02.12.2005 muscle-wp5&7 meeting priors, syntax and semantics in variational level-set approachs...

02.12.2005 MUSCLE-WP5&7 meeting

Priors, Syntax and Semantics

in Variational Level-Set

Approachs

TAU-VISUAL: Nir Sochen & Nahum Kiryati

Based on works with Tammy Riklin-Raviv


Segmentation as an inference problem

Given: a visual data – an image

Infer: (projection of) the scene structure


Maximum Likelihood

ws= World’s State

Then: find the ws that best

explains the data

Ws=argmax P(data|ws)

Problem: UFO


Bayesian formalism

ws= World’s State

Then: Find a ws that explains

the data and has high

probability

ws=argmax P(ws|data)=

argmax P(data|ws)P(ws)


Image Segmentation: General

Task:

Partition the image to its significant domains

Or

Find the relevant shapes


Segmentation Functional

:f is the input imagewhere

C is the segmenting boundary

u is the average color

fidelity

xxx dcdufdufEC

outC

in22

outside

2

inside

shape evolving-prior)(length

inu

outu

Compatibility

with the prior

Smoothnessand length


Image Segmentation: Assumptions

Assumptions:

There are objects and background .

Object pixels: Normal probability dist .

Background pixels: Another normal dist .

Boundary: Close smooth curve


Image Segmentation: Modelling1

)()()(

))(())(()|(

semanticsPSyntaxPCP

xfPxfPCfP

CC

xout

xin

0

0

: Foreground

: Background

C : Separating curve


Image Segmentation: Modelling2

)},(exp{)(

)}(exp{)(

}2/))((exp{))(( 22//

pC

C

outinoutin

CCdNsemanticsP

ClengthNsyntaxP

uxfNxIP


Image Segmentation: steps

Step 0: InitializeStep 1: Updating the constants u_in, u_out as the meanValues of the respective regions

Step 2: Maximization of the posterior w.r.t. C.

Denote E = -log P then

argmax P(C, u_in, u_out) = argmin E(C, u_in, u_out)

Step 3: while not converge go to step 1


Image Segmentation: math

(no shape prior yet)

0 0

)(())((log))((log

),,(

x

out

x

in

outin

ClengthxfPxfP

uuCE


LS Framework for MS

Let : be a level set function which embeds

the contour: 0 xxC

A piecewise constant segmentation of an input image f can be

obtained by minimizing the functional:

dxHHufHufuuECV

1,, 22 (3)

Where H denotes the Heaviside function:

othewise

H0

01 (4)

Chan & Vese Trans. IP 01


LS Framework for MS (Cont.)

Chan & Vese 01

Let

z

zH arctan2

12

1 be a smooth approximation of the

Heaviside function. It’s derivative z

zHz

takes the form: 22

1

zz

(7)


LS Framework for MS (Cont)

The Euler-Lagrange equation for this functional can be implemented

by the following gradient descent :

22 ufufdiv

t

E

(5)

where the scalars uu , are updated in alternation

with the level set evolution:

dxH

dxHxfu

dxH

dxHxfu

1

1

uu , are the mean gray value in the input image inside

and outside C respectively.

(6)


Shape prior + transformation

pT

priorshapeC

outC

in EcdufdufE )(length2

outside

2

inside

xx

),( pTD priorshapeE


Segmentation with Prior


Summary:

There is much to gain from exploiting

The relation statistics <-> variational

approaches

F I N

02.12.2005 muscle-wp5&7 meeting priors, syntax and semantics in variational level-set approachs...

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