2025Activity report​​Project-TeamGEOMERIX

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​‌ 111, 44,​​ 45, 121.​​​‌ 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 100​​, 89, 72​​. 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 112,‌​‌ 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 119,​​​‌ 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​‌ 120 or a length​​ cross-ratio 96 per edge)​​​‌ later formed the basis‌ of circle patterns 48‌​‌, 94 as well​​ as conformal representations 126​​​‌; the discrete Laplace-Beltrami‌ cotan formula 115 also‌​‌ determines the edge lengths​​ of a triangle mesh​​​‌ (and thus its discrete‌ metric) up to a‌​‌ global scaling 138.​​ More recently, generalized notions​​​‌ of metrics were proposed;‌ for instance, 86 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​​​‌ 85. 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 108.​​ 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 49.
• 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 73.​​​‌ 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​ 54. 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​ 131 and Taimiņa's crochet​‌ model). We are hoping​​ that this intrinsic metric​​​‌ characterization can be investigated​ through recent discrete hyperbolic​‌ parametrization tools 81,​​ 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 84 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.

• 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 129​​​‌, 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 68‌ or FEEC 42),‌​‌ where $k$-dimensional integrals​​ can be directly evaluated​​​‌ in $k$-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​‌ 83 to provide a​​ powerful Isogeometric Analysis (IGA).​​​‌ Recent developments 101,‌ 82 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 65​​, 101 (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.

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​​​‌ 124: 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​‌ 98. 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 50​​​‌, 64 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 62‌) or flux limiters‌​‌ in high resolution schemes​​ 133 (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 40), Large-Eddy Simulation‌​‌ (LES 92), or​​ Detached-Eddy Simulation (DES 125​​​‌) 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​ 104 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 117​ through such a geometric​‌ treatment. Now that a​​ variational formulation of nonequilibrium​​​‌ thermodynamics extending Hamilton's principle​ to include irreversible processes​‌ has been proposed 79​​, we are particularly​​​‌ interested in advancing further​ the arsenal of computational​‌ methods for physical simulation.​​

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 128.​​​‌ 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​​ 114. 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 67, 71​​​‌. 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 53,‌​‌ observer-based 106, and​​ probabilistic 77 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 130, 41​​​‌, 137 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​‌ 63 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 74,​​ recent advances 46,​​​‌ 99 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 69,​​ 70, 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.

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 58,‌​‌ 136, 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 121​, 113.

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 66,​​ 80, who proved​​​‌ that, in the special​ case where the data​‌ are sampled from some​​ subanalytic compact sets in​​​‌ Euclidean space ${ℝ}^{\mathrm{n}}$, 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 123‌, 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 90​​​‌, 103, 110‌. 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 55,​​​‌ 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 47‌​‌ and from sheaf theory​​ 93. 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 76,​​ differential and integral invariants​​​‌ 116, heat or‌ wave kernel signatures 43‌​‌, 127, 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​‌ 109. 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​​​‌ 97, as well​ as towards optimizing its​‌ parameters using classical stochastic​​ gradient descent 56.​​​‌ 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​​ 59, which integrates​​​‌ a TDA-based feedback loop​ into density based algorithms​‌ to improve their stability​​ and robustness; topological regularizers​​​‌ 61, 88,​ which add topology-based regularization​‌ terms to the loss​​ in supervised statistical learning;​​​‌ topological layers 57,​ 78, 95,​‌ which are meant to​​ be incorporated into neural​​​‌ networks. Meanwhile, geometry and​ topology have been used​‌ to analyze the behavior​​ of neural networks 122​​​‌, 52. 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​​ 135, 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 39‌​‌, 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 107​​ 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.

6.1.1 LUbie

• Name:‌​‌
  Lightning-fast Boundary Element Method​​
• Keywords:
  3D, Boundary element​​​‌ method
• Functional Description:
  The‌ boundary element method (BEM)‌​‌ is commonly used due​​ to the reduced dimensionality​​​‌ it offers: for three-dimensional‌ linear problems, it only‌​‌ requires variables on the​​ domain boundary to solve​​​‌ and evaluate the solution‌ throughout space, making it‌​‌ a valuable tool in​​ a wide variety of​​​‌ applications. However, BEM has‌ 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 generalized minimal residual​‌ (GMRES) algorithm greatly improves​​ the efficiency of BEM​​​‌ solves, up to four​ orders of magnitude in​‌ our series of tests.​​
• URL:
  https://­gitlab.­inria.­fr/­geomerix/­public/­lubie
• Publication:
  hal-05063377​​​‌
• Contact:
  Jiong Chen
• Participants:​
  Mathieu Desbrun, Jiong Chen​‌

6.1.2 PoNQ: a Neural​​ QEM-based Mesh Representation

• Name:​​​‌
  A Neural QEM-based Mesh​ Representation
• Keyword:
  Neural representation​‌
• Functional Description:
  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 neural​ shape representation, our contribution​‌ guarantees both topological and​​ geometrical properties by ensuring​​​‌ that a PoNQ mesh​ does not self-intersect and​‌ is always the boundary​​ of a volume. Notably,​​​‌ our representation does not​ rely on a regular​‌ grid, is supervised directly​​ by the target surface​​​‌ 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.
• Contact:
  Pierre​​ Alliez
• Participants:
  Nissim Maruani,​​​‌ Mathieu Desbrun, Pierre Alliez​

7.1.1 Efficient​​​‌ and Scalable Spatial Regularization​ of Optimal Transport

Participants:​‌ Mathieu Desbrun, Lucas​​ Brifault.

In collaboration​​​‌ with David Coehne-Steiner from​ Inria Sophia-Antipolis.

7.1.2 Discrete Torsion​‌ of Connection Forms on​​ Simplicial Meshes

Participants: Mathieu​​​‌ Desbrun, Mark Gillespie​, Theo Braune.​‌

In collaboration with Yiying​​ Tong (Michigan State University,​​​‌ USA).

7.1.3 ShapeShifter: 3D‌ Variations Using Multiscale and‌​‌ Sparse Point-Voxel Diffusion

Participants:​​ Mathieu Desbrun, Nissim​​​‌ Maruani.

In collaboration‌ with Adone Research (San‌​‌ Francisco, USA) and Inria​​ Sophia-Antipolis.

7.1.4‌​‌ Linear-Time Transport with Rectified​​ Flows

Participants: Pooran Memari​​​‌.

In collaboration with‌ Université Lyon 1.

7.1.5 MILo: Mesh-In-the-Loop Gaussian​​ Splatting for Detailed and​​​‌ Efficient Surface Reconstruction

Participants:‌ Antoine Guédon, Nissim‌​‌ Maruani, Maks Ovsjanikov​​​‌. While recent advances​ in Gaussian Splatting have​‌ enabled fast reconstruction of​​ high-quality 3D scenes from​​​‌ images, extracting accurate surface​ meshes remains a challenge.​‌ Current approaches extract the​​ surface through costly post-processing​​​‌ steps, resulting in the​ loss of fine geometric​‌ details or requiring significant​​ time and leading to​​​‌ very dense meshes with​ millions of vertices. More​‌ fundamentally, the a posteriori​​ conversion from a volumetric​​​‌ to a surface representation​ limits the ability of​‌ the final mesh to​​ preserve all geometric structures​​​‌ captured during training. We​ present MILo in 25​‌, a novel Gaussian​​ Splatting framework that bridges​​​‌ the gap between volumetric​ and surface representations by​‌ differentiably extracting a mesh​​ from the 3D Gaussians.​​​‌ We design a fully​ differentiable procedure that constructs​‌ the mesh-including both vertex​​ locations and connectivity-at every​​​‌ iteration directly from the​ parameters of the Gaussians,​‌ which are the only​​ quantities optimized during training.​​​‌ Our method introduces three​ key technical contributions: (1)​‌ a bidirectional consistency framework​​ ensuring both representations-Gaussians and​​​‌ the extracted mesh-capture the​ same underlying geometry during​‌ training; (2) an adaptive​​ mesh extraction process performed​​​‌ at each training iteration,​ which uses Gaussians as​‌ differentiable pivots for Delaunay​​ triangulation; (3) a novel​​​‌ method for computing signed​ distance values from the​‌ 3D Gaussians that enables​​ precise surface extraction while​​​‌ avoiding geometric erosion. Our​ approach can reconstruct complete​‌ scenes, including backgrounds, with​​ state-of-the-art quality while requiring​​​‌ an order of magnitude​ fewer mesh vertices than​‌ previous methods. Due to​​ their light weight and​​​‌ empty interior, our meshes​ are well suited for​‌ downstream applications such as​​ physics simulations and animation.​​​‌

7.1.6 DiffuMatch: Category-Agnostic Spectral​ Diffusion Priors for Robust​‌ Non-rigid Shape Matching

Participants:​​ Emery Pierson, Lei​​​‌ Li, Maks Ovsjanikov​. Deep functional maps​‌ have recently emerged as​​ a powerful tool for​​​‌ solving non-rigid shape correspondence​ tasks. Methods that use​‌ this approach combine the​​ power and flexibility of​​​‌ the functional map framework,​ with data-driven learning for​‌ improved accuracy and generality.​​ However, most existing methods​​​‌ in this area restrict​ the learning aspect only​‌ to the feature functions​​ and still rely on​​​‌ axiomatic modeling for formulating​ the training loss or​‌ for functional map regularization​​ inside the networks. This​​​‌ limits both the accuracy​ and the applicability of​‌ the resulting approaches only​​ to scenarios where assumptions​​​‌ of the axiomatic models​ hold. In this work​‌ 38, we show,​​ for the first time,​​​‌ that both in-network regularization​ and functional map training​‌ can be replaced with​​ data-driven methods. For this,​​​‌ we first train a​ generative model of functional​‌ maps in the spectral​​ domain using score-based generative​​​‌ modeling, built from a​ large collection of high-quality​‌ maps. We then exploit​​ the resulting model to​​​‌ promote the structural properties​ of ground truth functional​‌ maps on new shape​​ collections. Remarkably, we demonstrate​​​‌ that the learned models​ are category-agnostic, and can​‌ fully replace commonly used​​ strategies such as enforcing​​​‌ Laplacian commutativity or orthogonality​ of functional maps. Our​‌ key technical contribution is​​ a novel distillation strategy​​ from diffusion models in​​​‌ the spectral domain. Experiments‌ demonstrate that our learned‌​‌ regularization leads to better​​ results than axiomatic approaches​​​‌ for zero-shot non-rigid shape‌ matching.

7.1.7 ZeroKey: Point-Level‌​‌ Reasoning and Zero-Shot 3D​​ Keypoint Detection from Large​​​‌ Language Models

Participants: Bingchen‌ Gong, Diego Gomez‌​‌, Maks Ovsjanikov.​​ We propose in 32​​​‌ a novel zero-shot approach‌ for keypoint detection on‌​‌ 3D shapes. Point-level reasoning​​ on visual data is​​​‌ challenging as it requires‌ precise localization capability, posing‌​‌ problems even for powerful​​ models like DINO or​​​‌ CLIP. Traditional methods for‌ 3D keypoint detection rely‌​‌ heavily on annotated 3D​​ datasets and extensive supervised​​​‌ training, limiting their scalability‌ and applicability to new‌​‌ categories or domains. In​​ contrast, our method utilizes​​​‌ the rich knowledge embedded‌ within Multi-Modal Large Language‌​‌ Models (MLLMs). Specifically, we​​ demonstrate, for the first​​​‌ time, that pixel-level annotations‌ used to train recent‌​‌ MLLMs can be exploited​​ for both extracting and​​​‌ naming salient keypoints on‌ 3D models without any‌​‌ ground truth labels or​​ supervision. Experimental evaluations demonstrate​​​‌ that our approach achieves‌ competitive performance on standard‌​‌ benchmarks compared to supervised​​ methods, despite not requiring​​​‌ any 3D keypoint annotations‌ during training. Our results‌​‌ highlight the potential of​​ integrating language models for​​​‌ localized 3D shape understanding.‌ This work opens new‌​‌ avenues for cross-modal learning​​ and underscores the effectiveness​​​‌ of MLLMs in contributing‌ to 3D computer vision‌​‌ challenges.

7.1.8 NAM: Neural​​ Adjoint Maps for refining​​​‌ shape correspondences

Participants: Maks‌ Ovsjanikov. In this‌​‌ paper 27, we​​ propose a novel approach​​​‌ to refine 3D shape‌ correspondences by leveraging multi-layer‌​‌ perceptions within the framework​​ of functional maps. Central​​​‌ to our contribution is‌ the concept of Neural‌​‌ Adjoint Maps, a novel​​ neural representation that generalizes​​​‌ the traditional solution of‌ functional maps for estimating‌​‌ correspondence between manifolds. Fostering​​ our neural representation, we​​​‌ propose an iterative algorithm‌ explicitly designed to enhance‌​‌ the precision and robustness​​ of shape correspondence across​​​‌ diverse modalities such as‌ meshes and point clouds.‌​‌ By harnessing the expressive​​ power of non-linear solutions,​​​‌ our method captures intricate‌ geometric details and feature‌​‌ correspondences that conventional linear​​ approaches often overlook. Extensive​​​‌ evaluations on standard benchmarks‌ and challenging datasets demonstrate‌​‌ that our approach achieves​​ state-of-the-art accuracy for both​​​‌ isometric and non-isometric meshes‌ and for point clouds‌​‌ where traditional methods frequently​​ struggle. Moreover, we show​​​‌ the versatility of our‌ method in tasks such‌​‌ as signal and neural​​ field transfer, highlighting its​​​‌ broad applicability to domains‌ including computer graphics, medical‌​‌ imaging, and other fields​​ demanding precise transfer of​​​‌ information among 3D shapes.‌ Our work sets a‌​‌ new standard for shape​​ correspondence refinement, offering robust​​​‌ tools across various applications.‌

7.1.9 Escaping Plato's Cave:‌​‌ Towards the Alignment of​​ 3D and Text Latent​​​‌ Spaces

Participants: Souhail Hadgi‌, Diego Gomez,‌​‌ Maks Ovsjanikov. Recent​​ works have shown that,​​​‌ when trained at scale,‌ unimodal 2D vision and‌​‌ text encoders converge to​​ learned features that share​​​‌ remarkable structural properties, despite‌ arising from different representations.‌​‌ However, the role of​​​‌ 3D encoders with respect​ to other modalities remains​‌ unexplored. Furthermore, existing 3D​​ foundation models that leverage​​​‌ large datasets are typically​ trained with explicit alignment​‌ objectives with respect to​​ frozen encoders from other​​​‌ representations. In this work​ 33, we investigate​‌ the possibility of a​​ posteriori alignment of representations​​​‌ obtained from uni-modal 3D​ encoders compared to text-based​‌ feature spaces. We show​​ that naive post-training feature​​​‌ alignment of uni-modal text​ and 3D encoders results​‌ in limited performance. We​​ then focus on extracting​​​‌ subspaces of the corresponding​ feature spaces and discover​‌ that by projecting learned​​ representations onto well-chosen lowerdimensional​​​‌ subspaces the quality of​ alignment becomes significantly higher,​‌ leading to improved accuracy​​ on matching and retrieval​​​‌ tasks. Our analysis further​ sheds light on the​‌ nature of these shared​​ subspaces, which roughly separate​​​‌ between semantic and geometric​ data representations. Overall, ours​‌ is the first work​​ that helps to establish​​​‌ a baseline for post-training​ alignment of 3D unimodal​‌ and text feature spaces,​​ and helps to highlight​​​‌ both the shared and​ unique properties of 3D​‌ data compared to other​​ representations.

7.1.10 FourieRF: Few-Shot​​​‌ NeRFs via Progressive Fourier​ Frequency Control

Participants: Diego​‌ Gomez, Bingchen Gong​​, Maks Ovsjanikov.​​​‌ We present in 31​ a novel approach for​‌ few-shot NeRF estimation, aimed​​ at avoiding local artifacts​​​‌ and capable of efficiently​ reconstructing real scenes. In​‌ contrast to previous methods​​ that rely on pre-trained​​​‌ modules or various data-driven​ priors that only work​‌ well in specific scenarios,​​ our method is fully​​​‌ generic and is based​ on controlling the frequency​‌ of the learned signal​​ in the Fourier domain.​​​‌ We observe that in​ NeRF learning methods, highfrequency​‌ artifacts often show up​​ early in the optimization​​​‌ process, and the network​ struggles to correct them​‌ due to the lack​​ of dense supervision in​​​‌ few-shot cases. To counter​ this, we introduce an​‌ explicit curriculum training procedure,​​ which progressively adds higher​​​‌ frequencies throughout optimization, thus​ favoring global, low-frequency signals​‌ initially, and only adding​​ details later. We represent​​​‌ the radiance fields using​ a grid-based model and​‌ introduce an efficient approach​​ to control the frequency​​​‌ band of the learned​ signal in the Fourier​‌ domain. Therefore our method​​ achieves faster reconstruction and​​​‌ better rendering quality than​ purely MLP-based methods. We​‌ show that our approach​​ is general and is​​​‌ capable of producing high-quality​ results on real scenes,​‌ at a fraction of​​ the cost of competing​​​‌ methods. Our method opens​ the door to efficient​‌ and accurate scene acquisition​​ in the few-shot NeRF​​​‌ setting.

7.1.11 GANFusion: Feed-Forward​ Text-to-3D with Diffusion in​‌ GAN Space

Participants: Souhail​​ Attaiki, Maks Ovsjanikov​​​‌. In this work​ 29, we train​‌ a feed-forward text-to-3D diffusion​​ generator for human characters​​​‌ using only single-view 2D​ data for supervision. Existing​‌ 3D generative models cannot​​ yet match the fidelity​​​‌ of image or video​ generative models. State-of-the-art 3D​‌ generators are either trained​​ with explicit 3D supervision​​​‌ and are thus limited​ by the volume and​‌ diversity of existing 3D​​ data. Meanwhile, generators that​​ can be trained with​​​‌ only 2D data as‌ supervision typically produce coarser‌​‌ results, cannot be text-conditioned,​​ or must revert to​​​‌ test-time optimization. We observe‌ that GAN- and diffusion-based‌​‌ generators have complementary qualities:​​ GANs can be trained​​​‌ efficiently with 2D supervision‌ to produce high-quality 3D‌​‌ objects but are hard​​ to condition on text.​​​‌ In contrast, denoising diffusion‌ models can be conditioned‌​‌ efficiently but tend to​​ be hard to train​​​‌ with only 2D supervision.‌ We introduce GANFusion, which‌​‌ starts by generating unconditional​​ triplane features for 3D​​​‌ data using a GAN‌ architecture trained with only‌​‌ single-view 2D data. We​​ then generate random samples​​​‌ from the GAN, caption‌ them, and train a‌​‌ text-conditioned diffusion model that​​ directly learns to sample​​​‌ from the space of‌ good triplane features that‌​‌ can be decoded into​​ 3D objects.

7.2.1 Kinetic‌ Free-Surface Flows and Foams‌​‌ with Sharp Interfaces

Participants:​​ Mathieu Desbrun.

In​​​‌ collaboration with Jiao Tong‌ University, Shanghai.

7.2.2 Lightning-fast Boundary​​​‌ Element Method

Participants: Jiong‌ Chen, Mathieu Desbrun‌​‌.

In collaboration with​​ Florian Schäfer (Courant Institute,​​​‌ NYC, NY, USA).

7.2.3​​​‌ Immersed boundary-lattice Boltzmann mesoscale​ method for wetting problems​‌

Participants: Mathieu Desbrun.​​

In collaboration with Kiwon​​​‌ Um (Telecom Paris) and​ a bunch of colleagues​‌ from AQTIVATE (European project).​​

7.2.4 Solver-in-the-loop approach to​ closure of shell models​‌ of turbulence

Participants: Mathieu​​ Desbrun.

In collaboration​​​‌ with Kiwon Um (Telecom​ Paris) and a bunch​‌ of colleagues from AQTIVATE​​ (European project).

7.3.1​​ LACONIC: A 3D Layout​​​‌ Adapter for Controllable Image​ Creation

Participants: Léopold Maillard​‌, Maks Ovsjanikov.​​ Existing generative approaches for​​ guided image synthesis of​​​‌ multi-object scenes typically rely‌ on 2D controls in‌​‌ the image or text​​ space. As a result,​​​‌ these methods struggle to‌ maintain and respect consistent‌​‌ three-dimensional geometric structure, underlying​​ the scene. In this​​​‌ paper 34, we‌ propose a novel conditioning‌​‌ approach, training method and​​ adapter network that can​​​‌ be plugged into pretrained‌ text-to-image diffusion models. Our‌​‌ approach provides a way​​ to endow such models​​​‌ with 3D-awareness, while leveraging‌ their rich prior knowledge.‌​‌ Our method supports camera​​ control, conditioning on explicit​​​‌ 3D geometries and, for‌ the first time, accounts‌​‌ for the entire context​​ of a scene, i.e.,​​​‌ both on and off-screen‌ items, to synthesize plausible‌​‌ and semantically rich images.​​ Despite its multi-modal nature,​​​‌ our model is lightweight,‌ requires a reasonable number‌​‌ of data for supervised​​ learning and shows remarkable​​​‌ generalization power. We also‌ introduce methods for intuitive‌​‌ and consistent image editing​​ and restyling, e.g., by​​​‌ positioning, rotating or resizing‌ individual objects in a‌​‌ scene. Our method integrates​​ well within various image​​​‌ creation workflows and enables‌ a richer set of‌​‌ applications compared to previous​​ approaches.

7.3.2 AtomSurf :​​​‌ Surface Representation for Learning‌ on Protein Structures

Participants:‌​‌ Souhail Hadgi, Vincent​​ Mallet, Maks Ovsjanikov​​​‌. While there has‌ been significant progress in‌​‌ evaluating and comparing different​​ representations for learning on​​​‌ protein data, the role‌ of surface-based learning approaches‌​‌ remains not well-understood. In​​ particular, there is a​​​‌ lack of direct and‌ fair benchmark comparison between‌​‌ the best available surface-based​​ learning methods against alternative​​​‌ representations such as graphs.‌ Moreover, the few existing‌​‌ surface-based approaches either use​​ surface information in isolation​​​‌ or, at best, perform‌ global pooling between surface‌​‌ and graph-based architectures. In​​ this work 35,​​​‌ we fill this gap‌ by first adapting a‌​‌ state-of-the-art surface encoder for​​ protein learning tasks. We​​​‌ then perform a direct‌ and fair comparison of‌​‌ the resulting method against​​ alternative approaches within the​​​‌ Atom3D benchmark, highlighting the‌ limitations of pure surface-based‌​‌ learning. Finally, we propose​​ an integrated approach, which​​​‌ allows learned feature sharing‌ between graphs and surface‌​‌ representations on the level​​ of nodes and vertices​​​‌ across all layers. We‌ demonstrate that the resulting‌​‌ architecture achieves state-of-the-art results​​ on all tasks in​​​‌ the Atom3D benchmark, while‌ adhering to the strict‌​‌ benchmark protocol, as well​​ as more broadly on​​​‌ binding site identification and‌ binding pocket classification. Furthermore,‌​‌ we use coarsened surfaces​​ and optimize our approach​​​‌ for efficiency, making our‌ tool competitive in training‌​‌ and inference time with​​ existing techniques.

7.3.3 Finding​​​‌ antibodies in cryo-EM maps‌ with CrAI

Participants: Maks‌​‌ Ovsjanikov. Therapeutic antibodies​​ have emerged as a​​​‌ prominent class of new‌ drugs due to their‌​‌ high specificity and their​​ ability to bind to​​​‌ several protein targets. Once‌ an initial antibody has‌​‌ been identified, its design​​ and characteristics are refined​​​‌ using structural information, when‌ it is available. Cryo-EM‌​‌ is currently the most​​ effective method to obtain​​​‌ 3D structures. It relies‌ on well-established methods to‌​‌ process raw data into​​​‌ a 3D map, which​ may, however, be noisy​‌ and contain artifacts. To​​ fully interpret these maps​​​‌ the number, position, and​ structure of antibodies and​‌ other proteins present must​​ be determined. Unfortunately, existing​​​‌ automated methods addressing this​ step have limited accuracy,​‌ require additional inputs and​​ high-resolution maps, and exhibit​​​‌ long running times. We​ propose in 26 the​‌ first fully automatic and​​ efficient method dedicated to​​​‌ finding antibodies in cryo-EM​ maps: CrAI. This machine​‌ learning approach leverages the​​ conserved structure of antibodies​​​‌ and a dedicated novel​ database that we built​‌ to solve this problem.​​ Running a prediction takes​​​‌ only a few seconds,​ instead of hours, and​‌ requires nothing but the​​ cryo-EM map, seamlessly integrating​​​‌ within automated analysis pipelines.​ Our method can find​‌ the location and pose​​ of both Fabs and​​​‌ VHHs at resolutions up​ to 10 Å and​‌ is significantly more reliable​​ than existing approaches.

7.3.4​​​‌ Signed Barcodes for Multi-parameter​ Persistence via Rank Decompositions​‌ and Rank-Exact Resolutions

Participants:​​ Steve Oudot.

In​​​‌ collaboration with Magnus Botnan​ (VU Amsterdam) and Steffen​‌ Oppermann (NTNU).

7.3.5 Computation​ of gamma-linear projected barcodes​‌ for multiparameter persistence

Participants:​​ Steve Oudot.

In​​​‌ collaboration with Alex Fernandez​ and François Petit (CRESS,​‌ Inserm).

7.3.6 Intrinsic Interleaving Distance‌ for Merge Trees

Participants:‌​‌ Steve Oudot.

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 (UCSD).

7.3.7 T-REGS: Minimum Spanning​​ Tree Regularization for Self-Supervised​​​‌ Learning

Participants: Julie Mordacq‌, David Loiseaux,‌​‌ Steve Oudot.

In​​ collaboration with Vicky Kalogeiton​​​‌ (Vista team, LIX, École‌ polytechnique).

8.1.1 Contract​​​‌ with DASSAULT SYSTEMES

Participants:‌ Maks Ovsjanikov.

• Title:‌​‌
  Generative Models for the​​ Guided Synthesis of Complex​​​‌ and Functional 3D Scenes‌
• Partner Institution(s):
  DASSAULT SYSTEMES‌​‌
• Date/Duration:
  2023-2026
• 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.

8.1.2​​​‌ MEDITWIN with DASSAULT SYSTEMES​

Participants: Mathieu Desbrun,​‌ Maks Ovsjanikov.

• Title:​​
  MEDITWIN: Virtual human twins​​​‌ for medical applications
• Partner​ Institution(s):
  DASSAULT SYSTEMES
• Date/Duration:​‌
  2023-2028
• Additionnal info/keywords:
  In​​ the context of the​​​‌ consortium MEDITWIN, Geomerix has​ started working on geometric​‌ measure theory and reduced​​ models (Desbrun) and non-rigid​​​‌ registration (Ovsjanikov), with two​ students and two postdocs.​‌

9.1.1 Visits of international​ scientists

9.1.2 Visits​​ to international teams

9.2.1​​ Horizon Europe

9.2.2 Other european programs/initiatives​​

Mathieu Desbrun is part​​​‌ of AQTIVATE, a European​ project with funding from​‌ the European Union's research​​ and innovation programme under​​​‌ the Marie Skłodowska-Curie Doctoral​ Networks action and Grant​‌ Agreement No 101072344. He​​ is coadvising (with Kiwon​​​‌ Um, from Telecom Paris)​ three PhD candidates in​‌ this project.

Contrat de recherche​​​‌ Inria - SHOM

Participants:​ Steve Oudot.

• Title:​‌
  Traitement de nuage de​​ points bathyémtriques (SMF et​​​‌ Lidar) par l'approche apprentissage​ automatique
• Partner Institution(s):
  • Service​‌ Hydrographique et Océanographique de​​ la Marine (SHOM), Brest,​​​‌ France
• Date/Duration:
  2024-2028
• 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

Participants: Steve​‌ Oudot.

• Title:
  Precision​​ Medicine using Topology
• Partner​​​‌ Institution(s):
  • CRESS, Hôtel-Dieu, France​
• Date/Duration:
  2022-2025
• 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.​​

10.1.1 Scientific‌ events: organisation

10.1.2 Scientific events:‌ selection

10.1.3 Journal

10.1.4 Invited‌​‌ talks

• Steve Oudot Invited​​ speaker at the London-Oxford-Paris​​​‌ TDA seminar, University of‌ London, November 2025 (see‌​‌ https://sites.google.com/view/london-oxford-paris-seminar/).
• Steve Oudot​​ Invited speaker at the​​​‌ Math4AI/AI4Math Workshop, MPI (Leipzig),‌ March 2025 (see https://www.mis.mpg.de/events/series/math4ai-ai4math-workshop‌​‌).
• Steve Oudot Invited​​ speaker at the TDA​​​‌ week, University of Kyoto,‌ June 2025 (see https://sites.google.com/kyoto-u.ac.jp/tdaweek2025/home‌​‌).
• Steve Oudot speaker​​ at the AATRN online​​​‌ seminar, July 2025 (see‌ https://www.aatrn.net/seminar).
• Pooran Memari‌​‌ was invited as keynote​​ speaker at the thematic​​​‌ semester “Infinite-dimensional Geometry: Theory‌ and Applications” at the‌​‌ Erwin Schrödinger Institute (ESI​​ Wien), in Feb 2025,​​​‌ to deliver three mini-courses‌ on Geometric Modeling and‌​‌ Applications.
• Mathieu Desbrun was​​ invited as keynote speaker​​​‌ at the Workshop on‌ Geometry, Topology, and Machine‌​‌ Learning (GTML 2025) at​​ the MPI Leipzig in​​​‌ Germany, in November 2025.‌
• Mathieu Desbrun was invited‌​‌ as keynote speaker at​​ the Woudschoten Conference 2025​​​‌, the annual meeting‌ of the Dutch-Flemish Numerical‌​‌ Mathematics community.
• Mathieu Desbrun​​ was invited at the​​​‌ Institute for Mathematical and‌ Statistical Innovation in Chicago‌​‌ for a workshop on​​ Discrete Exterior Calculus.​​​‌
• Maks Ovsjanikov gave an‌ invited talk in the‌​‌ MBZUAI Workshop on the​​ Foundations and Advances in​​​‌ Generative AI held in‌ Paris.
• Maks Ovsjanikov gave‌​‌ a keynote talk in​​ the GDR IASIS workday​​​‌ on Deformable Object Modelling‌ Trends: from Perception to‌​‌ Applications.
• Maks Ovsjanikov gave​​ an invited talk in​​​‌ the CVPR 2025 workshop‌ on Enforcing Inductive Bias‌​‌ in 3D Generation.

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, for the academic​ year 2024-2025.
• Steve Oudot​‌ member of the Conseil​​ Académique of IP Paris,​​​‌ representing Inria, since 2024​
• Pooran Memari Member of​‌ the LIX laboratory council,​​ Ecole Polytechnique, since 2020​​​‌
• Mathieu Desbrun is on​ the Conseil de Direction​‌ for the LIX lab,​​ Ecole Polytechnique
• Mathieu Desbrun​​​‌ is the "Modeling, Simulation​ and Learning" pole leader​‌ for the LIX lab,​​ Ecole Polytechnique

10.2.1 Teaching

• Master:​​ Steve Oudot and Pooran​​​‌ Memari , Computational Geometry​ and Topology, 24h 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 , Geometric​​ Algorithms for Point Patterns​​​‌ and 2D embedded Structures,​ Across Applications in Visual​‌ Computing, Graduate Degree M2-​​ Artificial Intelligence and Advanced​​​‌ Visual Computing, Ecole Polytechnique​ (CSC-54445-EP), France;
• Master: Maks​‌ Ovsjanikov, Artificial Intelligence and​​ Advanced Visual Computing, École​​​‌ polytechnique, France;
• Undergrad-Master: Steve​ Oudot, Algorithms for data​‌ analysis in Python, 22.5h​​ eq-TD, L3/M1, École Polytechnique,​​​‌ France.
• Pooran Memari Bachelor:​ Computer Science Projects Coordinator,​‌ 3rt Year of Bachelor​​ Polytechnique, IP-Paris, since 2019.​​​‌

10.2.2 Supervision

• PhD in​ progress: Théo Prosper, Institut​‌ Polytechnique de Paris. Analysis​​ of multi-parameter Reeb spaces​​​‌ and Mappers. Started Oct.​ 2025. Steve Oudot and​‌ Mathieu Carrière (Inria Sophia-Antipolis).​​
• 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 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​‌ in progress: Nissim Maruani,​​ École Polytechnique, Palaiseau. Mathieu​​​‌ Desbrun.
• PhD in progress:​ Lucas Brifault, École Polytechnique,​‌ Palaiseau. Mathieu Desbrun.
• PhD​​ in progress: Souhail Hadgi,​​​‌ École Polytechnique, Palaiseau. Maks​ Ovsjanikov.
• PhD in progress:​‌ Leopold Maillard, Dassault Systèmes.​​ Maks Ovsjanikov.
• PhD in​​​‌ progress: Diego Gomez, École​ Polytechnique, Palaiseau. Maks Ovsjanikov.​‌
• PhD in progress: Erkan​​ Turan, École Polytechnique, Palaiseau.​​​‌ Maks Ovsjanikov.
• PhD in​ progress: Tamara Künzle, Sorbonne​‌ University, co-supervised by Maks​​ Ovsjanikov.
• PhD in progress:​​​‌ Julien Gaubil, supervised by​ Maks Ovsjanikov.
• PhD in​‌ progress: Aude Bouillé, Inria,​​ Computational Aspects of Generalized​​ Voronoi Diagrams, Pooran Memari​​​‌ and Mathieu Desbrun
• PhD‌ in progress: Emilien Ganier,‌​‌ Ecole Polytechnique, Geometric Proximity​​ graphs with Higher-Order Statistical​​​‌ Integration, Pooran Memari and‌ Steve Oudot.

10.2.3 Juries‌​‌

• Steve Oudot member of​​ the Commission Recherche of​​​‌ Institut Polytechnique de Paris,‌ in charge of evaluating‌​‌ applications to Habilitation à​​ Diriger des Recherches, since​​​‌ 2025.
• Steve Oudot president‌ of the hiring committee‌​‌ for 3 Assistant Professor​​ positions at the LIX​​​‌ (École Polytechnique), Spring 2025.‌
• Pooran Memari member of‌​‌ hiring comittee for Université​​ de Strasbourg (20/05/2025),
• Pooran​​​‌ Memari member of IP-Paris‌ IGD (Interaction, Graphics &‌​‌ Design) Master and PhD​​ Track Selection Committee, since​​​‌ 2021.
• Pooran Memari Member‌ of PhD Jury, Clément‌​‌ Poull, Université Bourgogne (11/12/2025).​​
• Pooran Memari Member of​​​‌ PhD Jury, Grégoire Grzeczkowicz‌ (IGN), IGN (13/01/2025).

10.3.1 Others science​​ outreach relevant activities

• Exhibition​​​‌ stand run by PhD‌ student Nasim Bagheri Shouraki‌​‌ , November 17–21, 2025,​​ at École Polytechnique, as​​​‌ part of the 29th‌ edition of the European‌​‌ Week for the Employment​​ of People with Disabilities​​​‌ (EWEPD), focused on our‌ project on the application‌​‌ of geometric tactile patterns​​ to accessibility, with particular​​​‌ emphasis on tactile maps‌ for navigation and education‌​‌ for blind and visually​​ impaired (BVI) people.
• Pooran​​​‌ Memari participated in the‌ middle school students work-experience‌​‌ week (stage de troisième)​​ at Inria, week of​​​‌ December 15th 2025.

International journals

• 17​‌ articleE.Elisa Bellantoni​​, F.Fabio Guglietta​​​‌, F.Francesca Pelusi​, M.Mathieu Desbrun​‌, K.Kiwon Um​​, M.Mihalis Nicolaou​​​‌, N.Nikos Savva​ and M.Mauro Sbragaglia​‌. Immersed boundary–lattice Boltzmann​​ mesoscale method for wetting​​​‌ problems.Physical Review​ E 1122August​‌ 2025, 025305 (1-15)​​HALDOIback to​​​‌ text
• 18 articleM.​ B.Magnus Bakke Botnan​‌, S.Steffen Oppermann​​ and S.Steve Oudot​​​‌. Signed Barcodes for​ Multi-parameter Persistence via Rank​‌ Decompositions and Rank-Exact Resolutions​​.Foundations of Computational​​​‌ Mathematics255October​ 2025, 1815-1874HAL​‌DOIback to text​​
• 19 articleT.Theo​​​‌ Braune, M.Mark​ Gillespie, Y.Yiying​‌ Tong and M.Mathieu​​ Desbrun. Discrete Torsion​​​‌ of Connection Forms on​ Simplicial Meshes.ACM​‌ Transactions on Graphics44​​4August 2025.​​ In press. HALDOI​​​‌back to text
• 20‌ articleJ.Jiong Chen‌​‌, F. T.Florian​​ T. Schäfer and M.​​​‌Mathieu Desbrun. Lightning-fast‌ Boundary Element Method.‌​‌ACM Transactions on Graphics​​444September 2025​​​‌. In press. HAL‌DOIback to text‌​‌
• 21 articleK.Khoa​​ Do, D.David​​​‌ Coeurjolly, P.Pooran‌ Memari and N.Nicolas‌​‌ Bonneel. Linear-Time Transport​​ with Rectified Flows.​​​‌ACM Transactions on Graphics‌444August 2025‌​‌HALback to text​​
• 22 articleA.Alex​​​‌ Fernandes, S.Steve‌ Oudot and F.François‌​‌ Petit. Computation of​​ gamma-linear projected barcodes for​​​‌ multiparameter persistence.Journal‌ of Applied and Computational‌​‌ Topology92May​​ 2025, 12HAL​​​‌DOIback to text‌
• 23 articleA.André‌​‌ Freitas, K.Kiwon​​ Um, M.Mathieu​​​‌ Desbrun, M.Michele‌ Buzzicotti and L.Luca‌​‌ Biferale. Solver-in-the-loop approach​​ to closure of shell​​​‌ models of turbulence.‌Physical Review Fluids10‌​‌4April 2025,​​ 044602HALDOIback​​​‌ to text
• 24 article‌E.Ellen Gasparovic,‌​‌ E.Elizabeth Munch,​​ S.Steve Oudot,​​​‌ K.Katharine Turner,‌ B.Bei Wang and‌​‌ Y.Yusu Wang.​​ Intrinsic Interleaving Distance for​​​‌ Merge Trees.La‌ Matematica41March‌​‌ 2025, 40-65HAL​​DOIback to text​​​‌
• 25 articleA.Antoine‌ Guédon, D.Diego‌​‌ Gomez, N.Nissim​​ Maruani, B.Bingchen​​​‌ Gong, G.George‌ Drettakis and M.Maks‌​‌ Ovsjanikov. MILo: Mesh-In-the-Loop​​ Gaussian Splatting for Detailed​​​‌ and Efficient Surface Reconstruction‌.ACM Transactions on‌​‌ GraphicsDecember 2025HAL​​back to text
• 26​​​‌ articleV.Vincent Mallet‌, C.Chiara Rapisarda‌​‌, H.Hervé Minoux​​ and M.Maks Ovsjanikov​​​‌. Finding antibodies in‌ cryo-EM maps with CrAI‌​‌.Bioinformatics415​​May 2025, btaf157​​​‌HALDOIback to‌ text
• 27 articleG.‌​‌Giulio Viganò, M.​​Maks Ovsjanikov and S.​​​‌Simone Melzi. NAM:‌ Neural Adjoint Maps for‌​‌ refining shape correspondences.​​ACM Transactions on Graphics​​​‌4460July 2025‌, 1-15HALDOI‌​‌back to text
• 28​​ articleH.Haoxiang Wang​​​‌, K.Kui Wu‌, H.Hui Qiao‌​‌, M.Mathieu Desbrun​​ and W.Wei Li​​​‌. Kinetic Free-Surface Flows‌ and Foams with Sharp‌​‌ Interfaces.ACM Transactions​​ on GraphicsDecember 2025​​​‌. In press. HAL‌back to text

International‌​‌ peer-reviewed conferences

• 29 inproceedings​​S.Souhaib Attaiki,​​​‌ P.Paul Guerrero,‌ D.Duygu Ceylan,‌​‌ N. J.Niloy J​​ Mitra and M.Maks​​​‌ Ovsjanikov. GANFusion: Feed-Forward‌ Text-to-3D with Diffusion in‌​‌ GAN Space.WACV​​ 2025 - IEEE/CVF Winter​​​‌ Conference on Applications of‌ Computer VisionTucson, United‌​‌ StatesFebruary 2025HAL​​DOIback to text​​​‌
• 30 inproceedingsL.Lucas‌ Brifault, D.David‌​‌ Cohen-Steiner and M.Mathieu​​ Desbrun. Efficient and​​​‌ Scalable Spatial Regularization of‌ Optimal Transport.SA‌​‌ Conference Papers '25: SIGGRAPH​​ Asia 2025 Conference Papers​​​‌SA Conference Papers '25:‌ SIGGRAPH Asia 2025 Conference‌​‌ PapersHong Kong Hong​​​‌ Kong, ChinaACMDecember​ 2025, 1-10HAL​‌DOIback to text​​back to text
• 31​​​‌ inproceedingsD.Diego Gomez​, B.Bingchen Gong​‌ and M.Maks Ovsjanikov​​. FourieRF: Few-Shot NeRFs​​​‌ via Progressive Fourier Frequency​ Control.3DV 2025​‌ - International Conference on​​ 3D VisionSingapore, Singapore​​​‌IEEEFebruary 2025,​ 607-615HALDOIback​‌ to text
• 32 inproceedings​​B.Bingchen Gong,​​​‌ D.Diego Gomez,​ A.Abdullah Hamdi,​‌ A.Abdelrahman Eldesokey,​​ A.Ahmed Abdelreheem,​​​‌ P.Peter Wonka and​ M.Maks Ovsjanikov.​‌ ZeroKey: Point-Level Reasoning and​​ Zero-Shot 3D Keypoint Detection​​​‌ from Large Language Models​.ICCV 2025 -​‌ International Conference on Computer​​ VisionHonolulu, United States​​​‌arXiv2025HALDOI​back to text
• 33​‌ inproceedingsS.Souhail Hadgi​​, L.Luca Moschella​​​‌, A.Andrea Santilli​, D.Diego Gomez​‌, Q.Qixing Huang​​, E.Emanuele Rodolà​​​‌, S.Simone Melzi​ and M.Maks Ovsjanikov​‌. Escaping Plato’s Cave:​​ Towards the Alignment of​​​‌ 3D and Text Latent​ Spaces.CVPR 2025​‌ - Conference on Computer​​ Vision and Pattern Recognition​​​‌Nashville, United StatesIEEE​June 2025, 19825-19835​‌HALDOIback to​​ text
• 34 inproceedingsL.​​​‌Léopold Maillard, T.​Tom Durand, A.​‌ R.Adrien Ramanana Rahary​​ and M.Maks Ovsjanikov​​​‌. LACONIC: A 3D​ Layout Adapter for Controllable​‌ Image Creation.International​​ Conference on Computer Vision,​​​‌ ICCV 2025Honolulu, United​ StatesarXiv2025HAL​‌DOIback to text​​
• 35 inproceedingsV.Vincent​​​‌ Mallet, S.Souhaib​ Attaiki, Y.Yangyang​‌ Miao, B.Bruno​​ Correia and M.Maks​​​‌ Ovsjanikov. AtomSurf :​ Surface Representation for Learning​‌ on Protein Structures.​​ICLR 2025 - The​​​‌ Thirteenth International Conference on​ Learning RepresentationsSingapore, Singapore​‌April 2025HALDOI​​back to text
• 36​​​‌ inproceedingsN.Nissim Maruani​, W.Wang Yifan​‌, M.Matthew Fisher​​, P.Pierre Alliez​​​‌ and M.Mathieu Desbrun​. ShapeShifter: 3D Variations​‌ Using Multiscale and Sparse​​ Point-Voxel Diffusion.CVPR​​​‌ 2025 - IEEE/CVF Conference​ on Computer Vision and​‌ Pattern RecognitionNashville (Tenessee),​​ United StatesJune 2025​​​‌HALback to text​
• 37 inproceedingsJ.Julie​‌ Mordacq, D.David​​ Loiseaux, V.Vicky​​​‌ Kalogeiton and S.Steve​ Oudot. T-REGS: Minimum​‌ Spanning Tree Regularization for​​ Self-Supervised Learning.Advances​​​‌ in Neural Information Processing​ Systems (NeurIPS)NeurIPS 2025​‌ - Annual Conference on​​ Neural Information Processing Systems​​​‌San Diego (CA), United​ StatesOctober 2025HAL​‌back to textback​​ to text
• 38 inproceedings​​​‌E.Emery Pierson,​ L.Lei Li,​‌ A.Angela Dai and​​ M.Maks Ovsjanikov.​​​‌ DiffuMatch: Category-Agnostic Spectral Diffusion​ Priors for Robust Non-rigid​‌ Shape Matching.ICCV​​ 2025 - International Conference​​​‌ on Computer VisionHonolulu,​ United StatesarXiv2025​‌HALDOIback to​​ text

Cited publications

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• 40 article​​G.Giancarlo Alfonsi.​​ Reynolds-averaged Navier--Stokes equations for​​​‌ turbulence modeling.Applied‌ Mechanics Reviews624‌​‌2009back to text​​
• 41 articleM. R.​​​‌Michael R. Allshouse and‌ J.-L.Jean-Luc Thiffeault.‌​‌ Detecting coherent structures using​​ braids.Physica D:​​​‌ Nonlinear Phenomena2412012‌, 95--105back to‌​‌ text
• 42 articleD.​​ N.Douglas N. Arnold​​​‌, R. S.Richard‌ S. Falk and R.‌​‌Ragnar Winther. Finite​​ element exterior calculus, homological​​​‌ techniques, and applications.‌Acta Numerica152006‌​‌, 1–155back to​​ text
• 43 inproceedingsM.​​​‌Mathieu Aubry, U.‌Ulrich Schlickewei and D.‌​‌Daniel Cremers. The​​ wave kernel signature: A​​​‌ quantum mechanical approach to‌ shape analysis.2011‌​‌ IEEE international conference on​​ computer vision workshops (ICCV​​​‌ workshops)IEEE2011,‌ 1626--1633back to text‌​‌
• 44 inproceedingsO.Omri​​ Azencot, M.Mirela​​​‌ Ben-Chen, F.Fédéric‌ Chazal and M.Maks‌​‌ Ovsjanikov. An operator​​ approach to tangent vector​​​‌ field processing.Computer‌ Graphics Forum325‌​‌Wiley Online Library2013​​, 73--82back to​​​‌ text
• 45 articleO.‌Omri Azencot, M.‌​‌Maks Ovsjanikov, F.​​Frédéric Chazal and M.​​​‌Mirela Ben-Chen. Discrete‌ derivatives of vector fields‌​‌ on surfaces--an operator approach​​.ACM Trans. Graph.​​​‌3432015,‌ 1--13back to text‌​‌
• 46 articleB.Bogdan​​ Batko, T.Tomasz​​​‌ Kaczynski, M.Marian‌ Mrozek and T.Thomas‌​‌ Wanner. Linking combinatorial​​ and classical dynamics: Conley​​​‌ index and Morse decompositions‌.Foundations of Computational‌​‌ Mathematics2052020​​, 967--1012back to​​​‌ text
• 47 articleU.‌Ulrich Bauer, M.‌​‌ B.Magnus B Botnan​​, S.Steffen Oppermann​​​‌ and J.Johan Steen‌. Cotorsion torsion triples‌​‌ and the representation theory​​ of filtered hierarchical clustering​​​‌.Advances in Mathematics‌3692020, 107171‌​‌back to text
• 48​​ articleA. I.Alexander​​​‌ I. Bobenko and B.‌ A.Boris A. Springborn‌​‌. Variational Principles for​​ Circle Patterns and Koebe's​​​‌ Theorem.Trans. Amer.‌ Math. Soc.3562003‌​‌, 659--689back to​​ text
• 49 articleJ.-D.​​​‌Jean-Daniel Boissonnat, R.‌Ramsay Dyer and A.‌​‌Arijit Ghosh. Delaunay​​ triangulation of manifolds.​​​‌Foundations of Computational Mathematics‌1822018,‌​‌ 399--431back to text​​
• 50 articleF.F.​​​‌ Brogi, O.O.‌ Malaspinas, B.B.‌​‌ Chopard and C.C.​​ Bonadonna. Hermite regularization​​​‌ of the lattice Boltzmann‌ method for open source‌​‌ computational aeroacoustics.Journal​​ of the Acoustical Society​​​‌ of America1424‌2017, 2332--2345back‌​‌ to text
• 51 article​​M. M.Michael M.​​​‌ Bronstein, J.Joan‌ Bruna, Y.Yann‌​‌ LeCun, A.Arthur​​ Szlam and P.Pierre​​​‌ Vandergheynst. Geometric Deep‌ Learning: Going beyond Euclidean‌​‌ data.IEEE Signal​​ Processing Magazine344​​​‌2017, 18-42DOI‌back to text
• 52‌​‌ inproceedingsR.Rickard Brüel​​ Gabrielsson and G.Gunnar​​​‌ Carlsson. Exposition and‌ Interpretation of the Topology‌​‌ of Neural Networks.​​2019 18th IEEE International​​​‌ Conference On Machine Learning‌ And Applications (ICMLA)2019‌​‌, 1069--1076back to​​​‌ text
• 53 articleM.​Marko Budiši ć and​‌ I.Igor Mezić.​​ Geometry of the ergodic​​​‌ quotient reveals coherent structures​ in flows.Physica​‌ D: Nonlinear Phenomena241​​152012, 1255--1269​​​‌back to text
• 54​ articleA.Alberto Cacciola​‌, A.Alessandro Muscoloni​​, V.Vaibhav Narula​​​‌, A.Alessandro Calamuneri​, S.Salvatore Nigro​‌, E. A.Emeran​​ A Mayer, J.​​​‌ S.Jennifer S Labus​, G.Giuseppe Anastasi​‌, A.Aldo Quattrone​​, A.Angelo Quartarone​​​‌ and others. Coalescent​ embedding in the hyperbolic​‌ space unsupervisedly discloses the​​ hidden geometry of the​​​‌ brain.arXiv preprint​ arXiv:1705.041922017back to​‌ text
• 55 articleG.​​Gunnar Carlsson and A.​​​‌Afra Zomorodian. The​ Theory of Multidimensional Persistence​‌.Discrete and Computational​​ Geometry4212009​​​‌, 71--93back to​ text
• 56 inproceedingsM.​‌Mathieu Carriere, F.​​Frédéric Chazal, M.​​​‌Marc Glisse, Y.​Yuichi Ike and H.​‌Hariprasad Kannan. Optimizing​​ persistent homology based functions​​​‌.Proc. International Conference​ on Machine Learning2021​‌back to text
• 57​​ inproceedingsM.Mathieu Carrière​​​‌, F.Frédéric Chazal​, Y.Yuichi Ike​‌, T.Théo Lacombe​​, M.Martin Royer​​​‌ and Y.Yuhei Umeda​. PersLay: a neural​‌ network layer for persistence​​ diagrams and new graph​​​‌ topological signatures.International​ Conference on Artificial Intelligence​‌ and Statistics (PMLR)2020​​, 2786--2796back to​​​‌ text
• 58 articleA.​ X.Angel X Chang​‌, T.Thomas Funkhouser​​, L.Leonidas Guibas​​​‌, P.Pat Hanrahan​, Q.Qixing Huang​‌, Z.Zimo Li​​, S.Silvio Savarese​​​‌, M.Manolis Savva​, S.Shuran Song​‌, H.Hao Su​​ and others. Shapenet:​​​‌ An information-rich 3d model​ repository.arXiv preprint​‌ arXiv:1512.030122015back to​​ text
• 59 articleF.​​​‌ d.Fré déric Chazal​, L. J.Leonidas​‌ J. Guibas, S.​​Steve Oudot and P.​​​‌Primoz Skraba. Persistence-Based​ Clustering in Riemannian Manifolds​‌.\bf Journal of​​ the ACM606​​​‌2013, 1--38URL:​ http://doi.acm.org/10.1145/2535927DOIback to​‌ textback to text​​
• 60 articleY.-C.Yen-Chi​​​‌ Chen, C. R.​Christopher R. Genovese and​‌ L.Larry Wasserman.​​ Statistical inference using the​​​‌ Morse-Smale complex.Electronic​ Journal of Statistics11​‌12017, 1390--1433​​back to text
• 61​​​‌ inproceedingsC.Chao Chen​, X.Xiuyan Ni​‌, Q.Qinxun Bai​​ and Y.Yusu Wang​​​‌. A topological regularizer​ for classifiers via persistent​‌ homology.The 22nd​​ International Conference on Artificial​​​‌ Intelligence and Statistics (PMLR)​2019, 2573--2582back​‌ to text
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