International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Proceedings of the Third International Workshop on Mathematical Foundations of Computational Anatomy - Geometrical and Statistical Methods for Modelling Biological Shape Variability

International audienceComputational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model the organ development across time (growth or aging), to establish their variability, and to correlate this variability information with other functional, genetic or structural information. The Mathematical Foundations of Computational Anatomy (MFCA) workshop aims at fostering the interactions between the mathematical community around shapes and the MICCAI community in view of computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop is a forum for the exchange of the theoretical ideas and aims at being a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical developments, applications and results being welcomed as illustrations. Following the successful rst edition of this workshop in 20061 and second edition in New-York in 20082, the third edition was held in Toronto on September 22 20113. Contributions were solicited in Riemannian and group theoretical methods, geometric measurements of the anatomy, advanced statistics on deformations and shapes, metrics for computational anatomy, statistics of surfaces, modeling of growth and longitudinal shape changes. 22 submissions were reviewed by three members of the program committee. To guaranty a high level program, 11 papers only were selected for oral presentation in 4 sessions. Two of these sessions regroups classical themes of the workshop: statistics on manifolds and diff eomorphisms for surface or longitudinal registration. One session gathers papers exploring new mathematical structures beyond Riemannian geometry while the last oral session deals with the emerging theme of statistics on graphs and trees. Finally, a poster session of 5 papers addresses more application oriented works on computational anatomy

Surface Registration for Pharyngeal Radiation Treatment Planning

Endoscopy is an in-body examination procedure that enables direct visualization of tumor spread on tissue surfaces. In the context of radiation treatment planning for throat cancer, there have been attempts to fuse this endoscopic information into the planning CT space for better tumor localization. One way to achieve this CT/Endoscope fusion is to first reconstruct a full 3D surface model from the endoscopic video and then register that surface into the CT space. These two steps both require an algorithm that can accurately register two or more surfaces. In this dissertation, I present a surface registration method I have developed, called Thin Shell Demons (TSD), for achieving the two goals mentioned above. There are two key aspects in TSD: geometry and mechanics. First, I develop a novel surface geometric feature descriptor based on multi-scale curvatures that can accurately capture local shape information. I show that the descriptor can be effectively used in TSD and other surface registration frameworks, such as spectral graph matching. Second, I adopt a physical thin shell model in TSD to produce realistic surface deformation in the registration process. I also extend this physical model for orthotropic thin shells and propose a probabilistic framework to learn orthotropic stiffness parameters from a group of known deformations. The anisotropic stiffness learning opens up a new perspective to shape analysis and allows more accurate surface deformation and registration in the TSD framework. Finally, I show that TSD can also be extended into a novel groupwise registration framework. The advantages of Thin Shell Demons allow us to build a complete 3D model of the throat, called an endoscopogram, from a group of single-frame-based reconstructions. It also allows us to register an endoscopogram to a CT segmentation surface, thereby allowing information transfer for treatment planning.Doctor of Philosoph

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

Part decomposition of 3D surfaces

This dissertation describes a general algorithm that automatically decomposes realworld scenes and objects into visual parts. The input to the algorithm is a 3 D triangle mesh that approximates the surfaces of a scene or object. This geometric mesh completely specifies the shape of interest. The output of the algorithm is a set of boundary contours that dissect the mesh into parts where these parts agree with human perception. In this algorithm, shape alone defines the location of a bom1dary contour for a part. The algorithm leverages a human vision theory known as the minima rule that states that human visual perception tends to decompose shapes into parts along lines of negative curvature minima. Specifically, the minima rule governs the location of part boundaries, and as a result the algorithm is known as the Minima Rule Algorithm. Previous computer vision methods have attempted to implement this rule but have used pseudo measures of surface curvature. Thus, these prior methods are not true implementations of the rule. The Minima Rule Algorithm is a three step process that consists of curvature estimation, mesh segmentation, and quality evaluation. These steps have led to three novel algorithms known as Normal Vector Voting, Fast Marching Watersheds, and Part Saliency Metric, respectively. For each algorithm, this dissertation presents both the supporting theory and experimental results. The results demonstrate the effectiveness of the algorithm using both synthetic and real data and include comparisons with previous methods from the research literature. Finally, the dissertation concludes with a summary of the contributions to the state of the art

Optimization and machine learning methods for Computational Protein Docking

Computational Protein Docking (CPD) is defined as determining the stable complex of docked proteins given information about two individual partners, called receptor and ligand. The problem is often formulated as an energy/score minimization where the decision variables are the 6 rigid body transformation variables for the ligand in addition to more variables corresponding to flexibilities in the protein structures. The scoring functions used in CPD are highly nonlinear and nonconvex with a very large number of local minima, making the optimization problem particularly challenging. Consequently, most docking procedures employ a multistage strategy of (i) Global Sampling using a coarse scoring function to identify promising areas followed by (ii) a Refinement stage using more accurate scoring functions and possibly allowing more degrees of freedom. In the first part of this work, the problem of local optimization in the refinement stage is addressed. The goal of local optimization is to remove steric clashes between protein partners and obtain more realistic score values. The problem is formulated as optimization on the space of rigid motions of the ligand. Employing a recently introduced representation of the space of rigid motions as a manifold, a new Riemannian metric is introduced that is closely related to the Root Mean Square Deviation (RMSD) distance measure widely used in Protein Docking. It is argued that the new metric puts rotational and translational variables on equal footing as far local changes of RMSD is concerned. The implications and modifications for gradient-based local optimization algorithms are discussed. In the second part, a new methodology for resampling and refinement of ligand conformations is introduced. The algorithm is a refinement method where the inputs to the algorithm are ensembles of ligand conformations and the goal is to generate new ensembles of refined conformations, closer to the native complex. The algorithm builds upon a previous work and introduces multiple new innovations: Clustering the input conformations, performing dimensionality reduction using Principle Component Analysis (PCA), underestimating the scoring function and resampling and refinement of new conformations. The performance of the algorithm on a comprehensive benchmark of protein complexes is reported. The third part of this work focuses on using machine learning framework for addressing two specific problems in Protein Docking: (i) Constructing a machine learning model in order to predict whether a given receptor and ligand pair interact. This is of significant importance for constructing the so-called protein interaction networks, an critical step in the Drug Discovery process. The success of the algorithm is verified on a benchmark for discrimination between Biological and Crystallographic Dimers. (ii) A ranking scheme for output predictions of a protein docking server is devised. The machine learning model employs the features of the docking server predictions to produce a ranked list with the top ranked predictions having higher probability of being close to the native solution. Two state-of-the-art approaches to the ranking problem are presented and compared in detail and the implications of using the superior approach for a structural docking server is discussed