Exact and Robust (Self‐)Intersections for Polygonal Meshes

We present a new technique to implement operators that modify the topology of polygonal meshes at intersections and self‐intersections. Depending on the modification strategy, this effectively results in operators for Boolean combinations or for the construction of outer hulls that are suited for mesh repair tasks and accurate mesh‐based front tracking of deformable materials that split and merge. By combining an adaptive octree with nested binary space partitions (BSP), we can guarantee exactness (= correctness) and robustness (= completeness) of the algorithm while still achieving higher performance and less memory consumption than previous approaches. The efficiency and scalability in terms of runtime and memory is obtained by an operation localization scheme. We restrict the essential computations to those cells in the adaptive octree where intersections actually occur. Within those critical cells, we convert the input geometry into a plane‐based BSP‐representation which allows us to perform all computations exactly even with fixed precision arithmetics. We carefully analyze the precision requirements of the involved geometric data and predicates in order to guarantee correctness and show how minimal input mesh quantization can be used to safely rely on computations with standard floating point numbers. We properly evaluate our method with respect to precision, robustness, and efficiency.     

Global Correspondence Optimization for Non‐Rigid Registration of Depth Scans

We present a registration algorithm for pairs of deforming and partial range scans that addresses the challenges of non‐rigid registration within a single non‐linear optimization. Our algorithm simultaneously solves for correspondences between points on source and target scans, confidence weights that measure the reliability of each correspondence and identify non‐overlapping areas, and a warping field that brings the source scan into alignment with the target geometry. The optimization maximizes the region of overlap and the spatial coherence of the deformation while minimizing registration error. All optimization parameters are chosen automatically; hand‐tuning is not necessary. Our method is not restricted to part‐in‐whole matching, but addresses the general problem of partial matching, and requires no explicit prior correspondences or feature points. We evaluate the performance and robustness of our method using scan data acquired by a structured light scanner and compare our method with existing non‐rigid registration algorithms.     

Non‐Rigid Registration Under Isometric Deformations

We present a robust and efficient algorithm for the pairwise non‐rigid registration of partially overlapping 3D surfaces. Our approach treats non‐rigid registration as an optimization problem and solves it by alternating between correspondence and deformation optimization. Assuming approximately isometric deformations, robust correspondences are generated using a pruning mechanism based on geodesic consistency. We iteratively learn an appropriate deformation discretization from the current set of correspondences and use it to update the correspondences in the next iteration. Our algorithm is able to register partially similar point clouds that undergo large deformations, in just a few seconds. We demonstrate the potential of our algorithm in various applications such as example based articulated segmentation, and shape interpolation.     

Reconstructing Animated Meshes from Time‐Varying Point Clouds

In this paper, we describe a novel approach for the reconstruction of animated meshes from a series of time‐deforming point clouds. Given a set of unordered point clouds that have been captured by a fast 3‐D scanner, our algorithm is able to compute coherent meshes which approximate the input data at arbitrary time instances. Our method is based on the computation of an implicit function in ?⁴ that approximates the time‐space surface of the time‐varying point cloud. We then use the four‐dimensional implicit function to reconstruct a polygonal model for the first time‐step. By sliding this template mesh along the time‐space surface in an as‐rigid‐as‐possible manner, we obtain reconstructions for further time‐steps which have the same connectivity as the previously extracted mesh while recovering rigid motion exactly. The resulting animated meshes allow accurate motion tracking of arbitrary points and are well suited for animation compression. We demonstrate the qualities of the proposed method by applying it to several data sets acquired by real‐time 3‐D scanners.     

Provably Good 2D Shape Reconstruction from Unorganized Cross‐Sections

This paper deals with the reconstruction of 2‐dimensional geometric shapes from unorganized 1‐dimensional cross‐sections. We study the problem in its full generality following the approach of Boissonnat and Memari [ [BM07] ] for the analogous 3D problem. We propose a new variant of this method and provide sampling conditions to guarantee that the output of the algorithm has the same topology as the original object and is close to it (for the Hausdorff distance).     

PackMerger: A 3D Print Volume Optimizer

We propose an optimization framework for 3D printing that seeks to save printing time and the support material required to print 3D shapes. Three‐dimensional printing technology is rapidly maturing and may revolutionize how we manufacture objects. The total cost of printing, however, is governed by numerous factors which include not only the price of the printer but also the amount of material and time to fabricate the shape. Our PackMerger framework converts the input 3D watertight mesh into a shell by hollowing its inner parts. The shell is then divided into segments. The location of splits is controlled based on several parameters, including the size of the connection areas or volume of each segment. The pieces are then tightly packed using optimization. The optimization attempts to minimize the amount of support material and the bounding box volume of the packed segments while keeping the number of segments minimal. The final packed configuration can be printed with substantial time and material savings, while also allowing printing of objects that would not fit into the printer volume. We have tested our system on three different printers and it shows a reduction of 5–30% of the printing time while simultaneously saving 15–65% of the support material. The optimization time was approximately 1 min. Once the segments are printed, they need to be assembled.     

Automatic Registration for Articulated Shapes

We present an unsupervised algorithm for aligning a pair of shapes in the presence of significant articulated motion and missing data, while assuming no knowledge of a template, user‐placed markers, segmentation, or the skeletal structure of the shape. We explicitly sample the motion, which gives a priori the set of possible rigid transformations between parts of the shapes. This transforms the problem into a discrete labeling problem, where the goal is to find an optimal assignment of transformations for aligning the shapes. We then apply graph cuts to optimize a novel cost function, which encodes a preference for a consistent motion assignment from both source to target and target to source. We demonstrate the robustness of our method by aligning several synthetic and real‐world datasets.     

Functional webs for freeform architecture

Rationalization and construction‐aware design dominate the issue of realizability of freeform architecture. The former means the decomposition of an intended shape into parts which are sufficiently simple and efficient to manufacture; the latter refers to a design procedure which already incorporates rationalization. Recent contributions to this topic have been concerned mostly with small‐scale parts, for instance with planar faces of meshes. The present paper deals with another important aspect, namely long‐range parts and supporting structures. It turns out that from the pure geometry viewpoint this means studying families of curves which cover surfaces in certain well‐defined ways. Depending on the application one has in mind, different combinatorial arrangements of curves are required. We here restrict ourselves to so‐called hexagonal webs which correspond to a triangular or tri‐hex decomposition of a surface. The individual curve may have certain special properties, like being planar, being a geodesic, or being part of a circle. Each of these properties is motivated by manufacturability considerations and imposes constraints on the shape of the surface. We investigate the available degrees of freedom, show numerical methods of optimization, and demonstrate the effectivity of our approach and the variability of construction solutions derived from webs by means of actual architectural designs.?     

Real‐time Bas‐Relief Generation from Depth‐and‐Normal Maps on GPU

To design a bas‐relief from a 3D scene is an inherently interactive task in many scenarios. The user normally needs to get instant feedback to select a proper viewpoint. However, current methods are too slow to facilitate this interaction. This paper proposes a two‐scale bas‐relief modeling method, which is computationally efficient and easy to produce different styles of bas‐reliefs. The input 3D scene is first rendered into two textures, one recording the depth information and the other recording the normal information. The depth map is then compressed to produce a base surface with level‐of‐depth, and the normal map is used to extract local details with two different schemes. One scheme provides certain freedom to design bas‐reliefs with different visual appearances, and the other provides a control over the level of detail. Finally, the local feature details are added into the base surface to produce the final result. Our approach allows for real‐time computation due to its implementation on graphics hardware. Experiments with a wide range of 3D models and scenes show that our approach can effectively generate digital bas‐reliefs in real time.     

Discrete Line Congruences for Shading and Lighting

Two‐parameter families of straight lines (line congruences) are implicitly present in graphics and geometry processing in several important ways including lighting and shape analysis. In this paper we make them accessible to optimization and geometric computing, by introducing a general discrete version of congruences based on piecewise‐linear correspondences between triangle meshes. Our applications of congruences are based on the extraction of a so‐called torsion‐free support structure, which is a procedure analogous to remeshing a surface along its principal curvature lines. A particular application of such structures are freeform shading and lighting systems for architecture. We combine interactive design of such systems with global optimization in order to satisfy geometric constraints. In this way we explore a new area where architecture can greatly benefit from graphics.     

Variational 3D Shape Segmentation for Bounding Volume Computation

We propose a variational approach to computing an optimal segmentation of a 3D shape for computing a union of tight bounding volumes. Based on an affine invariant measure of e-tightness, the resemblance to ellipsoid, a novel functional is formulated that governs an optimization process to obtain a partition with multiple components. Refinement of segmentation is driven by application-specific error measures, so that the final bounding volume meets pre-specified user requirement. We present examples to demonstrate the effectiveness of our method and show that it works well for computing ellipsoidal bounding volumes as well as oriented bounding boxes.     

Positional, Metric, and Curvature Control for Constraint-Based Surface Deformation

We present a geometry processing framework that allows direct manipulation or preservation of positional, metric, and curvature constraints anywhere on the surface of a geometric model. Target values for these properties can be specified point-wise or as integrated quantities over curves and surface patches embedded in the shape. For example, the user can draw several curves on the surface and specify desired target lengths, manipulate the normal curvature along these curves, or modify the area or principal curvature distribution of arbitrary surface patches. This user input is converted into a set of non-linear constraints. A global optimization finds the new deformed surface that best satisfies the constraints, while minimizing adaptable measures for metric and curvature distortion that provide explicit control of the deformation semantics. We illustrate how this approach enables flexible surface processing and shape editing operations not available in current systems.     

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.     

Discrete Distortion in Triangulated 3‐Manifolds

We introduce a novel notion, that we call discrete distortion, for a triangulated 3‐manifold. Discrete distortion naturally generalizes the notion of concentrated curvature defined for triangulated surfaces and provides a powerful tool to understand the local geometry and topology of 3‐manifolds. Discrete distortion can be viewed as a discrete approach to Ricci curvature for singular flat manifolds. We distinguish between two kinds of distortion, namely, vertex distortion, which is associated with the vertices of the tetrahedral mesh decomposing the 3‐manifold, and bond distortion, which is associated with the edges of the tetrahedral mesh. We investigate properties of vertex and bond distortions. As an example, we visualize vertex distortion on manifold hypersurfaces in R⁴ defined by a scalar field on a 3D mesh. distance fields.     

Streaming Surface Reconstruction Using Wavelets

We present a streaming method for reconstructing surfaces from large data sets generated by a laser range scanner using wavelets. Wavelets provide a localized, multiresolution representation of functions and this makes them ideal candidates for streaming surface reconstruction algorithms. We show how wavelets can be used to reconstruct the indicator function of a shape from a cloud of points with associated normals. Our method proceeds in several steps. We first compute a low‐resolution approximation of the indicator function using an octree followed by a second pass that incrementally adds fine resolution details. The indicator function is then smoothed using a modified octree convolution step and contoured to produce the final surface. Due to the local, multiresolution nature of wavelets, our approach results in an algorithm over 10 times faster than previous methods and can process extremely large data sets in the order of several hundred million points in only an hour.     

Global Intrinsic Symmetries of Shapes

Although considerable attention in recent years has been given to the problem of symmetry detection in general shapes, few methods have been developed that aim to detect and quantify the intrinsic symmetry of a shape rather than its extrinsic, or pose‐dependent symmetry. In this paper, we present a novel approach for efficiently computing symmetries of a shape which are invariant up to isometry preserving transformations. We show that the intrinsic symmetries of a shape are transformed into the Euclidean symmetries in the signature space defined by the eigenfunctions of the Laplace‐Beltrami operator. Based on this observation, we devise an algorithm which detects and computes the isometric mappings from the shape onto itself. We show that our approach is both computationally efficient and robust with respect to small non‐isometric deformations, even if they include topological changes.     

An Operator Approach to Tangent Vector Field Processing

In this paper, we introduce a novel coordinate‐free method for manipulating and analyzing vector fields on discrete surfaces. Unlike the commonly used representations of a vector field as an assignment of vectors to the faces of the mesh, or as real values on edges, we argue that vector fields can also be naturally viewed as operators whose domain and range are functions defined on the mesh. Although this point of view is common in differential geometry it has so far not been adopted in geometry processing applications. We recall the theoretical properties of vector fields represented as operators, and show that composition of vector fields with other functional operators is natural in this setup. This leads to the characterization of vector field properties through commutativity with other operators such as the Laplace‐Beltrami and symmetry operators, as well as to a straight‐forward definition of differential properties such as the Lie derivative. Finally, we demonstrate a range of applications, such as Killing vector field design, symmetric vector field estimation and joint design on multiple surfaces.     

Practical Anisotropic Geodesy

The computation of intrinsic, geodesic distances and geodesic paths on surfaces is a fundamental low‐level building block in countless Computer Graphics and Geometry Processing applications. This demand led to the development of numerous algorithms – some for the exact, others for the approximative computation, some focussing on speed, others providing strict guarantees. Most of these methods are designed for computing distances according to the standard Riemannian metric induced by the surface's embedding in Euclidean space. Generalization to other, especially anisotropic, metrics – which more recently gained interest in several application areas – is not rarely hampered by fundamental problems. We explore and discuss possibilities for the generalization and extension of well‐known methods to the anisotropic case, evaluate their relative performance in terms of accuracy and speed, and propose a novel algorithm, the Short‐Term Vector Dijkstra. This algorithm is strikingly simple to implement and proves to provide practical accuracy at a higher speed than generalized previous methods.     

SAFE: Structure‐aware facade editing

Many man‐made objects, in particular building facades, exhibit dominant structural relations such as symmetry and regularity. When editing these shapes, a common objective is to preserve these relations. However, often there are numerous plausible editing results that all preserve the desired structural relations of the input, creating ambiguity. We propose an interactive facade editing framework that explores this structural ambiguity. We first analyze the input in a semi‐automatic manner to detect different groupings of the facade elements and the relations among them. We then provide an incremental editing process where a set of variations that preserve the detected relations in a particular grouping are generated at each step. Starting from one input example, our system can quickly generate various facade configurations.