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Optimal Cycle

Optimal Cycle. Vida Movahedi Elder Lab, January 2008. Grouping Problem. We have a set of line segments We want the sequence (ordered subset) corresponding to an object boundary in image. Model. Virtual links Alternate paths Alternate Cycle

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Optimal Cycle

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  1. Optimal Cycle Vida Movahedi Elder Lab, January 2008

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

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

  4. 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. 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. Minimum Ratio Weight Algorithm • Given a graph G=<V,E> with all edges e in E doubly weighted with l(e)Z and t(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. Minimum Ratio Weight Algorithm (cont.) • Problem B: Given a graph G=<V,E> with all edges e in E weighted with l(e), find the minimum “total weight” cycle

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

  9. Proof

  10. 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 wt has zero weight • Finding the largest value of t such that G weighted by wt has no negative cycle • Linear search

  11. 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. Negative Weight CycleFirst Algorithm (Huang)

  13. Negative Weight CycleSecond 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. Negative Weight CycleWang’s algorithm- Cont.

  15. Negative Weight CycleWong’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. References [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 .

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