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


• Chan Vese method intuition:

I = 1

I = 0

in out in outc c I c I c N


• Chan Vese in Action:


• 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…


• Chan Vese Main Problem Demonstration:


• 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.


• 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.


• 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


• 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


• 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!!!


• 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


• Hand Drawing With Added Noise



• Church



• Boy At The Beach



• Flower


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!

usage of sobolev metric to detect an object’s boundaries supervisor: arie nahkmani students: yoav...

Documents

sobolev slide

sobolev gradient

energy gradient slide

external energy slide

suggested sobolev improvement

sobolev main idea

sobolev inner products

usage of sobolev metric