# Discrete Geometry and Mathematical Morphology

Head: | TAJINE Mohamed |

Permanent staff: | BAUDRIER Etienne, DA COL-JACOB Marie Andrée, MAZO Loïc, NAEGEL Benoît, RONSE Christian, TAJINE Mohamed |

Doctoral students: | Yves Michels |

Post-doctoral researcher: | Odyssée Merveille |

## Objectives

This theme regroups activities on geometrical, topological, algebraic and discrete models in imaging.

In Discrete Geometry, we study digital geometries and topologies, as well as discrete tomography. The mathematical tools that we develop are adapted to image analysis and synthesis. Our aims are, on the one hand to build a robust and performing algorithmic in imaging, controlling processing errors related to the use of real numbers, and on the other hand to develop tools for studying different properties (differential, geometrical and topological) of both discrete and Euclidean objects. More generally, we study the transfers of properties between Euclidean spaces representing "reality" and discrete spaces belonging to the computer.

In Mathematical Morphology, we study the construction of new operators for morphological image processing, and the extension of the morphology to new types of objects. Work is also conducted on connected operators, component trees and connective image segmentation, as well as the application of mathematical morphology to color or multispectral images.

The preferred application area is currently biomedical imaging, but we are also pursuing work in remote sensing imagery.

## Research topics

The research activities of the theme are structured along the following axes:

- Discretization models for objects and operators: Extension of the Hausdorff discretization to grey-level images and study of quasi-affine operators.
- Reconstruction of characteristics and objects:
- Estimation of geometrical parameters: The ultimate objective is to design an effective "theory" of measure and integration for discrete spaces, compatible with those of the "continuous" spaces, allowing thus to obtain robust estimators of geometrical parameters (perimeter, area, etc.).
- Reconstruction of Euclidean properties: Design of an axiomatization of digital geometries in the likeness of those of Euclid and Hilbert in the "continuous" space.
- Discrete tomography: Study of classical models of discrete tomography and of the model based on point sources, on the notion of convexity by quadrants, as well as the estimation and reconstruction of characteristics of an object directly from its projections.

- Digital topology: Study of topological models, notably for label images, with a view to applications in medical imaging.
- Image analysis: Study of connected operators and of hierarchical structures.
- Image segmentation: Study of optimization criteria for partial partitions, in relation with order relations on them.
- Pattern recognition: Integration of supervised learning methods in mathematical morphology.

## Illustrations

**Discretization models for objects and operators:**

**Reconstruction of characteristics and objects:**

**Digital topology:**

**Image segmentation:**

**Connected operators:**

## Publications