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.     

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.     

Deformable 3D Shape Registration Based on Local Similarity Transforms

In this paper, a new method for deformable 3D shape registration is proposed. The algorithm computes shape transitions based on local similarity transforms which allows to model not only as‐rigid‐as‐possible deformations but also local and global scale. We formulate an ordinary differential equation (ODE) which describes the transition of a source shape towards a target shape. We assume that both shapes are roughly pre‐aligned (e.g., frames of a motion sequence). The ODE consists of two terms. The first one causes the deformation by pulling the source shape points towards corresponding points on the target shape. Initial correspondences are estimated by closest‐point search and then refined by an efficient smoothing scheme. The second term regularizes the deformation by drawing the points towards locally defined rest positions. These are given by the optimal similarity transform which matches the initial (undeformed) neighborhood of a source point to its current (deformed) neighborhood. The proposed ODE allows for a very efficient explicit numerical integration. This avoids the repeated solution of large linear systems usually done when solving the registration problem within general‐purpose non‐linear optimization frameworks. We experimentally validate the proposed method on a variety of real data and perform a comparison with several state‐of‐the‐art approaches.     

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.     

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.     

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).     

A Local/Global Approach to Mesh Parameterization

We present a novel approach to parameterize a mesh with disk topology to the plane in a shape‐preserving manner. Our key contribution is a local/global algorithm, which combines a local mapping of each 3D triangle to the plane, using transformations taken from a restricted set, with a global “stitch” operation of all triangles, involving a sparse linear system. The local transformations can be taken from a variety of families, e.g. similarities or rotations, generating different types of parameterizations. In the first case, the parameterization tries to force each 2D triangle to be an as‐similar‐as‐possible version of its 3D counterpart. This is shown to yield results identical to those of the LSCM algorithm. In the second case, the parameterization tries to force each 2D triangle to be an as‐rigid‐as‐possible version of its 3D counterpart. This approach preserves shape as much as possible. It is simple, effective, and fast, due to pre‐factoring of the linear system involved in the global phase. Experimental results show that our approach provides almost isometric parameterizations and obtains more shape‐preserving results than other state‐of‐the‐art approaches. We present also a more general “hybrid” parameterization model which provides a continuous spectrum of possibilities, controlled by a single parameter. The two cases described above lie at the two ends of the spectrum. We generalize our local/global algorithm to compute these parameterizations. The local phase may also be accelerated by parallelizing the independent computations per triangle.     

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.     

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.     

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.     

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.     

Probably Approximately Symmetric: Fast Rigid Symmetry Detection With Global Guarantees

We present a fast algorithm for global rigid symmetry detection with approximation guarantees. The algorithm is guaranteed to find the best approximate symmetry of a given shape, to within a user‐specified threshold, with very high probability. Our method uses a carefully designed sampling of the transformation space, where each transformation is efficiently evaluated using a sublinear algorithm. We prove that the density of the sampling depends on the total variation of the shape, allowing us to derive formal bounds on the algorithm's complexity and approximation quality. We further investigate different volumetric shape representations (in the form of truncated distance transforms), and in such a way control the total variation of the shape and hence the sampling density and the runtime of the algorithm. A comprehensive set of experiments assesses the proposed method, including an evaluation on the eight categories of the COSEG data set. This is the first large‐scale evaluation of any symmetry detection technique that we are aware of.     

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.     

Sketch-Based Procedural Surface Modeling and Compositing Using Surface Trees

We present a system for creating and manipulating layered procedural surface editing operations, which is motivated by the limited support for iterative design in free‐form modeling. A combination of sketch‐based and traditional modeling tools are used to design soft displacements, sharp creases, extrusions along 3D paths, and topological holes and handles. Using local parameterizations, these edits are combined in a dynamic hierarchy, enabling procedural operations like linked copy‐and‐paste and drag‐and‐drop layer‐based editing. Such dynamic, layered “surface compositing” is formalized as a Surface Tree, an analog of CSG trees which generalizes previous hierarchical surface modeling techniques. By “anchoring” tree nodes in the parameter space of lower layers, our surface tree implementation can better preserve the semantics of an edit as the underlying surface changes. Details of our implementation are described, including an efficient procedural mesh data structure.     

Learning Line Features in 3D Geometry

Feature detection in geometric datasets is a fundamental tool for solving shape matching problems such as partial symmetry detection. Traditional techniques usually employ a priori models such as crease lines that are unspecific to the actual application. Our paper examines the idea of learning geometric features. We introduce a formal model for a class of linear feature constellations based on a Markov chain model and propose a novel, efficient algorithm for detecting a large number of features simultaneously. After a short user‐guided training stage, in which one or a few example lines are sketched directly onto the input data, our algorithm automatically finds all pieces of geometry similar to the marked areas. In particular, the algorithm is able recognize larger classes of semantically similar but geometrically varying features, which is very difficult using unsupervised techniques. In a number of experiments, we apply our technique to point cloud data from 3D scanners. The algorithm is able to detect features with very low rates of false positives and negatives and to recognize broader classes of similar geometry (such as “windows” in a building scan) even from few training examples, thereby significantly improving over previous unsupervised techniques.     

Adaptive Space Deformations Based on Rigid Cells

We propose a new adaptive space deformation method for interactive shape modeling. A novel energy formulation based on elastically coupled volumetric cells yields intuitive detail preservation even under large deformations. By enforcing rigidity of the cells, we obtain an extremely robust numerical solver for the resulting nonlinear optimization problem. Scalability is achieved using an adaptive spatial discretization that is decoupled from the resolution of the embedded object. Our approach is versatile and easy to implement, supports thin-shell and solid deformations of 2D and 3D objects, and is applicable to arbitrary sample-based representations, such as meshes, triangle soups, or point clouds.     

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.     

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.     

Interactive Cover Design Considering Physical Constraints

We developed an interactive system to design a customized cover for a given three‐dimensional (3D) object such as a camera, teapot, or car. The system first computes the convex hull of the input geometry. The user segments it into several cloth patches by drawing on the 3D surface. This paper provides two technical contributions. First, it introduces a specialized flattening algorithm for cover patches. It makes each two‐dimensional edge in the flattened pattern equal to or longer than the original 3D edge; a smaller patch would fail to cover the object, and a larger patch would result in extra wrinkles. Second, it introduces a mechanism to verify that the user‐specified opening would be large enough for the object to be removed. Starting with the initial configuration, the system virtually “pulls” the object out of the cover while avoiding excessive stretching of cloth patches. We used the system to design real covers and confirmed that it functions as intended.     

Range Scan Registration Using Reduced Deformable Models

We present an unsupervised method for registering range scans of deforming, articulated shapes. The key idea is to model the motion of the underlying object using a reduced deformable model. We use a linear skinning model for its simplicity and represent the weight functions on a regular grid localized to the surface geometry. This decouples the deformation model from the surface representation and allows us to deal with the severe occlusion and missing data that is inherent in range scan data. We formulate the registration problem using an objective function that enforces close alignment of the 3D data and includes an intuitive notion of joints. This leads to an optimization problem that we solve using an efficient EM-type algorithm. With our algorithm we obtain smooth deformations that accurately register pairs of range scans with significant motion and occlusion. The main advantages of our approach are that it does not require user specified markers, a template, nor manual segmentation of the surface geometry into rigid parts.