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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
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
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)
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
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)
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
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)
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
-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
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
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)
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
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*
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
Lac
h
k =2 k =3 k =5
Istituto per le Applicazioni del Calcolo "M. Picone"
Lac
hCircle64x64
log ( resz )
log(resu)
Istituto per le Applicazioni del Calcolo "M. Picone"
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
Istituto per le Applicazioni del Calcolo "M. Picone"
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