Stable Region Correspondences Between Non‐Isometric Shapes

We consider the problem of finding meaningful correspondences between 3D models that are related but not necessarily very similar. When the shapes are quite different, a point‐to‐point map is not always appropriate, so our focus in this paper is a method to build a set of correspondences between shape regions or parts. The proposed approach exploits a variety of feature functions on the shapes and makes use of the key observation that points in matching parts have similar ranks in the sorting of the corresponding feature values. Our algorithm proceeds in two steps. We first build an affinity matrix between points on the two shapes, based on feature rank similarity over many feature functions. We then define a notion of stability of a pair of regions, with respect to this affinity matrix, obtained as a fixed point of a nonlinear operator. Our method yields a family of corresponding maximally stable regions between the two shapes that can be used to define shape parts. We observe that this is an instance of the biclustering problem and that it is related to solving a constrained maximal eigenvalue problem. We provide an algorithm to solve this problem that mimics the power method. We show the robustness of its output to noisy input features as well its convergence properties. The obtained part correspondences are shown to be almost perfect matches in the isometric case, and also semantically appropriate even in non‐isometric cases. We provide numerous examples and applications of this technique, for example to sharpening correspondences in traditional shape matching algorithms.     

The shape variational autoencoder: A deep generative model of part‐segmented 3D objects

We introduce a generative model of part‐segmented 3D objects: the shape variational auto‐encoder (ShapeVAE). The ShapeVAE describes a joint distribution over the existence of object parts, the locations of a dense set of surface points, and over surface normals associated with these points. Our model makes use of a deep encoder‐decoder architecture that leverages the part‐decomposability of 3D objects to embed high‐dimensional shape representations and sample novel instances. Given an input collection of part‐segmented objects with dense point correspondences the ShapeVAE is capable of synthesizing novel, realistic shapes, and by performing conditional inference enables imputation of missing parts or surface normals. In addition, by generating both points and surface normals, our model allows for the use of powerful surface‐reconstruction methods for mesh synthesis. We provide a quantitative evaluation of the ShapeVAE on shape‐completion and test‐set log‐likelihood tasks and demonstrate that the model performs favourably against strong baselines. We demonstrate qualitatively that the ShapeVAE produces plausible shape samples, and that it captures a semantically meaningful shape‐embedding. In addition we show that the ShapeVAE facilitates mesh reconstruction by sampling consistent surface normals.     

Perfect Matching Quad Layouts for Manifold Meshes

This paper introduces a new approach to automatically generate pure quadrilateral patch layouts on manifold meshes. The algorithm is based on a careful construction of a singularity graph of a given input frame field or a given periodic global parameterization. A pure quadrilateral patch layout is then derived as a constrained minimum weight perfect matching of that graph. The resulting layout is optimal relative to a balance between coarseness and geometric feature alignment. We formulate the problem of finding pure quadrilateral patch layouts as a global optimization problem related to a well‐known concept in graph theory. The main advantage of the new method is its simplicity and its computation speed. Patch layouts generated by the present algorithm are high quality and are very competitive compared to current state of the art.     

Diffusion Diagrams: Voronoi Cells and Centroids from Diffusion

We define Voronoi cells and centroids based on heat diffusion. These heat cells and heat centroids coincide with the common definitions in Euclidean spaces. On curved surfaces they compare favorably with definitions based on geodesics: they are smooth and can be computed in a stable way with a single linear solve. We analyze the numerics of this approach and can show that diffusion diagrams converge quadratically against the smooth case under mesh refinement, which is better than other common discretization of distance measures in curved spaces. By factorizing the system matrix in a preprocess, computing Voronoi diagrams or centroids amounts to just back‐substitution. We show how to localize this operation so that the complexity is linear in the size of the cells and not the underlying mesh. We provide several example applications that show how to benefit from this approach.     

Freeform Honeycomb Structures

Motivated by requirements of freeform architecture, and inspired by the geometry of hexagonal combs in beehives, this paper addresses torsion‐free structures aligned with hexagonal meshes. Since repetitive geometry is a very important contribution to the reduction of production costs, we study in detail “honeycomb structures”, which are defined as torsion‐free structures where the walls of cells meet at 120 degrees. Interestingly, the Gauss‐Bonnet theorem is useful in deriving information on the global distribution of node axes in such honeycombs. This paper discusses the computation and modeling of honeycomb structures as well as applications, e.g. for shading systems, or for quad meshing. We consider this paper as a contribution to the wider topic of freeform patterns, polyhedral or otherwise. Such patterns require new approaches on the technical level, e.g. in the treatment of smoothness, but they also extend our view of what constitutes aesthetic freeform geometry.     

Fully Spectral Partial Shape Matching

We propose an efficient procedure for calculating partial dense intrinsic correspondence between deformable shapes performed entirely in the spectral domain. Our technique relies on the recently introduced partial functional maps formalism and on the joint approximate diagonalization (JAD) of the Laplace‐Beltrami operators previously introduced for matching non‐isometric shapes. We show that a variant of the JAD problem with an appropriately modified coupling term (surprisingly) allows to construct quasi‐harmonic bases localized on the latent corresponding parts. This circumvents the need to explicitly compute the unknown parts by means of the cumbersome alternating minimization used in the previous approaches, and allows performing all the calculations in the spectral domain with constant complexity independent of the number of shape vertices. We provide an extensive evaluation of the proposed technique on standard non‐rigid correspondence benchmarks and show state‐of‐the‐art performance in various settings, including partiality and the presence of topological noise.     

Robust Region Detection via Consensus Segmentation of Deformable Shapes

We consider the problem of stable region detection and segmentation of deformable shapes. We pursue this goal by determining a consensus segmentation from a heterogeneous ensemble of putative segmentations, which are generated by a clustering process on an intrinsic embedding of the shape. The intuition is that the consensus segmentation, which relies on aggregate statistics gathered from the segmentations in the ensemble, can reveal components in the shape that are more stable to deformations than the single baseline segmentations. Compared to the existing approaches, our solution exhibits higher robustness and repeatability throughout a wide spectrum of non‐rigid transformations. It is computationally efficient, naturally extendible to point clouds, and remains semantically stable even across different object classes. A quantitative evaluation on standard datasets confirms the potentiality of our method as a valid tool for deformable shape analysis.     

Non‐Rigid Puzzles

Shape correspondence is a fundamental problem in computer graphics and vision, with applications in various problems including animation, texture mapping, robotic vision, medical imaging, archaeology and many more. In settings where the shapes are allowed to undergo non‐rigid deformations and only partial views are available, the problem becomes very challenging. To this end, we present a non‐rigid multi‐part shape matching algorithm. We assume to be given a reference shape and its multiple parts undergoing a non‐rigid deformation. Each of these query parts can be additionally contaminated by clutter, may overlap with other parts, and there might be missing parts or redundant ones. Our method simultaneously solves for the segmentation of the reference model, and for a dense correspondence to (subsets of) the parts. Experimental results on synthetic as well as real scans demonstrate the effectiveness of our method in dealing with this challenging matching scenario.     

Fiber Surfaces: Generalizing Isosurfaces to Bivariate Data

Scientific visualization has many effective methods for examining and exploring scalar and vector fields, but rather fewer for bivariate fields. We report the first general purpose approach for the interactive extraction of geometric separating surfaces in bivariate fields. This method is based on fiber surfaces: surfaces constructed from sets of fibers, the multivariate analogues of isolines. We show simple methods for fiber surface definition and extraction. In particular, we show a simple and efficient fiber surface extraction algorithm based on Marching Cubes. We also show how to construct fiber surfaces interactively with geometric primitives in the range of the function. We then extend this to build user interfaces that generate parameterized families of fiber surfaces with respect to arbitrary polygons. In the special case of isovalue‐gradient plots, fiber surfaces capture features geometrically for quantitative analysis that have previously only been analysed visually and qualitatively using multi‐dimensional transfer functions in volume rendering. We also demonstrate fiber surface extraction on a variety of bivariate data.     

Scale‐Invariant Directional Alignment of Surface Parametrizations

Various applications of global surface parametrization benefit from the alignment of parametrization isolines with principal curvature directions. This is particularly true for recent parametrization‐based meshing approaches, where this directly translates into a shape‐aware edge flow, better approximation quality, and reduced meshing artifacts. Existing methods to influence a parametrization based on principal curvature directions suffer from scale‐dependence, which implies the necessity of parameter variation, or try to capture complex directional shape features using simple 1D curves. Especially for non‐sharp features, such as chamfers, fillets, blends, and even more for organic variants thereof, these abstractions can be unfit. We present a novel approach which respects and exploits the 2D nature of such directional feature regions, detects them based on coherence and homogeneity properties, and controls the parametrization process accordingly. This approach enables us to provide an intuitive, scale‐invariant control parameter to the user. It also allows us to consider non‐local aspects like the topology of a feature, enabling further improvements. We demonstrate that, compared to previous approaches, global parametrizations of higher quality can be generated without user intervention.     

Learning 3D Scene Synthesis from Annotated RGB‐D Images

We present a data‐driven method for synthesizing 3D indoor scenes by inserting objects progressively into an initial, possibly, empty scene. Instead of relying on few hundreds of hand‐crafted 3D scenes, we take advantage of existing large‐scale annotated RGB‐D datasets, in particular, the SUN RGB‐D database consisting of 10,000+ depth images of real scenes, to form the prior knowledge for our synthesis task. Our object insertion scheme follows a co‐occurrence model and an arrangement model, both learned from the SUN dataset. The former elects a highly probable combination of object categories along with the number of instances per category while a plausible placement is defined by the latter model. Compared to previous works on probabilistic learning for object placement, we make two contributions. First, we learn various classes of higher‐order object‐object relations including symmetry, distinct orientation, and proximity from the database. These relations effectively enable considering objects in semantically formed groups rather than by individuals. Second, while our algorithm inserts objects one at a time, it attains holistic plausibility of the whole current scene while offering controllability through progressive synthesis. We conducted several user studies to compare our scene synthesis performance to results obtained by manual synthesis, state‐of‐the‐art object placement schemes, and variations of parameter settings for the arrangement model.     

Directional Field Synthesis,Design, and Processing

Direction fields and vector fields play an increasingly important role in computer graphics and geometry processing. The synthesis of directional fields on surfaces, or other spatial domains, is a fundamental step in numerous applications, such as mesh generation, deformation, texture mapping, and many more. The wide range of applications resulted in definitions for many types of directional fields: from vector and tensor fields, over line and cross fields, to frame and vector‐set fields. Depending on the application at hand, researchers have used various notions of objectives and constraints to synthesize such fields. These notions are defined in terms of fairness, feature alignment, symmetry, or field topology, to mention just a few. To facilitate these objectives, various representations, discretizations, and optimization strategies have been developed. These choices come with varying strengths and weaknesses. This report provides a systematic overview of directional field synthesis for graphics applications, the challenges it poses, and the methods developed in recent years to address these challenges.     

Retargeting 3D Objects and Scenes with a General Framework

In this paper, we introduce an interactive method suitable for retargeting both 3D objects and scenes. Initially, the input object or scene is decomposed into a collection of constituent components enclosed by corresponding control bounding volumes which capture the intra‐structures of the object or semantic grouping of objects in the 3D scene. The overall retargeting is accomplished through a constrained optimization by manipulating the control bounding volumes. Without inferring the intricate dependencies between the components, we define a minimal set of constraints that maintain the spatial arrangement and connectivity between the components to regularize the valid retargeting results. The default retargeting behavior can then be easily altered by additional semantic constraints imposed by users. This strategy makes the proposed method highly flexible to process a wide variety of 3D objects and scenes under an unified framework. In addition, the proposed method achieved more general structure‐preserving pattern synthesis in both object and scene levels. We demonstrate the effectiveness of our method by applying it to several complicated 3D objects and scenes.     

Automatic 3D Indoor Scene Updating with RGBD Cameras

Since indoor scenes are frequently changed in daily life, such as re‐layout of furniture, the 3D reconstructions for them should be flexible and easy to update. We present an automatic 3D scene update algorithm to indoor scenes by capturing scene variation with RGBD cameras. We assume an initial scene has been reconstructed in advance in manual or other semi‐automatic way before the change, and automatically update the reconstruction according to the newly captured RGBD images of the real scene update. It starts with an automatic segmentation process without manual interaction, which benefits from accurate labeling training from the initial 3D scene. After the segmentation, objects captured by RGBD camera are extracted to form a local updated scene. We formulate an optimization problem to compare to the initial scene to locate moved objects. The moved objects are then integrated with static objects in the initial scene to generate a new 3D scene. We demonstrate the efficiency and robustness of our approach by updating the 3D scene of several real‐world scenes.