Self-Intersected Boundary Detection and

Prevention Methods

Joachim Stahl

4/26/2004

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Introduction

Image segmentation and most salient boundary detection. Why?

• Simulate human vision system.

• Object detection within an image.

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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.

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WKS Method in a nutshell

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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.

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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.

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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.

OriginalW = 5.5

W = 7 W = 7.6

W = 8 W = 10 W = 9 W = 9

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Self-intersection problem #1 (cont)

Sample result of applying the branching method.

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Self-intersection problem #2

Second case. Given two edges, the stochastic-completion-fields gap-filling method returns a self-intersecting segment.

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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).

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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.

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Self-intersection problem #2 (cont)

• Given the four points as p = [p1, p2, p3, p4]. We have vector u = [1 u u2 u3].

• We can calculate the a point in the approximation by doing:

p(u) = u.MB.pT where MB is the Bezier matrix

1 0 0 0

-3 3 0 0

3 -6 3 0

-1 3 -3 1

MB = Note: Approximation doneto a recursion depth of 10.Balance between fast andsmooth.

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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.

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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.

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Self-intersection problem #2 (cont)

The special case of parallel tangents needs to be addressed separately.

In general, they are discarded.

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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.

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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.

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The End

Questions?