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Usage of Sobolev metric to detect an object’s boundaries Supervisor: Arie Nahkmani Students: Yoav Ben-Raphael Itzik Ben-Basat

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Agenda Introduction Project Definitions & Goals Motivation & Background Theory: –Chan Vese –Sobolev Sobolev Algorithm Stages Suggested Sobolev Improvement Implementation Results Conclusion & Future Development

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Introduction Active Contour –AKA Snakes –Framework for separating an object outline from a possibly noisy 2D image –Minimize an energy associated with the contour as a sum of an internal and external energy

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Project Definitions & Goals Implement the Sobolev Algorithm in Matlab using the article “Sobolev Active Contours” by Ganesh Sundaramoorthi. Optimize code through Vectorization Test a suggested optimization for Sobolev

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Motivation & Background Theory Chan-Vese method –Region Based Active Contour –Minimization of an energy based-segmentation: –Move Contour according to the Energy Gradient

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Motivation & Background Theory Chan Vese method intuition: I = 1 I = 0

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Motivation & Background Theory Chan Vese in Action:

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Motivation & Background Theory Chan Vese Main Problem - Noise: –Contour becomes non-smooth instantly –Gradient depends on local derivatives: non-smooth curve inaccurate derivatives. –Points evolve independently, not collectively Possible Solution: –Add a penalty to the curve’s length in the Energy function –But then the Energy Function is altered…

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Motivation & Background Theory Chan Vese Main Problem Demonstration:

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Motivation & Background Theory Sobolev Method –A new way of doing Active Contours –Existing methods can benefit Sobolev main idea: –Represent the set of all smooth curves as an abstract space M. –A path on M looks like a morph between two contours.

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Motivation & Background Theory

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Region Based Active Contours -The Energy Gradient is the most efficient curve evolution. -Define Sobolev Inner Products based on the abstract space M. -Develop the Sobolev Gradient from the Sobolev Inner Product. -Non smooth contours unlikely due to derivatives in the Sobolev Gradient.

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Motivation & Background Theory Minimize the energy defined on contours The gradient is the most efficient perturbation:

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Motivation & Background Theory Sobolev Inner Product: Sobolev Gradient:

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Motivation & Background Theory Properties of the Sobolev Gradient –Like Chan-Vese, it does not depend on a particular parameterization of the curve. –less sensitive to noise. –Can be implemented on existing methods!!!

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Motivation & Background Theory Sobolev implemented on Chan Vese:

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Sobolev Algorithm Stages

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Suggested Sobolev Improvement After Matlab code was written and verified an improvement to the Sobolev algorithm was suggested. Hypothesis: Sorting the arc length vector, improves the Sobolev method.

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Implementation Results Square With Added Noise Original ModelSorted Model

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Implementation Results Hand Drawing With Added Noise Original ModelSorted Model

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Implementation Results Church Original ModelSorted Model

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Implementation Results Boy At The Beach Original ModelSorted Model

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Implementation Results Flower Original ModelSorted Model

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Conclusion & Future Development Conclusion: Not much noticeable difference between the two models But… –Original Sobolev Model handles noisy images better –Sorted Sobolev model is better tuned to the edges of real images (pointy edges).

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Conclusion & Future Development Future Development –Test both models in video tracking –Add level sets to the Chan Vese model

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THANK YOU!

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