Istituto per le Applicazioni del Calcolo "M. Picone" Multigrid Computation for Variational Image Segmentation Problems: Euler equations and approximation.

Istituto per le Applicazioni del Calcolo "M. Picone"

Multigrid Computation for Variational Image Segmentation Problems:

Euler equations and approximation

Rosa Maria Spitaleri

Istituto per le Applicazioni del Calcolo-CNRViale del Policlinico 137, 00161 Rome, Italye-mail: [email protected]

Advances in Numerical Algorithms -Graz, September 10-13, 2003


Variational Image Segmentationand Computational Approach

minimization of the Mumford-Shah functional definition of a sequence of -convergent functionals solution of associated Euler equations

finite difference approximation nonlinear system solution multigrid computation geometric and synthetic images visualization : computed solution (reconstructed

image and edge ), convergence histories


Segmentation Problem

appropriate decomposition of the domain

of a function (computer vision )

is the strength of the light signal striking a plane domain at the point with coordinates

the function is called image

f(x,y)Ω

f(x,y)Ω

(x, y)

f(x,y)


Discontinuity Causes

light reflected off surfaces of solid objects ,

seen from , the camera or eye point, will strike the

domain (retina or film) in various open subsets

which could have common boundaries (“edges” of the

objects in foreground),

surfaces with different orientation (“edges” of a cube),

discontinuity in illumination (shadows),

textured, partially transparent, highly-reflecting objects, ...

Si OiP

R iΩ


the segmentation problem consists in computing a decomposition of

such that the image varies smoothly and/or slowly

within each the image varies discontinuously and /or

rapidly across most of the boundary between different

computing optimal approximations of by piece-wise smooth functions

(restrictions to the pieces differentiables)

D =R1∪...∪Rn

f(x,y)

Ω

f(x,y)R i

R if(x,y)

u(x,y )ui R i


Mumford-Shah Functional

Given the image let be a real function

defined on a domain and

a decomposition of

such that ,

where

and the boundary of

D ={R1(u), ...,R

n(u)}

u

Ωf(x,y)

Ω

Ω = Ri

(u)∪Su

i = 1

nU

S

u= i

(u)

i =1

nU

i(u) R i

(u)


MSF Definition

the MSF is defined in the following form:

where ,

the Hausdorff measure of ,

assigned parameters.

E(u)= (Ω∫∫u- f)2dxdy+

λ ∇uΩ /S

u∫∫

2dxdy+αH1 (Su)

∇u2

=(∂u/∂x)2+(∂u/∂y)2

H1 (Su) Su

λ ,α > 0


MSF Minimization approximation of by

smooth on each

the boundary as short as

possible

(Ω∫∫u - f)2dxdy

f(x,y) u

u R i(u)

S

u= i

(u)

i =1

nU

λ ∇uΩ/S

u∫∫

2dxdy

αH1 (Su)


Parameters

can be calibrated to eliminate

“false edges” , created by noise, and save the actual

image;

is a scaling parameter, controls the noise

effects;

defines the threshold to detect the edge

λ ,α > 0

λ α

2αλ


Interest and Expectation

is a cartoon of the actual image : s a new image in which the edges are drawn sharply and

precisely and the objects are drawn smoothly without texture,

s idealization of a complicated image, representing essentially the same scene

(u,Su ) f(x,y)

droping any of the three terms : s without the first: ,

s without the second: ,

s without the third:

inf E = 0u =0,Su =∅u =f,Su =∅

u =meanRif ,Su ⇒ N ×N grid,

D ⇒ N2 small squares


Variational Convergence the problem of minimizing has been conjectured to

be well posed (open problem), these functionals have minimizers in the spaces of

Special functions of Bounded Variation (SBV), the minimization problem is difficult for the presence of

the set of discontinuity contours as unknown variational convergence to solve minimization of

functional depending on discontinuities:– approximation of a variational problem by a sequence

of more tractable problems

E

Su


the sequence of functionals on a metric space is -convergent to the functional if, :

(i) sequence converging to

(ii) a sequence converging to such

that

-Convergence

{Fk

(u)}U F(u)

∀

limk→∞

infFk(uk)≥F(u0)

∃

limk→∞

supFk(uk)≤F(u0)

{uk}

{uk}

u0

u0

∀ u0 ∈ U


Properties variational property- sequence of

functionals on -convergent to

-if a sequence of minimizers of converges, then the limit is a minimizer of

and converges to the minimal value of

k the -convergence is a variational convergence

-convergence stability under continuous perturbations- Let be a continuous functional

{Fk

(u)}F(u)U

{uk*} {F

k(u)}F(u)

{Fk(uk* )}

F(u)

Fk ⏐ → ⏐ F⇒ Fk +G ⏐ → ⏐ F +G ∀GG


-Convergent Functionals

sequence of -convergent functionals

or

stability property:

Ek(u, z) = (Ω∫∫u- f)

2dxdy+

λ z2∇uΩ∫∫

2dxdy+α (

Ω∫∫∇z 2

k + k(1-z)24 )dxdy

Ek(u, z) = (Ω∫∫u - f)2dxdy+ ˆ Ek(u,z)

Ek ⏐ → ⏐ E


Discontinuity Curves by Control Function the function controls the gradient of

and has values ranging between 0 and 1, the minimizer is close to 0 in a

neighbourhood of the set , which shrinks as

, and close to 1 in the continuity regions the gradient of thus is permitted to become

arbitrarily large along (jumps in the solution) the minimizers converge to a function equal

to 0 along and 1 everywhere else :

the -limit does not depend on

z (x,y ) u

zk*

Su

uk*

Su

zk*

Su

k→ ∞

E(u) z


minimizers of are the solutions of the following

coupled Euler equations

Neumann boundary conditions local minimum of the associated functional more accurate approximation as k increases

Euler Equations

Ek(u, z)

∂∂x

(z2ux)+ ∂∂y

(z2uy)=1λ

(u−f)

Δz=kλα z∇u2−k2

4 (1−z)


the discretization of with grid spacing h andthe finite difference operator ( complete approximation of u and z )

Lac

h

1λ + 4

h2z2 ⎛

⎝ ⎞ ⎠u− 4

h2z2˜ u−

1

2h2z zx ux + zy uy( ) =1

λ f

z kλ4αh2

ux2 + uy

2( ) + 4

h2+ k2

4 ⎛ ⎝

⎞ ⎠ − 4

h2˜ z−k2

4 =0

Ω

˜ w =14 w(x+ h,y) +w(x−h,y) +w(x,y+h) +w(x,y−h)( ) wx =w(x+h,y)−w(x−h,y) wy =w(x,y+ h)−w(x,y−h) wherew=(u,z)

Finite Difference Approximation


Solution Algorithm

equation systems on the grid , with mesh size h and covering the domain :

onegrid ( , l is the grid level ) computation of the solution

LacM wM =FM

ΩG

M

w k* =RelaxM wk

0 ,La,F( )

l =M


w k0

initial guess for GS-relaxation applied to each system associated to the functional for a fixed value of the index k

observations: the discretization step of the finite difference method

should decrease as k increases optimal choice of the parameters λ and α is a

delicate problem

x1,y1( )

x1 <x2 or x1 =x2,y1 < y2{ }x2, y2( )

Ek(u, z)

Gauss-Seidel relaxation rotated lexicographical ordering: a grid point

precedes another point if and only if


Image Segmentation a given image, and

geometrical and realistic problems: – one or more squares and circles, a vase

computed results: smoothed image and control function (discontinuity contours)

convergence histories: residuals, norms, logarithm values, iteration numbers

result visualization

f(x,y)Su = x,y( ) ∈Ω:z x,y( ) =0{ }

uk*

zk*


Experimental Evaluation

experimental choice of the parameters:

, image resolution:

64x64, 128x128, 256x256, 101x101 brightness measurements : 256 levels initial guess: equal to the input image,

equal to 1 everywhere even “small” values of k can be used

λ =3. α =0.05

uk0

zk0


Lac

h

k =2 k =3 k =5


Lac

hCircle64x64

log ( resz )

log(resu)


k and h link

discontinuity set, we can define

in ,

where is the distance of from and

(convergence in ) between 1 and 0 in , :

we have mh =10 l = 20/k where m is the node number in this interval for a given h

by p = hk we can control the “gap” approximation

Su

zk (x,y )

z (x,y )=σ(τ(x,y))τ (x,y )

Bk = x,y( ): τ(x,y) < ηk{ }(x, y) Su

σk ( t) =1 − e− kt

2

−5 l,5l[ ] l =2k

Bk


Conclusion We have defined a multigrid finite difference method

able to improve numerical solution of Euler equations in variational image segmentation

Application to segmentation problems shows the capabilities of the method in computing solutions and providing satisfactory convergence histories

Future research deals with improving performances of multigrid computation

Istituto per le Applicazioni del Calcolo "M. Picone" Multigrid Computation for Variational Image Segmentation Problems: Euler equations and approximation.

Documents

image slide

convergence slide

picone variational convergence

picone minimizers

boundary of slide

picone parameters

edge slide

possible slide