34-Issue 5

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https://diglib.eg.org/handle/10.2312/12669

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A One-dimensional Homologically Persistent Skeleton of an Unstructured Point Cloud in any Metric Space
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Kurlin, Vitaliy; Mirela Ben-Chen and Ligang Liu
Real data are often given as a noisy unstructured point cloud, which is hard to visualize. The important problem is to represent topological structures hidden in a cloud by using skeletons with cycles. All past skeletonization methods require extra parameters such as a scale or a noise bound. We define a homologically persistent skeleton, which depends only on a cloud of points and contains optimal subgraphs representing 1-dimensional cycles in the cloud across all scales. The full skeleton is a universal structure encoding topological persistence of cycles directly on the cloud. Hence a 1-dimensional shape of a cloud can be now easily predicted by visualizing our skeleton instead of guessing a scale for the original unstructured cloud. We derive more subgraphs to reconstruct provably close approximations to an unknown graph given only by a noisy sample in any metric space. For a cloud of n points in the plane, the full skeleton and all its important subgraphs can be computed in time O(n log n).
Quaternion Julia Set Shape Optimization
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Kim, Theodore; Mirela Ben-Chen and Ligang Liu
We present the first 3D algorithm capable of answering the question: what would a Mandelbrot-like set in the shape of a bunny look like? More concretely, can we find an iterated quaternion rational map whose potential field contains an isocontour with a desired shape? We show that it is possible to answer this question by casting it as a shape optimization that discovers novel, highly complex shapes. The problem can be written as an energy minimization, the optimization can be made practical by using an efficient method for gradient evaluation, and convergence can be accelerated by using a variety of multi-resolution strategies. The resulting shapes are not invariant under common operations such as translation, and instead undergo intricate, non-linear transformations.
Analysis and Synthesis of 3D Shape Families via Deep-learned Generative Models of Surfaces
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Huang, Haibin; Kalogerakis, Evangelos; Marlin, Benjamin; Mirela Ben-Chen and Ligang Liu
We present a method for joint analysis and synthesis of geometrically diverse 3D shape families. Our method first learns part-based templates such that an optimal set of fuzzy point and part correspondences is computed between the shapes of an input collection based on a probabilistic deformation model. In contrast to previous template-based approaches, the geometry and deformation parameters of our part-based templates are learned from scratch. Based on the estimated shape correspondence, our method also learns a probabilistic generative model that hierarchically captures statistical relationships of corresponding surface point positions and parts as well as their existence in the input shapes. A deep learning procedure is used to capture these hierarchical relationships. The resulting generative model is used to produce control point arrangements that drive shape synthesis by combining and deforming parts from the input collection. The generative model also yields compact shape descriptors that are used to perform fine-grained classification. Finally, it can be also coupled with the probabilistic deformation model to further improve shape correspondence. We provide qualitative and quantitative evaluations of our method for shape correspondence, segmentation, fine-grained classification and synthesis. Our experiments demonstrate superior correspondence and segmentation results than previous state-of-the-art approaches.
Fast and Exact (Poisson) Solvers on Symmetric Geometries
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Kazhdan, Misha; Mirela Ben-Chen and Ligang Liu
In computer graphics, numerous geometry processing applications reduce to the solution of a Poisson equation. When considering geometries with symmetry, a natural question to consider is whether and how the symmetry can be leveraged to derive an efficient solver for the underlying system of linear equations. In this work we provide a simple representation-theoretic analysis that demonstrates how symmetries of the geometry translate into block diagonalization of the linear operators and we show how this results in efficient linear solvers for surfaces of revolution with and without angular boundaries.
Example Based Repetitive Structure Synthesis
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Roveri, Riccardo; Öztireli, A. Cengiz; Martin, Sebastian; Solenthaler, Barbara; Gross, Markus; Mirela Ben-Chen and Ligang Liu
We present an example based geometry synthesis approach for generating general repetitive structures. Our model is based on a meshless representation, unifying and extending previous synthesis methods. Structures in the example and output are converted into a functional representation, where the functions are defined by point locations and attributes. We then formulate synthesis as a minimization problem where patches from the output function are matched to those of the example. As compared to existing repetitive structure synthesis methods, the new algorithm offers several advantages. It handles general discrete and continuous structures, and their mixtures in the same framework. The smooth formulation leads to employing robust optimization procedures in the algorithm. Equipped with an accurate patch similarity measure and dedicated sampling control, the algorithm preserves local structures accurately, regardless of the initial distribution of output points. It can also progressively synthesize output structures in given subspaces, allowing users to interactively control and guide the synthesis in real-time. We present various results for continuous/discrete structures and their mixtures, residing on curves, submanifolds, volumes, and general subspaces, some of which are generated interactively.
Can Bi-cubic Surfaces be Class A?
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Karciauskas, Kestutis; Peters, Jörg; Mirela Ben-Chen and Ligang Liu
Class A surface' is a term in the automotive design industry, describing spline surfaces with aesthetic, non- oscillating highlight lines. Tensor-product B-splines of degree bi-3 (bicubic) are routinely used to generate smooth design surfaces and are often the de facto standard for downstream processing. To bridge the gap, this paper explores and gives a concrete suggestion, how to achieve good highlight line distributions for irregular bi-3 tensor-product patch layout by allowing, along some seams, a slight mismatch of normals below the industry- accepted tolerance of one tenth of a degree. Near the irregularities, the solution can be viewed as transforming a higher-degree, high-quality formally smooth surface into a bi-3 spline surface with few pieces, sacrificing formal smoothness but qualitatively retaining the shape.
An Image Degradation Model for Depth-augmented Image Editing
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Hennessey, James W.; Mitra, Niloy J.; Mirela Ben-Chen and Ligang Liu
Images remain the most popular medium to capture our surroundings. Although significant advances have been made in developing image editing tools, the key challenge is to intelligently account for missing depth information. The growing popularity of depth images offers a new avenue to revisit image editing tasks. In this work, we investigate how even coarse depth information can be exploited to address some of the fundamental challenges in image editing namely producing correct perspective, handling occlusion, and obtaining segmentation. To this end, we propose a novel image degradation model that predicts how well an image edit can be performed in presence of coarse depth information. Technically, we create proxy geometry to summarize available depth information, and use it to predict occlusions and ordering between image patches, complete occluded regions, and anticipate image-level changes under camera movement. We evaluate the proposed image degradation model in the context of parallax photography from single depth images.
Dynamic SfM: Detecting Scene Changes from Image Pairs
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Wang, Tuanfeng Y.; Kohli, Pushmeet; Mitra, Niloy J.; Mirela Ben-Chen and Ligang Liu
Detecting changes in scenes is important in many scene understanding tasks. In this paper, we pursue this goal simply from a pair of image recordings. Specifically, our goal is to infer what the objects are, how they are structured, and how they moved between the images. The problem is challenging as large changes make point-level correspondence establishment difficult, which in turn breaks the assumptions of standard Structure-from-Motion (SfM). We propose a novel algorithm for dynamic SfM wherein we first generate a pool of potential corresponding points by hypothesizing over possible movements, and then use a continuous optimization formulation to obtain a low complexity solution that best explains the scene recordings, i.e., the input image pairs. We test the algorithm on a variety of examples to recover the multiple object structures and their changes.
Homotopic Morphing of Planar Curves
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Dym, Nadav; Shtengel, Anna; Lipman, Yaron; Mirela Ben-Chen and Ligang Liu
This paper presents an algorithm for morphing between closed, planar piecewise-C1 curves. The morph is guaranteed to be a regular homotopy, meaning that pinching will not occur in the intermediate curves. The algorithm is based on a novel convex characterization of the space of regular closed curves and a suitable symmetric length-deviation energy. The intermediate curves constructed by the morphing algorithm are guaranteed to be regular due to the convexity and feasibility of the problem. We show that our method compares favorably with standard curve morphing techniques, and that these methods sometimes fail to produce a regular homotopy, and as a result produce an undesirable morph. We explore several applications and extensions of our approach, including morphing networks of curves with simple connectivity, morphing of curves with different turning numbers with minimal pinching, convex combination of several curves, and homotopic morphing of b-spline curves via their control polygon.
Hierarchical Multiview Rigid Registration
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Tang, Yizhi; Feng, Jieqing; Mirela Ben-Chen and Ligang Liu
Registration is a key step in the 3D reconstruction of real-world objects. In this paper, we propose a hierarchical method for the rigid registration of multiple views. The multiview registration problem is solved via hierarchical optimization defined on an undirected graph. Each node or edge in this graph represents a single view or a connection between two overlapped views, respectively. The optimizations are performed hierarchically on the edges, the loops, and the entire graph. First, each overlapped pair of views is locally aligned. Then, a loop-based incremental registration algorithm is introduced to refine the initial pairwise alignments. After a loop is registered, the views in the loop are merged into a metaview in the graph. Finally, global error diffusion is applied to the entire graph to evenly distribute the accumulated errors to all views. In addition, a new objective function is defined to describe the loop closure problem; it improves the accuracy and robustness of registration by simultaneously considering transformation and registration errors. The experimental results show that the proposed hierarchical approach is accurate, efficient and robust for initial view states that are not well posed.
Robust Articulated-ICP for Real-Time Hand Tracking
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Tagliasacchi, Andrea; Schröder, Matthias; Tkach, Anastasia; Bouaziz, Sofien; Botsch, Mario; Pauly, Mark; Mirela Ben-Chen and Ligang Liu
We present a robust method for capturing articulated hand motions in realtime using a single depth camera. Our system is based on a realtime registration process that accurately reconstructs hand poses by fitting a 3D articulated hand model to depth images. We register the hand model using depth, silhouette, and temporal information. To effectively map low-quality depth maps to realistic hand poses, we regularize the registration with kinematic and temporal priors, as well as a data-driven prior built from a database of realistic hand poses. We present a principled way of integrating such priors into our registration optimization to enable robust tracking without severely restricting the freedom of motion. A core technical contribution is a new method for computing tracking correspondences that directly models occlusions typical of single-camera setups. To ensure reproducibility of our results and facilitate future research, we fully disclose the source code of our implementation.
Continuous Matching via Vector Field Flow
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Corman, Etienne; Ovsjanikov, Maks; Chambolle, Antonin; Mirela Ben-Chen and Ligang Liu
We present a new method for non-rigid shape matching designed to enforce continuity of the resulting correspondence. Our method is based on the recently proposed functional map representation, which allows efficient manipulation and inference but often fails to provide a continuous point-to-point mapping. We address this problem by exploiting the connection between the operator representation of mappings and flows of vector fields. In particular, starting from an arbitrary continuous map between two surfaces we find an optimal flow that makes the final correspondence operator as close as possible to the initial functional map. Our method also helps to address the symmetric ambiguity problem inherent in many intrinsic correspondence methods when matching symmetric shapes. We provide practical and theoretical results showing that our method can be used to obtain an orientation preserving or reversing map starting from a functional map that represents the mixture of the two. We also show how this method can be used to improve the quality of maps produced by existing shape matching methods, and compare the resulting map's continuity with results obtained by other operator-based techniques.
Texture Mapping Real-World Objects with Hydrographics
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Panozzo, Daniele; Diamanti, Olga; Paris, Sylvain; Tarini, Marco; Sorkine, Evgeni; Sorkine-Hornung, Olga; Mirela Ben-Chen and Ligang Liu
In the digital world, assigning arbitrary colors to an object is a simple operation thanks to texture mapping. However, in the real world, the same basic function of applying colors onto an object is far from trivial. One can specify colors during the fabrication process using a color 3D printer, but this does not apply to already existing objects. Paint and decals can be used during post-fabrication, but they are challenging to apply on complex shapes. In this paper, we develop a method to enable texture mapping of physical objects, that is, we allow one to map an arbitrary color image onto a three-dimensional object. Our approach builds upon hydrographics, a technique to transfer pigments printed on a sheet of polymer onto curved surfaces. We first describe a setup that makes the traditional water transfer printing process more accurate and consistent across prints. We then simulate the transfer process using a specialized parameterization to estimate the mapping between the planar color map and the object surface. We demonstrate that our approach enables the application of detailed color maps onto complex shapes such as 3D models of faces and anatomical casts.
Learning Class-specific Descriptors for Deformable Shapes Using Localized Spectral Convolutional Networks
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Boscaini, Davide; Masci, Jonathan; Melzi, Simone; Bronstein, Michael M.; Castellani, Umberto; Vandergheynst, Pierre; Mirela Ben-Chen and Ligang Liu
In this paper, we propose a generalization of convolutional neural networks (CNN) to non-Euclidean domains for the analysis of deformable shapes. Our construction is based on localized frequency analysis (a generalization of the windowed Fourier transform to manifolds) that is used to extract the local behavior of some dense intrinsic descriptor, roughly acting as an analogy to patches in images. The resulting local frequency representations are then passed through a bank of filters whose coefficient are determined by a learning procedure minimizing a task-specific cost. Our approach generalizes several previous methods such as HKS, WKS, spectral CNN, and GPS embeddings. Experimental results show that the proposed approach allows learning class-specific shape descriptors significantly outperforming recent state-of-the-art methods on standard benchmarks.
Tight Relaxation of Quadratic Matching
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Kezurer, Itay; Kovalsky, Shahar Z.; Basri, Ronen; Lipman, Yaron; Mirela Ben-Chen and Ligang Liu
Establishing point correspondences between shapes is extremely challenging as it involves both finding sets of semantically persistent feature points, as well as their combinatorial matching.We focus on the latter and consider the Quadratic Assignment Matching (QAM) model.We suggest a novel convex relaxation for this NP-hard problem that builds upon a rank-one reformulation of the problem in a higher dimension, followed by relaxation into a semidefinite program (SDP). Our method is shown to be a certain hybrid of the popular spectral and doublystochastic relaxations of QAM and in particular we prove that it is tighter than both. Experimental evaluation shows that the proposed relaxation is extremely tight: in the majority of our experiments it achieved the certified global optimum solution for the problem, while other relaxations tend to produce suboptimal solutions. This, however, comes at the price of solving an SDP in a higher dimension. Our approach is further generalized to the problem of Consistent Collection Matching (CCM), where we solve the QAM on a collection of shapes while simultaneously incorporating a global consistency constraint. Lastly, we demonstrate an application to metric learning of collections of shapes.
Sparse Non-rigid Registration of 3D Shapes
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Yang, Jingyu; Li, Ke; Li, Kun; Lai, Yu-Kun; Mirela Ben-Chen and Ligang Liu
Non-rigid registration of 3D shapes is an essential task of increasing importance as commodity depth sensors become more widely available for scanning dynamic scenes. Non-rigid registration is much more challenging than rigid registration as it estimates a set of local transformations instead of a single global transformation, and hence is prone to the overfitting issue due to underdetermination. The common wisdom in previous methods is to impose an l2-norm regularization on the local transformation differences. However, the l2-norm regularization tends to bias the solution towards outliers and noise with heavy-tailed distribution, which is verified by the poor goodnessof- fit of the Gaussian distribution over transformation differences. On the contrary, Laplacian distribution fits well with the transformation differences, suggesting the use of a sparsity prior. We propose a sparse non-rigid registration (SNR) method with an l1-norm regularized model for transformation estimation, which is effectively solved by an alternate direction method (ADM) under the augmented Lagrangian framework. We also devise a multi-resolution scheme for robust and progressive registration. Results on both public datasets and our scanned datasets show the superiority of our method, particularly in handling large-scale deformations as well as outliers and noise.
Perfect Laplacians for Polygon Meshes
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Herholz, Philipp; Kyprianidis, Jan Eric; Alexa, Marc; Mirela Ben-Chen and Ligang Liu
A discrete Laplace-Beltrami operator is called perfect if it possesses all the important properties of its smooth counterpart. It is known which triangle meshes admit perfect Laplace operators and how to fix any other mesh by changing the combinatorics. We extend the characterization of meshes that admit perfect Laplacians to general polygon meshes. More importantly, we provide an algorithm that computes a perfect Laplace operator for any polygon mesh without changing the combinatorics, although, possibly changing the embedding. We evaluate this algorithm and demonstrate it at applications.
Reforming Shapes for Material-aware Fabrication
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Yang, Yong-Liang; Wang, Jun; Mitra, Niloy J.; Mirela Ben-Chen and Ligang Liu
As humans, we regularly associate shape of an object with its built material. In the context of geometric modeling, however, this inter-relation between form and material is rarely explored. In this work, we propose a novel datadriven reforming (i.e., reshaping) algorithm that adapts an input multi-component model for a target fabrication material. The algorithm adapts both the part geometry and the inter-part topology of the input shape to better align with material-aware fabrication requirements. As output, we produce the reshaped model along with respective part dimensions and inter-part junction specifications. We evaluate our algorithm on a range of man-made models and demonstrate a variety of model reshaping examples focusing only on metal and wooden materials.
Stable Topological Signatures for Points on 3D Shapes
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Carrière, Mathieu; Oudot, Steve Y.; Ovsjanikov, Maks; Mirela Ben-Chen and Ligang Liu
Comparing points on 3D shapes is among the fundamental operations in shape analysis. To facilitate this task, a great number of local point signatures or descriptors have been proposed in the past decades. However, the vast majority of these descriptors concentrate on the local geometry of the shape around the point, and thus are insensitive to its connectivity structure. By contrast, several global signatures have been proposed that successfully capture the overall topology of the shape and thus characterize the shape as a whole. In this paper, we propose the first point descriptor that captures the topology structure of the shape as 'seen' from a single point, in a multiscale and provably stable way. We also demonstrate how a large class of topological signatures, including ours, can be mapped to vectors, opening the door to many classical analysis and learning methods. We illustrate the performance of this approach on the problems of supervised shape labeling and shape matching. We show that our signatures provide complementary information to existing ones and allow to achieve better performance with less training data in both applications.
Frontmatter: Symposium on Geometry Processing 2015
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Mirela Ben-Chen; Ligang Liu; -

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