Region coloring using the Welsh-Powell algorithm 6/2010

1. Region coloring using the Welsh-Powell algorithm6/2010 Jacob D’Avy

2. Region coloring • The output image from a segmentation method can be viewed as connected regions. • Coloring the regions can be used to help visualize the output. • The optimal coloring scheme would paint all adjacent regions in different colors and use the minimum required colors. D.Page, “Part decomposition of 3D models,” Ph.D. dissertation, University of Tennessee, Knoxville, TN 2003.

3. Region coloring • There is no method that can find the optimal solution in a reasonable amount of time.(min colors) • Welsh and Powell created a method to approximate the optimal solution. • The Welsh-Powell method guarantees that no adjacent regions will be painted the same color. • The number of colors used is not guaranteed to be minimal, but it is usually close. D.J.A. Welsh and M.B. Powell, “An upper bound on the chromatic number of a graph and its application to timetabling problems", The Computer Journal, volume 10, no. 1, pp. 85-86, 1967

4. Algorithm Definitions: Process: D.J.A. Welsh and M.B. Powell, “An upper bound on the chromatic number of a graph and its application to timetabling problems", The Computer Journal, volume 10, no. 1, pp. 85-86, 1967

5. Results Mean-shift segmentation segmentation region boundaries original image Welsh-Powell coloring of segmentation sb=8 rb=25.6 min=128 resolution: 240x240 3 colors 5 regions

6. Results Mean-shift segmentation segmentation region boundaries Welsh-Powell coloring of segmentation original image sb=8 rb=12.8 min=32 resolution: 376x490 5 colors 127 regions

7. Results • A basic region coloring scheme would be to define a set of distinguishable colors.(~10) • Color the regions by assigning them a random color from the set. • The benefit is fast execution, but its possible that two adjacent regions are colored the same. • Random coloring will be used as a baseline for execution time. Comparison with random region color assignment:

8. Results Comparison with random region color assignment: original image Random color Welsh-Powell 5 colors 127 regions 45 secs runtime 10 colors 127 regions 22 secs runtime resolution 376x490

9. Results Comparison with random region color assignment: original image Welsh-Powell Random color 3 colors 5 regions 17 secs runtime 5 colors 5 regions 10 secs runtime resolution: 240x240

10. Results Comparison with random region color assignment: original image segmented region boundaries Random color Welsh-Powell 5 colors 3237 regions 100 secs runtime 10 colors 3237 regions 5 secs runtime Resolution: 240x240

11. Results This table shows the difference in execution time for the two methods. original seg2 original seg3 original seg1 Execution time is for the whole process, from segmentation to final output.

12. Conclusions • Using the Welsh-Powell method for coloring a segmentation output guarantees that adjacent regions will not be painted with the same color. • It also uses a near minimal amount of different colors which makes regions easier to distinguish. • Using Welsh-Powell does require additional computation time. • For a small number of regions, the increase in computation time is not large. As the number of regions increase, the required time increases substantially.

13. End