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Usage of Sobolev metric to detect an object’s boundaries Supervisor: Arie Nahkmani Students: Yoav...
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Transcript of Usage of Sobolev metric to detect an object’s boundaries Supervisor: Arie Nahkmani Students: Yoav...
Usage of Sobolev metric to detect an object’s boundaries
Supervisor:
Arie Nahkmani
Students:
Yoav Ben-Raphael
Itzik Ben-Basat
Agenda
• Introduction• Project Definitions & Goals• Motivation & Background Theory:
– Chan Vese– Sobolev
• Sobolev Algorithm Stages• Suggested Sobolev Improvement• Implementation Results• Conclusion & Future Development
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
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
Motivation & Background Theory
2 2
( ) ( ), ,in out in out in outinside C outside C
E C c c F C F C I c dxdy I c dxdy
• Chan-Vese method– Region Based Active Contour– Minimization of an energy based-segmentation:
– Move Contour according to the Energy Gradient
2 2
( ) ( ), ,in out in outinside C outside C
E C c c I c dxdy I c dxdy
2 2
t C in out
in out in out
C E I c N I c N
c c I c I c N
Motivation & Background Theory
• Chan Vese method intuition:
I = 1
I = 0
in out in outc c I c I c N
Motivation & Background Theory
• Chan Vese in Action:
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…
Motivation & Background Theory
• Chan Vese Main Problem Demonstration:
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.
Motivation & Background Theory
Motivation & Background Theory
• 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.
Motivation & Background Theory
• Minimize the energy defined on contours
• The gradient is the most efficient perturbation:
maximizes sup
Change in E in moving in direction h
The cost of moving in direction h
C
C
dE C hh E
h
dE C h
h
Motivation & Background Theory
• Sobolev Inner Product:
• Sobolev Gradient:
1
2 22
,'
C HC
h h s h s ds
1 0
0
*H H
H
E K E
Where K is the smoothing Kernel
E is the energy gradient
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!!!
Motivation & Background Theory
• Sobolev implemented on Chan Vese:
Sobolev Algorithm Stages
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.
Implementation Results
• Square With Added Noise
Original Model Sorted Model
Implementation Results
• Hand Drawing With Added Noise
Original Model Sorted Model
Implementation Results
• Church
Original Model Sorted Model
Implementation Results
• Boy At The Beach
Original Model Sorted Model
Implementation Results
• Flower
Original Model Sorted Model
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).
Conclusion & Future Development
• Future Development– Test both models in video tracking– Add level sets to the Chan Vese model
THANK YOU!