Fronts propagation and normalized p-laplacien on graphs : Algorithms and applications to images and data processing

• Main Author: Desquesnes, Xavier
• Other Authors: Equipe Image - Laboratoire GREYC - UMR6072, Groupe de Recherche en Informatique, Image et Instrumentation de Caen (GREYC), Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU)-École Nationale Supérieure d'Ingénieurs de Caen (ENSICAEN), Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Université de Caen, Abderrahim Elmoataz
• Format: Doctoral or Postdoctoral Thesis
• Language: French
• Published: HAL CCSD 2012
• Subjects: Image Processing; Digital Techniques; Difference equations; Laplacian operator; Level set methods; Graphs; Ensemble de niveaux méthodes d; Équations aux différences; Laplacien; Ensemble de niveaux méthodes d'; Graphes; Traitement d'images; Techniques numériques; [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV]; [INFO.INFO-IM]Computer Science [cs]/Medical Imaging; [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH]; Antarctic; The Antarctic; Antarc*
• Online Access: https://theses.hal.science/tel-00773434; https://theses.hal.science/tel-00773434/document; https://theses.hal.science/tel-00773434/file/desquesnes-these2012.pdf

This work deals with the transcription of continuous partial derivative equations to arbitrary discrete domains by exploiting the formalism of partial difference equations defined on weighted graphs. In the first part, we propose a transcription of the normalized p-Laplacian operator to the graph domains as a linear combination between the non-local infinity Laplacian and the normalized Laplacian (both in their discrete version). This adaptation can be considered as a new class of p-Laplacian operators on graphs that interpolate between non-local infinity Laplacian and normalized Laplacian. In the second part, we present an adaptation of fronts propagation equations on weighted graphs. These equations are obtained by the transcription of the continuous level sets method to a discrete formulation on the graphs domain. Beyond the transcription in itself, we propose a very general formulation and efficient algorithms for the simultaneous propagation of several fronts on a single graph. Both transcription of the p-Laplacian operator and level sets method enable many applications in image segmentation and data clustering that are illustrated in this manuscript. Finally, in the third part, we present a concrete application of the different tools proposed in the two previous parts for computer aided diagnosis. We also present the Antarctic software that was developed during this PhD. Cette thèse s'intéresse à la transcription d'équations aux dérivées partielles vers des domaines discrets en exploitant le formalisme des équations aux différences partielles définies sur des graphes pondérés. Dans une première partie, nous proposons une transcription de l'opérateur p-laplacien normalisé au domaine des graphes comme une combinaison linéaire entre le laplacien infini non-local et le laplacien normalisé (ces deux opérateurs étant discrets). Cette adaptation peut être considérée comme une nouvelle classe d'opérateurs p-laplaciens sur graphes, qui interpolent entre le laplacien infini non-local et le laplacien normalisé. ...