3D Gaussian Splatting (GS) is one of the most promising novel 3D representations that has received great interest in computer graphics and computer vision. While various systems have introduced editing capabilities for 3D GS, such as those guided by text prompts, fine-grained control over deformation remains an open challenge. In this work, we present a novel sketch-guided 3D GS deformation system that allows users to intuitively modify the geometry of a 3D GS model by drawing a silhouette sketch from a single view-point. Our approach introduces a new deformation method that combines cage-based deformations with a variant of Neural Jacobian Fields, enabling fine-grained control. Additionally, it leverages 2D diffusion priors and ControlNet to ensure the generated deformations are semantically plausible. Through a series of experiments, we demonstrate the effectiveness of our method and showcase its ability to animate static 3D GS models as one of its applications.

Abstract Cutout animation is one of the earliest forms of animation, and to this day remains a popular technique featured in numerous films including Monty Python and South Park series. Most computer animation systems, however, focus on different styles, including cel animation, making cutout animation somewhat underexplored. As creating cutouts is meticulous, we propose a novel generative cutout animation system. Taking a skeletal animation and a text prompt as input, we automatically generate a 2.5D cutout rig ready for production in films and games. Our system optimizes cutout images with an SDS (Score Distillation Sampling) loss with a LoRA (Low‐Rank Adaptation) prior, in multiple target poses. Naïvely optimizing an SDS loss, however, would lead to inconsistent target pose images, and, as a result, blurry or transparent cutouts. To address this, we introduce a novel optimization with techniques targeting pose and noise consistency, resulting in coherent target images and sharp cutouts. We validate our system by demonstrating a gallery of results, comparing with previous works, ablations, and other analyses. Once generated, our cutout rigs can be used both for the given input animation and repurposed for other animations or edited as independent assets.

This paper tackles the task of learning to generate signals over triangle meshes in a triangulation-agnostic manner, meaning the trained model can be applied to different meshes and triangulations effectively. Practically, the paper adapts the flow matching (FM) paradigm to a mesh-based, triangulation-agnostic setting. Theoretically, it proposes a specific noise distribution which is triangulation agnostic, to be used inside the FM model's denoising process. While noise distributions are usually trivial to devise for, e.g., images, devising a triangulation-agnostic distribution proves to be a much more difficult task. We formulate a mathematical definition of triangulation agnosticism of distributions, via their spectrum. We then show that a discretization of a specific Gaussian random field called a Matérn process holds these desired properties, and provides a simple and efficient sampling algorithm. We use it as our noise model, and adapt FM to the triangulation-agnostic setting by using a state-of-the-art approach for learning signals on meshes in the gradient domain—PoissonNet—as the denoiser. We conduct experiments on elaborate tasks such as sampling elastic rest states, and generating poses of humanoids. Our method is shown to be capable of producing highly realistic results for meshes of over one million triangles, significantly exceeding the state-of-the-art in quality and diversity.

We propose Mesh Up, a technique that deforms a 3D mesh towards multiple target concepts, and intuitively controls the region where each concept is expressed. Conveniently, the concepts can be defined as either text queries, e.g., “a dog” and “a turtle,” or inspirational images, and the local regions can be selected as any number of vertices on the mesh. We can effectively control the influence of the concepts and mix them together using a novel score distillation approach, referred to as the Blended Score Distillation (BSD). BSD operates on each attention layer of the denoising U-Net of a diffusion model as it extracts and injects the per-objective activations into a unified denoising pipeline from which the deformation gradients are calculated. To lo-calize the expression of these activations, we create a probabilistic Region of Interest (ROI) map on the surface of the mesh, and turn it into 3D-consistent masks that we use to control the expression of these activations. We demonstrate the effectiveness of BSD empirically and show that it can deform various meshes towards multiple objectives. Our project page is at https://threedle.github.io/meshup/.

We propose a generative technique to edit 3D shapes, represented as meshes, NeRFs, or Gaussian Splats, in ≈ 3 seconds, without the need for running an SDS type of optimization. Our key insight is to cast 3D editing as a multiview image inpainting problem, as this representation is generic and can be mapped back to any 3D representation using the bank of available Large Reconstruction Models. We explore different fine-tuning strategies to obtain both multiview generation and inpainting capabilities within the same diffusion model. In particular, the design of the inpainting mask is an important factor of training an inpainting model, and we propose several masking strategies to mimic the types of edits a user would perform on a 3D shape. Our approach takes 3D generative editing from hours to seconds and produces higher-quality results compared to previous works.

Differentiable optimization has attracted significant research interest, particularly for quadratic programming (QP). Existing approaches for differentiating the solution of a QP with respect to its defining parameters often rely on specific integrated solvers. This integration limits their applicability, including their use in neural network architectures and bi-level optimization tasks, restricting users to a narrow selection of solver choices. To address this limitation, we introduce **dQP**, a modular and solver-agnostic framework for plug-and-play differentiation of virtually any QP solver. A key insight we leverage to achieve modularity is that, once the active set of inequality constraints is known, both the solution and its derivative can be expressed using simplified linear systems that share the same matrix. This formulation fully decouples the computation of the QP solution from its differentiation. Building on this result, we provide a minimal-overhead, open-source implementation (<https://github.com/cwmagoon/dQP>) that seamlessly integrates with over 15 state-of-the-art solvers. Comprehensive benchmark experiments demonstrate dQP’s robustness and scalability, particularly highlighting its advantages in large-scale sparse problems.

We propose a method for unsupervised reconstruction of a temporally-consistent sequence of surfaces from a sequence of time-evolving point clouds. It yields dense and semantically meaningful correspondences between frames. We represent the reconstructed surfaces as atlases computed by a neural network, which enables us to establish correspondences between frames. The key to making these correspondences semantically meaningful is to guarantee that the metric tensors computed at corresponding points are as similar as possible. We have devised an optimization strategy that makes our method robust to noise and global motions, without a priori correspondences or pre-alignment steps. As a result, our approach outperforms state-of-the-art ones on several challenging datasets.

Many network architectures exist for learning on meshes, yet their constructions entail delicate trade-offs between difficulty learning high-frequency features, insufficient receptive field, sensitivity to discretization, and inefficient computational overhead. Drawing from classic local-global approaches in mesh processing, we introduce PoissonNet, a novel neural architecture that overcomes all of these deficiencies by formulating a local-global learning scheme, which uses Poisson's equation as the primary mechanism for feature propagation. Our core network block is simple; we apply learned local feature transformations in the gradient domain of the mesh, then solve a Poisson system to propagate scalar feature updates across the surface globally. Our local-global learning framework preserves the features's full frequency spectrum and provides a truly global receptive field, while remaining agnostic to mesh triangulation. Our construction is efficient, requiring far less compute overhead than comparable methods, which enables scalability—both in the size of our datasets, and the size of individual training samples. These qualities are validated on various experiments where, compared to previous intrinsic architectures, we attain state-of-the-art performance on semantic segmentation and parameterizing highly-detailed animated surfaces. Finally, as a central application of PoissonNet, we show its ability to learn deformations, significantly outperforming state-of-the-art architectures that learn on surfaces. https://github.com/ArmanMaesumi/poissonnet

This work proposes a novel representation of injective deformations of 3D space, which overcomes existing limi-tations of injective methods, namely inaccuracy, lack of ro-bustness, and incompatibility with general learning and op-timization frameworks. Our core idea is to reduce the prob-lem to a “deep” composition of multiple 2D mesh-based piecewise-linear maps. Namely, we build differentiable lay-ers that produce mesh deformations through Tutte's embed-ding (guaranteed to be injective in 2D), and compose these layers over different planes to create complex 3D injective deformations of the 3D volume. We show our method pro-vides the ability to efficiently and accurately optimize and learn complex deformations, outperforming other injective approaches. As a main application, we produce complex and artifact-free NeRF and SDF deformations.

Abstract We present an automated technique for computing a map between two genus‐zero shapes, which matches semantically corresponding regions to one another. Lack of annotated data prohibits direct inference of 3D semantic priors; instead, current state‐of‐the‐art methods predominantly optimize geometric properties or require varying amounts of manual annotation. To overcome the lack of annotated training data, we distill semantic matches from pre‐trained vision models: our method renders the pair of untextured 3D shapes from multiple viewpoints; the resulting renders are then fed into an off‐the‐shelf image‐matching strategy that leverages a pre‐trained visual model to produce feature points. This yields semantic correspondences, which are projected back to the 3D shapes, producing a raw matching that is inaccurate and inconsistent across different viewpoints. These correspondences are refined and distilled into an inter‐surface map by a dedicated optimization scheme, which promotes bijectivity and continuity of the output map. We illustrate that our approach can generate semantic surface‐to‐surface maps, eliminating manual annotations or any 3D training data requirement. Furthermore, it proves effective in scenarios with high semantic complexity, where objects are non‐isometrically related, as well as in situations where they are nearly isometric.

We present a neural technique for learning to select a local sub-region around a point which can be used for mesh parameterization. The motivation for our framework is driven by interactive workflows used for decaling, texturing, or painting on surfaces. Our key idea is to incorporate segmentation probabilities as weights of a classical parameterization method, implemented as a novel differentiable parameterization layer within a neural network framework. We train a segmentation network to select 3D regions that are parameterized into 2D and penalized by the resulting distortion, giving rise to segmentations which are distortion-aware. Following training, a user can use our system to interactively select a point on the mesh and obtain a large, meaningful region around the selection which induces a low-distortion parameterization. Our code11https://github.com/threedle/DA-Wand and project22https://threedle.github.io/DA-Wand/ are publicly available.

Exploring variations of 3D shapes is a time-consuming process in traditional 3D modeling tools. Deep generative models of 3D shapes often feature continuous latent spaces that can, in principle, be used to explore potential variations starting from a set of input shapes; in practice, doing so can be problematic—latent spaces are high dimensional and hard to visualize, contain shapes that are not relevant to the input shapes, and linear paths through them often lead to sub-optimal shape transitions. Furthermore, one would ideally be able to explore variations in the original high-quality meshes used to train the generative model, not its lower-quality output geometry. In this paper, we present a method to explore variations among a given set of landmark shapes by constructing a mapping from an easily-navigable 2D exploration space to a subspace of a pre-trained generative model. We first describe how to find a mapping that spans the set of input landmark shapes and exhibits smooth variations between them. We then show how to turn the variations in this subspace into deformation fields, to transfer those variations to high-quality meshes for the landmark shapes. Our results show that our method can produce visually-pleasing and easily-navigable 2D exploration spaces for several different shape categories, especially as compared to prior work on learning deformation spaces for 3D shapes.

This work is concerned with a representation of shapes that disentangles fine, local and possibly repeating geometry, from global, coarse structures. Achieving such disentanglement leads to two unrelated advantages: i) a significant compression in the number of parameters required to represent a given geometry; ii) the ability to manipulate either global geometry, or local details, without harming the other. At the core of our approach lies a novel pipeline and neural architecture, which are optimized to represent one specific atlas, representing one 3D surface. Our pipeline and architecture are designed so that disentanglement of global geometry from local details is accomplished through optimization, in a completely unsupervised manner. We show that this approach achieves better neural shape compression than the state of the art, as well as enabling manipulation and transfer of shape details. Project page http://geometry.cs.ucl.ac.uk/projects/2022/cnnmaps/.

We investigate the problem of training generative models on very sparse collections of 3D models. Particularly, instead of using difficult-to-obtain large sets of 3D models, we demonstrate that geometrically-motivated energy functions can be used to effectively augment and boost only a sparse collection of example (training) models. Technically, we analyze the Hessian of the as-rigid-as-possible (ARAP) energy to adaptively sample from and project to the underlying (local) shape space, and use the augmented dataset to train a variational autoencoder (VAE). We iterate the process, of building latent spaces of VAE and augmenting the associated dataset, to progressively reveal a richer and more expressive generative space for creating geometrically and semantically valid samples. We evaluate our method against a set of strong baselines, provide ablation studies, and demonstrate application towards establishing shape correspondences. Glassproduces multiple interesting and meaningful shape variations even when starting from as few as 3–10 training shapes. Our code is available at https://sanjeevmk.github.io/glass_webpage/.

Computing injective maps with low distortions is a long-standing problem in computer graphics. Such maps are particularly challenging to obtain in the presence of positional constraints, because an injective initial map is often not available. Recently, several energies were proposed and shown to be highly successful in optimizing injectivity from non-injective initial maps while satisfying positional constraints. However, minimizing these energies tends to produce elements with significant isometric distortions. This paper presents simple variants of these energies that retain their desirable traits while promoting isometry. While our method is not guaranteed to provide an injective map, we observe that, on large-scale 2D and 3D data sets, minimizing the proposed isometric variants results in a similar level of success in recovering injectivity as the original energies but a significantly lower isometric distortion.

This paper introduces a framework designed to accurately predict piecewise linear mappings of arbitrary meshes via a neural network, enabling training and evaluating over heterogeneous collections of meshes that do not share a triangulation, as well as producing highly detail-preserving maps whose accuracy exceeds current state of the art. The framework is based on reducing the neural aspect to a prediction of a matrix for a single given point, conditioned on a global shape descriptor. The field of matrices is then projected onto the tangent bundle of the given mesh, and used as candidate jacobians for the predicted map. The map is computed by a standard Poisson solve, implemented as a differentiable layer with cached pre-factorization for efficient training. This construction is agnostic to the triangulation of the input, thereby enabling applications on datasets with varying triangulations. At the same time, by operating in the intrinsic gradient domain of each individual mesh, it allows the framework to predict highly-accurate mappings. We validate these properties by conducting experiments over a broad range of scenarios, from semantic ones such as morphing, registration, and deformation transfer, to optimization-based ones, such as emulating elastic deformations and contact correction, as well as being the first work, to our knowledge, to tackle the task of learning to compute UV parameterizations of arbitrary meshes. The results exhibit the high accuracy of the method as well as its versatility, as it is readily applied to the above scenarios without any changes to the framework.

We introduce a deep generative network for 3D shape detailization, akin to stylization with the style being geometric details. We address the challenge of creating large varieties of high-resolution and detailed 3D geometry from a small set of exemplars by treating the problem as that of geometric detail transfer. Given a low-resolution coarse voxel shape, our network refines it, via voxel upsampling, into a higher-resolution shape enriched with geometric details. The output shape preserves the overall structure (or content) of the input, while its detail generation is conditioned on an input "style code" corresponding to a detailed exemplar. Our 3D detailization via conditional refinement is realized by a generative adversarial network, coined DECOR-GAN. The network utilizes a 3D CNN generator for upsampling coarse voxels and a 3D PatchGAN discriminator to enforce local patches of the generated model to be similar to those in the training detailed shapes. During testing, a style code is fed into the generator to condition the refinement. We demonstrate that our method can refine a coarse shape into a variety of detailed shapes with different styles. The generated results are evaluated in terms of content preservation, plausibility, and diversity. Comprehensive ablation studies are conducted to validate our network designs. Code is available at https://github.com/czq142857/DECOR-GAN.

Maps are arguably one of the most fundamental concepts used to define and operate on manifold surfaces in differentiable geometry. Accordingly, in geometry processing, maps are ubiquitous and are used in many core applications, such as paramterization, shape analysis, remeshing, and deformation. Unfortunately, most computational representations of surface maps do not lend themselves to manipulation and optimization, usually entailing hard, discrete problems. While algorithms exist to solve these problems, they are problem-specific, and a general framework for surface maps is still in need.In this paper, we advocate considering neural networks as encoding surface maps. Since neural networks can be composed on one another and are differentiable, we show it is easy to use them to define surfaces via atlases, compose them for surface-to-surface mappings, and optimize differentiable objectives relating to them, such as any notion of distortion, in a trivial manner. In our experiments, we represent surfaces by generating a neural map that approximates a UV parameterization of a 3D model. Then, we compose this map with other neural maps which we optimize with respect to distortion measures. We show that our formulation enables trivial optimization of rather elusive mapping tasks, such as maps between a collection of surfaces.

We present a method for reconstructing triangle meshes from point clouds. Existing learning-based methods for mesh reconstruction mostly generate triangles individually, making it hard to create manifold meshes. We leverage the properties of 2D Delaunay triangulations to construct a mesh from manifold surface elements. Our method first estimates local geodesic neighborhoods around each point. We then perform a 2D projection of these neighborhoods using a learned logarithmic map. A Delaunay triangulation in this 2D domain is guaranteed to produce a manifold patch, which we call a Delaunay surface element. We synchronize the local 2D projections of neighboring elements to maximize the manifoldness of the reconstructed mesh. Our results show that we achieve better overall manifoldness of our reconstructed meshes than current methods to reconstruct meshes with arbitrary topology. Our code, data and pretrained models can be found online: https://github.com/mrakotosaon/dse-meshing

We propose a novel technique for producing high-quality 3D models that match a given target object image or scan. Our method is based on retrieving an existing shape from a database of 3D models and then deforming its parts to match the target shape. Unlike previous approaches that independently focus on either shape retrieval or deformation, we propose a joint learning procedure that simultaneously trains the neural deformation module along with the embedding space used by the retrieval module. This enables our network to learn a deformation-aware embedding space, so that retrieved models are more amenable to match the target after an appropriate deformation. In fact, we use the embedding space to guide the shape pairs used to train the deformation module, so that it invests its capacity in learning deformations between meaningful shape pairs. Furthermore, our novel part-aware deformation module can work with inconsistent and diverse part-structures on the source shapes. We demonstrate the benefits of our joint training not only on our novel framework, but also on other state- of-the-art neural deformation modules proposed in recent years. Lastly, we also show that our jointly-trained method outperforms various non-joint baselines.

We propose a method for the unsupervised reconstruction of a temporally-coherent sequence of surfaces from a sequence of time-evolving point clouds, yielding dense, semantically meaningful correspondences between all keyframes. We represent the reconstructed surface as an atlas, using a neural network. Using canonical correspondences defined via the atlas, we encourage the reconstruction to be as isometric as possible across frames, leading to semantically-meaningful reconstruction. Through experiments and comparisons, we empirically show that our method achieves results that exceed that state of the art in the accuracy of unsupervised correspondences and accuracy of surface reconstruction.

Injective parameterizations of triangulated meshes are critical across applications but remain challenging to compute. Existing algorithms to find injectivity either require initialization from an injective starting state, which is currently only possible without positional constraints, or else can only prevent triangle inversion, which is insufficient to ensure injectivity. Here we present, to our knowledge, the first algorithm for recovering a globally injective parameterization from an arbitrary non-injective initial mesh subject to stationary constraints. These initial meshes can be inverted, wound about interior vertices and/or overlapping. Our algorithm in turn enables globally injective mapping for meshes with arbitrary positional constraints. Our key contribution is a new energy, called smooth excess area (SEA), that measures non-injectivity in a map. This energy is well-defined across both injective and non-injective maps and is smooth almost everywhere, making it readily minimizable using standard gradient-based solvers starting from a non-injective initial state. Importantly, we show that maps minimizing SEA are guaranteed to be locally injective and almost globally injective, in the sense that the overlapping area can be made arbitrarily small. Analyzing SEA's behavior over a new benchmark set designed to test injective mapping, we find that optimizing SEA successfully recovers globally injective maps for 85% of the benchmark and obtains locally injective maps for 90%. In contrast, state-of-the-art methods for removing triangle inversion obtain locally injective maps for less than 6% of the benchmark, and achieve global injectivity (largely by chance as prior methods are not designed to recover it) on less than 4%.

Triangle meshes remain the most popular data representation for surface geometry. This ubiquitous representation is essentially a hybrid one that decouples continuous vertex locations from the discrete topological triangulation. Unfortunately, the combinatorial nature of the triangulation prevents taking derivatives over the space of possible meshings of any given surface. As a result, to date, mesh processing and optimization techniques have been unable to truly take advantage of modular gradient descent components of modern optimization frameworks. In this work, we present a differentiable surface triangulation that enables optimization for any per-vertex or per-face differentiable objective function over the space of underlying surface triangulations. Our method builds on the result that any 2D triangulation can be achieved by a suitably perturbed weighted Delaunay triangulation. We translate this result into a computational algorithm by proposing a soft relaxation of the classical weighted Delaunay triangulation and optimizing over vertex weights and vertex locations. We extend the algorithm to 3D by decomposing shapes into developable sets and differentiably meshing each set with suitable boundary constraints. We demonstrate the efficacy of our method on various planar and surface meshes on a range of difficult-to-optimize objective functions. Our code can be found online: https://github.com/mrakotosaon/diff-surface-triangulation.

Given a solid 3D shape and a trajectory of it over time, we compute its swept volume - the union of all points contained within the shape at some moment in time. We consider the representation of the input and output as implicit functions, and lift the problem to 4D spacetime, where we show the problem gains a continuous structure which avoids expensive global searches. We exploit this structure via a continuation method which marches and reconstructs the zero level set of the swept volume, using the temporal dimension to avoid erroneous solutions. We show that, compared to other methods, our approach is not restricted to a limited class of shapes or trajectories, is extremely robust, and its asymptotic complexity is an order lower than standards used in the industry, enabling its use in applications such as modeling, constructive solid geometry, and path planning.

We propose a novel learnable representation for detail preserving shape deformation. The goal of our method is to warp a source shape to match the general structure of a target shape, while preserving the surface details of the source. Our method extends a traditional cage-based deformation technique, where the source shape is enclosed by a coarse control mesh termed cage, and translations prescribed on the cage vertices are interpolated to any point on the source mesh via special weight functions. The use of this sparse cage scaffolding enables preserving surface details regardless of the shape's intricacy and topology. Our key contribution is a novel neural network architecture for predicting deformations by controlling the cage. We incorporate a differentiable cage-based deformation module in our architecture, and train our network end-to-end. Our method can be trained with common collections of 3D models in an unsupervised fashion, without any cage-specific annotations. We demonstrate the utility of our method for synthesizing shape variations and deformation transfer.

Mapping a source mesh into a target domain while preserving local injectivity is an important but highly non-trivial task. Existing methods either require an already-injective starting configuration, which is often not available, or rely on sophisticated solving schemes. We propose a novel energy form, called Total Lifted Content (TLC), that is equipped with theoretical properties desirable for injectivity optimization. By lifting the simplices of the mesh into a higher dimension and measuring their contents (2D area or 3D volume) there, TLC is smooth over the entire embedding space and its global minima are always injective. The energy is simple to minimize using standard gradient-based solvers. Our method achieved 100% success rate on an extensive benchmark of embedding problems for triangular and tetrahedral meshes, on which existing methods only have varied success.

This paper introduces Neural Subdivision , a novel framework for data-driven coarse-to-fine geometry modeling. During inference, our method takes a coarse triangle mesh as input and recursively subdivides it to a finer geometry by applying the fixed topological updates of Loop Subdivision, but predicting vertex positions using a neural network conditioned on the local geometry of a patch. This approach enables us to learn complex non-linear subdivision schemes, beyond simple linear averaging used in classical techniques. One of our key contributions is a novel self-supervised training setup that only requires a set of high-resolution meshes for learning network weights. For any training shape, we stochastically generate diverse low-resolution discretizations of coarse counterparts, while maintaining a bijective mapping that prescribes the exact target position of every new vertex during the subdivision process. This leads to a very efficient and accurate loss function for conditional mesh generation, and enables us to train a method that generalizes across discretizations and favors preserving the manifold structure of the output. During training we optimize for the same set of network weights across all local mesh patches, thus providing an architecture that is not constrained to a specific input mesh, fixed genus, or category. Our network encodes patch geometry in a local frame in a rotation- and translation-invariant manner. Jointly, these design choices enable our method to generalize well, and we demonstrate that even when trained on a single high-resolution mesh our method generates reasonable subdivisions for novel shapes.

This work concerns the computation and approximation of developable surfaces --- surfaces that are locally isometric to the two-dimensional plane. These surfaces are heavily studied in differential geometry, and are also of great interest to fabrication, architecture and fashion. We focus specifically on developability of heightfields. Our main observation is that developability can be cast as a rank constraint, which can then be plugged into theoretically-grounded rank-minimization techniques from the field of compressed sensing. This leads to a convex semidefinite optimization problem, which receives an input heightfield and recovers a similar heightfield which is developable. Due to the sparsifying nature of compressed sensing, the recovered surface is piecewise developable, with creases emerging between connected developable pieces. The convex program includes one user-specified parameter, balancing adherence to the original surface with developability and number of patches. We moreover show, that in contrast to previous techniques, our discretization does not introduce a bias and the same results are achieved across resolutions and orientations, and with no limit on the number of creases and patches. We solve this convex semidefinite optimization problem efficiently, by devising a tailor-made ADMM solver which leverages matrix-projection observations unique to our problem. We employ our method on a plethora of experiments, from denoising 3D scans of developable geometry such as documents and buildings, through approximating general heightfields with developable ones, and up to interpolating sparse annotations with a developable heightfield.

This work presents an algorithm for injectively parameterizing surfaces into spherical target domains called spherical orbifolds. Spherical orbifolds are cone surfaces that are generated from symmetry groups of the sphere. The surface is mapped the spherical orbifold via an extension of Tutte's embedding. This embedding is proven to be bijective under mild additional assumptions, which hold in all experiments performed. This work also completes the adaptation of Tutte's embedding to orbifolds of the three classic geometries - Euclidean, hyperbolic and spherical - where the first two were recently addressed. The spherical orbifold embeddings approximate conformal maps and require relatively low computational times. The constant positive curvature of the spherical orbifolds, along with the flexibility of their cone angles, enables producing embeddings with lower isometric distortion compared to their Euclidean counterparts, a fact that makes spherical orbifolds a natural candidate for surface parameterization.

The recent success of convolutional neural networks (CNNs) for image processing tasks is inspiring research efforts attempting to achieve similar success for geometric tasks. One of the main challenges in applying CNNs to surfaces is defining a natural convolution operator on surfaces. In this paper we present a method for applying deep learning to sphere-type shapes using a global seamless parameterization to a planar flat-torus, for which the convolution operator is well defined. As a result, the standard deep learning framework can be readily applied for learning semantic, high-level properties of the shape. An indication of our success in bridging the gap between images and surfaces is the fact that our algorithm succeeds in learning semantic information from an input of raw low-dimensional feature vectors. We demonstrate the usefulness of our approach by presenting two applications: human body segmentation, and automatic landmark detection on anatomical surfaces. We show that our algorithm compares favorably with competing geometric deep-learning algorithms for segmentation tasks, and is able to produce meaningful correspondences on anatomical surfaces where hand-crafted features are bound to fail.

Tutte's embedding is one of the most popular approaches for computing parameterizations of surface meshes in computer graphics and geometry processing. Its popularity can be attributed to its simplicity, the guaranteed bijectivity of the embedding, and its relation to continuous harmonic mappings. In this work we extend Tutte's embedding into hyperbolic cone-surfaces called orbifolds. Hyperbolic orbifolds are simple surfaces exhibiting different topologies and cone singularities and therefore provide a flexible and useful family of target domains. The hyperbolic Orbifold Tutte embedding is defined as a critical point of a Dirichlet energy with special boundary constraints and is proved to be bijective, while also satisfying a set of points-constraints. An efficient algorithm for computing these embeddings is developed. We demonstrate a powerful application of the hyperbolic Tutte embedding for computing a consistent set of bijective, seamless maps between all pairs in a collection of shapes, interpolating a set of user-prescribed landmarks, in a fast and robust manner.

Injective parameterizations of surface meshes are vital for many applications in Computer Graphics, Geometry Processing and related fields. Tutte's embedding, and its generalization to convex combination maps, are among the most popular approaches for computing parameterizations of surface meshes into the plane, as they guarantee injectivity, and their computation only requires solving a sparse linear system. However, they are only applicable to disk-type and toric surface meshes. In this paper we suggest a generalization of Tutte's embedding to other surface topologies, and in particular the common, yet untreated case, of sphere-type surfaces. The basic idea is to enforce certain boundary conditions on the parameterization so as to achieve a Euclidean orbifold structure. The orbifold-Tutte embedding is a seamless, globally bijective parameterization that, similarly to the classic Tutte embedding, only requires solving a sparse linear system for its computation. In case the cotangent weights are used, the orbifold-Tutte embedding globally minimizes the Dirichlet energy and is shown to approximate conformal and four-point quasiconformal mappings. As far as we are aware, this is the first fully-linear method that produces bijective approximations to conformal mappings. Aside from parameterizations, the orbifold-Tutte embedding can be used to generate bijective inter-surface mappings with three or four landmarks and symmetric patterns on sphere-type surfaces.

We introduce a method for computing seamless bijective mappings between two surface-meshes that interpolates a given set of correspondences. A common approach for computing a map between surfaces is to cut the surfaces to disks, flatten them to the plane, and extract the mapping from the flattenings by composing one flattening with the inverse of the other. So far, a significant drawback in this class of techniques is that the choice of cuts introduces a bias in the computation of the map that often causes visible artifacts and wrong correspondences. In this paper we develop a surface mapping technique that is indifferent to the particular cut choice. This is achieved by a novel type of surface flattenings that encodes this cut-invariance, and when optimized with a suitable energy functional results in a seamless surface-to-surface map. We show the algorithm enables producing high-quality seamless bijective maps for pairs of surfaces with a wide range of shape variability and from a small number of prescribed correspondences. We also used this framework to produce three-way, consistent and seamless mappings for triplets of surfaces.

We propose an efficient algorithm for computing large-scale bounded distortion maps of triangular and tetrahedral meshes. Specifically, given an initial map, we compute a similar map whose differentials are orientation preserving and have bounded condition number. Inspired by alternating optimization and Gauss-Newton approaches, we devise a first order method which combines the advantages of both. On the one hand, its iterations are as computationally efficient as those of alternating optimization. On the other hand, it enjoys preferable convergence properties, associated with Gauss-Newton like approaches. We demonstrate the utility of the proposed approach in efficiently solving geometry processing problems, focusing on challenging large-scale problems.

This paper introduces an algorithm for computing low-distortion, bijective mappings between surface meshes. The algorithm recieves as input a coarse set of corresponding pairs of points on the two surfaces, and follows three steps: (i) cutting the two meshes to disks in a consistent manner; (ii) jointly flattening the two disks via a novel formulation for minimizing isometric distortion while guaranteeing local injectivity (the flattenings can overlap, however); and (iii) computing a unique continuous bijection that is consistent with the flattenings. The construction of the algorithm stems from two novel observations: first, bijections between disk-type surfaces can be uniquely and efficiently represented via consistent locally injective flattenings that are allowed to be globally overlapping. This observation reduces the problem of computing bijective surface mappings to the task of computing locally injective flattenings, which is shown to be easier. Second, locally injective flattenings that minimize isometric distortion can be efficiently characterized and optimized in a convex framework. Experiments that map a wide baseline of pairs of surface meshes using the algorithm are provided. They demonstrate the ability of the algorithm to produce high-quality continuous bijective mappings between pairs of surfaces of varying isometric distortion levels.

Controlling the singular values of n -dimensional matrices is often required in geometric algorithms in graphics and engineering. This paper introduces a convex framework for problems that involve singular values. Specifically, it enables the optimization of functionals and constraints expressed in terms of the extremal singular values of matrices. Towards this end, we introduce a family of convex sets of matrices whose singular values are bounded. These sets are formulated using Linear Matrix Inequalities (LMI), allowing optimization with standard convex Semidefinite Programming (SDP) solvers. We further show that these sets are optimal, in the sense that there exist no larger convex sets that bound singular values. A number of geometry processing problems are naturally described in terms of singular values. We employ the proposed framework to optimize and improve upon standard approaches. We experiment with this new framework in several applications: volumetric mesh deformations, extremal quasi-conformal mappings in three dimensions, non-rigid shape registration and averaging of rotations. We show that in all applications the proposed approach leads to algorithms that compare favorably to state-of-art algorithms.

We introduce an efficient algorithm for producing provably injective mappings of tetrahedral meshes with strict bounds on their tetrahedra aspect-ratio distortion. The algorithm takes as input a simplicial map (e.g., produced by some common deformation or volumetric parameterization technique) andprojectsit on the space of injective and bounded-distortion simplicial maps. Namely, finds a similar map that is both bijective and bounded-distortion. As far as we are aware, this is the first algorithm to produce injective or bounded-distortion simplicial maps of tetrahedral meshes. The construction of the algorithm was made possible due to a novel closed-form solution to the problem of finding the closest orientation-preserving bounded-distortion matrix to an arbitrary matrix in three (and higher) dimensions. The algorithm is shown to have quadratic convergence, usually not requiring more than a handful of iterations to converge. Furthermore, it is readily generalized to simplicial maps of any dimension, including mixed dimensions. Finally, it can deal with different distortion spaces, such as bounded isometric distortion. During experiments we found the algorithm useful for producing bijective and bounded-distortion volume parameterizations and deformations of tetrahedral meshes, and improving tetrahedral meshes, increasing the tetrahedra quality produced by state-of-the-art techniques.