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21
Symmetry in 3D Geometry: Extraction and Applications
, 2012
"... The concept of symmetry has received significant attention in computer graphics and computer vision research in recent years. Numerous methods have been proposed to find and extract geometric symmetries and exploit such highlevel structural information for a wide variety of geometry processing task ..."
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Cited by 29 (7 self)
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The concept of symmetry has received significant attention in computer graphics and computer vision research in recent years. Numerous methods have been proposed to find and extract geometric symmetries and exploit such highlevel structural information for a wide variety of geometry processing tasks. This report surveys and classifies recent developments in symmetry detection. We focus on elucidating the similarities and differences between existing methods to gain a better understanding of a fundamental problem in digital geometry processing and shape understanding in general. We discuss a variety of applications in computer graphics and geometry that benefit from symmetry information for more effective processing. An analysis of the strengths and limitations of existing algorithms highlights the plenitude of opportunities for future research both in terms of theory and applications.
Inverse Procedural Modeling by Automatic Generation of Lsystems
"... We present an important step towards the solution of the problem of inverse procedural modeling by generating parametric contextfree Lsystems that represent an input 2D model. The Lsystem rules efficiently code the regular structures and the parameters represent the properties of the structure tr ..."
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We present an important step towards the solution of the problem of inverse procedural modeling by generating parametric contextfree Lsystems that represent an input 2D model. The Lsystem rules efficiently code the regular structures and the parameters represent the properties of the structure transformations. The algorithm takes as input a 2D vector image that is composed of atomic elements, such as curves and polylines. Similar elements are recognized and assigned terminal symbols of an Lsystem alphabet. The terminal symbols ’ position and orientation are pairwise compared and the transformations are stored as points in multiple 4D transformation spaces. By careful analysis of the clusters in the transformation spaces, we detect sequences of elements and code them as Lsystem rules. The coded elements are then removed from the clusters, the clusters are updated, and then the analysis attempts to code groups of elements in (hierarchies) the same way. The analysis ends with a single group of elements that is coded as an Lsystem axiom. We recognize and code branching sequences of linearly translated, scaled, and rotated elements and their hierarchies. The Lsystem not only represents the input image, but it can also be used for various editing operations. By changing the Lsystem parameters, the image can be randomized, symmetrized, and groups of elements and regular structures can be edited. By changing the terminal and nonterminal symbols, elements or groups of elements can be replaced.
SymmetryAware Mesh Processing
"... Abstract. Perfect, partial, and approximate symmetries are pervasive in 3D surface meshes of realworld objects. However, current digital geometry processing algorithms generally ignore them, instead focusing on local shape features and differential surface properties. This paper investigates how de ..."
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Cited by 15 (2 self)
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Abstract. Perfect, partial, and approximate symmetries are pervasive in 3D surface meshes of realworld objects. However, current digital geometry processing algorithms generally ignore them, instead focusing on local shape features and differential surface properties. This paper investigates how detection of largescale symmetries can be used to guide processing of 3D meshes. It investigates a framework for mesh processing that includes steps for symmetrization (applying a warp to make a surface more symmetric) and symmetric remeshing (approximating a surface with a mesh having symmetric topology). These steps can be used to enhance the symmetries of a mesh, to decompose a mesh into its symmetric parts and asymmetric residuals, and to establish correspondences between symmetric mesh features. Applications are demonstrated for modeling, beautification, and simplification of nearly symmetric surfaces. Key words: symmetry analysis, mesh processing 1
Symmetry detection using feature lines
 Comput. Graph. Forum
, 2009
"... In this paper, we describe a new algorithm for detecting structural redundancy in geometric data sets. Our algorithm computes rigid symmetries, i.e., subsets of a surface model that reoccur several times within the model differing only by translation, rotation or mirroring. Our algorithm is based on ..."
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Cited by 13 (0 self)
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In this paper, we describe a new algorithm for detecting structural redundancy in geometric data sets. Our algorithm computes rigid symmetries, i.e., subsets of a surface model that reoccur several times within the model differing only by translation, rotation or mirroring. Our algorithm is based on matching locally coherent constellations of feature lines on the object surfaces. In comparison to previous work, the new algorithm is able to detect a large number of symmetric parts without restrictions to regular patterns or nested hierarchies. In addition, working on relevant features only leads to a strong reduction in memory and processing costs such that very large data sets can be handled. We apply the algorithm to a number of real world 3D scanner data sets, demonstrating high recognition rates for general patterns of symmetry.
Factored Facade Acquisition using Symmetric Line Arrangements
"... We introduce a novel framework for imagebased 3D reconstruction of urban buildings based on symmetry priors. Starting from imagelevel edges, we generate a sparse and approximate set of consistent 3D lines. These lines are then used to simultaneously detect symmetric line arrangements while refinin ..."
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Cited by 11 (5 self)
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We introduce a novel framework for imagebased 3D reconstruction of urban buildings based on symmetry priors. Starting from imagelevel edges, we generate a sparse and approximate set of consistent 3D lines. These lines are then used to simultaneously detect symmetric line arrangements while refining the estimated 3D model. Operating both on 2D image data and intermediate 3D feature representations, we perform iterative feature consolidation and effective outlier pruning, thus eliminating reconstruction artifacts arising from ambiguous or wrong stereo matches. We exploit nonlocal coherence of symmetric elements to generate precise model reconstructions, even in the presence of a significant amount of outlier imageedges arising from reflections, shadows, outlier objects, etc. We evaluate our algorithm on several challenging test scenarios, both synthetic and real. Beyond reconstruction, the extracted symmetry patterns are useful towards interactive and intuitive model manipulations.
Closedform Blending of Local Symmetries
, 2010
"... We present a closedform solution for the symmetrization problem, solving for the optimal deformation that reconciles aset of local bilateral symmetries.Given as input aset of pointpairs which should be symmetric,we first compute for each local neighborhood a transformation which wouldproduce anapp ..."
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Cited by 7 (0 self)
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We present a closedform solution for the symmetrization problem, solving for the optimal deformation that reconciles aset of local bilateral symmetries.Given as input aset of pointpairs which should be symmetric,we first compute for each local neighborhood a transformation which wouldproduce anapproximate bilateral symmetry. We then solve forasingle global symmetry whichincludes all of these local symmetries,while minimizingthe deformation within each local neighborhood. Our main motivation is the symmetrization of digitized fossils, which areoftendeformedbyacombinationofcompressionandbending. Inaddition, we use the technique tosymmetrize articulated models.
Landmarkguided elastic shape analysis of sphericallyparameterized surfaces
 Computer Graphics Forum
, 2013
"... Figure 1: Top: Geodesic path between two surfaces with missing parts. Our approach can compute dense correspondences (illustrated with matching colors) and geodesics efficiently. Bottom: symmetrization of a highly articulated shape. The length of the geodesic provides a measure of asymmetry of the s ..."
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Cited by 4 (2 self)
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Figure 1: Top: Geodesic path between two surfaces with missing parts. Our approach can compute dense correspondences (illustrated with matching colors) and geodesics efficiently. Bottom: symmetrization of a highly articulated shape. The length of the geodesic provides a measure of asymmetry of the shape. Its midpoint is a symmetric version of the initial shape. We argue that full surface correspondence (registration) and optimal deformations (geodesics) are two related problems and propose a framework that solves them simultaneously. We build on the Riemannian shape analysis of anatomical and starshaped surfaces of Kurtek et al. and focus on articulated complex shapes that undergo elastic deformations and that may contain missing parts. Our core contribution is the reformulation of Kurtek et al.’s approach as a constrained optimization over all possible reparameterizations of the surfaces, using a sparse set of corresponding landmarks. We introduce a landmarkconstrained basis, which we use to numerically solve this optimization and therefore establish full surface registration and geodesic deformation between two surfaces. The length of the geodesic provides a measure of dissimilarity between surfaces. The advantages of this approach are: (1) simultaneous computation of full correspondence and geodesic between two surfaces, given a sparse set of matching landmarks (2) ability to handle more comprehensive deformations than nearly isometric, and (3) the geodesics and the geodesic lengths can be further used for symmetrizing 3D shapes and for computing their
SymmetryGuided Texture Synthesis and Manipulation
"... This paper presents a framework for symmetryguided texture synthesis and processing. It is motivated by the longstanding problem of how to optimize, transfer, and control the spatial patterns in textures. The key idea is that symmetry representations that measure autocorrelations with respect to a ..."
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Cited by 3 (2 self)
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This paper presents a framework for symmetryguided texture synthesis and processing. It is motivated by the longstanding problem of how to optimize, transfer, and control the spatial patterns in textures. The key idea is that symmetry representations that measure autocorrelations with respect to all transformations of a group are a natural way to describe spatial patterns in many realworld textures. To leverage this idea, we provide methods to transfer symmetry representations from one texture to another, process the symmetries of a texture, and optimize textures with respect to properties of their symmetry representations. These methods are automatic and robust, as they don’t require explicit detection of discrete symmetries. Applications are investigated for optimizing, processing and transferring symmetries and textures.
Pairwise Harmonics for Shape Analysis
"... This paper introduces a simple yet effective shape analysis mechanism for geometry processing. Unlike traditional shape analysis techniques which compute descriptors per surface point up to certain neighborhoods, we introduce a shape analysis framework in which the descriptors are based on pairs of ..."
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Cited by 3 (0 self)
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This paper introduces a simple yet effective shape analysis mechanism for geometry processing. Unlike traditional shape analysis techniques which compute descriptors per surface point up to certain neighborhoods, we introduce a shape analysis framework in which the descriptors are based on pairs of surface points. Such a pairwise analysis approach leads to a new class of shape descriptors that are more global, discriminative and can effectively capture the variations in the underlying geometry. Specifically, we introduce new shape descriptors based on the isocurves of harmonic functions whose global maximum and minimum occur at the point pair. We show that these shape descriptors can infer shape structures and consistently lead to simpler and more efficient algorithms than the stateoftheart methods for three applications: intrinsic reflectional symmetry axis computation, matching shape extremities, and simultaneous surface segmentation and skeletonization.
Single Viewpoint Model Completion of Symmetric Objects for Digital Inspection
"... The ability to create complete 3D models of real world objects is an important task for various applications. In digital inspection, complete models allow users to analyze the entirety of an object. However, various difficulties arise for imagebased acquisition techniques. First, the viewpoint plann ..."
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Cited by 2 (0 self)
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The ability to create complete 3D models of real world objects is an important task for various applications. In digital inspection, complete models allow users to analyze the entirety of an object. However, various difficulties arise for imagebased acquisition techniques. First, the viewpoint planning problem must be solved. Second, each of the resulting viewpoint captures must be combined with either zippering or 3D triangulation, both difficult problems. We observe that if an object is symmetric, then the object’s symmetry can be exploited so that a single viewpoint capture is sufficient to generate a complete, 3D triangulated model. In our work, three problems of previous approaches to generating complete models are avoided or minimized: 1) we avoid 3D triangulation, 2) we avoid searches for geometry to extend our models, and 3) we minimize viewpoint planning to the selection of a single viewpoint. Our approach also includes algorithms to mitigate global deformations due to capture error. We demonstrate our approach by capturing, reconstructing, and completing several scenes of one or more objects and illustrating several digital inspection methods with these scenes.