We are interested in the study within the area of Discrete Geometry of discrete objects having a geometric (planar or spatial) interpretation. The general goal of Discrete Geometry is to define a theoretical framework to translate to Z^n basic notions of Euclidean geometry (such as distance, length, convexity, ...) as “faithfully” as possible. Several approaches exist to pursue this goal.
In our studies, we follow an arithmetical approach, where discrete
objects, such as straight lines or planes, are defined with
arithmetical definitions. These analytical definitions allow us to
represent in a compact way any elementary digital object, to study some
objects that are intrinsically discrete (and are not only
approximations of continuous objects), and to define infinite discrete
objects.
This geometry used by computers is in full expansion nowadays as
attested by the international conferences DGCI (Discrete Geometry for
Computer Imagery) and IWCIA (Workshop on Combinatorial Image Analysis).
Moreover a technical committee on discrete geometry (TC18) of the
International Association of Pattern Recognition (IAPR) has been
created in order to promote this research area.
The study of the properties of discrete objects such as straight lines,
circles, planes, curves and discrete surfaces always remains a topical
subject in the last conferences of the domain. As for the topic of
reconstruction, polyhedrization, i.e. seeking an approximation or a
polyhedrical coding of a discrete 3D object, is a subject for which a
lot of works are still in progress.
These topics are studied by our team and more particularly one of them
which is the study of noisy discrete curves and surfaces. Our aim is to
determine, in the framework of discrete geometry, a paradigm adapted to
these objects, taking into account the noise associated with
acquisition tools and methods. To study noisy discrete curves and
surfaces, we analytically define discrete objects, by extending the
notions of segments of discrete straight lines and pieces of discrete
planes. The study of arithmetical, geometrical and combinatorial
properties of these objects leads to powerful algorithms (recognition,
scanning, ...) and to the extraction of geometrical parameters
(perimeter, curvature, normal, area, ...) of the noisy curves and
surfaces from which we may propose, for instance, reconstruction
algorithms of continuous objects corresponding to the studied discrete
structures.
On the other hand, we expect to enrich the list of discrete objects,
not only by proposing analytical definitions of new objects, but also
by developing some techniques of generation of discrete objects from
their properties.