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16
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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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.
Intrinsic Regularity Detection in 3D Geometry
"... Abstract. Automatic detection of symmetries, regularity, and repetitive structures in 3D geometry is a fundamental problem in shape analysis and pattern recognition with applications in computer vision and graphics. Especially challenging is to detect intrinsic regularity, where the repetitions are ..."
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Abstract. Automatic detection of symmetries, regularity, and repetitive structures in 3D geometry is a fundamental problem in shape analysis and pattern recognition with applications in computer vision and graphics. Especially challenging is to detect intrinsic regularity, where the repetitions are on an intrinsic grid, without any apparent Euclidean pattern to describe the shape, but rising out of (near) isometric deformation of the underlying surface. In this paper, we employ multidimensional scaling to reduce the problem of intrinsic structure detection to a simpler problem of 2D grid detection. Potential 2D grids are then identified using an autocorrelation analysis, refined using local fitting, validated, and finally projected back to the spatial domain. We test the detection algorithm on a variety of scanned plaster models in presence of imperfections like missing data, noise and outliers. We also present a range of applications including scan completion, shape editing, superresolution, and structural correspondence. 1
MultiScale Partial Intrinsic Symmetry Detection
"... shown in uniform color. Note the detection of inter and intraobject symmetries, as well as cylindrical symmetry of the limbs. We present an algorithm for multiscale partial intrinsic symmetry detection over 2D and 3D shapes, where the scale of a symmetric region is defined by intrinsic distances ..."
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Cited by 5 (1 self)
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shown in uniform color. Note the detection of inter and intraobject symmetries, as well as cylindrical symmetry of the limbs. We present an algorithm for multiscale partial intrinsic symmetry detection over 2D and 3D shapes, where the scale of a symmetric region is defined by intrinsic distances between symmetric points over the region. To identify prominent symmetric regions which overlap and vary in form and scale, we decouple scale extraction and symmetry extraction by performing two levels of clustering. First, significant symmetry scales are identified by clustering sample point pairs from an input shape. Since different point pairs can share a common point, shape regions covered by points in different scale clusters can overlap. We introduce the symmetry scale matrix (SSM), where each entry estimates the likelihood two point pairs belong to symmetries at the same scale. The pairtopair symmetry affinity is computed based on a pair signature which encodes scales. We perform spectral clustering using the SSM to obtain the scale clusters. Then for all points belonging to the same scale cluster, we perform the secondlevel spectral clustering, based on a novel pointtopoint symmetry affinity measure, to extract partial symmetries at that scale. We demonstrate our algorithm on complex shapes possessing rich symmetries at multiple scales. Links: DL PDF WEB DATA 1
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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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.
Hierarchical matching of nonrigid shapes
 International Conference on Scale Space and Variational Methods in Computer Vision (SSVM’11
, 2011
"... Abstract. Detecting similarity between nonrigid shapes is one of the fundamental problems in computer vision. While rigid alignment can be parameterized using a small number of unknowns representing rotations, reflections and translations, nonrigid alignment does not have this advantage. The major ..."
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Abstract. Detecting similarity between nonrigid shapes is one of the fundamental problems in computer vision. While rigid alignment can be parameterized using a small number of unknowns representing rotations, reflections and translations, nonrigid alignment does not have this advantage. The majority of the methods addressing this problem boil down to a minimization of a distortion measure. The complexity of a matching process is exponential by nature, but it can be heuristically reduced to a quadratic or even linear for shapes which are smooth twomanifolds. Here we model shapes using both local and global structures, and provide a hierarchical framework for the quadratic matching problem.
Möbius Transformations For Global Intrinsic Symmetry Analysis
"... The goal of our work is to develop an algorithm for automatic and robust detection of global intrinsic symmetries in 3D surface meshes. Our approach is based on two core observations. First, symmetry invariant point sets can be detected robustly using critical points of the Average Geodesic Distance ..."
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The goal of our work is to develop an algorithm for automatic and robust detection of global intrinsic symmetries in 3D surface meshes. Our approach is based on two core observations. First, symmetry invariant point sets can be detected robustly using critical points of the Average Geodesic Distance (AGD) function. Second, intrinsic symmetries are selfisometries of surfaces and as such are contained in the low dimensional group of Möbius transformations. Based on these observations, we propose an algorithm that: 1) generates a set of symmetric points by detecting critical points of the AGD function, 2) enumerates small subsets of those feature points to generate candidate Möbius transformations,and 3) selects among those candidate Möbius transformations the one(s) that best map the surface onto itself. The main advantages of this algorithm stem from the stability of the AGD in predicting potential symmetric point features and the low dimensionality of the Möbius group for enumerating potential selfmappings. During experiments with a benchmark set of meshes augmented with humanspecified symmetric correspondences, we find that the algorithm is able to find intrinsic symmetries for a wide variety of object types with moderate deviations from perfect symmetry. 1.
Y.: 3d symmetry detection and analysis using the pseudopolar fourier transform
 International Journal of Computer Vision
"... is a fundamental task in a gamut of scientific fields such as computer vision, medical imaging and pattern recognition to name a few. In this work, we present a computational approach to 3D symmetry detection and analysis. Our analysis is conducted in the Fourier domain using the pseudopolar Fourier ..."
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is a fundamental task in a gamut of scientific fields such as computer vision, medical imaging and pattern recognition to name a few. In this work, we present a computational approach to 3D symmetry detection and analysis. Our analysis is conducted in the Fourier domain using the pseudopolar Fourier transform. The pseudopolar representation enables to efficiently and accurately analyze angular volumetric properties such as rotational symmetries. Our algorithm is based on the analysis of the angular correspondence rate of the given volume and its rotated and rotatedinverted replicas in their pseudopolar representations. We also derive a novel rigorous analysis of the inherent constraints of 3D symmetries via groupstheory based analysis. Thus, our algorithm starts by detecting the rotational symmetry group of a given volume, and the rigorous analysis results pave the way to detect the rest of the symmetries. The complexity of the algorithm is O(N 3 log(N)), where N × N × N is the volumetric size in each direction. This complexity is independent of the number of the detected symmetries. We experimentally verified our approach by applying it to synthetic as well as real 3D objects.
Topology Robust Intrinsic Symmetries of nonrigid shapes based on Diffusion Distances
, 2009
"... Detection and modeling of selfsimilarity and symmetry is important in shape recognition, matching, synthesis, and reconstruction. While the detection of rigid shape symmetries is well established, the study of symmetries in nonrigid shapes is a much less researched problem. A particularly challeng ..."
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Detection and modeling of selfsimilarity and symmetry is important in shape recognition, matching, synthesis, and reconstruction. While the detection of rigid shape symmetries is well established, the study of symmetries in nonrigid shapes is a much less researched problem. A particularly challenging setting is the detection of symmetries in nonrigid shapes affected by topological noise and asymmetric connectivity. In this paper, we treat
Transformed Polynomials for Global Registration of Point Clouds
"... In this paper, we introduce a novel approach for global registration of partially overlapping point clouds. The approach identifies feature points of matching objects based on surfaceapproximating polynomials and finds an initial transformation depending on these polynomials. We compute an extended ..."
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In this paper, we introduce a novel approach for global registration of partially overlapping point clouds. The approach identifies feature points of matching objects based on surfaceapproximating polynomials and finds an initial transformation depending on these polynomials. We compute an extended set of rotationallyinvariant features for polynomials. In contrast to purely featurebased approaches, we do not only compute transformations based on the invariant properties of polynomials, but actually transform the polynomials into a common coordinate system and compare the transformed coefficients. This results in an improved correspondence analysis of local surfaces. Hence, using transformed polynomials, we gain more discriminating information about different structures. Therefore, the approach can handle partial scans of different objects simultaneously. Each partial scan is assigned to one of the objects and registered accordingly. Moreover, the approach is robust against noise and can process real data.
Shape Palindromes: Analysis of Intrinsic Symmetries in 2D Articulated Shapes
"... Abstract. Analysis of intrinsic symmetries of nonrigid and articulated shapes is an important problem in pattern recognition with numerous applications ranging from medicine to computational aesthetics. Considering articulated planar shapes as closed curves, we show how to represent their extrinsic ..."
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Abstract. Analysis of intrinsic symmetries of nonrigid and articulated shapes is an important problem in pattern recognition with numerous applications ranging from medicine to computational aesthetics. Considering articulated planar shapes as closed curves, we show how to represent their extrinsic and intrinsic symmetries as selfsimilarities of local descriptor sequences, which in turn have simple interpretation in the frequency domain. The problem of symmetry detection and analysis thus boils down to analysis of descriptor sequence patterns. For that purpose, we show two efficient computational methods: one based on Fourier analysis, and another on dynamic programming. Metaphorically, the later can be compared to finding palindromes in text sequences. 1