Public Thesis defense - ICTEAM

SST

22 septembre 2020

16h30

Louvain-la-Neuve

Auditoire BARB92 - Place Sainte Barbe

Interpolation and fitting on Riemannian manifolds by Pierre-Yves GOUSENBOURGER

Pour l’obtention du grade de Docteur en sciences de l’ingénieur et technologie

The access to constantly increasing computational capacities has revolutionized the way engineering is seen. We are now able to produce a large quantity of data, thanks to cheap sensors. However, processing such data remains costly both in computational time and in energy. One of the reasons is that the structure of the data is often omitted or unknown. The search space becomes so large that finding the solution to simple problems often turns out to be finding a needle in a haystack.

A classical problem in data processing is called the “fitting problem”. It consists in fitting a d-dimensional curve to a set of data points associated to d parameters. The curve must pass sufficiently close to the data points while being regular enough. When the underlying structure of the data points (i.e., the manifold) is known, one can impose to the curve to preserve this structure (i.e., to remain on the manifold), such that the search space is drastically reduced.

The goal of this thesis is to develop methods to (approximately) solve this fitting problem; the bet is to require “less” (less computational capabilities, power, storage, time) by leveraging “more” knowledge on the search space. The objective is the following: provide a toolbox that produces a differentiable fitting curve to data points on manifolds, based on very few and simple geometric tools, at low computational cost and storage capacity, all this while maintaining an acceptable quality of the solutions.

The algorithms are applied to different illustrative problems. In 3D shape reconstruction, the data points are organs contours acquired via MRI, and the parameter is the acquisition depth; in wind fields

estimation, the data points belong to the manifold of positive semi-definite matrices of given rank, and the parameters are the prevalent wind amplitudes and angles. We also show the performances of our algorithms in applications for parametric model order reduction.

Jury members :

  • Prof. Laurent Jacques (UCLouvain), supervisor
  • Prof. Pierre-Antoine Absil (UCLouvain), supervisor
  • Prof. Jean-Pierre Raskin (UCLouvain), chairperson
  • Prof. Jean-François Remacle (UCLouvain), secretary
  • Prof. Benedikt Wirth (Universität Münster, Germany)
  • Prof. Gabriel Peyré (Ecole Normale Supérieure Paris, France)

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