform • Consider one column on the 2D image. Column i B' B Fact 1: Among all sites in the same column, only the nearest matter. i3D 2010 - Washington DC, USA

7. Euclidean Distance Transform • Consider one column on the 2D image. A Column i B C Fact 2: The Voronoi region of A and C can dominatethat of B. i3D 2010 - Washington DC, USA

8. Euclidean Distance Transform • Consider one column on the 2D image. A Column i C Fact 3: If A is on top of C then A’s region is also on top of C’s region. i3D 2010 - Washington DC, USA

9. Parallel Banding Algorithm Phase 1 Final result Phase 3 Phase 2 i3D 2010 - Washington DC, USA

10. The naïve approach • Phase 1: • Left – Right sweep, each pixel updates with the pixel on its left. • Right – Left sweep. i3D 2010 - Washington DC, USA

11. The naïve approach • Phase 2: Finding proximate points in linear time. • Process sites from top to bottom • Store the current set of proximate sites in a stack dominated i3D 2010 - Washington DC, USA

12. The naïve approach • Phase 3: • For a column, pixels are “colored” from top to bottom s1 p-1 s2 p p is nearer to s1 than to s2 Yes No p belongs to s1 s1 can be removed i3D 2010 - Washington DC, USA

13. Parallel Banding Algorithm • Motivations: • Within a row/column, the computation is sequential. • Maximum number of threads used is n (512, 1024, etc). • We needs 104 – 105 threads to fully utilize the latest GPUs. i3D 2010 - Washington DC, USA

14. Parallel Banding Algorithm • Phase 1: • A row is equally divided into m1 bands. • Within a band, perform the sweeping similar to the naïve approach. • Propagate the information among the first and the last pixel of different bands. • Pixels update with the first/last pixel of its band Updated! i3D 2010 - Washington DC, USA

15. Parallel Banding Algorithm • Phase 2: • A column is equally divided into m2 bands • For each band, we compute the proximate sites using the naïve approach. • Bands are merged hierarchically i3D 2010 - Washington DC, USA

16. Parallel Banding Algorithm • Phase 3: • Color a band of m3 pixels at a time, 1 thread per pixel. • Move to the next band when the last pixel of the current band confirms its color. s1 s2 confirmed s3 remove s1 i3D 2010 - Washington DC, USA

17. Experiment results • CUDA, nVidia GeForce GTX280 Naïve v.s. Banding approach i3D 2010 - Washington DC, USA

18. Experiment results Scalability Different bands for Phase 2 i3D 2010 - Washington DC, USA

19. Experiment results • Compare with JFA [Rong et al. 2006]and SKW [Schneider et al. 2009] 2D 3D i3D 2010 - Washington DC, USA

20. Conclusions • Advantages: • Accurate result → Algorithm to computing weighted centroidal Voronoi Diagrams on GPU (see paper) • Optimal linear expected total work • High level of parallelism, GPU friendly • Limitations: • Load balancing in Phase 2 • Worst case not work-optimal in Phase 3 i3D 2010 - Washington DC, USA

21. The paper, video, presentation slides and source code are available at: http://www.comp.nus.edu.sg/~tants/pba.html THANK YOU!