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tions. Dynamic programming often focuses on the
study of specific lines in the image (i.e the epipolar
lines presented in the next section) doing a combi-
natorial search for matches. This search has to sat-
isfy a regularity constraint on these lines and between
them (Ohta and Kanade, 1985). The satisfaction of
this constraint, while preserving edges, remains a dif-
ficult task.
With the development of the Partial Derivative
Equations (PDE) and the increase of the computa-
tional speed, a new strategy based on a variational ap-
proach has been proposed by (Alvarez et al., 2000).
Their scheme naturally introduces a global regularity
constraint and a depth edge constraint into the varia-
tional formulation. After solving the Euler-Lagrange
equation they obtain a PDE composed of two terms: a
regularization term and an attachment term. As their
PDE is a gradient descent of an energy functional,
it is very important to have an energy “as convex as
possible”. For this reason, they use a coarse-to-fine
approach based on a multi-scale gaussian smoothing,
and at each scale, they compute a disparity map. At
the end of this process they obtain the final disparity
map.
Recently (Maier et al., 2003) has proposed a non
hierarchical approach which takes into account the
edges accurately. In (N.Slesareva et al., 2005) the
authors propose to use a Total Variation regularization
(Blomgren, 1998) and an attachment term coming
from the optical flow literature. This method seems
to give promising results especially for noisy images.
We propose a new scheme allowing to compute di-
rectly the disparity map without multi-scale approach.
As the other methods our approach involves two terms
of regularity and data attachment. As regularity term,
we use the regularity constraint of (Alvarez et al.,
2000), but for the attachment term, in order to over-
come the problem of numerous local minima (one of
the characteristics of the dense matching), we use the
signed distance to the local minima given by a sim-
ilarity measure. With this modification, our scheme
is no more deduced from a variational method but,
the iterative process that we have defined has heuris-
tic justifications:
• All pixels minimizing a given similarity measure
are candidates for matching and have the same im-
portance. For the non ambiguous pixels, the match-
ing candidate is often unique and will be the first
match found by our iterative process. For ambigu-
ous ones (as in textured region) the regularity con-
straint is used to improve the matches.
• Local minima of the similarity measure which are
close to each other collapse into one in the set
of matching candidates for our attachment term.
The resulting winning position is the one with the
P
p1
C1
l1
l2
p2
Figure 1: Epipolar constraint: plane (P, C
1
,C
2
) intersects
l
1
and l
2
and p
1
and p
2
are projections of P on the images.
p
1
and p
2
lie on the epipolar lines.
smallest disparity measure.
Our regularisation term is continuous, while the at-
tachement term can be considered as combinatorial
by its discrete valuations. In the following of this pa-
per our method will be referenced as Iterative Scheme
to Local Minima (ISLM). Note that the combinatorial
aspect of our method is not implemented as a kind of
relaxation, but by properties of the valuation of our
attachment term.
The paper is organized as follows: section 2
presents the epipolar constraint and the disparity map.
Section 3 presents PDE based on variational approach
(Alvarez et al., 2000). The role of Nagel-Enkelmann
operator, used in our work, as a diffusion-reaction
term preserved discontinuity will be explained here.
Section 4 presents our attachment term and its prop-
erties, we discuss the differences with the Alvararez
variational approach. Our generic attachment term
can be computed with different similarity measures,
and we present here a simple square difference on
a correlation mask. Section 5 develops the compu-
tation of our attachment term. Finally, in section
6 experimentations are presented. We use synthetic
data where the true disparity is known and also real
images. The pertinence of our approach is demon-
strated by quantitative results when the true disparity
is known, and by qualitative results otherwise.
2 EPIPOLAR GEOMETRY AND
DISPARITY ESTIMATION
We note by I
1
and I
2
the two different views of a rigid
scene. Each three-dimensional point P of the scene
form a plane with the two optical centers C
1
and C
2
.
The intersection of this plane and the retinas are
lines, respectively l
1
and l
2
called epipolar lines. The
epipolar constraint expresses the fact that pixels p
1
and p
2
, which are respectively the projections of P on
I
1
and I
2
, lie respectively on l
1
and l
2
(see Figure (1)).
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