Mathematics and Image Analysis

Roberto Ardon, joint work with Laurent Cohen
Philips
and
CEREMADE, University Paris Dauphine, France
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We present a novel method for extracting objects from 3D images under user
given constrains. Our approach is based on an energy minimization technique.
The constrains are introduced as curves or points into the 3D image. Our
approach exploits our capability of extracting globally minimal curves in 3D
when fixing their end points. The differential system permitting to build
the set of minimal paths joining the constraining objects, is used to
generate the minimal surface. Through a geometrical approach we derive a
simple partial differential equation that leads to an efficient numerical
construction of this surface. In opposition to most active
models, our surface is not concerned with local minima traps and its
initialization is derived from the constraints objects given by the user.
Our paper describes a fast construction obtained by exploiting Fast Marching
algorithm and ENO schemes for conservation laws. Our algorithm has been
successfully applied to synthetic and 3D medical images.

Here we suggest treating faces as deformable surfaces in the context of
Riemannian geometry. We show that facial expressions can be modeled as
near-isometric transformations (i.e. transformations that preserve the
geodesic distances) of the facial surface. This observation allows
constructing a geometric invariant of the face under different expressions.
As an example, we can convert the Riemannian structure of the facial surface
into a Euclidean one by embedding the surface into a low-dimensional flat
space and replacing the geodesic distances by Euclidean ones.

In many domains such as medical imagery, it is important to be able to match
shapes and to retrieve a deformation between two shapes. Here we will assume
that the shapes are defined by points (polygons (2D) or triangulations (3D)
vertices). The straightforward computation of the correspondences between
points may be numerically untractable from a combinative point of view. That's
why we decide to match quantizations of the shapes. We compute both the
quantization and the deformation at the same time so that we are assured that
the quantization in both shapes is adapted to the deformation and that there
isn't any combinatory issue. The matching consists in the minimization of an
energy composed of two terms : a quantization energy and a deformation energy,

        Segmentation may be formulated as the computation of an optimal domain with regards to a global criterion including both region-based and boundary-based terms. A local shape minimizer of this criterion can be reached using deformable domains, namely region-based active contours. The basic idea is to obtain, from the derivation of the criterion, a Partial Differential Equation (PDE) that drives an initial region towards a local shape minimum of the error criterion. Since the set of image regions does not have a structure of vector space, the main difficulty lies in the derivation of the criterion according to the domain. We show that shape derivation tools, coming from shape optimization theory, can be used to deal with the problem.
        A general framework is proposed for the computation of the PDE from a global criterion. We focus more particularly on the minimization of region-dependent functionals and give some results for the associated PDE. Among region-dependent functionals, we consider a class of functionals based on non parametric probability density functions of image features for comparison to a prototype, or estimation of information measures (PhD thesis of A. Herbulot). Various image and video segmentation problems may then be treated including face or video object segmentation as well as region matching or tracking.