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# SGP11: Eurographics Symposium on Geometry Processing

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### Browsing SGP11: Eurographics Symposium on Geometry Processing by Subject "Computational Geometry and Object Modeling"

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Item A Condition Number for Non-Rigid Shape Matching(The Eurographics Association and Blackwell Publishing Ltd., 2011) Ovsjanikov, Maks; Huang, Qi-Xing; Guibas, Leonidas; Mario Botsch and Scott SchaeferShow more Despite the large amount of work devoted in recent years to the problem of non-rigid shape matching, practical methods that can successfully be used for arbitrary pairs of shapes remain elusive. In this paper, we study the hardness of the problem of shape matching, and introduce the notion of the shape condition number, which captures the intuition that some shapes are inherently more difficult to match against than others. In particular, we make a connection between the symmetry of a given shape and the stability of any method used to match it while optimizing a given distortion measure. We analyze two commonly used classes of methods in deformable shape matching, and show that the stability of both types of techniques can be captured by the appropriate notion of a condition number. We also provide a practical way to estimate the shape condition number and show how it can be used to guide the selection of landmark correspondences between shapes. Thus we shed some light on the reasons why general shape matching remains difficult and provide a way to detect and mitigate such difficulties in practice.Show more Item CubeCover - Parameterization of 3D Volumes(The Eurographics Association and Blackwell Publishing Ltd., 2011) Nieser, Matthias; Reitebuch, Ulrich; Polthier, Konrad; Mario Botsch and Scott SchaeferShow more Despite the success of quad-based 2D surface parameterization methods, effective parameterization algorithms for 3D volumes with cubes, i.e. hexahedral elements, are still missing. CUBECOVER is a first approach for generating a hexahedral tessellation of a given volume with boundary aligned cubes which are guided by a frame field. The input of CUBECOVER is a tetrahedral volume mesh. First, a frame field is designed with manual input from the designer. It guides the interior and boundary layout of the parameterization. Then, the parameterization and the hexahedral mesh are computed so as to align with the given frame field. CUBECOVER has similarities to the QUADCOVER algorithm and extends it from 2D surfaces to 3D volumes. The paper also provides theoretical results for 3D hexahedral parameterizations and analyses topological properties of the appropriate function space.Show more Item Deformable 3D Shape Registration Based on Local Similarity Transforms(The Eurographics Association and Blackwell Publishing Ltd., 2011) Papazov, Chavdar; Burschka, Darius; Mario Botsch and Scott SchaeferShow more 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 closestpoint 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.Show more Item Functional Webs for Freeform Architecture(The Eurographics Association and Blackwell Publishing Ltd., 2011) Deng, B.; Pottmann, Helmut; Wallner, Johannes; Mario Botsch and Scott SchaeferShow more 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.Show more Item A Hierarchical Grid Based Framework for Fast Collision Detection(The Eurographics Association and Blackwell Publishing Ltd., 2011) Fan, Wenshan; Wang, Bin; Paul, Jean-Claude; Sun, Jiaguang; Mario Botsch and Scott SchaeferShow more We present a novel hierarchical grid based method for fast collision detection (CD) for deformable models on GPU architecture. A two-level grid is employed to accommodate the non-uniform distribution of practical scene geometry. A bottom-to-top method is implemented to assign the triangles into the hierarchical grid without any iteration while a deferred scheme is introduced to efficiently update the data structure. To address the issue of load balancing, which greatly influences the performance in SIMD parallelism, a propagation scheme which utilizes a parallel scan and a segmented scan is presented, distributing workloads evenly across all concurrent threads. The proposed method supports both discrete collision detection (DCD) and continuous collision detection (CCD) with self-collision. Some typical benchmarks are tested to verify the effectiveness of our method. The results highlight our speedups over prior algorithms on different commodity GPUs.Show more Item Large-Scale Integer Linear Programming for Orientation Preserving 3D Shape Matching(The Eurographics Association and Blackwell Publishing Ltd., 2011) Windheuser, Thomas; Schlickewei, Ulrich; Schmidt, Frank R.; Cremers, Daniel; Mario Botsch and Scott SchaeferShow more 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.Show more Item A Multiscale Approach to Optimal Transport(The Eurographics Association and Blackwell Publishing Ltd., 2011) MÃ©rigot, Quentin; Mario Botsch and Scott SchaeferShow more In this paper, we propose an improvement of an algorithm of Aurenhammer, Hoffmann and Aronov to find a least square matching between a probability density and finite set of sites with mass constraints, in the Euclidean plane. Our algorithm exploits the multiscale nature of this optimal transport problem. We iteratively simplify the target using Lloyd's algorithm, and use the solution of the simplified problem as a rough initial solution to the more complex one. This approach allows for fast estimation of distances between measures related to optimal transport (known as Earth-mover or Wasserstein distances). We also discuss the implementation of these algorithms, and compare the original one to its multiscale counterpart.Show more Item A Multiscale Metric for 3D Mesh Visual Quality Assessment(The Eurographics Association and Blackwell Publishing Ltd., 2011) LavouÃ©, Guillaume; Mario Botsch and Scott SchaeferShow more Many processing operations are nowadays applied on 3D meshes like compression, watermarking, remeshing and so forth; these processes are mostly driven and/or evaluated using simple distortion measures like the Hausdorff distance and the root mean square error, however these measures do not correlate with the human visual perception while the visual quality of the processed meshes is a crucial issue. In that context we introduce a full-reference 3D mesh quality metric; this metric can compare two meshes with arbitrary connectivity or sampling density and produces a score that predicts the distortion visibility between them; a visual distortion map is also created. Our metric outperforms its counterparts from the state of the art, in term of correlation with mean opinion scores coming from subjective experiments on three existing databases. Additionally, we present an application of this new metric to the improvement of rate-distortion evaluation of recent progressive compression algorithms.Show more Item On Approximation of the Laplace-Beltrami Operator and the Willmore Energy of Surfaces(The Eurographics Association and Blackwell Publishing Ltd., 2011) Hildebrandt, Klaus; Polthier, Konrad; Mario Botsch and Scott SchaeferShow more Discrete Laplace Beltrami operators on polyhedral surfaces play an important role for various applications in geometry processing and related areas like physical simulation or computer graphics. While discretizations of the weak Laplace Beltrami operator are well-studied, less is known about the strong form. We present a principle for constructing strongly consistent discrete Laplace Beltrami operators based on the cotan weights. The consistency order we obtain, improves previous results reported for the mesh Laplacian. Furthermore, we prove consistency of the discrete Willmore energies corresponding to the discrete Laplace Beltrami operators.Show more Item On the Shape of a Set of Points and Lines in the Plane(The Eurographics Association and Blackwell Publishing Ltd., 2011) Kreveld, Marc van; Lankveld, Thijs van; Veltkamp, Remco C.; Mario Botsch and Scott SchaeferShow more Detailed geometric models of the real world are in increasing demand. LiDAR data is appropriate to reconstruct urban models. In urban scenes, the individual surfaces can be reconstructed and connected to form the scene geometry. There are various methods for reconstructing the free-form shape of a point sample on a single surface. However, these methods do not take the context of the surface into account. We present the guided a-shape: an extension of the well known a-shape that uses lines (guides) to indicate preferred locations for the boundary of the shape. The guided a-shape uses (parts of) these lines as boundary where the points suggest that this is appropriate. We prove that the guided a-shape can be constructed in O((n+m) log(n+m)) time, from an input of n points and m guides. We apply guided a-shapes to urban reconstruction from LiDAR, where neighboring surfaces can be connected conveniently along their intersection lines into adjacent surfaces of a 3D model. We analyze guided a-shapes of both synthetic and real data and show they are consistently better than a-shapes for this application.Show more Item Skeleton Computation of Orthogonal Polyhedra(The Eurographics Association and Blackwell Publishing Ltd., 2011) Martinez, Jonas; Vigo, Marc; Pla-Garcia, Nuria; Mario Botsch and Scott SchaeferShow more Skeletons are powerful geometric abstractions that provide useful representations for a number of geometric operations. The straight skeleton has a lower combinatorial complexity compared with the medial axis. Moreover, while the medial axis of a polyhedron is composed of quadric surfaces the straight skeleton just consist of planar faces. Although there exist several methods to compute the straight skeleton of a polygon, the straight skeleton of polyhedra has been paid much less attention. We require to compute the skeleton of very large datasets storing orthogonal polyhedra. Furthermore, we need to treat geometric degeneracies that usually arise when dealing with orthogonal polyhedra. We present a new approach so as to robustly compute the straight skeleton of orthogonal polyhedra. We follow a geometric technique that works directly with the boundary of an orthogonal polyhedron. Our approach is output sensitive with respect to the number of vertices of the skeleton and solves geometric degeneracies. Unlike the existing straight skeleton algorithms that shrink the object boundary to obtain the skeleton, our algorithm relies on the plane sweep paradigm. The resulting skeleton is only composed of axis-aligned and 45 rotated planar faces and edges.Show more Item Surface Patches from Unorganized Space Curves(The Eurographics Association and Blackwell Publishing Ltd., 2011) Abbasinejad, Fatemeh; Joshi, Pushkar; Amenta, Nina; Mario Botsch and Scott SchaeferShow more Recent 3D sketch tools produce networks of three-space curves that suggest the contours of shapes. The shapes may be non-manifold, closed three-dimensional, open two-dimensional, or mixed. We describe a system that automatically generates intuitively appealing piecewise-smooth surfaces from such a curve network, and an intelligent user interface for modifying the automatically chosen surface patches. Both the automatic and the semi-automatic parts of the system use a linear algebra representation of the set of surface patches to track the topology. On complicated inputs from ILoveSketch [BBS08], our system allows the user to build the desired surface with just a few mouse-clicks.Show more