DISCRETE GEOMETRY AND MORPHOLOGY MATHEMATICS

Main lecturer

Mail address

Phone number

Mohamed TAJINE, Professor in Computer Science

tajine@unistra.fr, office XXX,

+33 (0)3 68 85 45 73

Other instructor(s)

B. NAEGEL

APOGEE code

Track - Year - Option - Semester

Coefficient = ECTS

Duration

EP013M90

Engineer - 3Y G ISSD - S9

Master - 2Y ID G / HCI - S3

1.25 / 1.5

14h CM

EXAMS

Duration

Authorized documents

      If yes, which ones :

School calculator authorized

Session 1

CT 1h

Yes

All documents (handouts, course notes)

Yes

Session 2

to complete

Yes / No

Yes / No

Prerequisites

Basic concepts of geometry and linear algebra and elementary analysis.

Lecture goals

Introduction to methods of topology and discrete geometry and its applications in image analysis and synthesis.

Detailed outline

Notions on general topology. Introduction to discrete topologies: pixels, voxels, and adjacency, connectivity, connected components; discrete curve, duality figure / background, Jordan's theorem, tree components, Euler number; Reconstruction of connected components.

Simple pixel, number of Yokoi. Topological Skeletons. Discrete distances, masks chamfer distance transform algorithm.

Different models of discretization and frameworks of discretization of objects and operators; Freeman's code. Properties of discretizations of linear and quadratic primitives (lines, planes, hyperplanes, conicals) and algorithms for discretization of these primitives. Reconstruction of qualitative and quantitative informations: Reconstruction of topological and geometrical properties; Reconstruction of polygons and polyhedrons, Estimators of differential characteristics, Estimators of measures. Algorithms for solving and simplification of Diophantine equations and constraints in relation to reconstruction. Introduction to axiomatization of discrete geometries. Passages between the Euclidean spaces and Discrete spaces via the "fractal geometry".

Applications

Examples, software or hardware demos, visits, ...

Acquired skills

A vision of imaging based on discrete topology and geometry. Techniques of discrete topology and geometry for image analysis and synthesis.

Bibliography

•    Chassery J.M., Montanvert A., Géométrie Discrète en Analyse d'Images, Hermès (1991)

•    Coeurjolly D., Montanvert A., Chassery J. M. (eds.), Géométrie discrète et imagerie numérique, Collection IC2 Signal et Image, Hermès 2007.

•    Klette R., Rosenfeld A., Digital Geometry : Geometric Methods for Digital Picture Analysis, Morgan Kaufmann Series in Computer Graphics and Geometric Modeling, San Francisco 2004.