Eigenpair extractions are crucial for various applications in geometry processing and graphics. State of the Art libraries like ARPACK or Spectra rely on the implicitly restarted Lanczos iteration to extract eigenpairs efficiently. However for some large scale problems they lack convergence speed and robustness. In this paper we present a simple multigrid extension to accelerate the convergence and robustness of the implicitly restarted Lanczos method, and we demonstrate the efficiency of our method on a variety of problems commonly found in geometry processing and graphics.

Geometric features between the micro and macro scales produce an expressive family of visual effects grouped under the term "glints". Efficiently rendering these effects amounts to finding the highlights caused by the geometry under each pixel. To allow for fast rendering, we represent our faceted geometry as a 4D point process on an implicit multiscale grid, designed to efficiently find the facets most likely to cause a highlight. The facets' normals are generated to match a given micro-facet normal distribution such as Trowbridge-Reitz (GGX) or Beckmann, to which our model converges under increasing surface area. Our method is simple to implement, memory-and-precomputation-free, allows for importance sampling and covers a wide range of different appearances such as anisotropic as well as individually colored particles. We provide a base implementation as a standalone fragment shader.

We present closed-form expressions for Green and biharmonic coordinates with respect to polynomial curved 2D cages, enabling reliable cage-based image deformation both to and from a curved cage. We further provide closed-form expressions for first- and second-order derivatives of these coordinates with respect to the encoded position. This enables the use of variational solvers for interacting with the 2D shape at arbitrary points while keeping the fast decoding strength of cage-based deformation, which we illustrate for a variety of elastic deformation energies.

Abstract We propose MatUp, an upsampling filter for material super‐resolution. Our method takes as input a low‐resolution SVBRDF and upscales its maps so that their rendering under various lighting conditions fits upsampled renderings inferred in the radiance domain with pre‐trained RGB upsamplers. We formulate our local filter as a compact Multilayer Perceptron (MLP), which acts on a small window of the input SVBRDF and is optimized using a data‐fitting loss defined over upsampled radiance at various locations. This optimization is entirely performed at the scale of a single, independent material. Doing so, MatUpleverages the reconstruction capabilities acquired over large collections of natural images by pre‐trained RGB models and provides regularization over self‐similar structures. In particular, our light‐weight neural filter avoids retraining complex architectures from scratch or accessing any large collection of low/high resolution material pairs – which do not actually exist at the scale RGB upsamplers are trained with. As a result, MatUpprovides fine and coherent details in the upscaled material maps, as shown in the extensive evaluation we provide.

Abstract Interactive deformation via control handles is essential in computer graphics for the modeling of 3D geometry. Deformation control structures include lattices for free‐form deformation and skeletons for character articulation, but this report focuses on cage‐based deformation. Cages for deformation control are coarse polygonal meshes that encase the to‐be‐deformed geometry, enabling high‐resolution deformation. Cage‐based deformation enables users to quickly manipulate 3D geometry by deforming the cage. Due to their utility, cage‐based deformation techniques increasingly appear in many geometry modeling applications. For this reason, the computer graphics community has invested a great deal of effort in the past decade and beyond into improving automatic cage generation and cage‐based deformation. Recent advances have significantly extended the practical capabilities of cage‐based deformation methods. As a result, there is a large body of research on cage‐based deformation. In this report, we provide a comprehensive overview of the current state of the art in cage‐based deformation of 3D geometry. We discuss current methods in terms of deformation quality, practicality, and precomputation demands. In addition, we highlight potential future research directions that overcome current issues and extend the set of practical applications. In conjunction with this survey, we publish an application to unify the most relevant deformation methods. Our report is intended for computer graphics researchers, developers of interactive geometry modeling applications, and 3D modeling and character animation artists.

As a natural extension to the harmonic coordinates, the biharmonic coordinates have been found superior for planar shape and image manipulation with an enriched deformation space. However, the 3D biharmonic coordinates and their derivatives have remained unexplored. In this work, we derive closed-form expressions for biharmonic coordinates and their derivatives for 3D triangular cages. The core of our derivation lies in computing the closed-form expressions for the integral of the Euclidean distance over a triangle and its derivatives. The derived 3D biharmonic coordinates not only fill a missing component in methods of generalized barycentric coordinates but also pave the way for various interesting applications in practice, including producing a family of biharmonic deformations, solving variational shape deformations, and even unlocking the closed-form expressions for recently-introduced Somigliana coordinates for both fast and accurate evaluations.

We introduce Green coordinates for triquad cages in 3D. Based on Green’s third identity, Green coordinates allow defining the harmonic deformation of a 3D point inside a cage as a linear combination of its vertices and face normals. Using appropriate Neumann boundary conditions, the resulting deformations are quasi-conformal in 3D, and thus best-preserve the local deformed geometry, in that volumetric conformal 3D deformations do not exist unless rigid. Most coordinate systems use cages made of triangles, yet quads are in general favored by artists as those align naturally onto important geometric features of the 3D shapes, such as the limbs of a character, without introducing arbitrary asymmetric deformations and representation. While triangle cages admit per-face constant normals and result in a single Green normal-coordinate per triangle, the case of quad cages is at the same time more involved (as the normal varies along non-planar quads) and more flexible (as many different mathematical models allow defining the smooth geometry of a quad interpolating its four edges). We consider bilinear quads, and we introduce a new Neumann boundary condition resulting in a simple set of four additional normal-coordinates per quad. Our coordinates remain quasi-conformal in 3D, and we demonstrate their superior behavior under non-trivial deformations of realistic triquad cages.

We present MIPNet, a novel approach for SVBRDF mipmapping which preserves material appearance under varying view distances and lighting conditions. As in classical mipmapping, our method explicitly encodes the multiscale appearance of materials in a SVBRDF mipmap pyramid. To do so, we use a tensor-based representation, coping with gradient-based optimization, for encoding anisotropy which is compatible with existing real-time rendering engines. Instead of relying on a simple texture patch average for each channel independently, we propose a cascaded architecture of multilayer perceptrons to approximate the material appearance using only the fixed material channels. Our neural model learns simple mipmapping filters using a differentiable rendering pipeline based on a rendering loss and is able to transfer signal from normal to anisotropic roughness. As a result, we obtain a drop-in replacement for standard material mipmapping, offering a significant improvement in appearance preservation while still boiling down to a single per-pixel mipmap texture fetch. We report extensive experiments on two distinct BRDF models.

We present a simple, fast, and smooth scheme to approximate Algebraic Point Set Surfaces using non-compact kernels, which is particularly suited for filtering and reconstructing point sets presenting large missing parts. Our key idea is to consider a moving level-of-detail of the input point set which is adaptive w.r.t. to the evaluation location, just such as the samples weights are output sensitive in the traditional moving least squares scheme. We also introduce an adaptive progressive octree refinement scheme, driven by the resulting implicit surface, to properly capture the modeled geometry even far away from the input samples. Similarly to typical compactly-supported approximations, our operator runs in logarithmic time while defining high quality surfaces even on challenging inputs for which only global optimizations achieve reasonable results. We demonstrate our technique on a variety of point sets featuring geometric noise as well as large holes.

Abstract Many global implicit surface reconstruction algorithms formulate the problem as a volumetric energy minimization, trading data fitting for geometric regularization. As a result, the output surfaces may be located arbitrarily far away from the input samples. This is amplified when considering i) strong regularization terms, ii) sparsely distributed samples or iii) missing data. This breaks the strong assumption commonly used by popular octree‐based and triangulation‐based approaches that the output surface should be located near the input samples. As these approaches refine during a pre‐process, their cells near the input samples, the implicit solver deals with a domain discretization not fully adapted to the final isosurface. We relax this assumption and propose a progressive coarse‐to‐fine approach that jointly refines the implicit function and its representation domain, through iterating solver, optimization and refinement steps applied to a 3D Delaunay triangulation. There are several advantages to this approach: the discretized domain is adapted near the isosurface and optimized to improve both the solver conditioning and the quality of the output surface mesh contoured via marching tetrahedra.

Abstract Skeleton‐based and cage‐based deformation techniques represent the two most popular approaches to control real‐time deformations of digital shapes and are, to a vast extent, complementary to one another. Despite their complementary roles, high‐end modelling packages do not allow for seamless integration of such control structures, thus inducing a considerable burden on the user to maintain them synchronized. In this paper, we propose a framework that seamlessly combines rigging skeletons and deformation cages, granting artists with a real‐time deformation system that operates using any smooth combination of the two approaches. By coupling the deformation spaces of cages and skeletons, we access a much larger space, containing poses that are impossible to obtain by acting solely on a skeleton or a cage. Our method is oblivious to the specific techniques used to perform skinning and cage‐based deformation, securing it compatible with pre‐existing tools. We demonstrate the usefulness of our hybrid approach on a variety of examples.

Abstract The spectrum of the Laplace‐Beltrami operator is instrumental for a number of geometric modeling applications, from processing to analysis. Recently, multiple methods were developed to retrieve an approximation of a shape that preserves its eigenvectors as much as possible, but these techniques output a subset of input points with no connectivity, which limits their potential applications. Furthermore, the obtained Laplacian results from an optimization procedure, implying its storage alongside the selected points. Focusing on keeping a mesh instead of an operator would allow to retrieve the latter using the standard cotangent formulation, enabling easier processing afterwards. Instead, we propose to simplify the input mesh using a spectrum‐preserving mesh decimation scheme, so that the Laplacian computed on the simplified mesh is spectrally close to the one of the input mesh. We illustrate the benefit of our approach for quickly approximating spectral distances and functional maps on low resolution proxies of potentially high resolution input meshes.

Abstract Modern 3D capture pipelines produce dense surface meshes at high speed, which challenge geometric operators to process such massive data on‐the‐fly. In particular, aiming at instantaneous feature‐preserving smoothing and clustering disqualifies global variational optimizers and one usually relies on high‐performance parallel kernels based on simple measures performed on the positions and normal vectors associated with the surface vertices. Although these operators are effective on small supports, they fail at properly capturing larger scale surface structures. To cope with this problem, we propose to enrich the surface representation with filtered quadrics, a compact and discriminating range space to guide processing. Compared to normal‐based approaches, this additional vertex attribute significantly improves feature preservation for fast bilateral filtering and mode‐seeking clustering, while exhibiting a linear memory cost in the number of vertices and retaining the simplicity of convolutional filters. In particular, the overall performance of our approach stems from its natural compatibility with modern fine‐grained parallel computing architectures such as graphics processor units (GPU). As a result, filtered quadrics offer a superior ability to handle a broad spectrum of frequencies and preserve large salient structures, delivering meshes on‐the‐fly for interactive and streaming applications, as well as quickly processing large data collections, instrumental in learning‐based geometry analysis.

Abstract Dictionaries are very useful objects for data analysis, as they enable a compact representation of large sets of objects through the combination of atoms. Dictionary‐based techniques have also particularly benefited from the recent advances in machine learning, which has allowed for data‐driven algorithms to take advantage of the redundancy in the input dataset and discover relations between objects without human supervision or hard‐coded rules. Despite the success of dictionary‐based techniques on a wide range of tasks in geometric modeling and geometry processing, the literature is missing a principled state‐of‐the‐art of the current knowledge in this field. To fill this gap, we provide in this survey an overview of data‐driven dictionary‐based methods in geometric modeling. We structure our discussion by application domain: surface reconstruction, compression, and synthesis. Contrary to previous surveys, we place special emphasis on dictionary‐based methods suitable for 3D data synthesis, with applications in geometric modeling and design. Our ultimate goal is to enlight the fact that these techniques can be used to combine the data‐driven paradigm with design intent to synthesize new plausible objects with minimal human intervention. This is the main motivation to restrict the scope of the present survey to techniques handling point clouds and meshes, making use of dictionaries whose definition depends on the input data, and enabling shape reconstruction or synthesis through the combination of atoms.

Abstract We present a fast, robust and high‐quality technique to skin a mesh with reference to a skeleton. We consider the space of possible skeleton deformations (based on skeletal constraints, or skeletal animations), and compute skinning weights based on an optimization scheme to obtain as‐rigid‐as‐possible (ARAP) corresponding mesh deformations. We support stretchable‐and‐twistable bones (STBs) and spines by generalizing the ARAP deformations to stretchable deformers. In addition, our approach can optimize joint placements. If wanted, a user can guide and interact with the results, which is facilitated by an interactive feedback, reached via an efficient sparsification scheme. We demonstrate our technique on challenging inputs (STBs and spines, triangle and tetrahedral meshes featuring missing elements, boundaries, self‐intersections or wire edges).

Space coordinates offer an elegant, scalable and versatile framework to propagate (multi-)scalar functions from the boundary vertices of a 3-manifold, often called a cage , within its volume. These generalizations of the barycentric coordinate system have progressively expanded the range of eligible cages to triangle and planar polygon surface meshes with arbitrary topology, concave regions and a spatially-varying sampling ratio, while preserving a smooth diffusion of the prescribed on-surface functions. In spite of their potential for major computer graphics applications such as freeform deformation or volume texturing, current space coordinate systems have only found a moderate impact in applications. This follows from the constraint of having only triangles in the cage most of the time, while many application scenarios favor arbitrary (non-planar) quad meshes for their ability to align the surface structure with features and to naturally cope with anisotropic sampling. In order to use space coordinates with arbitrary quad cages currently, one must triangulate them, which results in large propagation distortion. Instead, we propose a generalization of a popular coordinate system - Mean Value Coordinates - to quad and tri-quad cages, bridging the gap between high-quality coarse meshing and volume diffusion through space coordinates. Our method can process non-planar quads, comes with a closed-form solution free from global optimization and reproduces the expected behavior of Mean Value Coordinates, namely smoothness within the cage volume and continuity everywhere. As a result, we show how these coordinates compare favorably to classical space coordinates on triangulated quad cages, in particular for freeform deformation.

We propose a novel framework for photometric stereo (PS) under low-light conditions using uncalibrated near-light illumination. It operates on free-form video sequences captured with a minimalistic and affordable setup. We address issues such as albedo variations, shadowing, perspective projections, and camera noise. Our method uses specular spheres detected with a perspective-correcting Hough transform to robustly triangulate light positions in the presence of outliers via a least-squares approach. Furthermore, we propose an iterative reweighting scheme in combination with an ℓp-norm minimizer to robustly solve the calibrated near-light PS problem. In contrast to other approaches, our framework reconstructs depth, albedo (relative to light source intensity), and normals simultaneously and is demonstrated on synthetic and real-world scenes.

Abstract Voxel‐based approaches are today's standard to encode volume data. Recently, directed acyclic graphs (DAGs) were successfully used for compressing sparse voxel scenes as well, but they are restricted to a single bit of (geometry) information per voxel. We present a method to compressarbitrarydata, such as colors, normals, or reflectance information. By decoupling geometry and voxel data via a novel mapping scheme, we are able to apply the DAG principle to encode the topology, while using a palette‐based compression for the voxel attributes, leading to a drastic memory reduction. Our method outperforms existing state‐of‐the‐art techniques and is well‐suited for GPU architectures. We achieve real‐time performance on commodity hardware for colored scenes with up to 17 hierarchical levels (a 128K3voxel resolution), which are stored fully in core.

Performance capture systems are used to acquire high-quality animated 3D surfaces, usually in form of a dense 3D triangle mesh. Extracting a more compact yet faithful representation is often desirable, but existing solutions for animated sequences are surface based, which leads to a limited approximation power in the case of extreme simplification. We introduce animated sphere-meshes, which are meshes indexing a set of animated spheres. Our solution is the first to output an animated volumetric structure to approximate animated 3D surfaces and optimizes for the sphere approximation, connectivity, and temporal coherence. As a result, our algorithm produces a multiresolution structure from which a level of simplification can be selected in real time, preserving a faithful approximation of the input, even at the coarsest levels. We demonstrate the use of animated sphere-meshes for low-cost approximate collision detection. Additionally, we propose a skinning decomposition, which automatically rigs the input mesh to the chosen level of detail. The resulting set of weights are smooth, compress the animation, and enable easy edits.

Abstract Surface selection is one of the fundamental interactions in shape modeling. In the case of complex models, this task is often tedious for at least two reasons: firstly the local geometry of a given region may be hard to manually select and needs great accuracy; secondly the selection process may have to be repeated a large number of times for similar regions requiring similar subsequent editing. We propose SimSelect, a new system for interactive selection on 3D surfaces addressing these two issues. We cope with the accuracy issue by classifying selections in different types, namely components, parts and patches for which we independently optimize the selection process. Second, we address the repetitiveness issue by introducing an expansion process based on shape recognition which automatically retrieves potential selections similar to the user‐defined one. As a result, our system provides the user with a compact set of simple interaction primitives providing a smooth select‐and‐edit workflow.

Shape approximation algorithms aim at computing simple geometric descriptions of dense surface meshes. Many such algorithms are based on mesh decimation techniques, generating coarse triangulations while optimizing for a particular metric which models the distance to the original shape. This approximation scheme is very efficient when enough polygons are allowed for the simplified model. However, as coarser approximations are reached, the intrinsic piecewise linear point interpolation which defines the decimated geometry fails at capturing even simple structures. We claim that when reaching such extreme simplification levels, highly instrumental in shape analysis, the approximating representation should explicitly and progressively model the volumetric extent of the original shape. In this paper, we propose Sphere-Meshes , a new shape representation designed for extreme approximations and substituting a sphere interpolation for the classic point interpolation of surface meshes. From a technical point-of-view, we propose a new shape approximation algorithm, generating a sphere-mesh at a prescribed level of detail from a classical polygon mesh. We also introduce a new metric to guide this approximation, the Spherical Quadric Error Metric in R 4 , whose minimizer finds the sphere that best approximates a set of tangent planes in the input and which is sensitive to surface orientation, thus distinguishing naturally between the inside and the outside of an object. We evaluate the performance of our algorithm on a collection of models covering a wide range of topological and geometric structures and compare it against alternate methods. Lastly, we propose an application to deformation control where a sphere-mesh hierarchy is used as a convenient rig for altering the input shape interactively.

Abstract We present a new curve skeleton model designed for surface modeling and processing. This skeleton is defined as the geometrical integration of a piecewise harmonic parameterization defined over a disk‐cylinder surface decomposition. This decomposition is computed using a progressive Region Graph reduction based on both geometric and topological criteria which can be iteratively optimized to improve region boundaries. The skeleton has an analytical form with regularity inherited from the surface one. Such a form offers well‐defined surface‐skeleton and skeleton‐surface projections. The resulting skeleton satisfies quality criteria which are relevant for skeleton‐based modeling and processing. We propose applications that benefit from our skeleton model, including local thickness editing, inset surface creation for shell mapping, as well as a new mid‐scale feature preserving smoothing.

Abstract We present CageR: A novel framework for converting animated 3D shape sequences into compact and stable cage‐based representations. Given a raw animated sequence with one‐to‐one point correspondences together with an initial cage embedding, our algorithm automatically generates smoothly varying cage embeddings which faithfully reconstruct the enclosed object deformation. Our technique is fast, automatic, oblivious to the cage coordinate system, provides controllable error and exploits a GPU implementation. At the core of our method, we introduce a new algebraic algorithm based on maximum volume sub‐matrices (maxvol) to speed up and stabilize the deformation inversion. We also present a new spectral regularization algorithm that can apply arbitrary regularization terms on selected subparts of the inversion spectrum. This step allows to enforce a highly localized cage regularization, guaranteeing its smooth variation along the sequence. We demonstrate the speed, accuracy and robustness of our framework on various synthetic and acquired data sets. The benefits of our approach are illustrated in applications such as animation compression and post‐editing.