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59
Partial and approximate symmetry detection for 3D geometry
 ACM TRANSACTIONS ON GRAPHICS
, 2006
"... “Symmetry is a complexityreducing concept [...]; seek it everywhere.” Alan J. Perlis Many natural and manmade objects exhibit significant symmetries or contain repeated substructures. This paper presents a new algorithm that processes geometric models and efficiently discovers and extracts a com ..."
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Cited by 180 (28 self)
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“Symmetry is a complexityreducing concept [...]; seek it everywhere.” Alan J. Perlis Many natural and manmade objects exhibit significant symmetries or contain repeated substructures. This paper presents a new algorithm that processes geometric models and efficiently discovers and extracts a compact representation of their Euclidean symmetries. These symmetries can be partial, approximate, or both. The method is based on matching simple local shape signatures in pairs and using these matches to accumulate evidence for symmetries in an appropriate transformation space. A clustering stage extracts potential significant symmetries of the object, followed by a verification step. Based on a statistical sampling analysis, we provide theoretical guarantees on the success rate of our algorithm. The extracted symmetry graph representation captures important highlevel information about the structure of a geometric model which in turn enables a large set of further processing operations, including shape compression, segmentation, consistent editing, symmetrization, indexing for retrieval, etc.
ExampleBased 3D Scan Completion
 EUROGRAPHICS SYMPOSIUM ON GEOMETRY PROCESSING
, 2005
"... Optical acquisition devices often produce noisy and incomplete data sets, due to occlusion, unfavorable surface reflectance properties, or geometric restrictions in the scanner setup. We present a novel approach for obtaining a complete and consistent 3D model representation from such incomplete sur ..."
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Cited by 87 (24 self)
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Optical acquisition devices often produce noisy and incomplete data sets, due to occlusion, unfavorable surface reflectance properties, or geometric restrictions in the scanner setup. We present a novel approach for obtaining a complete and consistent 3D model representation from such incomplete surface scans, using a database of 3D shapes to provide geometric priors for regions of missing data. Our method retrieves suitable context models from the database, warps the retrieved models to conform with the input data, and consistently blends the warped models to obtain the final consolidated 3D shape. We define a shape matching penalty function and corresponding optimization scheme for computing the nonrigid alignment of the context models with the input data. This allows a quantitative evaluation and comparison of the quality of the shape extrapolation provided by each model. Our algorithms are explicitly designed to accommodate uncertain data and can thus be applied directly to raw scanner output. We show on a variety of real data sets how consistent models can be obtained from highly incomplete input. The information gained during the shape completion process can be utilized for future scans, thus continuously simplifying the creation of complex 3D models.
4Points Congruent Sets for Robust Pairwise Surface Registration
 INTERNATIONAL CONFERENCE ON COMPUTER GRAPHICS AND INTERACTIVE TECHNIQUES
, 2008
"... We introduce 4PCS, a fast and robust alignment scheme for 3D point sets that uses wide bases, which are known to be resilient to noise and outliers. The algorithm allows registering raw noisy data, possibly contaminated with outliers, without prefiltering or denoising the data. Further, the method ..."
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Cited by 48 (2 self)
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We introduce 4PCS, a fast and robust alignment scheme for 3D point sets that uses wide bases, which are known to be resilient to noise and outliers. The algorithm allows registering raw noisy data, possibly contaminated with outliers, without prefiltering or denoising the data. Further, the method significantly reduces the number of trials required to establish a reliable registration between the underlying surfaces in the presence of noise, without any assumptions about starting alignment. Our method is based on a novel technique to extract all coplanar 4points sets from a 3D point set that are approximately congruent, under rigid transformation, to a given set of coplanar 4points. This extraction procedure runs in roughly O(n2 + k) time, where n is the number of candidate points and k is the number of reported 4points sets. In practice, when noise level is low and there is sufficient overlap, using local descriptors the time complexity reduces to O(n + k). We also propose an extension to handle similarity and affine transforms. Our technique achieves an order of magnitude asymptotic acceleration compared to common randomized alignment techniques. We demonstrate the robustness of our algorithm on several sets of multiple range scans with varying degree of noise, outliers, and extent of overlap.
Scan registration for autonomous mining vehicles using 3DNDT
 Journal of Field Robotics
, 2007
"... Scan registration is an essential subtask when building maps based on range finder data from mobile robots. The problem is to deduce how the robot has moved between consecutive scans, based on the shape of overlapping portions of the scans. This paper presents a new algorithm for registration of 3D ..."
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Cited by 46 (13 self)
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Scan registration is an essential subtask when building maps based on range finder data from mobile robots. The problem is to deduce how the robot has moved between consecutive scans, based on the shape of overlapping portions of the scans. This paper presents a new algorithm for registration of 3D data. The algorithm is a generalization and improvement of the normal distributions transform �NDT � for 2D data developed by Biber and Strasser, which allows for accurate registration using a memoryefficient representation of the scan surface. A detailed quantitative and qualitative comparison of the new algorithm with the 3D version of the popular ICP �iterative closest point � algorithm is presented. Results with actual mine data, some of which were collected with a new prototype 3D laser scanner, show that the presented algorithm is faster and slightly more reliable than the standard ICP algorithm for 3D registration, while using a more memoryefficient scan surface representation. © 2007 Wiley Periodicals, Inc. 1.
Exploring collections of 3D models using fuzzy correspondences
"... Large collections of 3D models from the same object class (e.g., chairs, cars, animals) are now commonly available via many public repositories, but exploring the range of shape variations across such collections remains a challenging task. In this work, we present a new exploration interface that a ..."
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Cited by 45 (17 self)
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Large collections of 3D models from the same object class (e.g., chairs, cars, animals) are now commonly available via many public repositories, but exploring the range of shape variations across such collections remains a challenging task. In this work, we present a new exploration interface that allows users to browse collections based on similarities and differences between shapes in userspecified regions of interest (ROIs). To support this interactive system, we introduce a novel analysis method for computing similarity relationships between points on 3D shapes across a collection. We encode the inherent ambiguity in these relationships using fuzzy point correspondences and propose a robust and efficient computational framework that estimates fuzzy correspondences using only a sparse set of pairwise model alignments. We evaluate our analysis method on a range of correspondence benchmarks and report substantial improvements in both speed and accuracy over existing alternatives. In addition, we demonstrate how fuzzy correspondences enable key features in our exploration tool, such as automated view alignment, ROIbased similarity search, and faceted browsing.
Geometry and convergence analysis of algorithms for registration of 3D shapes
, 2006
"... The computation of a rigid body transformation which optimally aligns a set of measurement points with a surface and related registration problems are studied from the viewpoint of geometry and optimization. We provide a convergence analysis for widely used registration algorithms such as ICP, using ..."
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Cited by 45 (6 self)
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The computation of a rigid body transformation which optimally aligns a set of measurement points with a surface and related registration problems are studied from the viewpoint of geometry and optimization. We provide a convergence analysis for widely used registration algorithms such as ICP, using either closest points (Besl and McKay [2]) or tangent planes at closest points (Chen and Medioni [4]), and for a recently developed approach based on quadratic approximants of the squared distance function [24]. ICP based on closest points exhibits local linear convergence only. Its counterpart which minimizes squared distances to the tangent planes at closest points is a GaussNewton iteration; it achieves local quadratic convergence for a zero residual problem and – if enhanced by regularization and step size control – comes close to quadratic convergence in many realistic scenarios. Quadratically convergent algorithms are based on the approach in [24]. The theoretical results are supported by a number of experiments; there, we also compare the algorithms with respect to global convergence behavior, stability and running time.
Integral Invariants for Robust Geometry Processing
 IN: ICCV ’95: PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON COMPUTER VISION. IEEE COMPUTER SOCIETY
, 2005
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TopologyInvariant Similarity of Nonrigid Shapes
, 2009
"... This paper explores the problem of similarity criteria between nonrigid shapes. Broadly speaking, such criteria are divided into intrinsic and extrinsic, the first referring to the metric structure of the object and the latter to how it is laid out in the Euclidean space. Both criteria have their ..."
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Cited by 33 (3 self)
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This paper explores the problem of similarity criteria between nonrigid shapes. Broadly speaking, such criteria are divided into intrinsic and extrinsic, the first referring to the metric structure of the object and the latter to how it is laid out in the Euclidean space. Both criteria have their advantages and disadvantages: extrinsic similarity is sensitive to nonrigid deformations, while intrinsic similarity is sensitive to topological noise. In this paper, we approach the problem from the perspective of metric geometry. We show that by unifying the extrinsic and intrinsic similarity criteria, it is possible to obtain a stronger topologyinvariant similarity, suitable for comparing deformed shapes with different topology. We construct this new joint criterion as a tradeoff between the extrinsic and intrinsic similarity and use it as a setvalued distance. Numerical results demonstrate the efficiency of our approach in cases where using either extrinsic or intrinsic criteria alone would fail.
Generalized Surface Flows for Mesh Processing
, 2007
"... Geometric flows are ubiquitous in mesh processing. Curve and surface evolutions based on functional minimization have been used in the context of surface diffusion, denoising, shape optimization, minimal surfaces, and geodesic paths to mention a few. Such gradient flows are nearly always, yet often ..."
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Cited by 32 (2 self)
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Geometric flows are ubiquitous in mesh processing. Curve and surface evolutions based on functional minimization have been used in the context of surface diffusion, denoising, shape optimization, minimal surfaces, and geodesic paths to mention a few. Such gradient flows are nearly always, yet often implicitly, based on the canonical L 2 inner product of vector fields. In this paper, we point out that changing this inner product provides a simple, powerful, and untapped approach to extend current flows. We demonstrate the value of such a norm alteration for regularization and volumepreservation purposes and in the context of shape matching, where deformation priors (ranging from rigid motion to articulated motion) can be incorporated into a gradient flow to drastically improve results. Implementation details, including a differentiable approximation of the Hausdorff distance between irregular meshes, are presented.
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.