Least squares for programmers

DoctoralThis course explains least squares optimization, nowadays a simple and well-mastered technology. We show how this simple method can solve a large number of problems that would be difficult to approach in any other way. This course provides a simple, understandable yet powerful tool that most coders can use, in the contrast with other algorithms sharing this paradigm (numerical simulation and deep learning) which are more complex to master.Linear regression is often underestimated being considered only as a sub-domain of statistics / data analysis, but it is much more than that. We propose to discover how the same method (least squares) applies to the manipulation of geometric objects. This first step into the numerical optimization world can be done without strong applied mathematics background; while being simple, this step suffices for many applications, and is a good starting point for learning more advanced algorithms. We strive to communicate the underlying intuitions through numerous examples of classic problems, we show different choices of variables and the ways the energies are built. Over the last two decades, the geometry processing community have used it for computing 2D maps, deformations, geodesic paths, frame fields, etc. Our examples provide many examples of applications that can be directly solved by the least squares method. Note that linear regression is an efficient tool that has deep connections to other scientific domains; we show a few such links to broaden reader's horizons.This course is intended for students/engineers/researchers who know how to program in the traditional way: by breaking down complex tasks into elementary operations that manipulate combinatorial structures (trees, graphs, meshes\dots). Here we present a different paradigm, in which we describe what a good result looks like, and let numerical optimization algorithms find it for us

Global parametrization based on Ginzburg-Landau functional

International audienceQuad meshing is a fundamental preprocessing task for many applications (subdivision surfaces, boundary layer simulation). State-of-the-art quad mesh generators proceed in three steps: first a guiding cross field is computed, then a parametrization representing the quads is generated, and finally a mesh is extracted from the parameterization. In this paper we show that in the case of a periodic global parameterization two first steps answer to the same equation and inherently face the same challenges. This new insight allows us to use recent cross field generation algorithms based on Ginzburg-Landau equations to accurately solve the parametrization step. We provide practical evidence that this formulation enables us to overcome common shortcomings in parametrization computation (inaccuracy away from the boundary, singular dipole placement)

Complex Functional Maps : a Conformal Link Between Tangent Bundles

International audienceIn this paper, we introduce complex functional maps, which extend the functional map framework to conformal maps between tangent vector fields on surfaces. A key property of these maps is their orientation awareness. More specifically, we demonstrate that unlike regular functional maps that link functional spaces of two manifolds, our complex functional maps establish a link between oriented tangent bundles, thus permitting robust and efficient transfer of tangent vector fields. By first endowing and then exploiting the tangent bundle of each shape with a complex structure, the resulting operations become naturally orientationaware, thus favoring orientation and angle preserving correspondence across shapes, without relying on descriptors or extra regularization. Finally, and perhaps more importantly, we demonstrate how these objects enable several practical applications within the functional map framework. We show that functional maps and their complex counterparts can be estimated jointly to promote orientation preservation, regularizing pipelines that previously suffered from orientation-reversing symmetry errors

Génération de maillage quadrangulaire d'un domaine du plan via les équations de Ginzburg-Landau

National audienceGénérer un maillage d'une surface est un pré-requis souvent indispensable à de nombreuses applications. Certaines (la subdivision de surfaces, la simulation de couches limites) nécessitent l'utilisation de maillage quadrangulaire. L'état de l'art procède en trois étapes. Il s'agit d'abord de calculer un champ de croix, puis de l'intégrer pour obtenir une paramétrisation et enfin d'extraire un maillage quadrangulaire à partir de la paramétrisation. Nous montrerons que les deux premières étapes réfèrent aux mêmes équations et peuvent donc être traitées de la même manière. Cette approche permet de résoudre des problèmes (imprécision loin des bords, mauvaise localisation des singularités) qui se posaient jusqu'alors

Champs de repères pour les modèles CAO

National audienceMaillage quadrangulaire de modèles CAO obtenus avec notre champ de repères. Le champ de repères le plus lisse, obtenu par une méthode usuelle, aurait une topologie incompatible avec un maillage quadrangulaire dans les zones montrées par les flèches en orange. L'impact de ces incohérences n'est pas local : cela produirait des quadrilatères dégénérés sur l'ensemble des zones surlignées en orange

Champs de repères 2D non orthogonaux

National audience

Frame Fields for CAD models

International audienceGiven a triangulated surface, a unit length tangent vector field can be used to orient entities located on the surface, such as glyphs or strokes. When these entities are invariant under a π/2 rotation (squares, or curvature hatching), the orientation can be represented by a frame field i.e. four orthogonal tangent unit vectors at each point of the surface. The generation of such fields is a key component of recent quad meshing algorithms based on global parameterization, as it defines the orientation of the final facets. State-of-the-art methods are able to generate smooth frame fields subject to some hard constraints (direction and topology) or smooth constraints (matching the curvature direction). When we have a surface triangular mesh, and a vector defined on each facet, we can't directly know if all the vectors are colinear. We first have to define the (so called) parallel transport of every edge to compare the vectors on a common plan. When dealing with CAD models, the field must be aligned with feature edges. A problem occurs when there is a low angle corner formed by two colliding feature edges. Our solution not only defines the parallel transport to obtain smoothed frame fields on a surface triangular mesh, it also redefines the parallel transport wherever there is a low angle corner, to smooth a frame field as if these corners' angles were π/2

Designing 2D and 3D Non-Orthogonal Frame Fields

International audienceWe present a method for direction field design on surface and volumetric meshes supporting non-orthogonality. Our approach is a generalization of the representation of 3D cross fields in spherical harmonic basis. As such it induces a geometrically meaningful measure of smoothness, allows orthogonality control by a simple parameter and enables orientation constraints of a single direction. To the best of our knowledge this is the first work to propose non-orthogonal 3D frame field design. We demonstrate the applicability of our method to generate anisotropic quadrangular and hexahedral meshes which are particularly useful for remeshing CAD models

The evolving SARS-CoV-2 epidemic in Africa: Insights from rapidly expanding genomic surveillance

INTRODUCTION Investment in Africa over the past year with regard to severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) sequencing has led to a massive increase in the number of sequences, which, to date, exceeds 100,000 sequences generated to track the pandemic on the continent. These sequences have profoundly affected how public health officials in Africa have navigated the COVID-19 pandemic. RATIONALE We demonstrate how the first 100,000 SARS-CoV-2 sequences from Africa have helped monitor the epidemic on the continent, how genomic surveillance expanded over the course of the pandemic, and how we adapted our sequencing methods to deal with an evolving virus. Finally, we also examine how viral lineages have spread across the continent in a phylogeographic framework to gain insights into the underlying temporal and spatial transmission dynamics for several variants of concern (VOCs). RESULTS Our results indicate that the number of countries in Africa that can sequence the virus within their own borders is growing and that this is coupled with a shorter turnaround time from the time of sampling to sequence submission. Ongoing evolution necessitated the continual updating of primer sets, and, as a result, eight primer sets were designed in tandem with viral evolution and used to ensure effective sequencing of the virus. The pandemic unfolded through multiple waves of infection that were each driven by distinct genetic lineages, with B.1-like ancestral strains associated with the first pandemic wave of infections in 2020. Successive waves on the continent were fueled by different VOCs, with Alpha and Beta cocirculating in distinct spatial patterns during the second wave and Delta and Omicron affecting the whole continent during the third and fourth waves, respectively. Phylogeographic reconstruction points toward distinct differences in viral importation and exportation patterns associated with the Alpha, Beta, Delta, and Omicron variants and subvariants, when considering both Africa versus the rest of the world and viral dissemination within the continent. Our epidemiological and phylogenetic inferences therefore underscore the heterogeneous nature of the pandemic on the continent and highlight key insights and challenges, for instance, recognizing the limitations of low testing proportions. We also highlight the early warning capacity that genomic surveillance in Africa has had for the rest of the world with the detection of new lineages and variants, the most recent being the characterization of various Omicron subvariants. CONCLUSION Sustained investment for diagnostics and genomic surveillance in Africa is needed as the virus continues to evolve. This is important not only to help combat SARS-CoV-2 on the continent but also because it can be used as a platform to help address the many emerging and reemerging infectious disease threats in Africa. In particular, capacity building for local sequencing within countries or within the continent should be prioritized because this is generally associated with shorter turnaround times, providing the most benefit to local public health authorities tasked with pandemic response and mitigation and allowing for the fastest reaction to localized outbreaks. These investments are crucial for pandemic preparedness and response and will serve the health of the continent well into the 21st century

Représentation fonctionnelle des surfaces déformables pour l’analyse et la synthèse géométrique

Creating and understanding deformations of surfaces is a recurring theme in geometry processing. As smooth surfaces can be represented in many ways from point clouds to triangle meshes, one of the challenges is being able to compare or deform consistently discrete shapes independently of their representation. A possible answer is choosing a flexible representation of deformable surfaces that can easily be transported from one structure to another.Toward this goal, the functional map framework proposes to represent maps between surfaces and, to further extents, deformation of surfaces as operators acting on functions. This approach has been recently introduced in geometry processing but has been extensively used in other fields such as differential geometry, operator theory and dynamical systems, to name just a few. The major advantage of such point of view is to deflect challenging problems, such as shape matching and deformation transfer, toward functional analysis whose discretization has been well studied in various cases. This thesis investigates further analysis and novel applications in this framework. Two aspects of the functional representation framework are discussed.First, given two surfaces, we analyze the underlying deformation. One way to do so is by finding correspondences that minimize the global distortion. To complete the analysis we identify the least and most reliable parts of the mapping by a learning procedure. Once spotted, the flaws in the map can be repaired in a smooth way using a consistent representation of tangent vector fields.The second development concerns the reverse problem: given a deformation represented as an operator how to deform a surface accordingly? In a first approach, we analyse a coordinate-free encoding of the intrinsic and extrinsic structure of a surface as functional operator. In this framework a deformed shape can be recovered up to rigid motion by solving a set of convex optimization problems. Second, we consider a linearized version of the previous method enabling us to understand deformation fields as acting on the underlying metric. This allows us to solve challenging problems such as deformation transfer are solved using simple linear systems of equations.La création et la compréhension des déformations de surfaces sont des thèmes récurrent pour le traitement de géométrie 3D. Comme les surfaces lisses peuvent être représentées de multiples façon allant du nuage ​​de points aux maillages polygonales, un enjeu important est de pouvoir comparer ou déformer des formes discrètes indépendamment de leur représentation. Une réponse possible est de choisir une représentation flexible des surfaces déformables qui peut facilement être transportées d'une structure de données à une autre.Dans ce but, les "functional map" proposent de représenter des applications entre les surfaces et, par extension, des déformations comme des opérateurs agissant sur des fonctions. Cette approche a été introduite récemment pour le traitement de modèle 3D, mais a été largement utilisé dans d'autres domaines tels que la géométrie différentielle, la théorie des opérateurs et les systèmes dynamiques, pour n'en citer que quelques-uns. Le principal avantage de ce point de vue est de détourner les problèmes encore non-résolus, tels que la correspondance forme et le transfert de déformations, vers l'analyse fonctionnelle dont l'étude et la discrétisation sont souvent mieux connues. Cette thèse approfondit l'analyse et fournit de nouvelles applications à ce cadre d'étude. Deux questions principales sont discutées.Premièrement, étant donné deux surfaces, nous analysons les déformations sous-jacentes. Une façon de procéder est de trouver des correspondances qui minimisent la distorsion globale. Pour compléter l'analyse, nous identifions les parties les moins fiables du difféomorphisme grâce une méthode d'apprentissage. Une fois repérés, les défauts peuvent être éliminés de façon différentiable à l'aide d'une représentation adéquate des champs de vecteurs tangents.Le deuxième développement concerne le problème inverse : étant donné une déformation représentée comme un opérateur, comment déformer une surface en conséquence ? Dans une première approche, nous analysons un encodage de la structure intrinsèque et extrinsèque d'une forme en tant qu'opérateur fonctionnel. Dans ce cadre, l'objet déformé peut être obtenu, à rotations et translations près, en résolvant une série de problèmes d'optimisation convexe. Deuxièmement, nous considérons une version linéarisée de la méthode précédente qui nous permet d'appréhender les champs de déformation comme agissant sur la métrique induite. En conséquence la résolution de problèmes difficiles, tel que le transfert de déformation, sont effectués à l'aide de simple systèmes linéaires d'équations