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.     

Multi‐style paper pop‐up designs from 3D models

Paper pop‐ups are interesting three‐dimensional books that fascinate people of all ages. The design and construction of these pop‐up books however are done manually and require a lot of time and effort. This has led to computer‐assisted or automated tools for designing paper pop‐ups. This paper proposes an approach for automatically converting a 3D model into a multi‐style paper pop‐up. Previous automated approaches have only focused on single‐style pop‐ups, where each is made of a single type of pop‐up mechanisms. In our work, we combine multiple styles in a pop‐up, which is more representative of actual artist's creations. Our method abstracts a 3D model using suitable primitive shapes that both facilitate the formation of the considered pop‐up mechanisms and closely approximate the input model. Each shape is then abstracted using a set of 2D patches that combine to form a valid pop‐up. We define geometric conditions that ensure the validity of the combined pop‐up structures. In addition, our method also employs an image‐based approach for producing the patches to preserve the textures, finer details and important contours of the input model. Finally, our system produces a printable design layout and decides an assembly order for the construction instructions. The feasibility of our results is verified by constructing the actual paper pop‐ups from the designs generated by our system.     

Dense Point‐to‐Point Correspondences Between Genus‐Zero Shapes

We describe a novel approach that addresses the problem of establishing correspondences between non‐rigidly deformed shapes by performing the registration over the unit sphere. In a pre‐processing step, each shape is conformally parametrized over the sphere, centered to remove Möbius inversion ambiguity, and authalically evolved to expand regions that are excessively compressed by the conformal parametrization. Then, for each pair of shapes, we perform fast SO(3) correlation to find the optimal rotational alignment and refine the registration using optical flow. We evaluate our approach on the TOSCA dataset, demonstrating that our approach compares favorably to state‐of‐the‐art methods.     

Remeshing‐assisted Optimization for Locally Injective Mappings

Constructing locally injective mappings for 2D triangular meshes is vital in applications such as deformations. In such a highly constrained optimization, the prescribed tessellation may impose strong restriction on the solution. As a consequence, the feasible region may be too small to contain an ideal solution, which leads to problems of slow convergence, poor solution, or even that no solution can be found. We propose to integrate adaptive remeshing into interior point method to solve this issue. We update the vertex positions via a parameter‐free relaxation enhanced geometry optimization, and then use edge‐flip operations to reduce the residual and keep a reasonable condition number for better convergence. For more robustness, when the iteration of interior point method terminates but leaves the positional constraints unsatisfied, we estimate the edges in the current tessellation that block vertices moving based on the convergence information of the optimization, and then split neighboring edges to break the restriction. The results show that our method has better performance than the solely geometric optimization approaches, especially for extreme deformations.     

Sets of Globally Optimal Stream Surfaces for Flow Visualization

Stream surfaces are a well‐studied and widely used tool for the visualization of 3D flow fields. Usually, stream surface seeding is carried out manually in time‐consuming trial and error procedures. Only recently automatic selection methods were proposed. Local methods support the selection of a set of stream surfaces, but, contrary to global selection methods, they evaluate only the quality of the seeding lines but not the quality of the whole stream surfaces. Global methods, on the other hand, only support the selection of a single optimal stream surface until now. However, for certain flow fields a single stream surface is not sufficient to represent all flow features. In our work, we overcome this limitation by introducing a global selection technique for a set of stream surfaces. All selected surfaces optimize global stream surface quality measures and are guaranteed to be mutually distant, such that they can convey different flow features. Our approach is an efficient extension of the most recent global selection method for single stream surfaces. We illustrate its effectiveness on a number of analytical and simulated flow fields and analyze the quality of the results in a user study.     

Polycube Simplification for Coarse Layouts of Surfaces and Volumes

Representing digital objects with structured meshes that embed a coarse block decomposition is a relevant problem in applications like computer animation, physically‐based simulation and Computer Aided Design (CAD). One of the key ingredients to produce coarse block structures is to achieve a good alignment between the mesh singularities (i.e., the corners of each block). In this paper we improve on the polycube‐based meshing pipeline to produce both surface and volumetric coarse block‐structured meshes of general shapes. To this aim we add a new step in the pipeline. Our goal is to optimize the positions of the polycube corners to produce as coarse as possible base complexes. We rely on re‐mapping the positions of the corners on an integer grid and then using integer numerical programming to reach the optimal. To the best of our knowledge this is the first attempt to solve the singularity misalignment problem directly in polycube space. Previous methods for polycube generation did not specifically address this issue. Our corner optimization strategy is efficient and requires a negligible extra running time for the meshing pipeline. In the paper we show that our optimized polycubes produce coarser block structured surface and volumetric meshes if compared with previous approaches. They also induce higher quality hexahedral meshes and are better suited for spline fitting because they reduce the number of splines necessary to cover the domain, thus improving both the efficiency and the overall level of smoothness throughout the volume.     

Near‐Isometric Level Set Tracking

Implicit representations of geometry have found applications in shape modeling, simulation, and other graphics pipelines. These representations, however, do not provide information about the paths of individual points as shapes move and undergo deformation. For this reason, we reconsider the problem of tracking points on level set surfaces, with the goal of designing an algorithm that — unlike previous work — can recover rotational motion and nearly isometric deformation. We track points on level sets of a time‐varying function using approximate Killing vector fields (AKVFs), the velocity fields of near‐isometric motions. To this end, we provide suitable theoretical and discrete constructions for computing AKVFs in a narrow band surrounding an animated level set surface. Furthermore, we propose time integrators well‐suited to integrating AKVFs in time to track points. We demonstrate the theoretical and practical advantages of our proposed algorithms on synthetic and practical tasks.     

Efficient Volumetric PolyCube‐Map Construction

PolyCubes provide compact representations for closed complex shapes and are essential to many computer graphics applications. Existing automatic PolyCube construction methods usually suffer from poor quality or time‐consuming computation. In this paper, we provide a highly efficient method to compute volumetric PolyCube‐maps. Given an input tetrahedral mesh, we utilize two novel normal‐driven volumetric deformation schemes and a polycube‐allowable mesh segmentation to drive the input to a volumetric PolyCube structure. Our method can robustly generate foldover‐free and low‐distortion PolyCube‐maps in practice, and provide a flexible control on the number of corners of Polycubes. Compared with state‐of‐the‐art methods, our method is at least one order of magnitude faster and has better mapping qualities. We demonstrate the efficiency and efficacy of our method in PolyCube construction and all‐hexahedral meshing on various complex models.     

Data‐Driven Sparse Priors of 3D Shapes

We present a sparse optimization framework for extracting sparse shape priors from a collection of 3D models. Shape priors are defined as point‐set neighborhoods sampled from shape surfaces which convey important information encompassing normals and local shape characterization. A 3D shape model can be considered to be formed with a set of 3D local shape priors, while most of them are likely to have similar geometry. Our key observation is that the local priors extracted from a family of 3D shapes lie in a very low‐dimensional manifold. Consequently, a compact and informative subset of priors can be learned to efficiently encode all shapes of the same family. A comprehensive library of local shape priors is first built with the given collection of 3D models of the same family. We then formulate a global, sparse optimization problem which enforces selecting representative priors while minimizing the reconstruction error. To solve the optimization problem, we design an efficient solver based on the Augmented Lagrangian Multipliers method (ALM). Extensive experiments exhibit the power of our data‐driven sparse priors in elegantly solving several high‐level shape analysis applications and geometry processing tasks, such as shape retrieval, style analysis and symmetry detection.     

Pattern‐Based Quadrangulation for N‐Sided Patches

We propose an algorithm to quadrangulate an N‐sided patch (2 ≤ N ≤ 6) with prescribed numbers of edge subdivisions at its boundary. Our algorithm is guaranteed to succeed for arbitrary valid input, which is proved using a canonical simplification of the input and a small set of topological patterns that are sufficient for supporting all possible cases. Our algorithm produces solutions with minimal number of irregular vertices by default, but it also allows the user to choose other feasible solutions by solving a set of small integer linear programs. We demonstrate the effectiveness of our algorithm by integrating it into a sketch‐based quad remeshing system. A reference C++ implementation of our algorithm is provided as a supplementary material.     

2D‐D Lifting for Shape Reconstruction

We present an algorithm for shape reconstruction from incomplete 3D scans by fusing together two acquisition modes: 2D photographs and 3D scans. The two modes exhibit complementary characteristics: scans have depth information, but are often sparse and incomplete; photographs, on the other hand, are dense and have high resolution, but lack important depth information. In this work we fuse the two modes, taking advantage of their complementary information, to enhance 3D shape reconstruction from an incomplete scan with a 2D photograph. We compute geometrical and topological shape properties in 2D photographs and use them to reconstruct a shape from an incomplete 3D scan in a principled manner. Our key observation is that shape properties such as boundaries, smooth patches and local connectivity, can be inferred with high confidence from 2D photographs. Thus, we register the 3D scan with the 2D photograph and use scanned points as 3D depth cues for lifting 2D shape structures into 3D. Our contribution is an algorithm which significantly regularizes and enhances the problem of 3D reconstruction from partial scans by lifting 2D shape structures into 3D. We evaluate our algorithm on various shapes which are loosely scanned and photographed from different views, and compare them with state‐of‐the‐art reconstruction methods.     

Single‐View Modeling of Layered Origami with Plausible Outer Shape

Modeling 3D origami pieces using conventional software is laborious due to the geometric constraints imposed by the complicated layered structure. Targeting origami models used in visual content such as CG illustrations and movies, we propose an interactive system that dramatically simplifies the modeling of 3D origami pieces with plausible outer shapes, while omitting accurate inner structures. By focusing on flat origami models with a front‐and‐back symmetry commonly found in traditional artworks, our system realizes easy and quick modeling via single‐view interface; given a reference image of the target origami piece, the user draws polygons of planar faces onto the image, and assigns annotations indicating the types of folding operations. Our system automatically rectifies the manually‐specified polygons, infers the folded structures that should yield the user‐specified polygons with reference to the depth order of layered polygons, and generates a plausible 3D model while accounting for gaps between layers. Our system is versatile enough for modeling pseudo‐origami models that are not realizable by folding a single sheet of paper. Our user study demonstrates that even novice users without the specialized knowledge and experience on origami and 3D modeling can create plausible origami models quickly.     

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.     

Design Transformations for Rule‐based Procedural Modeling

We introduce design transformations for rule‐based procedural models, e.g., for buildings and plants. Given two or more procedural designs, each specified by a grammar, a design transformation combines elements of the existing designs to generate new designs. We introduce two technical components to enable design transformations. First, we extend the concept of discrete rule switching to rule merging, leading to a very large shape space for combining procedural models. Second, we propose an algorithm to jointly derive two or more grammars, called grammar co‐derivation. We demonstrate two applications of our work: we show that our framework leads to a larger variety of models than previous work, and we show fine‐grained transformation sequences between two procedural models.     

Learning 3D Deformation of Animals from 2D Images

Understanding how an animal can deform and articulate is essential for a realistic modification of its 3D model. In this paper, we show that such information can be learned from user‐clicked 2D images and a template 3D model of the target animal. We present a volumetric deformation framework that produces a set of new 3D models by deforming a template 3D model according to a set of user‐clicked images. Our framework is based on a novel locally‐bounded deformation energy, where every local region has its own stiffness value that bounds how much distortion is allowed at that location. We jointly learn the local stiffness bounds as we deform the template 3D mesh to match each user‐clicked image. We show that this seemingly complex task can be solved as a sequence of convex optimization problems. We demonstrate the effectiveness of our approach on cats and horses, which are highly deformable and articulated animals. Our framework produces new 3D models of animals that are significantly more plausible than methods without learned stiffness.     

Template Assembly for Detailed Urban Reconstruction

We propose a new framework to reconstruct building details by automatically assembling 3D templates on coarse textured building models. In a preprocessing step, we generate an initial coarse model to approximate a point cloud computed using Structure from Motion and Multi View Stereo, and we model a set of 3D templates of facade details. Next, we optimize the initial coarse model to enforce consistency between geometry and appearance (texture images). Then, building details are reconstructed by assembling templates on the textured faces of the coarse model. The 3D templates are automatically chosen and located by our optimization‐based template assembly algorithm that balances image matching and structural regularity. In the results, we demonstrate how our framework can enrich the details of coarse models using various data sets.