Active Contours without Edges

Tony ChanLuminita Vese

Peter Horvath – University of Szeged 29/09/2006

Introduction

• Variational approach– The main problem is to minimise an integral

functional (e.g.):

– In the case f:, f’=0 gives the extremum(s)– In the case of functionals similary F’=0, where

F’=(F/u) is the first variation.– Most of the cases the solution is analyticly hard, in

these cases we use gradient descent to optimise.

VudxxuxuxfuF ,))(),(,()(inf

Introduction

• Active Contour (Snake Model)– Kass, Witkin and Terzopoulos [Kas88]

- tension - rigidity– Eext – external energy– Problem is: infxE

+ Fast evaluation- But difficult to handle

topological changes

1

0

22))(()('')('

2

1dssxEsxsxE ext

Introduction

• A typical external energy coming from the image:– Positive on homogeneous regions– Near zero on the sharp edges

p

0

0uG1

1ug

Intoduction

• Level Set methods– S.Osher and J. Sethian [Set89]– Embed the contour into a higher dimensional space

+Automatically handles the topological changes

- Slower evaluation (., t) level set function• Implicit contour (=0)• The contour is evolved

implicitly by moving the surface

Introduction

• The curve is moving with an F speed:

• The geometric active contour, based on a mean curvature (length) motion:

Ft

0ugt

Chan and Vese model position

Chan and Vese model

Energy functionals for image segmentation

Representation/optimization

Contour basedgradient descent Level set based

gradient descent

Important to distinguish the model and the representation

•Model: describing problems from the real world with equations

•Representation: type of the description

•Optimization: solving the equations

Chan and Vese model

• The model is based on trying to separate the image into regions based on intensities

• The minimization problem:

dxdyc)y,x(udxdyc)y,x(u

)C(inACLC,c,cF

)C(out

2

202

2

)C(in 101

21

C,c,cFinf 21C,c,c 21

Chan and Vese model

• c1 and c2 are the average intensity levels inside and outside of the contour

• Experiments:

Relation with the Mumford-Shah functional

• The Chan and Vese model is a special case of the Mumford Shah model (minimal partition problem) =0 and 1=2=

– u=average(u0 in/out)

– C is the CV active contour

• “Cartoon” model

C\

22

0MS duduu)C(L)C,u(F

Level set formulation

• Considering the disadvantages of the active contour representation the model is solved using level set formulation

• level set form -> no explicit contour

0)y,x(:y,x)C(out

0)y,x(:y,x)C(in

0)y,x(:y,xC

Replacing C with Φ

• Introducing the Heaviside (sign) and Dirac (PSF) functions

dxdyy,xH0A

dxdyy,xy,x0L

Replacing C with Φ

• The intensity terms

dxdy))y,x((Hc)y,x(u

dxdyc)y,x(udxdyc)y,x(u

2

10

0

2

10

)C(in

2

10

dxdy)))y,x((H1(c)y,x(u

dxdyc)y,x(udxdyc)y,x(u

2

20

0

2

20

)C(out

2

20

Average intensities

• We can calculate the average intensities using the step function

dxdy)))y,x((H1(

dxdy)))y,x((H1)(y,x(u

)(c

dxdy))y,x((H

dxdy))y,x((H)y,x(u

)(c

0

2

0

1

Level set formulation of the model

2202

2101 )cu()cu()()(

t

Combining the above presented energy terms we can write the Chan and Vese functional as a function of Φ.

Minimization F wrt. Φ -> gradient descent

The corresponding Euler-Lagrange equation:

Approximation of the Curvature

2/32y

2x

2xyyyxxy

2yxx

)(

2)(

s2

s2

m0p0y

0m0px

2

mmpmmpppxy

2

00m0p0yy

2

000m0pxx

s4

s

2s

2

The algorithm

• Initialization n=0• repeat

– n++– Computing c1 and c2

– Evolving the level-set function

• until the solution is stationary, or n>nmax

Initialization

• We set the values of the level set function– outside = -1– inside = 1

• Any shape can be the initialization shape

init() for all (x, y) in Phi if (x, y) is inside Phi(x, y)=1; else Phi(x, y)=-1; fi; end for

Computing c1 and c2

• The mean intensity of the image pixels inside and outside

colors()

out = find(Phi < 0);

in = find(Phi > 0);

c1 = sum(Img(in)) / size(in);

c2 = sum(Img(out)) / size(out);

Finite differences

for all (x, y)

fx(x, y) = (Phi(x+1, y)-Phi(x-1, y))/(2*delta_s);

fy(x, y) =…

fxx(x, y) =…

fyy(x, y) =…

fxy(x, y) =…

delta_s recommended between 0.1 and 1.0

Curvature

grad = (fx.^2.+fy.^2);

curvature = (fx.^2.*fyy + fy.^2.*fxx - 2.*fx.*fy.*fxy) ./ (grad.^1.5);

Be careful! Grad can be 0!

Force

gradient_m = (fx.^2.+fy.^2).^0.5; force = mu * curvature .* gradient_m - nu – lambda1 *

(image - c1).^2 + lambda2 * (image - c2).^2; We should normalize the force. abs(force) <= 1!

Main step:Phi=Phi+deltaT*force;

deltaT is recommended between 0.01 and 0.9. Be careful deltaT<1!

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

• Initialization n=0• repeat

– n++– Determination of the narrow band– Computing c1 and c2

– Evolving the level-set function on the narrow band

– Re-initialization• until the solution is stationary, or n>nmax

Re-initialization

• Optional step

;G*)(sign*T

0)y,x(,1)d,cmax()b,amax(

0)y,x(,1)d,cmax()b,amax(G

);...0,amin(a);0,amax(a

;h

)y,x()1y,x(d;

h

)1y,x()y,x(c

;h

)y,x()y,1x(b;

h

)y,1x()y,x(a

2222

2222

H is a normalizing term recommended between 0.1 and 2.deltaT time step see above!

Stop criteria

• Stop the iterations if:– The maximum iteration number were reached– Stationary solution:

• The energy is not changing• The contour is not moving• …

Demonstration of the program

• Thanks for your attention

MATLAB tutorial

• Imread, imwrite

• .*, .^, ./

• Find

• Size, length, max, min, mod, sum, zeros, ones

• A(x:y, z:v)

• Visualization: figure, plot, surf, imagesc