Source files of my Ph.D. thesis (/content) and slides of my defence (/slides).
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PhD Thesis of Jérémy Levallois

Estimateurs différentiels en géométrie discrète : applications à l'analyse de surfaces digitales

This repository contains source files of my Ph.D. thesis (/content) and slides of my defence (/slides).



Mean curvature (from blue to yellow)

Principal directions

Feature classification (red: singularities, blue: smooth, green: flat)

About me

I amwas* a PhD student in co-supervision with the M2DisCo team at the LIRIS laboratory (Université de Lyon) and the LIMD team at the LAMA laboratory (Université de Savoie Mont-Blanc) since the 15th of March 2012. I'm currently working under the advisorship of David Coeurjolly (@dcoeurjo) and Jacques-Olivier Lachaud (@JacquesOlivierLachaud). My research topic revolves around obtaining geometrical characteristics on digital objects.

(*) My Ph.D. defence was Thursday, 12th November, 2015, at Lyon - France.


3D image acquisition devices are now ubiquitous in many domains of science, including biomedical imaging, material science, or manufacturing. Most of these devices (MRI, Backscatter X-ray, micro-tomography, confocal microscopy, PET scans) produce a set of data organized on a regular grid, which we call digital data, commonly called pixels in 2D images and voxels in 3D images. Properly processed, these data approach the geometry of imaged shapes, like organs in biomedical imagery or objects in engineering.

In this thesis, we are interested in extracting the geometry of such digital data, and, more precisely, we focus on approaching geometrical differential quantities such as the curvature of these objects. These quantities are the critical ingredients of several applications like surface reconstruction or object recognition, matching or comparison. We focus on the proof of multigrid convergence of these estimators, which in turn guarantees the quality of estimations. More precisely, when the resolution of the acquisition device is increased, our geometric estimates are more accurate. Our method is based on integral invariants and on digital approximation of volumetric integrals.

Finally, we present a surface classification method, which analyzes digital data in a multiscale framework and classifies surface elements into three categories: smooth part, planar part, and singular part (tangent discontinuity). Such feature detection is used in several geometry pipelines, like mesh compression or object recognition. The stability to parameters and the robustness to noise are evaluated with respect to state-of-the-art methods. All our tools for analyzing digital data are applied to 3D X-ray tomography of snow microstructures and their relevance is evaluated and discussed.


digital geometry; multigrid convergence; differential quantities; curvature; normal vector; estimators; integral invariants; feature; surface classification;

Bibtex entry

@PHDTHESIS {jlevallois_PhD,
  author  = "Levallois, J{\'e}r{\'e}my",
  title   = "Estimateurs diff{\'e}rentiels en g{\'e}om{\'e}trie discr{\`e}te : applications {\`a} l'analyse de surfaces digitales",
  school  = "INSA-Lyon",
  year    = 2015,
  month   = 11

Jérémy Levallois. "Estimateurs différentiels en géométrie discrète : applications à l'analyse de surfaces digitales." PhD thesis, INSA-Lyon, 11 2015.

Tools used


All this work is under Creative Commons CC BY-NC-SA 4.0 license, see

This work has been done thanks to

LIRIS LAMA INSA-Lyon Université de Lyon Université de Savoie Mont-Blanc CNRS ANR DGtal