2024Activity reportProject-TeamGEOMERIX
RNSR: 202224337M- Research center Inria Saclay Centre at Institut Polytechnique de Paris
- In partnership with:CNRS, Institut Polytechnique de Paris
- Team name: Geometry-driven Numerics
- In collaboration with:Laboratoire d'informatique de l'école polytechnique (LIX)
- Domain:Perception, Cognition and Interaction
- Theme:Interaction and visualization
Keywords
Computer Science and Digital Science
- A3.4.1. Supervised learning
- A3.4.2. Unsupervised learning
- A3.4.4. Optimization and learning
- A3.4.6. Neural networks
- A5.5. Computer graphics
- A5.5.1. Geometrical modeling
- A5.5.4. Animation
- A6.1.4. Multiscale modeling
- A6.1.5. Multiphysics modeling
- A6.2.5. Numerical Linear Algebra
- A6.2.6. Optimization
- A6.2.8. Computational geometry and meshes
- A6.5.1. Solid mechanics
- A6.5.2. Fluid mechanics
- A8.3. Geometry, Topology
- A8.7. Graph theory
- A8.12. Optimal transport
- A9.2. Machine learning
Other Research Topics and Application Domains
- B9.2.2. Cinema, Television
- B9.2.3. Video games
- B9.5.1. Computer science
- B9.5.2. Mathematics
- B9.5.3. Physics
- B9.5.5. Mechanics
- B9.5.6. Data science
1 Team members, visitors, external collaborators
Research Scientists
- Steve Oudot [Team leader, INRIA, Senior Researcher]
- Jiong Chen [INRIA, Researcher, from Oct 2024]
- Jiong Chen [INRIA, Starting Research Position, until Sep 2024]
- Mathieu Desbrun [INRIA, Senior Researcher]
- Pooran Memari [CNRS, from Jun 2024]
Faculty Member
- Maksims Ovsjanikovs [LIX, from Jul 2024]
Post-Doctoral Fellows
- Maysam Behmanesh [ECOLE POLY PALAISEAU, from Apr 2024]
- Anouar Abdeldjaoued Ferfache [INRIA, Post-Doctoral Fellow, until Aug 2024]
- Mark Gillespie [INRIA, Post-Doctoral Fellow, from Sep 2024]
- Roman Klokov [ECOLE POLY PALAISEAU, from Feb 2024 until Oct 2024]
- Vincent Mallet [FX CONSEIL, from Feb 2024]
- Emery Pierson [LIX]
PhD Students
- Souhaib Attaiki [LIX, until Mar 2024]
- Nasim Bagheri Shouraki [IP PARIS, from Oct 2024]
- Theo Braune [ECOLE POLY PALAISEAU]
- Diego Gomez [ECOLE POLY PALAISEAU, until Sep 2024]
- Souhail Hadgi [ECOLE POLY PALAISEAU]
- Jingyi Li [IP PARIS]
- Robin Magnet [LIX, from Feb 2024 until May 2024]
- Leopold Maillard [DASSAULT SYSTEMES]
- Julie Mordacq [Ministère des Armées]
- Tim Scheller [ECOLE POLY PALAISEAU]
Interns and Apprentices
- Adrien Lanne [INRIA, Intern, from May 2024 until Oct 2024]
- Garance Perrot [ECOLE POLY PALAISEAU, Intern, until Mar 2024]
- Aristotelis Siozopoulos [ECOLE POLY PALAISEAU, Intern, from Jul 2024]
Administrative Assistant
- Michael Barbosa [INRIA]
Visiting Scientists
- Diana Marin [UNIV VIENNE, until Apr 2024]
- Matteo Pegoraro [UNIV AALBORG, until Feb 2024]
- Shreyas Samaga [Purdue University]
2 Overall objectives
Historical context. Geometry has been a unifying formalism for science: predictive models of the world around us have often been derived using geometric notions which formalize observable symmetries and experimental invariants. Tools such as differential geometry and tensor calculus quickly became invaluable in describing the complexity of natural phenomena and mechanical systems through concise equations, condensing local and global properties into simple relations between measurable quantities. Today, geometry (be it Euclidean or not) is at the core of many current physical theories: general relativity, electromagnetism (E&M), gauge theory, quantum mechanics, as well as solid and fluid mechanics, all have strong underlying structures that are best described and elucidated through geometric notions like differential forms, curvatures, vector bundles, connections, and covariant derivative. Geometry also creeps up in unexpected fields such as number theory and functional analysis, offering new insights and even breakthroughs, e.g., the use of algebraic geometry to address Fermat's last theorem.
Geometry in Digital Sciences. In sharp contrast, the role of geometry was mostly ignored at the inception of computer science. Yet, it has now become clear that digital sciences are imbued with an overwhelming amount of fundamentally geometric and topological concepts. Some are rather obvious, when dealing with the modeling of Euclidean shapes in computer graphics or the analysis of images in computer vision for instance; some are more subtle, such as the “manifold hypothesis” underlying a number of supervised or unsupervised learning techniques; and some are only nascent, such as the fields of Information Geometry (basically, the geometry used to understand probability distributions), Geometric Statistics (new statistical methodology for non-Euclidean entities), and Topological Data Analysis (where algebraic topology is used as a tool to enhance data analysis pipelines). In fact, even the discretization of physical theories needed to offer fast numerical simulation has brought geometry back to the forefront after it was understood that the loss of numerical fidelity in standard numerical methods is due to a fundamental failure to preserve geometric or topological structures of the underlying continuous models: partial differential equations (PDEs) modeling our physical world are typically encoding invariants and structures that are independent of the choice of coordinates used to express the equations and the tensors involved in them; but invariance to the choice of basis is often lost during discretization, as numerical approximations will in general not capture, let alone preserve, the key geometric structures that exist in the continuous case. Seeing these numerical issues through the lens of geometry is thus not just of academic interest: failure to maintain geometric invariants has serious consequences for the accuracy and stability of solutions.
Rationale. Given the unusual reach of geometry and its rich literature, there is an opportunity to assemble a team of experts in geometry and its vernacular, to help broadly impact digital science and technology. We thus propose the creation of a new project-team whose core scientific mission is to use geometry as a bedrock for the development of numerical tools and algorithms: we wish to exploit the properties of infinite-dimensional and finite-dimensional spaces that are related with distance, shape, size, and relative position, and bringing them to bear on computational discretizations and algorithms for analysis, processing, and simulation. Adhering to geometric structures and invariants as a guiding principle for computations is a rich source of both theoretical and practical challenges, allowing to combine concepts and results from different areas of geometry broadly construed to produce new computational tools with solid mathematical foundations. While our team will be very focused in terms of the mathematical foundations and tools upon which it builds, it will also be very broad in terms of applications given the pervasiveness of geometry in sciences and technology. This makes for an unusual, yet powerful scientific setup that will facilitate interdisciplinary projects through the common use of geometric foundations and their specialized terminology. It will also allow us to contribute sporadically to pure and computational mathematics when appropriate in order to push our scientific mission forward.
Positioning. We see GeomeriX as first and foremost Inria Saclay’s graphics team, but with wider objectives afforded by the broad relevance of geometry. It is worth noting that graphics has evolved to the point where it often intersects with applied mathematics, machine learning, vision, and computational science in some of its efforts, and GeomeriX intends to continue this trend.
Objectives. Our project-team's overall scientific objective is to contribute, through a geometric perspective, both foundational and practical methods for geometric data processing. In particular, we seek the development of predictive computational tools by drawing from the many facets of geometry and topology: whether it be discrete geometry, basic differential geometry or exterior calculus, symplectic geometry, persistent homology or sheaf theory, optimal transport, Riemannian or conformal geometry, these topics of geometry inform and guide both our discretizations and algorithmic designs towards computing. Note that we do not plan to merely adapt and exploit geometric concepts and understanding for numerical purposes, as our focus on digital data may even result in contributions to these mathematical fields, extending the current body of knowledge. While we intentionally leave the range of our mathematical foundations open so as not to restrict our potential team-wide explorations, we concentrate our research on four concrete themes, which we believe can be most significantly impacted by a geometric approach to developing new numerical tools:
- Euclidean shape processing: from computer graphics to geometry processing and vision, the analysis and manipulation of low-dimensional shapes (2D and 3D) is an important endeavor with applications covering a wide range of areas from entertainment and classical computer-aided design, to reverse engineering and biomedical engineering. Our project-team intends to lead efforts in this competitive field, with key contributions in shape matching, geometric analysis, and discrete calculus on meshes.
- Simulation: traditional finite-element treatments of various physical models have had tremendous success. Recently, a number of geometric integrators have upended the field, either through structure-preserving integration which offers improved statistical predictability by respecting the geometric properties of the exact flow of the differential equations, or through novel discretizations of the state space. We intend to continue introducing novel integration methods for increasingly complex multiphysics systems, as well as exploiting the use of learning methods to accelerate simulation.
- Dynamical systems: we intend to leverage the geometric nature of dynamical systems to investigate and promote high-dimensional data analysis for dynamics: the study of dynamical systems from a limited number of observations of the state of a given system (for example, time series or a sparse set of trajectories) offers a unique opportunity to develop scalable computational tools to detect or characterize unusual features and coherent structures. Meanwhile, the study of dynamical systems from a combinatorial point of view opens up the possibility of characterizing their invariant sets and assessing their stability.
- Data science: finally, we are intent on exploring the underlying role of geometry in machine learning and statistical analysis. This role has been put forward in the recent years, with the emergence of approaches such as geometric deep learning or topological data analysis, whose aim is to leverage the underlying geometry or topology of the data to enhance the performance, robustness, or explainability of the methods used for their analysis. We will pursue investigations toward this goal, concentrating our efforts on topics related to explainable feature design, geometric feature learning, geometry-driven learning, and geometry for categorical and mixed data types.
Evidently, our research efforts may at times lie across multiple of these themes given our multi-disciplinary objectives, and it is our hope that we will all eventually participate in the four themes.
3 Research program
Below we introduce the details of our four research themes, in four separate subsections. In each subsection, we first present the scientific focus and research objectives of the corresponding theme, then we detail the research topics we intend to address and how we plan to leverage topology and geometry for each one of them. For each theme, we list the most likely contributors, and organize the various subtopics within each theme from short to long-term goals, based on our current expectations and focus.
3.1 Geometry for Euclidean shape processing
Euclidean space is the default setting of classical geometry in two or three dimensions. Shapes in 3D space are of particular interest as they represent the typical objects we interact with. Geometry processing is an area of research focusing on these low-dimensional shapes in Euclidean space, with the goal to design algorithms, data structures, as well as analysis tools for their digital acquisition, reconstruction, analysis, manipulation, synthesis, classification, transmission, and animation. Digital shapes are typically discretized through either point clouds, triangle meshes, or polygonal meshes for surfaces, and through tetrahedron or polytopal meshes for volumes. Analyzing and manipulating these digital representations already involve fundamental difficulties in terms of efficiency, scalability, and robustness to arbitrary sampling, for which computational geometry and computer graphics have generated a number of key algorithms. Simple surface meshes in 3D also offer a simple context in which to define discrete notions of basic topological properties (quantities preserved through arbitrary stretching, such as Euler characteristic, genus, Betti numbers, etc) and relevant geometric properties (normal, curvatures, covariant derivatives, parallel transport, etc). Yet the digital counterpart of the low-dimensional case of Euclidean geometry is far from being settled or complete: it remains obviously relevant in a number of scientific fields on which we plan to focus. A few research directions of particular interest are described below.
Operator-based methods for shape analysis
We plan to develop novel approaches for representing and manipulating geometric concepts as linear functional operators. Specifically we will focus on tools for shape matching, design and analysis of differential quantities such as vector fields or cross fields, shape deformation and shape comparison, where functional approaches have recently been shown to provide a natural and discretization-agnostic representation 122, 55, 56, 132. This “functional” point of view is classical in many scientific areas, including dynamical systems (where the pullback with respect to a map is closely related to the Koopman or composition operator, allowing the study ergodicity or mixing property of non-linear maps through the spectral properties of a linear operator), differential geometry (where vector fields are often defined by their action on real-valued functions) and representation theory among others. However, it has only recently been adopted in geometry processing with tremendous and constantly growing potential in both axiomatic or even learning-based approaches 111, 100, 83. We will continue developing efficient and robust algorithms by considering shapes as functional spaces and by representing various geometric operations as linear operators acting on appropriate real-valued functions. In addition to the efficiency and robustness of methods obtained by considering this linear operator point of view of geometry processing and dynamical systems, another very significant advantage of these techniques is that they allow to express many different geometric operations in a common language. This means, for example, that it makes it easy to define the pushforward of a vector field with respect to a map by simply considering a composition of appropriate discrete operators. Despite the significant recent success of tools within this area, especially related to the functional map framework 123, there does not exist a unified coherent theoretical framework in which different geometric concepts can be represented and manipulated via their functional equivalents. Our main long-term goal therefore would be to establish a novel field within geometry processing by creating both a computational framework and a coherent theoretical formalism in which all of the different basic geometric operations can be expressed, and thus in which different concepts can “communicate” with one another. We believe that such a formalism and associated computational tools, already quite well developed, will not only greatly extend the scope of applicability of many existing geometry processing pipelines, but will also help expand this language to novel concepts, and ultimately help pave the way towards representation-agnostic geometric data manipulation.
Discrete metrics and applications.
While three-dimensional shapes are often encoded via their Euclidean embedding, numerous research efforts have focused on studying and discretizing their intrinsic metric. Regge calculus 130, an early approach to numerical relativity without coordinates, proposed the use of edge lengths to encode a piecewise-Euclidean metric per simplex, from which the Riemann curvature tensor can be easily computed to derive local areas or curvatures. This early work led to a series of alternative metric representations: tip angles, for instance, are known to encode the intrinsic geometry of a triangle mesh up to a scaling, while local measurements (an angle 131 or a length cross-ratio 107 per edge) later formed the basis of circle patterns 59, 105 as well as conformal representations 137; the discrete Laplace-Beltrami cotan formula 126 also determines the edge lengths of a triangle mesh (and thus its discrete metric) up to a global scaling 149. More recently, generalized notions of metrics were proposed; for instance, 97 presented a characterization of an augmented discrete metric resulting from the orthogonal primal-dual structure of weighted triangulations. Common to many of these various metric characterizations is the existence of convex energies which allow to efficiently compute these metrics from various boundary conditions. We intend to investigate the discrete treatment of metric for low-dimensional manifolds as a counterpart to the discretization of antisymmetric tensors (differential forms), which is far less studied — and a discrete theory unifying symmetric and anti-symmetric tensors remains elusive despite recent advances 96. Moreover, the metric of a surface is known in the continuous realm to induce Hodge stars and a canonical torsion-free Levi-Civita connection (or parallel transport), but this picture is far less clear for discrete manifolds, even if the construction of arbitrary-order discrete Hodge stars and metric connections are well understood by now. A few research directions on generalized metrics seem particularly interesting due to their likelihood of resulting in novel algorithmic and computational frameworks:
- Metric-dependent meshing: Given a set of metric-based operators, optimized mesh structures can be designed to offer optimal accuracy akin to Hodge-star mesh optimization for the augmented weighted metric proposed in 119. Another interesting research question is the existence and construction of intrinsic Delaunay triangulation, the most common discrete shape representation, with respect to a particular metric 60.
- Metric-aware sampling: Metric-dependent descriptors such as the pair correlation function are particularly efficient in characterizing statistical properties of point distributions for texture synthesis 84. Extending this framework to arbitrary non-flat domains through Multi-Dimensional Scaling (MDS) seem particularly promising.
- Shape characterization: Highly convoluted embeddings like the cortical surface of the brain and its functional connectivity graph are naturally hyperbolic in nature 65. However, investigating a link between cortical folding and the volumetric fiber bundle structure from a pure geometric viewpoint through a hyperbolic metric characterization has surprisingly not be done in brain analysis, despite striking visual similarities between brain folding and geometric realizations of the hyperbolic plane (see 142 and Taimiņa's crochet model). We are hoping that this intrinsic metric characterization can be investigated through recent discrete hyperbolic parametrization tools 92, which may also lead to other shape classification techniques in more general contexts.
- Piecewise-linear maps: We also wish to study the classification of the deformation of a triangle mesh through its induced metric change in the embedding space. Developing an approach to decompose such a diffeomorphic piecewise-linear map into canonical geometric transformations through either linear algebra or convex minimization could offer new discrete equivalences for conformal, equiareal, and curvature-preserving maps between triangulations, with direct applications to mesh parameterization and more general processing of discrete meshes.
- Geodesic abstractions: curve-network representations 95 based on a few geodesics to describe a shape provide a compact encoding of surfaces. While it is increasingly useful for artistic depictions, we also want to study its relevance as a compact compression scheme from which the shape and its metric can be derived with controllable precision.
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Metric-dependent cage: Finally, we also want to understand how to define optimized metric-dependent cages for intuitive & expressive deformation and animation of complex shapes 140, and how these cages can be understood as polygonal or polyhedral cells to locally simplify a simplicial complex.
Discrete differential and tensor calculus.
When working on low-dimensional spaces, the use of meshes (as opposed to just point clouds) pays dividends as it allows for the development of discrete versions of Exterior Calculus (see DEC 79 or FEEC 53), where -dimensional integrals can be directly evaluated in -cells, and differentiation can formally achieved through the boundary operator: the concept of chains and cochains from algebraic topology forms the basis of a discrete analog of Cartan's exterior calculus of differential forms, providing crucial numerical tools such as a discrete de Rham cohomology and a discrete Helmholtz-Hodge decomposition that precisely mimick their continuous counterparts. Moreover, finite elements of arbitrary order can be associated with these discrete forms through subdivision 94 to provide a powerful Isogeometric Analysis (IGA). Recent developments 112, 93 have offered also a discrete approach to tangent vector fields. While DEC encodes vector fields as 1-forms, processing tangent vectors and, more generally, directional fields sampled at vertices of discrete surfaces requires the development of discrete (metric) connections 76, 112 (which can be seen as discrete equivalent to the Christoffel symbols) to handle the non-linearity of non-flat domains. From these connections can be derived the usual continuous notions of covariant derivatives or Killing operator, and these discrete operators demonstrate the same intimate link between geometry and topology as exemplified by the hairy ball theorem (Hopf index theorem). While these operators apply equally well on discrete three-manifolds, much remains to do: properly defining the notion of curvature matrix-valued 2-form or torsion vector-valued 2-form in 3D and checking that these definitions provide consistent Bianchi identities (i.e., there exists an exterior covariant derivative satisfying fundamental geometric and topological properties) is an exciting research direction. Not only will it allow to deal with the line singularities in hexahedral meshing robustly, but it will also provide a Bochner Laplacian (also called the vector Laplacian) in 3D devoid of the type of spurious modes that discrete Laplacians over flat domains can introduce if one does not enforce a proper discrete deRham complex. Such a tensor calculus for three-manifolds may allow us to explore possible applications in the context of general relativity in the longer term. Finally, the design of simplicial or cell meshes that guarantee accurate computations while approximating a given domain well remains an important endeavor for practical applications.
3.2 Geometry for simulation
Mathematical models of the evolution in time of mechanical systems generally involve systems of differential equations. Simulating a physical system consists in figuring out how to move the system forward in time from a set of initial conditions, allowing the computation of an actual trajectory through classical methods such as fourth-order Runge-Kutta or Newmark schemes. However, a geometric — instead of a traditional numerical-analytic — approach to the problem of time integration is particularly pertinent 98: the very essence of a mechanical system is indeed characterized by its symmetries and invariants (e.g., momenta), thus preserving these geometric notions into the discrete computational setting is of paramount importance if one wants discrete time integration to properly capture the underlying continuous motion. Considering mechanics from a variational point of view goes back to Euler, Lagrange and Hamilton 86, and Poincaré famously stated that geometry and physics are “indissociable”. The variational principle most important for continuous mechanics is due to Hamilton, and is often called Hamilton’s principle or the least action principle: it states that a dynamical system always finds an optimal course from one position to another. One consequence is that we can recast the traditional way of thinking about an object accelerating in response to applied forces, into a geometric viewpoint: the path followed by the object between two space-time positions has optimal geometric properties, analogous to the notion of geodesics on curved surfaces. This point of view is equivalent to Newton’s laws in the context of classical mechanics, but is broad enough to encompass physical models ranging to E&M and quantum mechanics 115. While the idea of discretizing variational formulations of mechanics is standard for elliptic problems using Galerkin Finite Element methods for instance, only recently did it get used to derive variational time-stepping algorithms for mechanical systems 116. These variational integrators have been shown to be remarkably versatile, powerful, and general for simulations of physical phenomena when compared to traditional numerical time stepping methods: the symplectic character of variational integrators guarantees good statistical predictability through accurate preservation of the geometric properties of the exact flow of the differential equations. We endeavor to continue contributing to this particular application of geometry and extend it further, as we foresee a number of interesting scientific developments and industrial applications.
State-space discretization of statistical physics.
Kinetic equations are used to describe a variety of phenomena in various scientific fields, ranging from rarefied gas dynamics and plasma physics to biology and socio-economics, and appear naturally when one considers a statistical description of a large particle system evolving in time. In incompressible fluid simulation, kinetic solvers based on the lattice Boltzmann method (LBM) have generated growing interest due to their use of the Boltzmann transport equation and to its unusual state-space discretization based on a computationally-efficient lattice 135: compared to macroscopic solvers directly integrating Navier-Stokes equations, LBM totally bypasses the difficult issue of discretizing advection to high order, and absence of global pressure solves makes for extremely efficient parallel implementations, which are now surpassing alternative discretizations 109. However, the numerical treatment of the collision operator of the Boltmann equation has not reached maturity; most surprising is the complete absence of geometric approaches to deal with Boltzmann equations. One should be able to formulate a variational approach to LBM based on Hamilton's principle to derive a systematic integrator with guaranteed accuracy and structure-preserving properties. Moreover, while dealing with isothermal and incompressible flows is a good starting point, the kinetic standpoint of fluid dynamics is not theoretically restricted to this case: far more complex physical systems, from compressible flow (with shocks), to thermal conductivity, to even acoustics for example, can be handled; but far less is known on how to handle these more involved cases computationally, because no systematic numerical approach to handle Boltzmann equations is known. Success in our geometric approach to LBM should offer a much better handle to deal with these difficult cases: between new Hermite regularization tools 61, 75 and the recent introduction of variational integrators for non-equilibrium thermodynamical systems mentioned above should provide the necessary theoretical foundations to establish a geometric solver for this generalized case.
Learning-aided simulation.
Computational physics is experiencing a tectonic shift as data-driven approaches are quickly becoming mainstream. While we do not adhere to the idea being floated that numerical integration could be simply “learned” to improve current solvers, the fact is that many machine learning tools may have profound influence in practical applications using simulation. Long standing problems such as the design of perfectly matched layers (PML, an artificial absorbing layer for transport equations used to reduce the domain of simulation without suffering from reflected waves 73) or flux limiters in high resolution schemes 144 (to avoid the spurious oscillations (wiggles) that would otherwise occur due to shocks or sharp changes) could be found through training, and applied at very low numerical cost. We are curious to see if geometry can help design better architectures or approaches for this type of learning-aided simulation, by helping with better loss functions (with soft constraints) or better architectures (to enforce hard constraints) that account for the importance of structure preservation. Learning the highly non-linear and chaotic dynamics of fluids is also an interesting direction: we believe that one can infer predictive high-frequency details of a turbulent flow from a low-resolution simulation as it is an attractive alternative to non-linear turbulence modeling, extending the computationally-expensive Reynolds-Averaged Navier-Stokes (RANS 51), Large-Eddy Simulation (LES 103), or Detached-Eddy Simulation (DES 136) models used in CFD. Many other learning efforts in the domain of simulation are being explored, in particular towards the goal of allowing real-time design of shapes that satisfy some physical properties, such as lowest drag for improved aerodynamics or highest stiffness for a light cantilever.
Geometric integration of physical systems and multiphysics.
Although the use of geometric integrators for differential equations in computational physics has recently brought off many numerical improvements, the large body of knowledge in differential geometric mechanics remains vastly under-utilized in discrete mechanics. Many mechanical systems require geometric objects such as diffeomorphisms, vector fields, or (principal) connections for which no structure-preserving discretization exists. Hydrodynamics, for instance, has well established and rich differential geometric foundations, but rare are the numerical methods that take advantage of this rich body of knowledge as yet. Yet, satisfying a form of “particle relabeling” symmetry 115 on a discrete level could directly enforce Kelvin’s circulation theorem, a momentum preservation as important as angular momentum preservation for rigid bodies. Relativity is another example, albeit much more involved, where structure-preserving numerics would strongly impact the scientific community: having discretizations automatically enforcing Bianchi’s identities would not only simplify the numerical procedures involved in gravitational theory (as spectral accuracy would no longer be required to avoid spurious modes), but could in fact result in conservation of energy and angular momentum. Moreover, multiphysics (coupled mechanical systems involving more than one simultaneously occurring physical field) can be consistently described through constrained variational principles: a simple, yet already interesting example is the case of the equations of motion for the garden hose, where rod dynamics coupled with fluid motion was only fully modeled (along with its nonlinear solutions of traveling-wave type) a few years back 128 through such a geometric treatment. Now that a variational formulation of nonequilibrium thermodynamics extending Hamilton's principle to include irreversible processes has been proposed 90, we are particularly interested in advancing further the arsenal of computational methods for physical simulation.
3.3 Geometry for dynamical systems
Dynamical systems – whether physical, biological, chemical, or social – are ubiquitous in nature, and their study deals with the concept of change, rate of change, rate of rate of change, etc. Dynamical systems are often better elucidated and modeled through topology and geometry. Whether we consider a continuous-time dynamical system (flow) or discrete-time dynamical system (map), the geometric theory of dynamical systems studies phase portraits: on the state-space manifold (a geometric model for the set of all possible states of the system), the global behavior of the dynamical system is determined by a cellular structure of basins enclosed by separatrices, each basin being dominated by a different specific behavior or fate. A system's trajectories on the state-space manifold determine velocity vectors by differentiation; conversely, velocity vectors determine trajectories by integration. Bifurcations can also be understood as geometric models for the controlled change of one system into another, while the rate of divergence of trajectories in phase space measures a system's stability. Given this overwhelming relevance of geometry in dynamical systems, we intend to dedicate some of our activities to develop geometry-based computational tools to study time series and dynamical systems: while classic dynamical systems theory has established solid foundations to study structures in steady and time-periodic flows and maps, new tools are needed to analyze the complexity of time series or aperiodic large-scale flows from sampled trajectories, and to automatically generate a simplified skeleton of the overall dynamics of a system from input data. We discuss a few directions we are interested in further impacting next.
Time series.
Geometric methods play an important part in the study of time series. Of particular interest are time-delay embeddings, which are generically able to capture the underlying state space and dynamics from which the time series data have been acquired, by the Takens embedding theorem 139. Such embeddings transform discrete time series into point clouds in Euclidean space, so that the underlying geometry of the point cloud reflects the geometry of the phase space the data originate from. By doing so, questions related to the seasonality or anomalous behavior of the time series are naturally reformulated into questions about the geometry or topology of their embeddings 125. Beside this approach, other more direct methods apply geometric or topological tools in the original physical or frequency domain, which, despite its simplicity, has proven to be relevant in various contexts 78, 82. A common thread to all these developments is their restriction to numerical time series, including (but not restricted to) data for which geometry plays an obvious role—e.g. inertial or gyroscopic sensor data. With potential medical applications in mind, one of our main long-term goals will be to adapt and extend these approaches to handle categorical data, in connection to the item in the Geometry for data science theme. We also plan to find principled methods to tuning the various parameters involved in the techniques, e.g. the window size in time-delay embeddings: we will seek to optimize or learn these parameters automatically, in connection to the item Geometry-driven learning in the Geometry for data science theme. We will also seek to make these parameters adaptive, e.g. using time-varying window sizes in time-delay embeddings of irregular time series, in order to obtain more accurate data representations and improved learning performance.
Coherent structures.
Another interesting area in need of new numerical methods concerns coherent structures, i.e., persisting features of a flow over long periods that tend to favor or inhibit material transport between distinct flow regions. While there is no universally agreed-upon definition for coherent structures (there exist ergodicity-based 64, observer-based 117, and probabilistic 88 approaches to their definition), most variants and associated computational methods assume a fine knowledge of the Eulerian velocity field in space and time to deduce a good approximation of the flow. However, flows are often known only as a set of sparse particle trajectories in time (an example is the trajectory of buoys in the ocean). Such a sparse sampling of the dynamical system does not lend itself well to a geometric analysis of transport, so topological methods have recently been proposed to extract structures from a sparse set of trajectories by measuring their entanglement 141, 52, 148 based on the theory of braid groups, a classical area of topology. Coherent regions can then be defined as containing particles that possibly mix with other particles within the region itself but do not mix with particles outside the region; the set of trajectories arising from the particles within a coherent region forms a coherent bundle. Even if the use of braid groups offers sound foundations and numerical tools for the definition of coherent structures in 2D, there has been only limited efforts in developing practical and scalable computational tools for the efficient analysis of flow structures in 3D, offering a clear opportunity for us to try new geometric insights.
Invariant sets.
Much of the theory of dynamical systems revolves around the existence and structure of invariant sets, which by definition are subsets of the state space that are invariant under the action of the dynamics. Invariant sets come in many different forms (stationary solutions, periodic orbits, connecting orbits, chaotic invariant sets, etc), and their structure can be very complicated and can undergo drastic changes under perturbations of the system, thus making their study difficult. This is all the more true in practical applications, where the systems are only known through space and/or time discretizations. Conley index theory 74 overcomes these issues by restricting the focus to invariant sets that admit an isolating neighborhood, and by introducing a topological invariant—the Conley index—that characterizes whether such isolated invariant sets are attracting, repelling, or saddle-like. It is defined as the homotopy type of a pair of compact subsets of the neighborhood, and it is proven to be independent of the choice of neighborhood—thus characterizing the invariant set itself. We are interested in the study of invariant sets in the discrete space and continuous time setting, where the space is typically described by a simplicial complex and the dynamics by a combinatorial vector (or multivector) field. Building upon Forman's seminal work in combinatorial dynamical systems 85, recent advances 57, 110 have shown that isolated invariant sets and their Conley indices can be properly defined even in this setting, and that they can be related to the dynamics of some upper semicontinuous acyclic multivalued map defined on the geometric realization of the simplicial complex; in simpler terms, not only can Conley index theory be adapted to the combinatorial setting, but it also connects to its classical analog in the underlying space. Two important questions for applications arise from this line of work: (1) how to compute the invariant sets and their Conley indices (including choosing relevant isolating neighboroods) efficiently? (2) how do they behave under perturbations of the input vector field or simplicial complex? These questions have just started to be addressed 80, 81, mostly through the lens of single-parameter topological persistence theory, developed in the context of topological data analysis. We intend to push this direction further, notably using multi-parameter persistence theory to cope with some of the key difficulties such as the choice of isolating neighborhoods.
3.4 Geometry for data science
The last decade has seen the advent of machine learning (ML), and in particular deep learning (DL), in a large variety of fields, including some directly connected to geometry. For instance, DL-based approaches have become increasingly popular in geometry processing 129 due to their ability to outperform state-of-the-art, domain-specific methods by leveraging the ever-increasing amounts of available labeled data. On the downside, DL approaches suffer from a general lack of explainability. Moreover, their performances can be disappointing on small data due to their large numbers of parameters; this is especially true with end-to-end learning pipelines, which tend to require humongous amounts of training data to learn the right data representation. Finally, DL is by essence tied to Euclidean data representations, and as such it requires intermediate transforms in order to be applicable to non-Euclidean data types such as graphs or probability measures. Because of these limitations, we are seeing a rise of geometric and topological methods for data science in general, and for ML and DL in particular, whose aim is to help address the aforementioned challenges as well as others. For instance, geometric deep learning 62 tries to generalize deep neural models to non-Euclidean domains. This includes for instance using information geometry to apply deep neural models in probability spaces. Topological data analysis (TDA) 120 is another popular approach to enhance ML and DL methods. It contributes to data science in at least three different ways: first, by providing data mining tools that can help users uncover hidden structures in data; second, by providing generic descriptors for geometric data that can be turned into features for ML and DL with provable stability properties; third, by integrating itself deeply into existing ML methods or DL architectures to enhance their performances or to analyze their behavior 70, 113. Other contributions of geometry to data science at large include: the use of Forman’s Ricci curvature and its corresponding Ricci flow in networks, to understand the networks' properties and growth 145; the application of the Hodge-Hemholtz decomposition to statistical ranking problems with sparse response data, with theoretical connections to both PageRank and LASSO 102; the use of Reeb graphs or Morse-Smale complexes in statistical inference 71 as well as in data visualization 143. These important developments reinforce our argument that geometry and topology have their role to play in the elaboration of the next-generation data analysis tools. We plan to focus on a few research directions related to these developments, which are of particular interest in our view.
Deep learning for large-scale 3D geometric data analysis.
We first propose to develop efficient algorithms and mathematical tools for analyzing large geometric data collections using Deep Learning techniques. This includes 3D shapes represented as triangle or quad meshes, volumetric data, point clouds possibly embedded in high-dimensions, and graphs representing geometric (e.g. proximity) data. Our project is motivated by the fact that large annotated collections of geometric models have recently become available 69, 147, and that machine learning algorithms applied to such collections have shown promising initial results, both for data analysis as well as synthesis. We believe that these results can be significantly extended by building on recent advances in geometry processing, optimization and learning. Our ultimate goal is to design novel deep learning techniques capable both of handling geometric data directly and of combining and integrating different data sources into a unified analysis pipeline. A key challenge in this project is the fact that geometric data can come in a myriad different representations, such as point clouds and meshes among others, with variable sampling and discretization. Furthermore, geometric shapes can undergo both rigid and non-rigid deformations. Unfortunately, most existing deep learning approaches focus only on a particular type of representations and deformation classes (e.g., considering purely extrinsic or purely intrinsic methods). Instead we propose to place special focus on designing learning techniques capable of handing diverse multimodal data sources undergoing arbitrary deformations, in a coherent theoretical and practical framework. Moreover we propose to develop novel powerful interactive tools for analysis and annotation, to help harness user input, and also provide better mechanisms for exploration of variability in the data 132, 124.
Explainable geometric and topological features for data.
Another of our goals is to design geometric and topological features that can capture richer content from the data, while maintaining the robustness and stability properties that the existing features enjoy. If we can make our features rich enough so that they characterize the input data (or their underlying geometric structures, assuming such structures exist) completely, then we will be able to leverage them in the context of explainable AI, to compute pre-images with guarantees on the corresponding interpretations. In cases where our features cannot completely describe the data, we will study the geometry of the fibers of the feature extraction step, in order to quantify the discrepancy that may appear between different interpretations of the same feature. We envision two complementary approaches for this:
- The first approach relies on feature aggregation. In the context of TDA for instance, one may consider using multiple filtrations (or filter functions on a fixed simplicial complex), computing their corresponding topological descriptors, then aggregating these descriptors together to form a feature vector.
- The second approach relies on more elaborate geometric and topological tools to design the features. The idea is to encode the joint effect of multiple geometric and topological constructions on the data, in a more integrated way than just by aggregating the corresponding features. By encoding more complex effects, we hope to extract a richer content using smaller constructions.
Research on the first approach in TDA started with 77, 91, who proved that, in the special case where the data are sampled from some subanalytic compact sets in Euclidean space , the compact sets themselves are fully described by the aggregated features obtained by orthogonal projections onto lines. This follows from a more fundamental result on the invertibility of the Radon transforms of constructible functions 134, to which the above aggregated features belong. This initial result has sparked a thriving new direction of research, exploring larger and larger classes of compact sets 101, 114, 121. Many important questions arise from this line of work, some of which have been partially addressed, including: what kind of stability or robustness properties do these aggregated features enjoy? Can the size of the collection of filter functions used be reduced, to become finite and (more importantly) independent of the compact set under consideration? Can the aggregated features be computed efficiently? Can non-Euclidean compact sets, such as manifolds or length spaces, be considered as well, with similar guarantees?
The second approach is related to the development of multi-parameter persistence 66, which is undeniably the most widely open and long-standing research topic in TDA today. The core challenge is to define computationally tractable algebraic invariants that can capture as much of the joint structure of multiple topological constructions as possible. The notorious difficulty of this question comes from the fact that the algebraic objects underlying multi-parameter topological constructions are significantly more complicated than the ones underlying single-parameter constructions. The question also connects to notoriously hard problems in other areas of pure mathematics, such as the classification of isomorphism classes of indecomposable poset representations in quiver representation theory for instance. It can benefit from these connections, as mathematical tools that have been developed for those problems can be imported into the TDA literature—several promising such imports have been made in the recent past, including from representation theory 58 and from sheaf theory 104. In turn, mathematical and algorithmic advances made in multi-parameter persistence may benefit these other areas of mathematics as well. This is clearly a high-risk and long-term research topic, but if successful, it may eventually have an enormous impact on TDA and related areas.
Geometric feature learning.
Geometry and topology have played a key role in the design of feature extraction pipelines for certain types of data. The numerous existing geometric features for geometry processing (shape contexts 87, differential and integral invariants 127, heat or wave kernel signatures 54, 138, etc.) are a sign of the importance of this topic for the computer graphics community. Meanwhile, the TDA community has developed generic feature extraction pipelines, based on combinatorial constructions and their algebraic invariants, which have proven to be useful in a variety of application domains 120. All these approaches are, however, handcrafted, with hyperparameters being tuned via manual, grid, or random search. Our goal is to make these approaches transition from a paradigm of feature engineering to that of feature learning, in order to set up end-to-end learning pipelines for improved performances and adaptability. Two complementary directions are considered:
- designing piecewise-smooth variants of the existing pipelines, with a fine control over the underlying stratification. This will make it possible to apply variational optimization methods, typically stochastic (sub-)gradient descent, and to optimize the gradient sampling steps for improved convergence rates.
- designing novel pipelines based on a combination of geometric/topological tools and deep learning, in order to get the best out of both worlds.
Research in the first direction is still in its infancy. Promising theoretical advances were made recently, towards understanding the piecewise differentiability of the basic topological persistence operator in full generality 108, as well as towards optimizing its parameters using classical stochastic gradient descent 67. Can the knowledge gained in these studies about the underlying stratification of the operator be leveraged to optimize the gradient sampling step and thus improve the convergence rates? Can these results be extended to more advanced pipelines, such as the one for Mapper or for zigzags and multi-parameter persistence?
The idea behind the second direction is to integrate topological or geometric layers into neural network architectures such as auto-encoders or GANs for feature extraction — the challenge being to determine how to do it in the appropriate way, so that we can make the most of this combination. This question connects to the research topic described further down in this section.
Geometry-driven learning.
Most of the contributions of geometry and topology to machine learning until recently have been to the design of pre-processing steps (e.g. feature extraction) to enhance the performances of the learning pipeline. There is now a thriving effort of the community toward integrating geometric and/or topological computations deeper into the core of the pipeline. This includes for instance: ToMATo 70, which integrates a TDA-based feedback loop into density based algorithms to improve their stability and robustness; topological regularizers 72, 99, which add topology-based regularization terms to the loss in supervised statistical learning; topological layers 68, 89, 106, which are meant to be incorporated into neural networks. Meanwhile, geometry and topology have been used to analyze the behavior of neural networks 133, 63. This exciting line of work is just emerging, and our intent is to push this direction further, in particular to address the following important questions:
- How can we generalize the use of topological layers in neural networks? This question is connected to the differentiability of the TDA pipeline, addressed in the research topic Geometric feature learning. Inded, generalizing the current (nascent) framework for differential calculus and optimization with the TDA pipeline will be key to designing both generic and effective topological layers. Another more practical aspect of the question is to evaluate the contribution of topological layers as initial or intermediate layers, depending on the neural network architecture that they are combined with and on the data they are applied to.
- The same question arises for topological regularizers, with similar theoretical and practical challenges.
- The development of richer families of geometric and topological descriptors, undertaken in the item Richer geometric and topological features for data, will eventually lead to the question of generalizing the current differentiable framework to these new descriptors, in order to make them as widely applicable as the current descriptors, and also to the practical question of determining how to best combine them with existing loss functions, regularizers, or neural network architectures.
- The aforementioned contributions and research directions concern mostly supervised learning. Can we contribute as well to unsupervised learning problems, including clustering (as ToMATo does already for density-based clustering), dimensionality reduction, or unsupervised feature learning? This question connects also to the research topic Geometric feature learning described previously. One direction we may explore is the design of geometric or topological layers to be inserted in unsupervised neural network architectures such as auto-encoders or GANs.
- Finally, as TDA is concerned primarily with topology, an obvious (yet still wide open) question to ask is whether it can contribute to the current effort towards generating neural network architectures automatically.
Geometry for categorical and mixed data types.
Categorical data types are notoriously hard to deal with in the context of ML and AI. Indeed, most of the existing ML toolbox has been designed specifically to work with numerical variables, usually sitting in some vector or metric space. By contrast, spaces of categorical data do not naturally come equipped with a linear structure nor a metric. More importantly, these spaces are discrete by nature, so choices of metrics or (dis-)similarity measures can be scarce, with limited effects on the learning efficiency. To make things worse, categorical variables are often mixed with numerical variables, and choosing a proper weighting for them is a challenge in its own right. Meanwhile, categorical variables play an important part in many applications: for instance, in precision medicine, where the monitoring of patients relies on collected longitudinal data that include not only numerical variables such as temperature or blood pressure, but also categorical variables such as illness antecedents or symptoms lists. Thus, handling categorical and mixed data types represents an important challenge today. Unfortunately, with very few exceptions 146, it has been mostly overlooked so far in the development of topological methods for ML and AI, so our goal will be to help fix this situation. The standard approach for handling categorical variables is to define a proper vector representation, then to apply—either off-the-shelf or with minor adaptations—an analysis method designed for numerical variables to the new data representation. A prototypical instance of this approach is Multiple Correspondance Analysis for dimensionality reduction 50, which applies classical PCA to the one-hot encoding matrix of the input data. A variant of the approach replaces the vector representation by a suitable metric or (dis-)similarity measure on the initial categorical variables or on some transformed version of those. For instance, in clustering, one can define a metric on the input data, e.g. Jaccard or Hamming distance, then apply a hierarchical bottom-up clustering algorithm such as single-linkage to the resulting distance matrix. This variant seems quite appropriate for geometric or topological methods, since the latter typically work with metric or (dis-)similarity spaces. The challenge is to determine with which metrics or (dis-)similarity measures, and on which data types, geometric or topological methods will be provably better.
A more refined version of the approach learns the new data representation instead of engineering it, which is particularly relevant when end-to-end learning pipelines are sought for. The methods are usually taylored to a specific data type, for instance word2vec 118 computes word embeddings for text data using a two-layer neural network. Our developments in the research topic Geometry-driven learning will make it possible to combine TDA layers with such networks, and thus to benefit from the most recent advances on representation learning for these data types. The challenge will be to understand when and how to make the most of this combination.
4 Application domains
Our work aims at a wide range of applications covering 3D shape analysis and processing, simulation, and data science in general. While we typically focus on contributions that are of a fundamental, mathematical and algorithmic nature, we seek collaborations with academics and industrial from applied fields, who can use our tools on practical and concrete problems. Here are a few examples of collaborations:
- In the context of 3D geometry processing, we collaborate with Dassault Systèmes for a) the PhD of Lucas Brifault on the design of novel geometric representations for shapes through measure theory and b) the PhD of Mariem Mezghanni on the design of physical simulation layers for 3D modeling.
- In the context of personalized medicine, we collaborate with statisticians and medical doctors to incorporate our geometric and topological features into learning pipelines to design better dynamic treatment regimens (AEx PreMediT).
- In a collaboration with the French Ministry of Defense, we seek to develop tools to analyze multimodal time series data in order to predict the appearance of G-LOCs among fighter jet pilots in training or in operation (PhD of Julie Mordacq).
Beside these few illustrative examples, GeomeriX also maintains regular collaborations with Sanofi, EDF, Danone R&D, Immersion Tools, as well as with several key players in the world-wide tech industry, including Ansys, Adobe Research, Disney/Pixar, NVidia.
5 Highlights of the year
5.1 Thematic programs organization
- Mathieu Desbrun , in collaboration with Jacques-Olivier Lachaud (Université de Savoie Mont-Blanc), organized the Year of Geometry under the auspices of CNRS' GdR Informatique Fondamentale et ses Mathématique. Two workshops (one on AI for Geometry, one on Geometry in Industry), two Young Researchers in Geometry meetings, and a 5-day capstone conference at CIRM in Luminy were organized for this year-long effort.
5.2 Awards
- Jiong Chen and Mathieu Desbrun received a Best Paper award at the ACM SIGGRAPH 2024 conference in Denver, CO, the premier conference in graphics, for their paper on preconditioning of boundary integral equations 20.
5.3 Distinctions
- Steve Oudot was an Invited Speaker at the 9th European Congress of Mathematics (ECM), a quadriennal event that, together with the International Congress of Mathematicians, constitute the two main events of the mathematics community — see details here.
5.4 HdR
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Pooran Memari
defended her Habilitation à diriger des recherches at Institut Polytechnique de Paris on September 5, 2024.
Title: "Points, Patterns, and Shapes Towards Accessible Geometric Modeling." The corresponding HdR Jury was composed of:
- David Coeurjolly, Directeur de recherche au CNRS, Université de Lyon (Examiner)
- Stefanie Hahmann, Professor at Université de Grenoble - Ensimag (Examiner)
- Leif Kobbelt, Professor of Computer Science, RWTH Aachen University (Reviewer)
- Sylvain Lefebvre, Directeur de recherche, Inria Nancy (Reviewer)
- Daniele Panozzo, Associate Professor of Computer Science, New York University (Reviewer)
6 New software, platforms, open data
Although software production and maintenance is not a priority for our team, code is systematically used to develop proof-of-concept implementations, both for reproducibility and to facilitate technology transfer. We adopt an opportunistic approach to code development depending on the project being carried out and the will of the main developers of the software: while many projects limit their involvement in code sharing to a minimum just in order to prove the usefulness and reliability of their contributions, others have large applicability and therefore deserve more time and effort to be devoted in order to provide full-fledged software packages. In particular, our research sometimes yields new packages in well-established libraries such as Cgal in computational geometry or Gudhi in topological data analysis, to which we contribute either directly or indirectly.
6.1 Open Source Code
- Ballmerge Surface Reconstruction CGAL package with Telecom Paris, TU Wien, TU Delft, was released in 2024.
- We have released several packages in github associated with the papers published by Maks Ovsjanikov and Mathieu Desbrun (and their collaborators). All of these packages are freely available and provide open-source implementations for nearly all the published papers.
6.2 New software
6.2.1 MFS-chol
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Name:
Lightning-fast Method of Fundamental Solutions
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Keywords:
3D, Boundary element method
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Functional Description:
The method of fundamental solutions (MFS) and its associated boundary element method (BEM) are commonly used due to the reduced dimensionality they offer: for three-dimensional linear problems, they only require variables on the domain boundary to solve and evaluate the solution throughout space, making them a valuable tool in a wide variety of applications. However, MFS and BEM have poor computational scalability and huge memory requirements for large-scale problems, limiting their applicability and efficiency in practice. By leveraging connections with Gaussian Processes and exploiting the sparse structure of the inverses of boundary integral matrices, we introduce a variational preconditioner that can be computed via a sparse inverse-Cholesky factorization in a massively parallel manner. We show that applying our preconditioner to the Preconditioned Conjugate Gradient algorithm greatly improves the efficiency of MFS or BEM solves, up to four orders of magnitude in our series of tests.
-
Release Contributions:
N/A
- Publication:
-
Contact:
Mathieu Desbrun
-
Participants:
Jiong Chen, Mathieu Desbrun
7 New results
We list our new results for each of the four themes that our team is articulated around.
7.1 Geometry for Euclidean shape processing
7.1.1 SING: Stability-Incorporated Neighborhood Graph
In collaboration with Diana Marin, Stefan Ohrhallinger and Michael Wimmer (TU Wien) and Amal Dev Parakkat (Telecom, IP-Paris).
7.1.2 Stochastic Computation of Barycentric Coordinates
In collaboration with Fernando de Goes (Pixar).
In this work 22, we present a practical and general approach for computing barycentric coordinates through stochastic sampling. Our key insight is a reformulation of the kernel integral defining barycentric coordinates into a weighted least-squares minimization that enables Monte Carlo integration without sacrificing linear precision. Our method can thus compute barycentric coordinates directly at the points of interest, both inside and outside the cage, using just proximity queries to the cage such as closest points and ray intersections. As a result, we can evaluate barycentric coordinates for a large variety of cage representations (from quadrangulated surface meshes to parametric curves) seamlessly, bypassing any volumetric discretization or custom solves. To address the archetypal noise induced by sample-based estimates, we also introduce a denoising scheme tailored to barycentric coordinates. We demonstrate the efficiency and flexibility of our formulation by implementing a stochastic generation of harmonic coordinates, mean-value coordinates, and positive mean-value coordinates.
7.1.3 PoNQ: a Neural QEM-based Mesh Representation
In collaboration with Pierre Alliez (Inria Sophia).
Although polygon meshes have been a standard rep resentation in geometry processing, their irregular and combinatorial nature hinders their suitability for learning based applications. In this work 39, we introduce a novel learnable mesh representation through a set of local 3D sample Points and their associated Normals and Quadric error metrics (QEM) w.r.t. the underlying shape, which we denote PoNQ. A global mesh is directly derived from PoNQ by efficiently leveraging the knowledge of the local quadric errors. Besides marking the first use of QEM within a neu ral shape representation, our contribution guarantees both topological and geometrical properties by ensuring that a PoNQ mesh does not self-intersect and is always the bound ary of a volume. Notably, our representation does not rely on a regular grid, is supervised directly by the target sur face alone, and also handles open surfaces with boundaries and/or sharp features. We demonstrate the efficacy of PoNQ through a learning-based mesh prediction from SDF grids and show that our method surpasses recent state-of-the-art techniques in terms of both surface and edge-based metrics.
7.1.4 Biharmonic Coordinates and their Derivatives for Triangular 3D Cages
In collaboration with Jean-Marc Thiery and Élie Michel (Adobe).
This work 32 extends biharmonic coordinates, which were previously derived for 2D shape deformation, into three dimensions. The key contribution is deriving closed-form mathematical expressions for biharmonic coordinates and their derivatives specifically for 3D triangular cages. At the heart of this work is the development of closed-form expressions that calculate how the Euclidean distance integrates over a triangle, along with the derivatives of this integration. The significance of this advancement is twofold: it completes a gap in the theory of generalized barycentric coordinates, and it enables practical applications in 3D shape manipulation. These applications include creating various types of biharmonic deformations, solving shape deformation problems through variational methods, and providing efficient closed-form solutions for the recently developed Somigliana coordinates. This work ultimately bridges a theoretical gap while offering practical tools for 3D shape manipulation and deformation.
7.1.5 A Survey on Cage-based Deformation of 3D Models
In collaboration with Daniel Ströter, Johannes Sebastian Mueller-Roemer, Sebastian Besler, Andre Stork and Dieter W. Fellner (Technical University of Darmstadt), Jean-Marc Thiery and Tamy Boubekeur (Adobe), Kai Hormann and Qingjun Chang (University of Italian Switzerland).
In this work 30, we review the advancement of 3D cage-based deformation. Cage-based deformation enables users to quickly manipulate 3D geometry by deforming the cage. Due to their utility, cage-based deformation techniques increasingly appear in many geometry modeling applications. For this reason, the computer graphics community has invested a great deal of effort in the past decade and beyond into improving automatic cage generation and cage-based deformation. Recent advances have significantly extended the practical capabilities of cage-based deformation methods. As a result, there is a large body of research on cage-based deformation. In this report, we provide a comprehensive overview of the current state of the art in cage-based deformation of 3D geometry. We discuss current methods in terms of deformation quality, practicality, and precomputation demands. In addition, we highlight potential future research directions that overcome current issues and extend the set of practical applications. In conjunction with this survey, we publish an application to unify the most relevant deformation methods. Our report is intended for computer graphics researchers, developers of interactive geometry modeling applications, and 3D modeling and character animation artists.
7.1.6 BallMerge: High-quality Fast Surface Reconstruction via Voronoi Balls
In collaboration with Amal Dev Parakkat (Telecom, IP-Paris), Stefan Ohrhallinger and Michael Wimmer (TU Wien) and Elmar Eisemann (TU Delft).
7.1.7 DynBioSketch: A tool for sketching dynamic visual summaries in biology, and its application to infection phenomena
In collaboration with Pauline Olivier, Tara Butler, Pascal Guehl, Renaud Chabrier and Marie-Paule Cani (LIX, Ecole Polytechnique) and Jean-Luc Coll (Institute for Advanced Biosciences, Grenoble Alpes University).
7.1.8 Ricci flow-based brain surface covariance descriptors for diagnosing Alzheimer’s disease
In collaboration with Fatemeh Ahmadi, Mohamad-Ebrahim Shiri, Behroz Bidabad, Maral Sedaghat (Math department of Amirkabir University of Technology).
7.1.9
In collaboration with Charline Grenier and Basile Sauvage (Strasbourg University).
7.1.10 Versatile Curve Design by Level Set with Quadratic Convergence
In collaboration with Xiaohu Zhang, Shuang Wu, Hujun Bao and Jin Huang (Zhejiang University), and Yao Jin (Zhejiang Sci-Tech University).
In this work 33, we present an efficient and versatile approach to curve design based on an implicit representation known as the level set. While previous works have explored the use of the level set to generate curves with minimal length, they typically have limitations in accommodating additional conditions for rich and robust control. To address these challenges, we formulate curve editing with constraints like smoothness, interpolation, tangent control, etc., via a level set based variational problem by constraining the values or derivatives of the level set function. However, the widely used gradient flow strategy converges very slowly for this complicated variational problem compared to the classical geodesic one. Thus, we propose to solve it via Newton's method enhanced by local Hessian correction and a trust-region strategy. As a result, our method not only enables versatile control, but also excels in terms of performance due to nearly quadratic convergence and almost linear complexity in each iteration via narrow band acceleration. In practice, these advantages effectively benefit various applications, such as interactive curve manipulation, boundary smoothing for surface segmentation and path planning with obstacles as demonstrated.
7.2 Geometry for simulation
7.2.1 Hybrid LBM-FVM Solver for Two-phase Flow Simulation
In collaboration with Wei Li and Xiaopei Liu (Shanghaitech University)
In this work 25, we introduce a hybrid LBM-FVM solver for two-phase fluid flow simulations in which interface dynamics is modeled by a conservative phase-field equation. Integrating fluid equations over time is achieved through a velocity-based lattice Boltzmann solver which is improved by a central-moment multiple-relaxation-time collision model to reach higher accuracy. For interface evolution, we propose a finite-volume-based numerical treatment for the integration of the phase-field equation: we show that the second-order isotropic centered stencils for diffusive and separation fluxes combined with the WENO-5 stencils for advective fluxes achieve similar and sometimes even higher accuracy than the state-of-the-art double-distribution function LBM methods as well as the DUGKS-based method, while requiring less computations and a smaller amount of memory. Benchmark tests (such as the 2D diagonal translation of a circular interface), along with quantitative evaluations on more complex tests (such as the rising bubble and Rayleigh-Taylor instability simulations) allowing comparisons with prior numerical methods and/or experimental data, are presented to validate the advantage of our hybrid solver. Moreover, 3D simulations (including a dam break simulation) are also compared to the time-lapse photography of physical experiments in order to allow for more qualitative evaluations.
7.2.2 Kinetic Simulation of Turbulent Multifluid Flows
In collaboration with Wei Li (Tencent)
Despite its visual appeal, the simulation of separated multiphase flows (i.e., streams of fluids separated by interfaces) faces numerous challenges in accurately reproducing complex behaviors such as guggling, wetting, or bubbling. These difficulties are especially pronounced for high Reynolds numbers and large density variations between fluids, most likely explaining why they have received comparatively little attention in Computer Graphics compared to single- or two-phase flows. In this work 24, we present a full LBM solver for multifluid simulation. We derive a conservative phase field model with which the spatial presence of each fluid or phase is encoded to allow for the simulation of miscible, immiscible and even partially-miscible fluids, while the temporal evolution of the phases is performed using a D3Q7 lattice-Boltzmann discretization. The velocity field, handled through the recent high-order moment-encoded LBM (HOME-LBM) framework to minimize its memory footprint, is simulated via a velocity-based distribution stored on a D3Q27 or D3Q19 discretization to offer accuracy and stability to large density ratios even in turbulent scenarios, while coupling with the phases through pressure, viscosity, and interfacial forces is achieved by leveraging the diffuse encoding of interfaces. The resulting solver addresses a number of limitations of kinetic methods in both computational fluid dynamics and computer graphics: it offers a fast, accurate, and low-memory fluid solver enabling efficient turbulent multiphase simulations free of the typical oscillatory pressure behavior near boundaries. We present several numerical benchmarks, examples and comparisons of multiphase flows to demonstrate our solver's visual complexity, accuracy, and realism.
7.2.3 Lightning-fast Method of Fundamental Solutions
In collaboration with Florian Schaefer (Georgia Tech)
7.2.4 TwisterForge: Controllable and Efficient Animation of Virtual Tornadoes
In collaboration with James Gain (University of Cape Town), Jean-Marc Chomaz and Marie-Paule Cani (LIX, Ecole Polytechnique)
In this work 44, we introduce a layered approach for creating and animating realistic virtual tornadoes in computer graphics. The method centers on two types of cureves: a 3D curve to intitialize the tornado's core as a vortex filament and 2D profile curves to control the surrounding funnel shape. The core evolves dynamically subject to the Biot-Savart law, bending and twisting driven by its initial curvature, while the funnel profile represents the Stokes stream function and dictates the radial and axial air motion around the core. These two components, together, capture the rotation, sliding, and uplift of the tornado's air volume. Our method achieves visually plausible animations of tornadoes, capable of interacting with uneven terrain, destroying infrastructure, and transporting debris, offering a controllable and realistic solution for visual effects and interactive applications.
7.2.5 Volcanic Skies: coupling explosive eruptions with atmospheric simulation to create consistent skyscapes
In collaboration with Cilliers Pretorius and James Gain (University of Cape Town), Maud Lastic, Damien Rohmer and Marie-Paule Cani (LIX, Ecole Polytechnique), and Guillaume Cordonnier (Inria).
Explosive volcanic eruptions rank among the most terrifying natural phenomena, and are thus frequently depicted in films, games, and other media, usually with a bespoke once-off solution. In this work 29, we introduce the first general-purpose model for bi-directional interaction between the atmosphere and a volcano plume. In line with recent interactive volcano models, we approximate the plume dynamics with Lagrangian disks and spheres and the atmosphere with sparse layers of 2D Eulerian grids, enabling us to focus on the transfer of physical quantities such as temperature, ash, moisture, and wind velocity between these sub-models. We subsequently generate volumetric animations by noise-based procedural upsampling keyed to aspects of advection, convection, moisture, and ash content to generate a fully-realized volcanic skyscape. Our model captures most of the visually salient features emerging from volcano-sky interaction, such as windswept plumes, enmeshed cap, bell and skirt clouds, shockwave effects, ash rain, and sheathes of lightning visible in the dark.
7.3 Geometry for dynamical systems
7.3.1 ADAPT: Multimodal Learning for Detecting Physiological Changes under Missing Modalities
In collaboration with Vicky Kalogeiton (Vista, LIX), Leo Milecki and Maria Vakalopoulou (Centrale Supelec).
7.3.2 Multimodal Learning for Detecting Stress under Missing Modalities
In collaboration with Vicky Kalogeiton (Vista, LIX), Leo Milecki and Maria Vakalopoulou (Centrale Supelec).
7.4 Geometry for data science
7.4.1 On the bottleneck stability of rank decompositions of multi-parameter persistence modules
In collaboration with Magnus botnan (Vrije Universiteit Amsterdam), Steffen Oppermann (NTNU) and Luis Scoccola (University of Oxford).
7.4.2 Signed Barcodes for Multi-parameter Persistence via Rank Decompositions and Rank-Exact Resolutions
In collaboration with Magnus botnan (Vrije Universiteit Amsterdam) and Steffen Oppermann (NTNU).
7.4.3 Intrinsic Interleaving Distance for Merge Trees
In collaboration with Ellen Gasparovic (Union College), Elizabeth Munch (Michigan State University), Katharine Turner (Australian National University), Bei Wang (University of Utah) and Yusu Wang (University of California).
7.4.4 On the stability of multigraded Betti numbers and Hilbert functions
In collaboration with Luis Scoccola (University of Oxford).
7.4.5 Efficient computation of topological integral transforms
In collaboration with Vadim Lebovici (University of Oxford) and Hugo Passe (École Normale Supérieure de Lyon).
7.4.6 Differentiability and Optimization of Multiparameter Persistent Homology
In collaboration with Luis Scoccola (University of Oxford), Siddharth Setlur (ETH Zürich), David Loiseaux and Mathieu Carrière (Datashape, Inria Sophia-Antipolis).
7.4.7 Fine-tuning 3D foundation models for geometric object retrieval
In collaboration with Jarne van den Herrewegen, Tom Tourwé, and Francis Wyffels (from Oqton AI; and AI and Robotics Lab, IDLab-AIRO, Ghent University-imec, Belgium).
7.4.8 DeBaRA: Denoising-Based 3D Room Arrangement Generation
In collaboration with Nicolas Sereyjol-Garros, and Tom Durand (Dassault Systèmes).
7.4.9 Smoothed Graph Contrastive Learning via Seamless Proximity Integration
This work 34 introduces a smoothed graph contrastive learning (SGCL) technique for unsupervised representation learning on graphs, while leveraging proximity integration. Standard graph contrastive learning (GCL) aligns node representations by classifying node pairs as positives or negatives, typically treating all negatives equally in the contrastive loss. The Smoothed Graph Contrastive Learning model (SGCL) improves this by leveraging the geometric structure of augmented graphs to include proximity information for positive and negative pairs, regularizing the learning process. SGCL uses three smoothing techniques to adjust penalties in the contrastive loss and employs a batch-generating strategy to efficiently train on large-scale graphs. Extensive experiments show SGCL outperforms recent baselines in unsupervised settings across various benchmarks.
7.4.10 To Supervise or Not to Supervise: Understanding and Addressing the Key Challenges of Point Cloud Transfer Learning
In collaboration with Lei Li (Technical University of Munich).
7.4.11 Self-Supervised Dual Contouring
This work 42 introduces a self-supervised dual contouring framework. Learning-based isosurface extraction methods offer robust alternatives to axiomatic techniques but often rely on supervised training with axiomatically computed ground truths, inheriting their biases. To address this, Self-Supervised Dual Contouring (SDC) introduces a self-supervised training scheme for the Neural Dual Contouring framework. SDC employs novel self-supervised loss functions to optimize mesh vertices by enforcing consistency with distances to the generated mesh. SDC surpasses data-driven methods in capturing intricate details and handling input irregularities. Additionally, the self-supervised objective regularizes Deep Implicit Networks (DINs), improving the quality of implicit functions and detail preservation across input modalities. SDC also enhances single-view reconstruction by enabling joint training of the predicted SDF and output mesh.
7.4.12 Back to 3D: Few-Shot 3D Keypoint Detection with Back-Projected 2D Features
In collaboration with Thomas Wimmer and Peter Wonka (from Technical University of Munich and KAUST).
7.4.13 Unsupervised Representation Learning for Diverse Deformable Shape Collections
In collaboration with Sara Hahner and Jochen Garcke (Fraunhofer SCAI,
Sankt Augustin, Germany, and Institute for Numerical Simulation,
University of Bonn, Germany).
7.4.14 RIVQ-VAE: Discrete Rotation-Invariant 3D Representation Learning
In collaboration with Mariem Mezghanni and Malika Boulkenafed (Dassault Systèmes).
7.4.15 Deformation Recovery: Localized Learning for Detail-Preserving Deformations
In collaboration with Nicolas Donati (Ansys, France), Simone Melzi (University of Milano-Bicocca, Italy), Etienne Corman (Université de Lorraine, CNRS, Inria).
7.4.16 Memory-Scalable and Simplified Functional Map Learning
8 Bilateral contracts and grants with industry
8.1 Bilateral contracts with industry
8.1.1 Contract with Sanofi Inc.
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Title:
Machine learning approaches for cryo-EM
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Partner Institution(s):
Sanofi Inc.
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Date/Duration:
2023-2024
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Additionnal info/keywords:
Cryogenic electron microscopy (cryo-EM) allows the structure determination of antibody fragments bound to pharmaceutically relevant targets to accelerate drug discovery. The process of cryo-EM data analysis is time consuming and requires user input. To accelerate the rate of structure solution by cryo-EM, this project investigates machine learning approachesto fit and model the atomic coordinates of antibody fragments into the cryo-EM density.
The project funds one post-doctoral researcher for 2 years, jointly between Sanofi Inc., and Ecole Polytechnique (the employer of Maks Ovsjanikov).
8.1.2 Contract with DASSAULT SYSTEMES
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Title:
Generative Models for the Guided Synthesis of Complex and Functional 3D Scenes
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Partner Institution(s):
DASSAULT SYSTEMES
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Date/Duration:
2023-2026
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Additionnal info/keywords:
This thesis focuses on machine learning applied to 3D computer vision, specifically addressing challenges related to the automatic synthesis of 3D environments.
The project funds one PhD student for 3 years.
8.1.3 MEDITWIN with DASSAULT SYSTEMES
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Title:
MEDITWIN: Virtual human twins for medical applications
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Partner Institution(s):
DASSAULT SYSTEMES
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Date/Duration:
2023-2028
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Additionnal info/keywords:
In the context of IPCEI on Health called MEDITWIN, Geomerix has started working on geometric measure theory and reduced models (Desbrun) and non-rigid registration (Ovsjanikov), with one student and two postdocs to be hired soon.
9 Partnerships and cooperations
9.1 International research visitors
9.1.1 Visits of international scientists
Other international visits to the team
Diana Marin
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Status:
PhD
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Institution of origin:
TU Wien
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Country:
Austria
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Dates:
End of January – April 2024
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Context of the visit:
Research collaboration leading to a publication in ACM Siggraph Asia
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Mobility program/type of mobility:
Research stay, under the supervision of Pooran Memari
9.1.2 Visits to international teams
Sabbatical programme
- Maks Ovsjanikov Visiting Researcher, Google DeepMind, Paris.
9.2 European initiatives
9.2.1 Horizon Europe
ERC Consolidator grant VEGA
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Title:
VEGA: Universal Geometric Transfer Learning
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Partner Institution(s):
- European Research Concil (ERC)
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Date/Duration:
2024-2028
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Additionnal info/keywords:
In this project, we propose to develop a theoretical and practical framework for transfer learning with geometric 3D data. Most existing learning-based approaches, aimed at analyzing 3D data, are based on training neural networks from scratch for each data modality and application. Our main goal will be to develop universally-applicable methods by combining powerful pre-trainable modules with effective multi-scale analysis and fine-tuning, given minimal task-specific data. The overall key to our study will be analyzing rigorous ways, both theoretically and in practice, in which solutions can be transferred and adapted across problems, semantic categories and geometric data types.
9.3 National initiatives
Contrat de recherche Inria - SHOM
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Title:
Traitement de nuage de points bathyémtriques (SMF et Lidar) par l'approche apprentissage automatique
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Partner Institution(s):
- Service Hydrographique et Océanographique de la Marine (SHOM), Brest, France
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Date/Duration:
2024-2028
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Additionnal info/keywords:
Ce projet a pour objectif de mieux appréhender, à l’aide de l’apprentissage automatique, la donnée bathymétrique sous forme de nuages de points pour améliorer la description des fonds marins et des zones côtières. Ce sujet est en lien avec le traitement des erreurs ponctuelles de la donnée bathymétrique et également avec l’utilisation de cette donnée pour la génération de modèles numériques de terrain.
AEx PreMediT
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Title:
Precision Medicine using Topology
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Partner Institution(s):
- CRESS, Hôtel-Dieu, France
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Date/Duration:
2022-2025
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Additionnal info/keywords:
While recent advances in machine learning are opening promising prospects for precision medicine, the sometimes small size, sparsity, or partly categorical nature of the data involved pose some crucial challenges. The goal of PreMediT is to address these challenges by integrating information about the geometric and topological structure of the data into the machine learning pipelines.
ANR AI Chair AIGRETTE
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Title:
Analyzing Large Scale Geometric Data Collections
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Partner Institution(s):
- ANR
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Date/Duration:
2020-2024
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Additionnal info/keywords:
Motivated by the deluge of 3D data using geometric representations (point clouds, triangle, quad meshes, graphs...) that are ill-suited for modern applications, we are developing efficient algorithms and mathematical tools for analyzing diverse geometric data collections.
10 Dissemination
10.1 Promoting scientific activities
10.1.1 Scientific events: organisation
General chair, scientific chair
- Mathieu Desbrun (cochairing with Jacques-Olivier Lachaud) for Year of Geometry, funded by the GDR IFM, with its capstone Geometry and Computing conference at the CIRM in October 2024.
Member of the organizing committees
- Steve Oudot co-organizer (with Claire Amiot, Thomas Brüstle, Sergio Estrada and Luis Scoccola) of BIRS workshop Representation Theory and Topological Data Analysis (24w5241), April 7-12, Banff, Canada.
10.1.2 Scientific events: selection
Member of the conference program committees
- Pooran Memari for ACM Siggraph 2024
- Pooran Memari for Eurographics 2024
- Steve Oudot for International Symposium on Computational Geometry (SoCG) 2024
10.1.3 Journal
Member of the editorial boards
- Pooran Memari Associate Editor of Computer Graphics Forum (CGF), since April 2021.
- Pooran Memari Associate Editor of Graphical Models Journal, Elsevier, April 2023 – May 2024.
- Mathieu Desbrun Associate Editor of ournal of Geometric Mechanics, AIMS, 2024.
- Maks Ovsjanikov Associate Editor, Transactions on Visualization and Computer Graphics journal, since 2020.
- Steve Oudot Associate Editor of Journal of Computational Geometry.
10.1.4 Invited talks
- Pooran Memari Invited talk at the Geometry & Computing conference, CIRM Luminy, October 2024.
- Mathieu Desbrun , invited talk at the annual GDR IFM meeting in 2024.
- Steve Oudot Invited speaker at the 9th European Congress of Mathematics (9ECM), Seville, July 2024.
- Steve Oudot Plenary speaker at the 21st International Conference on Representations of Algebras (ICRA), Shanghai, August 2024.
- Steve Oudot Invited speaker at the workshop Representation theory - combinatorial aspects and applications to TDA, NTNU, Trondheim, December 2024.
- Steve Oudot Seminar speaker in the Mathematical Institute of the University of Oxford, January 2024.
10.1.5 Research administration
- Maks Ovsjanikov Fellow of ELLIS, senior member of the European society for top AI researchers, since 2023.
- Pooran Memari Member of the Board of the French Chapter of Eurographics (EGFR), since October 2024.
- Pooran Memari Co-responsible for the Interaction, Graphics & Design (IGD) master’s program at IP-Paris, since September 2024.
- Steve Oudot member of the Conseil Académique of IP Paris, representing Inria
10.2 Teaching - Supervision - Juries
10.2.1 Teaching
- Master: Steve Oudot, Computational Geometry and Topology, 18h eq-TD, M2, MPRI;
- Master: Maks Ovsjanikov, Geometry Processing and Geometric Deep Learning, M2, MVA;
- Master: Steve Oudot, Topological data analysis, 45h eq-TD, M1, École polytechnique, France;
- Master: Mathieu Desbrun , Digital Representation and Analysis of Shapes, M2, École polytechnique, France;
- Master: Mathieu Desbrun , Computer Animation, M2, École polytechnique, France;
- Master: Pooran Memari, Digital Representation and Analysis of Shapes, M2, École polytechnique, France;
- Master: Pooran Memari, Computer Science refresher course at Artificial Intelligence and Advanced Visual Computing Master Program, M2, École polytechnique, France;
- Master: Maks Ovsjanikov, Artificial Intelligence and Advanced Visual Computing, École polytechnique, France;
- Undergrad-Master: Steve Oudot, Algorithms for data analysis in C++, 22.5h eq-TD, L3/M1, École Polytechnique, France.
10.2.2 Supervision
- PhD in progress: Julie Mordacq, Analyse Topologique des Données et Apprentissage Machine pour analyser et prédire des transitions de phase en n-dimensions, Institut Polytechnique de Paris. Started Sept. 2022. Steve Oudot and Vicky Kalogeiton (Vista, LIX).
- PhD in progress: Jingyi Li, Invariants algébriques effectifs pour la persistance multi-paramètre, Institut Polytechnique de Paris. Started Nov. 2023. Steve Oudot.
- PhD: Souhaib Attaiki, LIX. defended: March 2024?. Maks Ovsjanikov.
- PhD in progress: Nasim Bagheri Shouraki, Application of neurocognition to study the effectiveness of geometric tactile 2D patterns in navigation maps and instructions for Visually Impaired Individuals, IP Paris. Start date: October 2024. Pooran Memari and Panos Mavros (Telecom Paris).
- PhD in progress: Theo Braune, École Polytechnique, Palaiseau. Mathieu Desbrun.
- PhD: Diego Gomez, École Polytechnique, Palaiseau. Defended: September 2024?. Maks Ovsjanikov.
- PhD in progress: Souhail Hadgi, École Polytechnique, Palaiseau. Maks Ovsjanikov.
- PhD: Robin Magnet, LIX. Defended: May 2024?. Maks Ovsjanikov.
- PhD in progress: Leopold Maillard, Dassault Systèmes. Maks Ovsjanikov.
- PhD in progress: Tim Scheller, École Polytechnique, Palaiseau. Maks Ovsjanikov.
10.2.3 Juries
- Mathieu Desbrun reviewer and jury member for Colin Weil-Duflos, Université Savoie Mont-Blanc.
- Pooran Memari Admission Jury for Masters & PhD-Track IGD (Interaction, Graphics & Design), IP-Paris, since 2020.
- Pooran Memari PhD Jury for Clément Chomicki, Université Gustave Eiffel, LIGM UMR8049 (28/11/2024).
- Pooran Memari PhD Jury for Bastien Doignies, Université Lyon 1 (25/11/2024).
- Pooran Memari Recruitment Committee for Assistant Professors at LORIA, Faculty of Sciences and Technologies, University of Lorraine (5/2024).
- Pooran Memari is a member of the Jury d’admission Masters & PhD Track IGD (Interaction, Graphics & Design), IP-Paris (Since 2020).
11 Scientific production
Major publications
11.1 Publications of the year
International journals
International peer-reviewed conferences
Conferences without proceedings
Reports & preprints
Cited publications