Variational Approaches and Image Segmentation Lecture #7

1

Variational Approaches and Image Segmentation

Lecture #7Lecture #7Hossam 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

kF 1Among several forms, the following speed function is used:

Contour characteristics:

Smoothes the evolution and the bending is quantized by ε

Image data (force):

+1 for expansion

-1 for contraction

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

Variational Edge-based SegmentationVariational Edge-based Segmentation

|)*(|1

1)(

IGIg

Where g is an indicator function of the image gradient:

Edge map

Variational Edge-based Segmentation(Cont…)

Variational Edge-based Segmentation(Cont…)

dHIgdIgE )()(||)()()(

Energy = Arc-Length + Enclosed Area:

By calculus of variation:

])||

()[( ggdivt

The amount of bending is controlled by λ>0.

The sign of ע depends on the position of the contour w.r.t. the object.

Variational Segmentation without EdgesChan-Vese Model

Variational Segmentation without EdgesChan-Vese Model

dHHcIHcIEcv |])(|)()()()[( 22

21

dH

IdHc

)(

)(1 Object

Mean

dH

IdHc

)(

)(2

Background Mean

Maximizes the distance between c1 and c2

Only one level set function is used

Variational Segmentation without EdgesChan-Vese Model (Cont…)

Variational Segmentation without EdgesChan-Vese Model (Cont…)

])()()||

()[( 22

21 cIcIdivt

The PDE will be:

For computational issues:

])||

()[(

divt

where:

0)()(1

0)()(12

22

1

22

21

cIcIif

cIcIif

Chan & Vese--ExamplesChan & Vese--Examples

Multi-phase EvolutionChan & Vese

Multi-phase EvolutionChan & Vese

Ф1>0Ф2>0 Ф2<0

Ф1<0

Ф1<0

Ф2<0

Ф1>0Ф2>0

In this example 2 functions are used.

Then 22=4 regions are considered.

The energy will be:

C2

C3

C1

C4

dHH

HHcI

HHcI

HHcI

HHcI

Ecv

|)])(||)((|

)()()(

)()()(

)()()(

)()()[(

21

212

4

212

3

212

2

212

1

Multi-phase EvolutionChan & Vese (Cont…)Multi-phase EvolutionChan & Vese (Cont…)

])()(

)()()()()())[((

122

4

22

322

222

111

HcI

HcIHcIHcIt

])()(

)()()()()())[((2

212

4

12

312

212

12

HcI

HcIHcIHcIt

Using calculus of variations will result in:

Multi-phase EvolutionChan & Vese (Example)Multi-phase Evolution

Chan & Vese (Example)

The given image contains 4 regions. Three different color boxes are represented in the foreground. The background is considered the fourth region.

Multi-phase Evolution8 Regions-3 Level setsMulti-phase Evolution8 Regions-3 Level sets

1

2

3 4

5 6

78

Chan & Vese (Cont…)Chan & Vese (Cont…)

The curvature is included with a coefficient μ which helps in segmenting images with noise but when the noise level is high, the weight needs to be increased. This affects the boundaries of the object and also increases the convergence time.

Number of regions are always 2n depending on the number of level set functions n.

No vacuum pixels appear because if any point does not belong to a certain region, it will go to another one.

Unless the region can be described by only its mean, the segmentation will fail.

Thank You&

Questions

Thank You&

Questions