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Variational Approaches and Image Segmentation

Lecture #6Lecture #6Hossam Abdelmunim1 & Aly A. Farag2

1Computer & Systems Engineering Department, Ain Shams University, Cairo, Egypt

2Electerical and Computer Engineering Department, University of Louisville, Louisville, KY, USA

ECE 643 – Fall 2010

The curvature and The Implicit Function FormThe curvature and The Implicit Function Form

)0(0)( 1 CorC

The level set function has the following relation with the embedded curve C:

0)( sTC

Us the following derivative equation w.r.t. the arc-length s:

To prove that: (Assignment)

Calculating Additional Quantities

||),2/())/cos(1()(

||,1)(

||)),/sin(1

1(5.0)(

H

HExample of a Level Set Function

iso-contours

H and Delta FunctionsApplying H FunctionApplying δ Function

,)( dxdyHA

,||)( dxdyL

• Enclosed Area

• Length of Interface

• Mainly used to track the Interface/contour:-

Narrow Banding

• Points of the interface/front/contour are only the points of interest.

• The points (highlighted) are called the narrow band.

• The change of the level set function at these points only are considered.

• Other points (outside the narrow band) are called far away points and take large positive or large negative values.

• This will expedite the processing later on.

Boundary Band Points.

Red line is the zero level set corresponding to

front.

Level Set PDELevel Set PDE

0),).(,(

dt

dy

dt

dx

yxt0.

||||

Vt

0),,( tyx

Curve Contracts with time

0

dyy

dxx

dtt

Level Set Function changes with time

0||

Ft

Fundamental Level Set Equation

The velocity vector V has a component F in the normal direction. The other tangential component has no effect because the gradient works in the normal direction.

Speed FunctionSpeed Function

1F

Among several forms, the following speed function can be used:

Contour characteristics:

Forces the contour to evolve smoothly. The bending is quantized by ε.

Image data (force):

+1 for expansion

-1 for contraction

It will be a function of the image (I).

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Need for Re-initializationNeed for Re-initializationNeed for Re-initializationNeed for Re-initialization• Solving the PDE of level set evolution does not keep the definition.• Keeping the definition is very necessary to hold the front between

the positive and negative regions.

|)|1)(sgn( 0 t

• Solving this equation frequently often in parallel with the main equation keeps the function close to the signed distance definition.

1|| x

x0x

Numerical Solution

Upwind Scheme and Discontinuous Solutions

Upwind Scheme and Discontinuous Solutions

Consider the following PDE:

It is one dimensional in x and can have the following numerical solution for different values of the speed a (can be of course a function of x):

Upwind Scheme and Discontinuous Solutions (Cont…)

Upwind Scheme and Discontinuous Solutions (Cont…)

So, we can define (as a notation):

To put the solution in the following general form:

First Order Upwind Scheme and Discontinuous Solutions

First Order Upwind Scheme and Discontinuous Solutions

Consider the Solution of the re-initialization PDE:-

a+=max(a,0) and a-=min(a,0)

First Order Upwind Scheme and Discontinuous Solutions whereFirst Order Upwind Scheme and Discontinuous Solutions where

)2

1)((2)( HS

And a smoothed version of the sign function is defined as follows:

Numerical Algorithm for the Level Set Evolution Equation – Higher Order Scheme

Numerical Algorithm for the Level Set Evolution Equation – Higher Order Scheme

We consider the numerical solution of the equation:

0|| Ft

Note that it is very similar to the 1D equation we showed above. Without proof, this equation will have the following numerical solution:

Numerical Algorithm for the Level Set Evolution Equation (Cont…) where

Numerical Algorithm for the Level Set Evolution Equation (Cont…) where

Numerical Algorithm for the Level Set Evolution Equation (Cont…) and

Numerical Algorithm for the Level Set Evolution Equation (Cont…) and

The switching function m is given by:

The speed function is given by:

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Examples..1Examples..1Examples..1Examples..1• Curvature flow with a curvature speed:

F

Parts of the curve with different curvature signs, move in opposite directions.

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Examples..2Examples..2Examples..2Examples..2• Curvature flow with a positive curvature speed:

)0,max(F

Parts of the curve with -ve curvature do not move

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Examples..3Examples..3Examples..3Examples..3• Curvature flow with a negative speed vector:

Parts of the curve with +ve curvature do not move

)0,min(F