Self-Intersected Boundary Detection and Prevention Methods Joachim Stahl 4/26/2004

2 Introduction Image segmentation and most salient boundary detection. Why? Simulate human vision system. Object detection within an image.

3 Wang, Kubota, Siskind Method Advantages of WKS method: Global Optimal. Not biased towards boundaries with fewer fragments. Reference: S. Wang, J. Wang, T. Kubota. From Fragments to Salient Closed Boundaries: An In-Depth Study, to appear in IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Washington, DC, 2004.

4 WKS Method in a nutshell

5 Self-intersection problem #1 First case of self-intersection. Two segments of the boundary intersect themselves. It is a closed boundary though. Shape of eight or infinity.

6 Self-intersection problem #1 (cont) Proposed solution: Branch & Bound First checks if an intersection occurred. If yes, branch execution. In each branch run the same set again, but ignore one of the segments. Repeat until you get non-intersected results. Pick the one with the least weight.

7 Self-intersection problem #1 (cont) Additionally: Establish a threshold. If the total weight of a boundary in a branch goes over it, reject. Do not go a level down if there is already a candidate with less weight in same level. Original W = 5.5 W = 7 W = 8W = 10 W = 7.6 W = 9

8 Self-intersection problem #1 (cont) Sample result of applying the branching method.

9 Self-intersection problem #2 Second case. Given two edges, the stochastic- completion-fields gap-filling method returns a self- intersecting segment.

10 Self-intersection problem #2 (cont) Proposed solution: Use instead a Bezier approximation. First check that the set of points satisfy minimum requirements. Then calculate the Bezier approximation. Else, return an artificial infinite long segment. (i.e. discard the segment).

11 Self-intersection problem #2 (cont) Bezier approximation works by calculating the middle points of segments. It needs four points, two for the origins and two to determine tangents at those points.

12 Self-intersection problem #2 (cont) Given the four points as p = [p 1, p 2, p 3, p 4 ]. We have vector u = [1 u u 2 u 3 ]. We can calculate the a point in the approximation by doing: p(u) = u.M B.p T where M B is the Bezier matrix M B = Note: Approximation done to a recursion depth of 10. Balance between fast and smooth.

13 Self-intersection problem #2 (cont) Proposed solution implementation. Extend the given tangents and find intersection between them. Use the intersection point for both tangent points of Bezier approximation.

14 Self-intersection problem #2 (cont) Cases where Bezier approximation does not work. But it is a case that is not desirable anyway. Can be detected easily, and return an infinite gap.

15 Self-intersection problem #2 (cont) The special case of parallel tangents needs to be addressed separately. In general, they are discarded.

16 Conclusion Both cases of self-intersecting boundaries can be overcome by implementing the proposed solutions. In the first case, the problem can be detect and corrected. In the second it is avoided.

17 Final Remarks This is a part of this research project. Other topics include: Dealing with open boundaries. Multiple boundaries. To be presented by Jun Wang.

18 The End Questions?