Detection and reconstruction of freeform sweeps

We study the difficult problem of deciding if parts of a freeform surface can be generated, or approximately generated, by the motion of a planar profile through space. While this task is basic for understanding the geometry of shapes as well as highly relevant for manufacturing and building construction, previous approaches were confined to special cases like kinematic surfaces or “moulding” surfaces. The general case remained unsolved so far. We approach this problem by a combination of local and global methods: curve analysis with regard to “movability”, curve comparison by common substring search in curvature plots, an exhaustive search through all planar cuts enhanced by quick rejection procedures, the ordering of candidate profiles and finally, global optimization. The main applications of our method are digital reconstruction of CAD models exhibiting sweep patches, and aiding in manufacturing freeform surfaces by pointing out those parts which can be approximated by sweeps.     

Projective Feature Learning for 3D Shapes with Multi‐View Depth Images

Feature learning for 3D shapes is challenging due to the lack of natural paramterization for 3D surface models. We adopt the multi‐view depth image representation and propose Multi‐View Deep Extreme Learning Machine (MVD‐ELM) to achieve fast and quality projective feature learning for 3D shapes. In contrast to existing multi‐view learning approaches, our method ensures the feature maps learned for different views are mutually dependent via shared weights and in each layer, their unprojections together form a valid 3D reconstruction of the input 3D shape through using normalized convolution kernels. These lead to a more accurate 3D feature learning as shown by the encouraging results in several applications. Moreover, the 3D reconstruction property enables clear visualization of the learned features, which further demonstrates the meaningfulness of our feature learning.     

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.     

Skeleton‐driven Adaptive Hexahedral Meshing of Tubular Shapes

We propose a novel method for the automatic generation of structured hexahedral meshes of articulated 3D shapes. We recast the complex problem of generating the connectivity of a hexahedral mesh of a general shape into the simpler problem of generating the connectivity of a tubular structure derived from its curve‐skeleton. We also provide volumetric subdivision schemes to nicely adapt the topology of the mesh to the local thickness of tubes, while regularizing per‐element size. Our method is fast, one‐click, easy to reproduce, and it generates structured meshes that better align to the branching structure of the input shape if compared to previous methods for hexa mesh generation.     

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.     

Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects

We introduce techniques for the processing of motion and animations of non‐rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in ?ⁿ to the processing of motion of non‐rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced‐order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non‐linear “Bézier curves” by executing de Casteljau's algorithm in shape space.     

Optimal Spline Approximation via ℓ0‐Minimization

Splines are part of the standard toolbox for the approximation of functions and curves in ?^d. Still, the problem of finding the spline that best approximates an input function or curve is ill‐posed, since in general this yields a “spline” with an infinite number of segments. The problem can be regularized by adding a penalty term for the number of spline segments. We show how this idea can be formulated as an ?₀‐regularized quadratic problem. This gives us a notion of optimal approximating splines that depend on one parameter, which weights the approximation error against the number of segments. We detail this concept for different types of splines including B‐splines and composite Bézier curves. Based on the latest development in the field of sparse approximation, we devise a solver for the resulting minimization problems and show applications to spline approximation of planar and space curves and to spline conversion of motion capture data.     

Sparse Non‐rigid Registration of 3D Shapes

Non‐rigid registration of 3D shapes is an essential task of increasing importance as commodity depth sensors become more widely available for scanning dynamic scenes. Non‐rigid registration is much more challenging than rigid registration as it estimates a set of local transformations instead of a single global transformation, and hence is prone to the overfitting issue due to underdetermination. The common wisdom in previous methods is to impose an ?₂‐norm regularization on the local transformation differences. However, the ?₂‐norm regularization tends to bias the solution towards outliers and noise with heavy‐tailed distribution, which is verified by the poor goodness‐of‐fit of the Gaussian distribution over transformation differences. On the contrary, Laplacian distribution fits well with the transformation differences, suggesting the use of a sparsity prior. We propose a sparse non‐rigid registration (SNR) method with an ?₁‐norm regularized model for transformation estimation, which is effectively solved by an alternate direction method (ADM) under the augmented Lagrangian framework. We also devise a multi‐resolution scheme for robust and progressive registration. Results on both public datasets and our scanned datasets show the superiority of our method, particularly in handling large‐scale deformations as well as outliers and noise.     

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.     

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.     

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.     

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.     

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.     

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.     

Exploration of Empty Space among Spherical Obstacles via Additively Weighted Voronoi Diagram

Properties of granular materials or molecular structures are often studied on a simple geometric model – a set of 3D balls. If the balls simultaneously change in size by a constant speed, topological properties of the empty space outside all these balls may also change. Capturing the changes and their subsequent classification may reveal useful information about the model. This has already been solved for balls of the same size, but only an approximate solution has been reported for balls of different sizes. These solutions work on simplicial complexes derived from the dual structure of the ordinary Voronoi diagram of ball centers and use the mathematical concept of simplicial homology groups. If the balls have different radii, it is more appropriate to use the additively weighted Voronoi diagram (also known as the Apollonius diagram) instead of the ordinary diagram, but the dual structure is no longer a simplicial complex, so the previous approaches cannot be used directly. In this paper, a method is proposed to overcome this problem. The method works with Voronoi edges and vertices instead of the dual structure. Additional bridge edges are introduced to overcome disconnected cases. The output is a tree graph of events where cavities are created or merged during a simulated shrinking of the balls. This graph is then reorganized and filtered according to some criteria to get a more concise information about the development of the empty space in the model.     

String‐Based Synthesis of Structured Shapes

We propose a novel method to synthesize geometric models from a given class of context‐aware structured shapes such as buildings and other man‐made objects. The central idea is to leverage powerful machine learning methods from the area of natural language processing for this task. To this end, we propose a technique that maps shapes to strings and vice versa, through an intermediate shape graph representation. We then convert procedurally generated shape repositories into text databases that, in turn, can be used to train a variational autoencoder. The autoencoder enables higher level shape manipulation and synthesis like, for example, interpolation and sampling via its continuous latent space. We provide project code and pre‐trained models.