active contour without edges - university of...

Active Contour without Edges

Tony Chan and Luminita VeseTony Chan and Luminita Vese

Basic idea in classical active contoursCurve evolution and deformation (internal forces):

Min Length(C)+Area(inside(C))Boundary detection: stopping edge-function (external forces)

0)(lim0 tggg

Example:

0)(lim , ,0

||11|)(| 0

puG ||1 00

Snake model (Kass, Witkin, Terzopoulos 88)

2 |))((||)(''||)('|)(inf dssCudssCdssCCFC

Geodesic model (Caselles, Kimmel, Sapiro 95)

|)))(((||)('|2)(inf dssCIgsCCFC

Basic idea in classical active contours

Curve evolution and deformation (internal forces):

Min Length(C)+Area(inside(C))Boundary detection: stopping edge-function (external forces)

0)(lim , ,0

1Example: puGug

||11|)(|

fihiG 00 ofversion smoother a is uuG

edgeson theonly0|)(| ug edges on theonly 0|)(| 0 ugedges fromfar 0constant|)(| 0 ug

Classical Snakes and Active Contour Models

)(Models in variational formulation

* Snake model (Kass, Witkin, Terzopoulos 88)

)(sCs a parameterization of C

* Geodesic model (Caselles, Kimmel, Sapiro 95)

2 |)))(((||)('|)(inf dssCIgdssCCFC

|)))(((||)('|2)(inf dssCIgsCCFC

* C ll C tt C ll Dib 93

Other related models:

* Caselles, Catte, Coll, Dibos 93

* Malladi, Sethian, Vemuri 93

* Caselles Kimmel Sapiro 95 Caselles, Kimmel, Sapiro 95

* Kichenassamy, Kumar, Olver, Tannenbaum, Yezzi ‘95

Some active contours with edge-functionModels in variational formulation

* Snake model: Kass, Witkin, Terzopoulos ’88, , p

* Geodesic model: Caselles, Kimmel, Sapiro ’95

M d l i l l t f l ti

* Kichenassamy, Kumar, Olver, Tannenbaum, Yezzi ‘95Models in level set formulation:

* Geometric model: Caselles, Catte, Coll, Dibos ‘93

M ll di S hi V i ‘93Malladi, Sethian, Vemuri ‘93

* Geodesic model: Caselles, Kimmel, Sapiro ‘95

A fitting term “without edges”

)(10 ||||

CoutsideCinside

dxdycudxdycu

whereCuaveragec

Cuaveragecoutside)(

inside )( 01

Cuaveragec outside)( 02

Fit > 0 Fit > 0 Fit > 0 Fit ~ 0

Minimize: (Fitting +Regularization)

Fitting not depending on gradient detects “contours without gradient”

An active contour model “without edges”

Fitting + Regularization terms (length, area)(C. + Vese 98)

))((||)(inf CinsideAreaCCccF

Fitting Regularization terms (length, area)

))((||),,(inf21 Ccc

d dd d

CinsideAreaCCccF

210 ||||

Cinside Coutside

dxdycudxdycu

C = boundary of an open and bounded domain

|C| = the length of the boundary-curve C

Advantages- detects objects without sharp edges

- detects cognitive contours

- robust to noise; no need for pre-smoothing

Examplesp

Relation with the Mumford-Shah segmentation model* The Mumford-Shah functional ‘87

CudxdyudxdyuuCCuF 22

0,||||||),(inf

- u is an optimal approximation of the initial image- u is an optimal approximation of the initial image

- C is the set of jumps or edges of u

i th t id th d C* The model is a particular case of the Mumford-Shah functional

- u is smooth outside the edges C

Cuc inside)(average

outside )(average inside )(average

u is the best approximation of taking only two values and1c 2cu- u is the best approximation of taking only two values and , and with one edge C

- the snake C is the boundary between the sets and

the snake C is the boundary between the sets and

- we do not need regularization in u, because are constant

1cu 2cu

Level Set Representation (S. Osher - J. Sethian ‘87)n 0),(|),( yxyxC

Inside C

Outside C0 0

),(|),( yy

Outside C0

C= boundary of an open domain

Normal

divKCurvaturen

Example: mean curvature motion* All t ti t l h i d b ki* Allows automatic topology changes, cusps, merging and breaking.

•Originally developed for tracking fluid interfaces.

Variational Formulations and Level Sets(Following Zhao, Chan, Merriman and Osher ’96)

h i id f i

0),( :),( yxyxC

HC |)(|||L thThe Heaviside function

0if00 if ,1

dxdyHCinsideArea

)())((

|)(|||Length

0 if ,0

dxdyHCinsideArea )())((

))),((1()),((),( 21 yxHcyxHcyxu The level set formulation of the active contour model

ccF ),,(inf 21

))),((()),((),( 21 yyy

dxdyHHccF

)(|)(|),,(

21,, 21

dxdyHcyxudxdyHcyxu ))(1(|),(|)(|),(| 220

HHFF )())(())()((

Existence of minimizers among characteristic functions of sets with finite perimeter (in any dim. N)

HHFccF

)(||)(

)( )),(()),(),(( }0{21

dxxcudxxcu ))(1(|)(|)(|)(| 220

z. Lipschit withbounded open, be Let : RTheorem N

.,.1,0)(),( ),(inf problem onminimizati following the then),( If 0

eadxxBVFLu

solution.a has

,,)(),(),(

i ib d dff if)(BV BVin scompactnes and TV of :s variationof calculus in standard :Proof

variationboundedoffunctions of space)(lsc

The Euler-Lagrange equations)(i f U i h i i f),,(inf 21, ,21

dfE ti

Using smooth approximations for the Heaviside and Delta functions

dxdyHudxdyHu ))(1()( 00 2 1 andfor Equations cc

dxdyHc

))(1()( ,

)()( 21

),,(for Equation yxt

)()(||

)( 220

210 cucudiv

),(),,0(

0 yxyx

Two approximations of the Heaviside and Delta functions

xxH arctan21

||ifsin111 - xif,0

if ,1)(

,-support compact small has -

of curves level allon acts -everywhere 0 -

||if,sin12

computed are minima localonly - of curve

level-zerothearround localy,only acts - lyautomatica contoursinterior detects -

minima global compute to tendency thehas -

AdvantagesExperimental Results

CofEvolution )(Averages cc AdvantagesAutomatically detects i t i t !

C of Evolution ),( Averages 21 cc

interior contours!

Works very well for concave objects

Robust w.r.t. noise

Detects blurred contours

Th i i i l bThe initial curve can be placed anywhere!

Allows for automatical change of topolgy

Contours without gradient (cognitive contours)

A spiral from an art picture(the initial curve is the border of the image)

Europe nightlightsTh d l d t t l t f i t ( iti t )The model detects clusters of points (cognitive contours)

A plane in a noisy environment

A galaxyg y

From Active Contours (2D) to Active Surfaces (3D)

MRI DATA FROM LONI-UCLA

Potential Application to Data Mining:pp g

(Data from V. Kumar et al: Chameleon, IEEE Computer 2000)

Extension to Vector-Valued ImagesTony F. CHAN & Luminita VESE & B. Yezrielev SANDBERGApplications

* Color images (RGB)

* M l i l (PET MRI CT)* Multi-spectral (PET, MRI, CT)

channels ),,...,,( ,02,01,00 Nuuuu N

Vector-Valued image:

( f h h l i idectorsconstant v are , 21 cc

The model

(averages of each channel inside and outside the curve)

),,(inf 21 CccF

21 ))(()(),,( CinsideAreaCLengthCccF

),,( 21,, 21 Ccc

2,1,0 ||||

Cinside Coutsideiiii

idxdycudxdycu

* The model can be solved using level sets, similar with the scalar case

Extension to Vector-Valued ImagesChannel 1 with occlusion Channel 2 with noise

Recovered object and averages

Color (RGB) picture Intensity (gray-level) pictureColor Images

Recovered objects and contours in RGB modeRecovered objects and contours in RGB mode

45 transforms of image with Gabor functionActive Contour for Texture

F F120 120

180 180

User Picked Gabor transforms yield final contour

Texture Segmentation with User picked Gabor transforms

Gabor Transforms

Original Image F=120

Initial Contour Final Contour Final Contour on the original image

U i d T S i S h iUnsupervised Texture Segmentation SyntheticAutomatically select small subset of Gabor frames for AC model

Final Contour on the original imageOriginal Image Final Contour

Original Image Final Contour on the original imageFinal Contour

PlanIntroduction and Motivations:

Active contours “without edges” Chan-Vese, Active Contours without Edges, SS ’99, IEEE IP

Generalization to the Mumford-Shah model:Chan-Sandberg-Vese, Active Contours without Edges for Vector-Valued Images, JVCI

Generalization to the Mumford Shah model:

The piecewise-constant case

The piecewise-smooth caseChan Vese I t ti i l l t d th i i t t M f dChan-Vese, Image segmentation using level sets and the piecewise-constant Mumford-Shah model, A level set algorithm for minimizing the Mumford-Shah functional in image processing, preprints 2000, to appear in Special Issue of IJCV, for VLSM/SS’01

Active contours “without edges”

Generalization to the Mumford-Shah model:Generalization to the Mumford Shah model:

The piecewise-smooth case

Variational Segmentation ModelsM f d Sh h D l M M l S li i i Sh h Ch b llMumford-Shah, Dal Maso-Morel-Solimini, Shah, Chambolle, Gobino, March, Koepfler-Lopez-Morel, Ambrosio-Tortorelli, Morel-Solimini BraidesMorel Solimini, Braides,

Mumford-Nitzberg-Shiota, Zhu-Lee-Yuille,

Chambolle-Dal Maso,

Chambolle-Bourdin, Cortesani-Toader,Chambolle Bourdin, Cortesani Toader,

Malladi-Sarti-Sethian, …

Works Related to Ours:

Samson-Feraud-Aubert-Zerubia, Sifakis-Garcia-Tziritas,, ,

Paragios-Deriche, Yezzi-Tsai-Willsky

Mumford-Shah Segmentation 89

dSuuu ))(||(min 22

SudSuuu ))(||(min

S=“edges”

Algorithms for Minimizing Mumford-Shah energy

Difficult problem in practice: non-convex; lower-dim. unknown C

Solutions proposed:

* Region growing and merging: use MS energy in greedy algorithm (Dal Maso - Morel - Solimini, Koepfler-Lopez-Morel)

* Elliptic approximations: embed C in 2D phase-field function (Ambrosio - Tortorelli, Chambolle, Braides, March)

* Statistical framework: (Zhu-Lee-Yuille)

Our work:

* provides an efficient level set formulation for the M-S modelp

* inherits the advantages of level sets

Relation C-Vese AC & Mumford-Shah segmentation models

* Our AC model is a particular case of the Mumford-Shah functional

u inside )(average1

u outside )(average2

- u is the best approximation of I taking only two values and , and with one edge C

- the contour C is the boundary between the sets and

- we do not need regularization in u, because are constant

1cu 2cu

Relation C-Vese AC & Mumford-Shah segmentation models* The Mumford-Shah functional 87

CudxdyudxdyIuCLengthCuF 22

,||||)(),(inf

- u is an optimal approximation of Iu is an optimal approximation of I

- C is the set of jumps or edges of u

- u is smooth outside the edges C

* Our AC model is a particular case of the Mumford-Shah functional

CIc inside)(average1

CIcCIc

u outside )(average

inside)(average

i th b t i ti f I t ki l t l dc c- u is the best approximation of I taking only two values and , and with one edge C

- the snake C is the boundary between the sets and

1cu 2cu

- we do not need regularization in u, because are constant21,cc

Generalizing CV-AC to MS

• From 2 segments (1 level set function) to multiple segments and level set functionsp g

• From p.w. constant to p.w. smoothapproximationsapproximations

Generalization to Multiple (>2) SegmentsThe Mumford Shah piecewise constant segmentation modelThe Mumford-Shah piecewise-constant segmentation model

N d 1 l l f i 2 i l j i

MS CdxdycuCuF || ||),( 20

Need > 1 level set function to represent > 2 segments, triple junctions, …

Classical methods for Multiphase Level Set Representation:

(Zhao-Chan-Merriman-Osher):

- associate one level set function to each phase, allows for triple junctions

hi h t ti l t- high computational cost

- needs additional constraints to prevent vacuum and overlap

New Multiphase Level Set Representation:segmentsor phases 2represent can wefunctions,set level using - nn

- reduced computational cost

New Multiphase Level Set Representation:

- by definition of the partition, no vacuum and no overlap

- allows for triple junctions and other complex topologies

Multiphase level set representations and partitionsallows for triple junctions, with no vacuum and no overlap of phases

phases2)( n 4-phase segmentation2 l l f i:Curves

phases2 ),...,( 1 n

2 level set functions

}0{:Curves

}0{}0{:Curves

2-phase segmentation

1 l l t f ti

1 level set function

Example: two level set functions and four phases:segmentsorphases4)(functionssetlevel2

functionssetlevelThe)( 0,0 ,0,0 ,0,0 ,0,0

:segmentsor phases4 ),(functionsset level 2

21212121

vectorConstant ),,,( functionsset level The ),(

00011011

21 ccccc

))(1))((1()())(1( ))(1)(()()(

21002101

21102111

HHcHHcHHcHHcu

))(1)((||)()(||),(

Energy ))())((()())((

100212

21002101

dxdyHHcudxdyHHcucFInf

))(1))((1(||)())(1(||

))()((||)()(||),(

000212

2110021110),(

dxdyHHcudxdyHHcu

|)(||)(| 21 HH

Existence of minimizers f

Theorem:

Existence of minimizers among characteristic functions of sets with finite perimeterp

Similar with the previous case

Remarks:

Non-convex minimization problem

No uniqueness of minimizers

The Euler-Lagrange equations

00in)( constants The

umeanc

Using smooth approximations for the Heaviside and Delta functions

00i)(0,0in )( 0,0in )( 0,0in )(

umeancumeancumeanc

),(in sPDE' The

0,0in )(

umeanc

))(1(||||)(||||

||)( 2

HcucuHcucudivt

))(1(||||)(||||

||)( 1

HcucuHcucudivt

),...,,(for Similarly 21 n

Results with 2 level set functions (4 phases)* t diff t i iti li ti* two different initializations

* interior contours are automatically detected

* model robust to noise

Faster

Energy Decrease With Iterations for 3 initial conditions(here, only with (a) and (c) it converge to the global minimum)y g g

( )(a)

Non-convex problem: different (I.C.) may converge to local/global min.

Detection of contours without gradient -cognitive contours

Triple junctions can be detected and represented using only two level set functions, without vacuum and without overlap (new).

0 0 21

An MRI brain image

Phase 11 Phase 10 Phase 01 Phase 00

mean(11)=45 mean(10)=159 mean(01)=9 mean(00)=103mean(11)=45 mean(10)=159 mean(01)=9 mean(00)=103

Numerical results on an MRI brain picture

Multiphase segmentation model using two level set functions and four phases

A real outdoor picture

mean(11)=159 mean(10)=205 mean(01)=23 mean(00)=97mean(11)=159 mean(10)=205 mean(01)=23 mean(00)=97

Three level set functions representing up to eight phasesSix phases are detected, together with the triple junctions.

Evolution of the 3 level sets Segmented image

Evolution of the 3 individual level sets.

Original gene chip data

(Terry Speed)

Detected contours

Segmented images for three different scale parameters

Original gene data (G) Detected contours Segmented image

1 Segment(11)=180 2 Segment(10)=93 3 Segment(01)=56

Gene Microarray4 Segment(00)=6 Gene Microarray

Expression Data

Narrow band

It is useful to compute the level set function not on the whole image domain but in a narrow band near to the contour. Abs()<d

Decreasing the computational complexity.

Narrow Band AlgorithmNarrow Band Algorithm

• Tag alive points in narrow bandTag alive points in narrow band

• Build land mines to indicate near edge

i i li f i id h• Initialize far away points outside the narrow band with large positive (negative) values if l id (i id ) h f i lfvalues are outside(inside) the front itself.

• Solve level set equation until land mine hit

• Rebuild and loop.

• Bandwidth can be set as 12Bandwidth can be set as 12

Active contours “without edges” Chan-Vese, Active Contours without Edges, SS ’99, and IEEE IP

Generalization to the Mumford-Shah model:Chan-Sandberg-Vese, Active Contours without Edges for Vector-Valued Images, JVCI

Generalization to the Mumford Shah model:

The piecewise-smooth caseChan Vese I t ti i l l t d th i i t t M f dChan-Vese, Image segmentation using level sets and the piecewise-constant Mumford-Shah model, A level set algorithm for minimizing the Mumford-Shah functional in image processing, preprints 2000, to appear in Special Issue of IJCV for VLSM/SS’01

Active contours “without edges”

Generalization to the Mumford-Shah model:Generalization to the Mumford Shah model:

The piecewise-smooth case

Generalization to Piecewise-Smooth Approximations by the Mumford-Shah segmentation modely f gThe Mumford-Shah segmentation model ‘89

),(inf CuF MS

MS CdxdyudxdyuuCuF ||||||),( 220

),(inf,

A level set algorithm for the M-S model

( fi h C i h b d f d i )

0if),,(

)(0)(|)(yxu

(assume first that C is the boundary of an open domain)

0 if ),,(0),,(

),( ,0),( |),(

yxuyxu

yxuyxyxC

))(1)(,()(),(),( HyxuHyxuyxu

),,(inf,,

2 ||||),,(

dxdyuudxdyuuuuF

2 ||||||

dxdyudxdyuC

Using the Heaviside function:

|)(|))(1(||)(||

))(1(||)(||),,(

HdxdyHudxdyHu

dxdyHuudxdyHuuuuF

|)(|))(1(||)(|| HdxdyHudxdyHu

The model is a common framework for activeThe model is a common framework for active contours, denoising, segmentation and edge detection.

The Euler-Lagrange equations

u 0on 00,on )( 02

unuuuu

0on 0 0,on )( 02

2 ||||||||)( uuuuuudiv

00 ||||||||

The model is a common framework for active

Curvature Jump of energy terms in u

The model is a common framework for active contours, denoising, segmentation and edge detection.

functions by 0 on and 0 on extend tohave We :Remark

Sbert) Morel,(Caselles, functions of ionInterpolat :direction) normal thein diffusion(by solutions Several

Aslam) (Fedkiw, method fluidGhost Cao) Crandall, (Jensen, extensions Lipschitzmizing Mini

),( ,p

),( Normal

yxyx nnN

:Examples||||

on 0 0,on 2 22

yyyxyyxxxxt nuununnunu

,...0in 0 :Or

yyxxt ununu

Combining p.w. smooth approx. with multiple LS’sWe consider (again) two level set functions with four phases

Remark: Based on Four Color Thm., 2 LS’s should suffice.

))(1))((1()())(1(

))(1)(()()( 2121

HHuHHu

HHuHHuu

))(1))((1()())(1( 2121 HHuHHu

)( ),( 21

The energy and the Euler-Lagrange equation can be written and solvedThe energy and the Euler Lagrange equation can be written and solved in a similar manner

To handle triple junctions and more general casesWe consider (again) two level set functions with four phases

Remark: Based on the Four Color Thm., it should suffice.

))(1))((1()())(1(

))(1)(()()( 2121

HHuHHu

HHuHHuu

))(1))((1()())(1( 2121 HHuHHu

)( ),( 21

The energy and the Euler-Lagrange equation can be written and solvedThe energy and the Euler Lagrange equation can be written and solved in a similar manner

Existence of minimizers in the space SBV

(Special functions of Bounded Variation)

LBVSBV NN )()()( 1/ Proof: standard in calculus of i i

BVu )(

variations

JuCudxxuDu )( Compactness and l.s.c. results on SBV by L. Ambrosio

JudxxuDuSBVu

)()( )(

Signal denoising - segmentation - jump detection(numerical results in 1D; note jumps well located and preserved)

Remark: Related works by Anne Gelb and Eitan Tadmor usingRemark: Related works by Anne Gelb and Eitan Tadmor using spectral methods

Piecewise-smooth approximationsActive Contours+Denoising+Segmentation g g

M S Th th bj t d t t d ith th t l !M-S: The three objects are detected with the correct values! Before (A-C): all objects were detected, but with the same mean

2 level sets (four phases) in the piecewise-smooth case

Initial Noisy Image Initial curves

Evolutions to the steady state (denoised - segmented image)

Future worksConsider other surface energies depending on the normal- Consider other surface energies, depending on the normal

or on the curvature (beyond image processing applications)

Th lti h b d f l i (P P )- The multiphases can be used for occlusion (P. Perona)

- Higher order regularizations, to detect jumps in the derivatives

- Study the convergence of the algorithm, when adding some y g g gregularizing/elliptic terms

- Use the method for related problems arising in fractureUse the method for related problems arising in fracture mechanics and optimal design

Viscosity solutions for these geometric curve evolution PDEs- Viscosity solutions for these geometric curve evolution PDEs

Features & Advantages of PDE Imaging ModelsConclusion

Features & Advantages of PDE Imaging Models

* Use PDE & Geometry concepts: gradients diffusion curvature level Use PDE & Geometry concepts: gradients, diffusion, curvature, level sets

* Exploit sophisticated PDE and CFD (e.g. shock capturing) techniquesp p ( g p g) q

Restoration:

sharper edges less edge artifacts often morphological- sharper edges, less edge artifacts, often morphological

Segmentation (using level set techniques):

l d i i b d ll d l i f b d i- scale adaptivity, geometry-based, controlled regularity of boundaries, segments can have complex topologies

Newer, less well developed/accepted.

OUTLINE

• Review of Variational Level Set Segmentation• Fast AlgorithmFast Algorithm

A Fast Algorithm for Level Set Based Optimization(Bing Song and Tony Chan, 2003)

Available at www math ucla edu/applied/cam/index htmlwww.math.ucla.edu/applied/cam/index.html

CAM 02-68(Related work by Gibou & Fedkiw)(Related work by Gibou & Fedkiw)

Outline

• Motivation• General AlgorithmGeneral Algorithm• Application to the Chan-Vese segmentation

modelmodel• Convergence of algorithm• Examples• Closely related work by Gibou and Fedkiwy y

Motivation

• Minimize level set based functional

))((min HF

• Examples:

))((min

– Image segmentation– Inverse problem– Shape analysis and optimizationShape analysis and optimization– Data Clustering– …

How to solve it?

))(( HFt

• Usually use gradient descent flow

• Drawback

))(( HFt

Drawback– Slow (CFL)– F need to be differentiable

An Insight

• Segmentation only needs sign of but not its value

Di t h ki f bj ti f ti d• Direct search making use of objective function does not require derivatives of functional

New Algorithm

1. Initialization. Partition the domain into and

2. Advance. For each point x in the domain, if the energy F lower when we change )(xthe energy F lower when we change to , then update this point.

3 Repeat step 2 until the energy F remains

3. Repeat step 2 until the energy F remains unchanged

An example: Chan-Vese model

)(|||)(|(),),(( 2

10121 HcuHccHF

))(1(|| 2202

Core Step of AlgorithmChange in F if is changed to at a pixel:0 0

nncuFF

lengthinchangemcuncuF 1

2212 gg

mn 1)(

1)( 1212

How to calculate length term?2

,,1 ))()(())()((|)(| jijijijiij HHHHH

Change in length term can be updated locally.

Only possible value for length term locally is 0 1 sqrt(2)Only possible value for length term locally is 0, 1, sqrt(2) depending the 3 points in the length term belong to the same region or not.

A 2-phase example

(a), (b), (c), (d) are four different initial condition.All of them converge in one sweep!All of them converge in one sweep!

Example with Noise

Converged in 4 steps.

(G di D E l L k 400 )(Gradient Descent on Euler-Lagrange took > 400 steps.)

Convergence of the algorithm

Theorem: For a two phase image, the algorithm will converge in one sweep, independent of sweeping order.

Proof considers the following 2 cases:Green part of inside of contour will flip sign. Only one of the 2 black parts can change sign

Why is 1-step convergence possible?

• Problem is global: usually cannot have finite step convergence based on local p gupdates only

• But, in our case, we can exactly calculate the global energy change via local updatethe global energy change via local update (can update global average locally)

Application to piecewise linear CV modelApplication to piecewise linear CV model(Vese 2002)

))(1(||

)(|||)(|(),),((2

22100121

Hybxbbu

HyaxaauHccHF

))(1(|| 21002 Hybxbbu

Original P.W.ConstantConverged in 4

P.W. LinearConverged in 6

Application to multiphase CV modelApplication to multiphase CV model(C.- Vese 2000)

• Using n level set to represent 2n phases.

IIIn HdxdycucF

20 |)(|||),(

mInI m 121

Multiphase CV model (cont’d)

Converged in 1 step.

Gibou and Fedkiw’s method (02)• First discard the stiff length term in E L:• First, discard the stiff length term in E-L:

)(:)()( 222

211 xVcucu

• Take large time step enough to change sign• If , then 0)()( xVx ))(()( xsignx ,• Followed by anisotropic diffusion to handle

noisenoise• Difference (Ours wrt GF):

A ifi d f k t h dl l th t– A unified framework to handle length term– F does not need to be differentiable

Conclusion• A fast algorithm to solve the Chan Vese• A fast algorithm to solve the Chan-Vese

segmentation models.• 1-step convergence for 2 phase images (no reg )1-step convergence for 2 phase images (no reg.).• Robust wrt noise: 3-4 step convergence.• The gradient of functional is not needed• The gradient of functional is not needed.• We think it can be applied to more general level

set based optimization problemsset based optimization problems.• In general, cannot expect finite step convergence

(need global change from local update). Can get(need global change from local update). Can get stuck in local minima.

That’s all.Th k f i !

That’s all.Th k f i !Thank you for your patience!Thank you for your patience!

Logic Operators on Multi-Channel Active Contour(Chan, Sandberg 2001)

• An Example for two channel logical segmentation:• An Example for two channel logical segmentation:1A 2A

21 (a) AA 21 (b) AA 21 ~ (c) AA

Redefining the fitting term of CV Model

dxdyyyyxxxfClengthccCF nn ),..,,,,..,,()(),,(min 21,21

using the logical variables:

otherwise1Cinsidefit good 0),( (x,y)yxix

th i1C outsidefit good 0),(

otherwise 1 (x,y)yxiy

otherwise 1

Logical variables for CV Model:

C inside 2||)(

2|0|),( (x,y)ici

ouiuyx

iciuyxix

C outside 0

),( iouyx

Coutside2|0|

C inside 0 ),(

(x y)iciu

Coutside2||),(

max0 (x,y)ici

ouiouyx

ii xx 1'And their complements:

ii yy 1'

I i f bjTruth Tables

Intersection of objects:intersection of xi inside C, and union of yi outside C.

Union of objects: (analogous)Truth Table using parameters :

Truth Table for 2 Channels

Union of objects: (analogous)

Truth Table for 2 Channels

inside C 1 1 0 0 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1

1x 2x 1y 2y 21 AA 21 AA 21 ~ AA

outside C 0 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0

Fitting the truth table for function f:

)),(),((2/)),(),((),(

2/)),(),(()),(),((),(

yxyyxyyxxyxxyxf

2/12121~ ))),(1)(,((2/))),(1(),((),(

yxyyxyyxxyxxyxf AA

Examples using logic operators1A 2A

1A 2A 3A

21 (a) AA

21 (b) AA 321(b) AAA 321(a) AAA 321 ~(c) AAA 321 ~(d) AAA 321( )321( ) 321( ) 321( )

Given the 3 channels 4 possible logic operations are implemented.

21 ~ (c) AA

Time evolution for original example

Example of an Application

Original image with contour overlapping

Contour only

Time evolution for finding the “tumor” in the first image that is not in the second.

A Fast Algorithm for Variational Level Set Segmentation

Tony F. Chan

Department of Mathematics, Univ. Calif. Los Angeles

SONAD 2003SONAD 2003

McMaster University, Ontario, Canada

May 2, 2003

Reprints: www.math.ucla.edu/~imagersp g

Supported by ONR and NSF

(J i k i h Bi S )(Joint work with Bing Song)

OUTLINE

• Review of Variational Level Set Segmentation• Fast AlgorithmFast Algorithm

Narrow band

• Initialization n=0• repeat

– n++– Determination of the narrow band– Computing c and c– Computing c1 and c2– Evolving the level-set function on the narrow band

– Re-initialization

• until the solution is stationary, or n>nor n>nmax

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