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Digital geometry outline

The goal of my research in the area of digital geometry is to be abble propose a complete discrete framework to describe digital images. Digital topology should allow me to have a consistent model to work with, as well to help the image analysis as to represent consistently objects segmented. Digital geometry would allow to characterize digital object with geometrical features coded exactly by integer or rationnal representation.

Digital geometry
Digital geometry Imprimer Envoyer

Digital geometry consists in studying definitions and properties of digital objects with a complete discrete point of view. It means working with integer (at least rational) representation to have exact coding and not handling double approximation: real number to float, float to integer matrix display.

In order to have a complete discrete framework to image analysis, I try to understand the discrete properties of digital objects and so being abble to better recognize them and to characterize the geometry of regions images by an exact (integer) representation. So coding efficiently (in time and space) the geometry of region contours, of surfaces and volumes, identified in images after analysis, will be possible.

We have chosen to use approach proposed by J.-P. Réveilles which defines digital objects analytically as solution of diophantine inequalities systems.

 

I first studied linear object as digital lines and digital planes. With V. Berthe, D. Jamet and F. Philippe, we showed that digital planes, whatever their thickness, have an intrinsic 2 dimensional structure. For that, we proved that all digital planes are functionnal, not only naive ones as already known. We call this property, the Generalized Functionnality ([BerFioJamPhi:IVC07], [BerFioJam:DGCI05]).

Digital Naive Plane Digital naive surface

Next we have started a study of non linear object, mainly hyperspheres, curves. This works are mainly those of J.-L. Toutant, a former phd student. We have shown that it is important to separate analytical representation of digital object to the one of the thickness. In fact, this last should be related to the discretization scheme chosen. Thickness is then expressed as a function. So we claimed thatthickness has to be variable and dependent on the location. It should not be constant. Note that for linear object, variability of the object being constant, thickness function gives a constant value. This approach is then general and includes digital lines and planes, but allow also to generalize all previous definition of circle. Moreover, it allows to garantee some digital topological properties by an accurate choice of the thickness function.

Naive Ellipsis Curves Part of digital naive sphere Digital Regular Spheres
Mise à jour le Samedi, 18 Avril 2009 17:16
 
Digital topology Imprimer Envoyer

My goal is to define a formal model of digital images that will allow to propose a coherent discrete framework for image segmentation, contour representation, and geometry of segmented objects in image.

T.Y. Kong et A. Rosenfeld have defined digital topology as:

Digital topology is the study of the topological properties of image arrays. Its results provide a sound mathematical basis for image processing operations such as image thinning, border following, countour filling and object counting.

This model would allow us to define notions such as interior, closing, borders, adjacency, connexity, curves, surfaces, ... Our approach is to no more consider image as an homogeneous space but as a cellular complex where interpixel elements ensure the consistency of regions.

2d interpixel elements 3d interpixel elements open star of a pointel

My approach is to use an order relation that characterizes topology on finite set, to associate a dimension to cells relatively to this order, and to finally define the digital topology as a quotient topology. So I have defined image as a polyhedral cellular complex associated to quotient topology of ℝn: the star-topology (see paper [AhrAubFio:DGCI95] or my thesis [Fiorio:HDR08],[Fiorio:th95]).

This topology is based on interpixel elements has axioms that garantee properties of segmented regions. Then, n-cells are the representative elements of regions as attended in image analysis, interpixel cells are only there to garantee consistency of connexity and adjacency.
Mise à jour le Samedi, 18 Avril 2009 17:18