Active Contours without Edges (PPT)

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!
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
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);
•Representation: type of the description •Optimization: solving the equations
Chan and Vese model
Representation/optimization
Contour based gradient descent
Introduction
• Active Contour (Snake Model)
– Kass, Witkin and Terzopoulos [Kas88]
– – – –
- tension - rigidity Eext – external energy Problem is: infxE
Introduction
• The curve is moving with an F speed:
• The geometric active contour, based on a mean curvature (length) motion:
F t
g u 0 d = (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;
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.
in (C) x, y : ( x, y) 0
C x, y : ( x, y) 0
out (C) x, y : ( x, y) 0
Replacing C with Φ
• Introducing the Heaviside (sign) and Dirac (PSF) functions
• “Cartoon” model
F (u, C) L(C) u 0 u d
MS
2
\C
u d
2
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
2
u 0 ( x, y) c1 H(( x, y))dxdy
u
out ( C )
0
( x , y) c 2 dxdy
2
2
0
u
0
( x , y) c 2 dxdy
2
u 0 ( x , y) c 2 (1 H(( x, y))) dxdy
Average intensities
2 y
2 yy x
( )
2 x
2 3/ 2 y
x y
p0 m0 2s 0 p 0 m 2s
xx yy xy
p 0 m 0 200 s 2 0 p 0 m 200 s pp mp pm mm

u
0

(1 H((x, y)))dxdy
Level set formulation of the model
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
Relation with the MumfordShah 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
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
inf F (u ) f ( x, u ( x), u ( x)) dx, u V
– 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.
– Positive on homogeneous regions – Near zero on the sharp edges
g u 0
1 1 G u 0
p
Intoduction
• Level Set methods
– S.Osher and J. Sethian [Set89] – Embed the contour into a higher dimensional space
• We can calculate the average intensities using the step function
c1 ()

u
0
( x , y)H(( x , y))dxdy

H((x, y))dxdy
( x, y)(1 H(( x , y))) dxdy
c 2 ()
in ( C )
u 0 ( x, y) c1 dxdy 2
2
out ( C )
u 0 ( x, y) c 2 dxdy
2
• The minimization problem:
c1 ,c 2 ,C
inf Fc1 , c 2 , C
Chan and Vese model
• c1 and c2 are the average intensity levels inside and outside of the contour • Experiments:
L 0 x, y x, y dxdy

A 0 Hx, y dxdy

Replacing C with Φ
• The intensity terms
u
in ( C )
0
( x, y) c1 dxdy
2
2
0
u
0
( x, y) c1 dxdy
+Automatically handles the topological changes - Slower evaluation • (., t) level set function • Implicit contour (=0) • The contour is evolved implicitly by moving the surface