Semi‐Supervised Learning in Reconstructed Manifold Space for 3D Caricature Generation

Recently, automatic 3D caricature generation has attracted much attention from both the research community and the game industry. Machine learning has been proven effective in the automatic generation of caricatures. However, the lack of 3D caricature samples makes it challenging to train a good model. This paper addresses this problem by two steps. First, the training set is enlarged by reconstructing 3D caricatures. We reconstruct 3D caricatures based on some 2D caricature samples with a Principal Component Analysis (PCA)‐based method. Secondly, between the 2D real faces and the enlarged 3D caricatures, a regressive model is learnt by the semi‐supervised manifold regularization (MR) method. We then predict 3D caricatures for 2D real faces with the learnt model. The experiments show that our novel approach synthesizes the 3D caricature more effectively than traditional methods. Moreover, our system has been applied successfully in a massive multi‐user educational game to provide human‐like avatars.     

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.     

Multi‐Scale Geometry Interpolation

Interpolating vertex positions among triangle meshes with identical vertex‐edge graphs is a fundamental part of many geometric modelling systems. Linear vertex interpolation is robust but fails to preserve local shape. Most recent approaches identify local affine transformations for parts of the mesh, model desired interpolations of the affine transformations, and then optimize vertex positions to conform with the desired transformations. However, the local interpolation of the rotational part is non‐trivial for more than two input configurations and ambiguous if the meshes are deformed significantly. We propose a solution to the vertex interpolation problem that starts from interpolating the local metric (edge lengths) and mean curvature (dihedral angles) and makes consistent choices of local affine transformations using shape matching applied to successively larger parts of the mesh. The local interpolation can be applied to any number of input vertex configurations and due to the hierarchical scheme for generating consolidated vertex positions, the approach is fast and can be applied to very large meshes.     

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.     

Compressed Manifold Modes for Mesh Processing

This paper introduces compressed eigenfunctions of the Laplace‐Beltrami operator on 3D manifold surfaces. They constitute a novel functional basis, called the compressed manifold basis, where each function has local support. We derive an algorithm, based on the alternating direction method of multipliers (ADMM), to compute this basis on a given triangulated mesh. We show that compressed manifold modes identify key shape features, yielding an intuitive understanding of the basis for a human observer, where a shape can be processed as a collection of parts. We evaluate compressed manifold modes for potential applications in shape matching and mesh abstraction. Our results show that this basis has distinct advantages over existing alternatives, indicating high potential for a wide range of use‐cases in mesh processing.     

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.     

Fast Parallel Construction of Smooth Surfaces from Meshes with Tri/Quad/Pent Facets

Polyhedral meshes consisting of triangles, quads, and pentagons and polar configurations cover all major sampling and modeling scenarios. We give an algorithm for efficient local, parallel conversion of such meshes to an everywhere smooth surface consisting of low‐degree polynomial pieces. Quadrilateral facets with 4‐valent vertices are ‘regular’ and are mapped to bi‐cubic patches so that adjacent bi‐cubics join C² as for cubic tensor‐product splines. The algorithm can be implemented in the vertex and geometry shaders of the GPU pipeline and does not use the fragment shader. Its implementation in DirectX 10 achieves conversion plus rendering at 659 frames per second with 42.5 million triangles per second on input of a model of 1300 facets of which 60% are not regular.     

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.     

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.     

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.     

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).     

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.     

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.     

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.     

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 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.     

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.     

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.