1. 1 Optimal Cycle Vida Movahedi Elder Lab, January 2008

2. 2/16 Grouping Problem We have a set of line segments We want the sequence (ordered subset) corresponding to an object boundary in image

3. 3/16Model Virtual links Alternate paths Alternate Cycle Each link is assigned a weight indicating the cost of being on boundary (Gestalt)

4. 4/16 Reduction to Graph Solid edges as nodes, Virtual links as edges Assigning the solid edges’ weights to neighboring virtual links –Wang had the solid edges as links, but looked for alternate paths Main tool in a weighted graph: Shortest Path Cycle: a path from a node to itself

5. 5/16 Graph Algorithms Finding a cycle with minimum sum of weights? Finding a cycle with minimum mean weight? –Could work if all line segments had same lengths, or else prefers more of shorter lines Finding a cycle with minimum ratio weight:

6. 6/16 Minimum Ratio Weight Algorithm Given a graph G= with all edges e in E doubly weighted with (e)Z and (e)Z + “ Ratio weight ” for a set of edges is defined as We want to find a cycle C* which minimizes W(C) among all cycles as W* Let ’ s call this problem A

7. 7/16 Minimum Ratio Weight Algorithm (cont.) Problem B: Given a graph G= with all edges e in E weighted with (e), find the minimum “total weight” cycle

8. 8/16 Minimum Ratio Weight Algorithm (cont.) First algorithm by Lawler (1966) Observation: We can define a new edge weight The solution t*, of w t (C t *)=0, where C t * is the solution to problem B with weights w t is equal to the minimum ratio weight W* in problem A, and C t * is equal to C*.

9. 9/16Proof

10. 10/16 Minimum Ratio Weight Algorithm (cont.) The problem is thus reduced to finding t*, or finding the value of t such that the minimum total weight cycle using w t has zero weight Finding the largest value of t such that G weighted by w t has no negative cycle Linear search

11. 11/16 Negative weight cycle algorithm X is the ratio weight of the negative cycle found Now we need an algorithm that can detect negative cycles in a graph

12. 12/16 Negative Weight Cycle First Algorithm (Huang)

13. 13/16 Negative Weight Cycle Second Algorithm (Wong) Finding a minimum weight perfect matching Edmond’s polynomial time algorithm All solid edges form a trivial perfect matching with zero total weight  the min. is nonpositive

14. 14/16 Negative Weight Cycle Wang’s algorithm- Cont.

15. 15/16 Negative Weight Cycle Wong’s algorithm- Cont. All cycles in a minimum weight perfect matching should have negative total weight Choose the one with minimum cycle ratio to return as X in previous linear search

16. 16/16References [1] S. Wang et al. (2005), “Salient Closed Boundary Extraction with Ratio Contour”, IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 27, no. 4, pp. 546- 561. [2] I.H. Jermyn and H. Ishikawa (2001),”Globally optimal Regions and Boundaries as Minimum Ratio weight Cycles”, IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 23, no. 10, pp. 1075-1088. [3] X. Huang (2006), “Negative-Weight Cycle Algorithms”, Int’l Conf. on Foundations of Computer Science, pp. 109 - 115.