6 v. Caselles et al.
we prove that 8C converges to the desired contour OO as t~ + oo provided
that we suppose it to be smooth (C 2, in fact), hence, justifying the issue vi)
above for the geometric PDE model. In Sect. 5 we discuss the numerical algo-
rithm and present the experimental results obtained using the geometric PDE
model (1.5a, b).
Let us mention that the basic mathematical tools employed are the theory
of viscosity solutions for second order degenerate elliptic equations (see [I5,
16, 17, 30]) and a lot of standard calculations.
The authors would like to thank P.L. Lions and J.M. Morel for their help
and encouragement during the preparation of this work.
2. Snake model can retrieve angles
Let us consider the image intensity function 1: [- 1, i] 2 ~]R given by
I(x,y)=y
for
y>=O,x+y>=O; -x
for
x<O,y+x<O;
0 for x~0, y~0; (x,y)e[-1,1] 2 .
I is a Lipschitz function such that
[VI(x,y)l=l
for x<0 or y>0; 0 for x>0, y<0
and it is discontinuous on the corner C= {(x, y)e[-1,112
:X=0, yE[--1,
0] or
x~[0, 1], y=0}. For technical reasons we make [VI(x, y)[ upper semicontinuous
by defining it to be 1 on the corner C. We want to study whether the minimiza-
tion of the functional
(2.1)
1 1
E(v;
o~, [3, 9~)= ~ (~ [v'(t)[ z + [3[v"(t)l 2
dt- )L ~ [VI(v(t))] dt
0 0
on a suitable space of admissible functions
Ad(~, fl)
will give an accurate descrip-
tion of the corner C. Our minimization problem can be written as
(2.2) Min
E(v; ~, fl,
2) on
Ad(~, [3)
where
Ad(~, O)= {veil 1
([0, 1], ]R2): v(0)= (0, - 1), v(1)=(1, 0), v(t)e] - 1, 1 [z},
Ad(O,
[3)-- {veH2([0, 1], ~,~1): v(0) = (0, -- 1), v(1)= (1, 0), if(0)= (0, 0), v'(1)
----(0, 0), v(t)~] -- 1, 1[ 2}
and
Ad(~,
[3) = Ad(0, [3), ~>0, [3>0. We are going to prove
the following
Proposition
2.1.
For 2 big enough, there exists a unique function cbeAd(a, [3)
such that E(cl$; ~, [3,
2)=Min
E(v; ~, [3,
2)
on Ad(~, [3), a >O, [3>=0. Moreover cb is
a parametrization of the corner.
The solution of the minimization problem (2.2) is an accurate parametrization
of the corner belonging to
Ad(~, [3).
The proof of Proposition 2.1 is elementary
but long. Therefore, we give the complete proof only in the cases ~ > 0, /7 = 0
and a = 0, [3 > 0. The case ~ > 0, [3 > 0 is rather tedious and we give only a sketch
of the proof. To simplify the presentation, this will be done in the appendix
(Sect. 6).
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