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Digital Topology (2019-2020)

Lecture instructor : Nicolas Passat
Laboratory session instructor: Yukiko Kenmochi

Description

Topology is concerned with the properties of geometric objects that are preserved under continuous deformations without tearing or gluing. On the other hand, an object is represented by a set of pixels/voxels in digital images. Thus, due to the discontinuities induced by the digitization process of a function, topological defects can be frequently observed in such a digital world. Digital topology aims at designing discrete models that allow topological calculations in the most reliable and efficient way possible for discrete objects. Its main application areas are image analysis and computer graphics.

This course will focus on computerized modeling of topological shapes in digital world. We will study algorithms and data structures for creating, manipulating and analyzing digital shapes.

Schedule and Detail

The course contains eight lectures given by Nicolas Passat and three laboratory sessions by Yukiko Kenmochi:

Lecture 1

Time: 8:30-10:30 on Monday 3rd February 2020
Place: Room 1305 (ESIEE Paris)
Topic: Image models for digital topology

Lecture 2

Time: 13:45-15:45 on Monday 3rd February 2020
Place: Room 1305 (ESIEE Paris)
Topic: Topological invariants

Laboratory session 1

Time: 8:30-10:30 on Tuesday 4th February 2020
Place: Room 5301V (ESIEE Paris)
Topic: Topological invariants and operators in 2D
Experiment subject: 2nd session of the tutorial of the image processing library Pink, developed by Michel Couprie.
Preparation: Download the source file of the latest version of Pink and compile it following this instruction.

Lecture 3

Time: 8:30-10:30 on Monday 17th February 2020
Place: Room 1305 (ESIEE Paris)
Topic: Homotopy, simple points

Lecture 4

Time: 13:45-15:45 on Monday 17th February 2020
Place: Room 1305 (ESIEE Paris)
Topic: Continuous vs discrete

Laboratory session 2

Time: 8:30-10:30 on Tuesday 18th February 2020
Place: Room 5309 (ESIEE Paris)
Topic: Topological operators in 3D
Experiment subject: 4th session of the tutorial of the image processing library Pink, developed by Michel Couprie.
Preparation: In addition to Pink, you need VTK for 3D visualization. On the PCs in the laboratory room, it is not installed by default. Thus you need to settle the working environment where Pink and VTK are pre-installed through the linnux virtual machine (ubuntu 18.04) whose image (2.7GB) is provided here. In order to play this virtual machine you may need to install VirtualBox (this should be already installed on the PCsin the laboratory room). Once your virtual machine is running, you can use the account whose user id is "user" and password is "DigiTopo".
Alternatively, you can also use your own PC, where both Pink and VTK are installed.

Lecture 5

Time: 8:30-10:30 on Monday 2nd March 2020
Place: Room 1401 (ESIEE Paris)
Topic: Thinning, skeletonization, segmentation

Lecture 6

Time: 13:45-15:45 on Monday 2nd March 2020
Place: Room 1401 (ESIEE Paris)
Topic: Parallel algorithms

Laboratory session 3

Time: 8:30-10:30 on Tuesday 18th February 2020
Place: Room 5309 (ESIEE Paris)
Topic: Operators modifying topology
Experiment subject: 5th session of the tutorial of the image processing library Pink, developed by Michel Couprie.

Lecture 7

Time: 8:30-10:30 on Monday 16th March 2020
Place: Room 1305 (ESIEE Paris)
Topic: Topological operators in multi-scale, gray-scale and color

Lecture 8

Time: 13:45-15:45 on Monday 16th March 2020
Place: Room 1305 (ESIEE Paris)
Topic: Applications

Textbooks

M. Couprie and G. Bertrand
Discrete topological transformation for image processing, in V. E. Brimkov and R. P. Barneva (Eds.), "Digital Geometry Algorithms: Theoretical Foundations and Applications to Computational Imaging", pp. 73-107, Springer, 2012.

D. Coeurjolly, A. Montanvert et J.-M. Chassery.
"Géométrie discrète et images numériques", Hermès, 2007.

R. Klette and A. Rosenfeld.
"Digital geometry: geometric methods for digital picture analysis", San Diego: Morgan Kaufmann, 2004.

Useful links

PINK: Image processing library in C

DGtal: Open source C++ library of Digital Geometry Tools and Algorithms

Fiji: Image processing package with Java

Data sets at IAPR - TC18 Discrete Geometry and Mathematical Morphology