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).     

Skeleton-based Variational Mesh Deformations

In this paper, a new free-form shape deformation approach is proposed. We combine a skeleton-based mesh deformation technique with discrete differential coordinates in order to create natural-looking global shape deformations. Given a triangle mesh, we first extract a skeletal mesh, a two-sided Voronoibased approximation of the medial axis. Next the skeletal mesh is modified by free-form deformations. Then a desired global shape deformation is obtained by reconstructing the shape corresponding to the deformed skeletal mesh. The reconstruction is based on using discrete differential coordinates. Our method preserves fine geometric details and original shape thickness because of using discrete differential coordinates and skeleton-based deformations. We also develop a new mesh evolution technique which allow us to eliminate possible global and local self-intersections of the deformed mesh while preserving fine geometric details. Finally, we present a multi-resolution version of our approach in order to simplify and accelerate the deformation process. In addition, interesting links between the proposed free-form shape deformation technique and classical and modern results in the differential geometry of sphere congruences are established and discussed.     

A Hierarchical Segmentation of Articulated Bodies

This paper presents a novel segmentation method to assist the rigging of articulated bodies. The method computes a coarse‐to‐fine hierarchy of segments ordered by the level of detail. The results are invariant to deformations, and numerically robust to noise, irregular tessellations, and topological short‐circuits. The segmentation is based on two key ideas. First, it exploits the multiscale properties of the diffusion distance on surfaces, and then it introduces a new definition of medial structures, composing a bijection between medial structures and segments. Our method computes this bijection through a simple and fast iterative approach, and applies it to triangulated meshes.     

Hierarchical Convex Approximation of 3D Shapes for Fast Region Selection

Given a 3D solid model S represented by a tetrahedral mesh, we describe a novel algorithm to compute a hierarchy of convex polyhedra that tightly enclose S. The hierarchy can be browsed at interactive speed on a modern PC and it is useful for implementing an intuitive feature selection paradigm for 3D editing environments. Convex parts often coincide with perceptually relevant shape components and, for their identification, existing methods rely on the boundary surface only. In contrast, we show that the notion of part concavity can be expressed and implemented more intuitively and efficiently by exploiting a tetrahedrization of the shape volume. The method proposed is completely automatic, and generates a tree of convex polyhedra in which the root is the convex hull of the whole shape, and the leaves are the tetrahedra of the input mesh. The algorithm proceeds bottom‐up by hierarchically clustering tetrahedra into nearly convex aggregations, and the whole process is significantly fast. We prove that, in the average case, for a mesh of n tetrahedra O(n log² n) operations are sufficient to compute the whole tree.     

Curvature-Domain Shape Processing

We propose a framework for 3D geometry processing that provides direct access to surface curvature to facilitate advanced shape editing, filtering, and synthesis algorithms. The central idea is to map a given surface to the curvature domain by evaluating its principle curvatures, apply filtering and editing operations to the curvature distribution, and reconstruct the resulting surface using an optimization approach. Our system allows the user to prescribe arbitrary principle curvature values anywhere on the surface. The optimization solves a nonlinear least‐squares problem to find the surface that best matches the desired target curvatures while preserving important properties of the original shape. We demonstrate the effectiveness of this processing metaphor with several applications, including anisotropic smoothing, feature enhancement, and multi‐scale curvature editing.     

High-Resolution Volumetric Computation of Offset Surfaces with Feature Preservation

We present a new algorithm for the efficient and reliable generation of offset surfaces for polygonal meshes. The algorithm is robust with respect to degenerate configurations and computes (self‐)intersection free offsets that do not miss small and thin components. The results are correct within a prescribed ε‐tolerance. This is achieved by using a volumetric approach where the offset surface is defined as the union of a set of spheres, cylinders, and prisms instead of surface‐based approaches that generally construct an offset surface by shifting the input mesh in normal direction. Since we are using the unsigned distance field, we can handle any type of topological inconsistencies including non‐manifold configurations and degenerate triangles. A simple but effective mesh operation allows us to detect and include sharp features (shocks) into the output mesh and to preserve them during post‐processing (decimation and smoothing). We discretize the distance function by an efficient multi‐level scheme on an adaptive octree data structure. The problem of limited voxel resolutions inherent to every volumetric approach is avoided by breaking the bounding volume into smaller tiles and processing them independently. This allows for almost arbitrarily high voxel resolutions on a commodity PC while keeping the output mesh complexity low. The quality and performance of our algorithm is demonstrated for a number of challenging examples.     

Sparse points matching by combining 3D mesh saliency with statistical descriptors

This paper proposes new methodology for the detection and matching of salient points over several views of an object. The process is composed by three main phases. In the first step, detection is carried out by adopting a new perceptually‐inspired 3D saliency measure. Such measure allows the detection of few sparse salient points that characterize distinctive portions of the surface. In the second step, a statistical learning approach is considered to describe salient points across different views. Each salient point is modelled by a Hidden Markov Model (HMM), which is trained in an unsupervised way by using contextual 3D neighborhood information, thus providing a robust and invariant point signature. Finally, in the third step, matching among points of different views is performed by evaluating a pairwise similarity measure among HMMs. An extensive and comparative experimental session has been carried out, considering real objects acquired by a 3D scanner from different points of view, where objects come from standard 3D databases. Results are promising, as the detection of salient points is reliable, and the matching is robust and accurate.     

Context-Aware Skeletal Shape Deformation

We describe a system for the animation of a skeleton-controlled articulated object that preserves the fine geometric details of the object skin and conforms to the characteristic shapes of the object specified through a set of examples. The system provides the animator with an intuitive user interface and produces compelling results even when presented with a very small set of examples. In addition it is able to generalize well by extrapolating far beyond the examples.     

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.?     

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.     

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.     

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.     

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.     

A Statistical Model of Human Pose and Body Shape

Generation and animation of realistic humans is an essential part of many projects in today's media industry. Especially, the games and special effects industry heavily depend on realistic human animation. In this work a unified model that describes both, human pose and body shape is introduced which allows us to accurately model muscle deformations not only as a function of pose but also dependent on the physique of the subject. Coupled with the model's ability to generate arbitrary human body shapes, it severely simplifies the generation of highly realistic character animations. A learning based approach is trained on approximately 550 full body 3D laser scans taken of 114 subjects. Scan registration is performed using a non-rigid deformation technique. Then, a rotation invariant encoding of the acquired exemplars permits the computation of a statistical model that simultaneously encodes pose and body shape. Finally, morphing or generating meshes according to several constraints simultaneously can be achieved by training semantically meaningful regressors.     

Large‐Scale Integer Linear Programming for Orientation Preserving 3D Shape Matching

We study an algorithmic framework for computing an elastic orientation‐preserving matching of non‐rigid 3D shapes. We outline an Integer Linear Programming formulation whose relaxed version can be minimized globally in polynomial time. Because of the high number of optimization variables, the key algorithmic challenge lies in efficiently solving the linear program. We present a performance analysis of several Linear Programming algorithms on our problem. Furthermore, we introduce a multiresolution strategy which allows the matching of higher resolution models.     

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.     

Stability of Curvature Measures

We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the Gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive μ-reach can be estimated by the same curvature measures of the offset of a compact set K' close to K in the Hausdorff sense. We show how these curvature measures can be computed for finite unions of balls. The curvature measures of the offset of a compact set with positive μ-reach can thus be approximated by the curvature measures of the offset of a point-cloud sample.     

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.