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102
RidgeValley Lines on Meshes via Implicit Surface Fitting
 ACM TRANS. GRAPH
, 2004
"... We propose a simple and effective method for detecting view and scaleindependent ridgevalley lines defined via first and secondorder curvature derivatives on shapes approximated by dense triangle meshes. A highquality estimation of highorder surface derivatives is achieved by combining multil ..."
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Cited by 101 (8 self)
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We propose a simple and effective method for detecting view and scaleindependent ridgevalley lines defined via first and secondorder curvature derivatives on shapes approximated by dense triangle meshes. A highquality estimation of highorder surface derivatives is achieved by combining multilevel implicit surface fitting and finite difference approximations. We demonstrate that the ridges and valleys are geometrically and perceptually salient surface features and, therefore, can be potentially used for shape recognition, coding, and quality evaluation purposes.
Estimating Curvatures and Their Derivatives on Triangle Meshes
, 2004
"... The computation of curvature and other differential properties of surfaces is essential for many techniques in analysis and rendering. We present a finitedifferences approach for estimating curvatures on irregular triangle meshes that may be thought of as an extension of a common method for estimat ..."
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Cited by 92 (1 self)
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The computation of curvature and other differential properties of surfaces is essential for many techniques in analysis and rendering. We present a finitedifferences approach for estimating curvatures on irregular triangle meshes that may be thought of as an extension of a common method for estimating pervertex normals. The technique is efficient in space and time, and results in significantly fewer outlier estimates while more broadly offering accuracy comparable to existing methods. It generalizes naturally to computing derivatives of curvature and higherorder surface differentials.
Discovering Structural Regularity in 3D Geometry
, 2008
"... We introduce a computational framework for discovering regular or repeated geometric structures in 3D shapes. We describe and classify possible regular structures and present an effective algorithm for detecting such repeated geometric patterns in point or meshbased models. Our method assumes no p ..."
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Cited by 82 (10 self)
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We introduce a computational framework for discovering regular or repeated geometric structures in 3D shapes. We describe and classify possible regular structures and present an effective algorithm for detecting such repeated geometric patterns in point or meshbased models. Our method assumes no prior knowledge of the geometry or spatial location of the individual elements that define the pattern. Structure discovery is made possible by a careful analysis of pairwise similarity transformations that reveals prominent lattice structures in a suitable model of transformation space. We introduce an optimization method for detecting such uniform grids specifically designed to deal with outliers and missing elements. This yields a robust algorithm that successfully discovers complex regular structures amidst clutter, noise, and missing geometry. The accuracy of the extracted generating transformations is further improved using a novel simultaneous registration method in the spatial domain. We demonstrate the effectiveness of our algorithm on a variety of examples and show applications to compression, model repair, and geometry synthesis.
Voronoibased Variational Reconstruction of Unoriented Point Sets
, 2007
"... We introduce an algorithm for reconstructing watertight surfaces from unoriented point sets. Using the Voronoi diagram of the input point set, we deduce a tensor field whose principal axes and eccentricities locally represent respectively the most likely direction of the normal to the surface, and t ..."
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Cited by 52 (8 self)
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We introduce an algorithm for reconstructing watertight surfaces from unoriented point sets. Using the Voronoi diagram of the input point set, we deduce a tensor field whose principal axes and eccentricities locally represent respectively the most likely direction of the normal to the surface, and the confidence in this direction estimation. An implicit function is then computed by solving a generalized eigenvalue problem such that its gradient is most aligned with the principal axes of the tensor field, providing a bestfitting isosurface reconstruction. Our approach possesses a number of distinguishing features. In particular, the implicit function optimization provides resilience to noise, adjustable fitting to the data, and controllable smoothness of the reconstructed surface. Finally, the use of simplicial meshes (possibly restricted to a thin crust around the input data) and (an)isotropic Laplace operators renders the numerical treatment simple and robust.
Registration of Point Cloud Data from a Geometric Optimization Perspective
, 2004
"... We propose a framework for pairwise registration of shapes represented by point cloud data (PCD). We assume that the points are sampled from a surface and formulate the problem of aligning two PCDs as a minimization of the squared distance between the underlying surfaces. Local quadratic approximant ..."
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Cited by 44 (11 self)
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We propose a framework for pairwise registration of shapes represented by point cloud data (PCD). We assume that the points are sampled from a surface and formulate the problem of aligning two PCDs as a minimization of the squared distance between the underlying surfaces. Local quadratic approximants of the squared distance function are used to develop a linear system whose solution gives the best aligning rigid transform for the given pair of point clouds. The rigid transform is applied and the linear system corresponding to the new orientation is build. This process is iterated until it converges. The pointtopoint and the pointtoplane Iterated Closest Point (ICP) algorithms can be treated as special cases in this framework. Our algorithm can align PCDs even when they are placed far apart, and is experimentally found to be more stable than pointtoplane ICP. We analyze the convergence behavior of our algorithm and of pointtopoint and pointtoplane ICP under our proposed framework, and derive bounds on their rate of convergence. We compare the stability and convergence properties of our algorithm with other registration algorithms on a variety of scanned data.
Smooth Feature Lines on Surface Meshes
, 2005
"... Feature lines are salient surface characteristics. Their definition involves third and fourth order surface derivatives. This often yields to unpleasantly rough and squiggly feature lines since third order derivatives are highly sensitive against unwanted surface noise. The present work proposes two ..."
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Cited by 42 (2 self)
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Feature lines are salient surface characteristics. Their definition involves third and fourth order surface derivatives. This often yields to unpleasantly rough and squiggly feature lines since third order derivatives are highly sensitive against unwanted surface noise. The present work proposes two novel concepts for a more stable algorithm producing visually more pleasing feature lines: First, a new computation scheme based on discrete differential geometry is presented, avoiding costly computations of higher order approximating surfaces. Secondly, this scheme is augmented by a filtering method for higher order surface derivatives to improve both the stability of the extraction of feature lines and the smoothness of their appearance.
Fast and robust detection of crest lines on meshes
 Proc. of ACM Symposium on Solid and Physical Modeling
, 2005
"... We propose a fast and robust method for detecting crest lines on surfaces approximated by dense triangle meshes. The crest lines, salient surface features defined via first and secondorder curvature derivatives, are widely used for shape matching and interrogation purposes. Their practical extract ..."
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Cited by 25 (2 self)
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We propose a fast and robust method for detecting crest lines on surfaces approximated by dense triangle meshes. The crest lines, salient surface features defined via first and secondorder curvature derivatives, are widely used for shape matching and interrogation purposes. Their practical extraction is difficult because it requires good estimation of highorder surface derivatives. Our approach to the crest line detection is based on estimating the curvature tensor and curvature derivatives via local polynomial fitting. Since the crest lines are not defined in the surface regions where the surface focal set (caustic) degenerates, we introduce a new thresholding scheme which exploits interesting relationships between curvature extrema, the socalled MVS functional of Moreton and Sequin, and Dupin cyclides, An application of the crest lines to adaptive mesh simplification is also considered.
Robust feature classification and editing
 IEEE Trans. Visualization and Computer Graphics
"... Abstract—Sharp edges, ridges, valleys, and prongs are critical for the appearance and an accurate representation of a 3D model. In this paper, we propose a novel approach that deals with the global shape of features in a robust way. Based on a remeshing algorithm which delivers an isotropic mesh in ..."
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Cited by 20 (7 self)
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Abstract—Sharp edges, ridges, valleys, and prongs are critical for the appearance and an accurate representation of a 3D model. In this paper, we propose a novel approach that deals with the global shape of features in a robust way. Based on a remeshing algorithm which delivers an isotropic mesh in a featuresensitive metric, features are recognized on multiple scales via integral invariants of local neighborhoods. Morphological and smoothing operations are then used for feature region extraction and classification into basic types such as ridges, valleys, and prongs. The resulting representation of feature regions is further used for featurespecific editing operations.
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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Robust Principal Curvatures on Multiple Scales
"... Geometry processing algorithms often require the robust extraction of curvature information. We propose to achieve this with principal component analysis (PCA) of local neighborhoods, defined via spherical kernels centered on the given surface Φ. Intersection of a kernel ball Br or its boundary sphe ..."
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Cited by 16 (8 self)
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Geometry processing algorithms often require the robust extraction of curvature information. We propose to achieve this with principal component analysis (PCA) of local neighborhoods, defined via spherical kernels centered on the given surface Φ. Intersection of a kernel ball Br or its boundary sphere Sr with the volume bounded by Φ leads to the socalled ball and sphere neighborhoods. Information obtained by PCA of these neighborhoods turns out to be more robust than PCA of the patch neighborhood Br ∩ Φ previously used. The relation of the quantities computed by PCA with the principal curvatures of Φ is revealed by an asymptotic analysis as the kernel radius r tends to zero. This also allows us to define principal curvatures “at scale r ” in a way which is consistent with the classical setting. The advantages of the new approach are discussed in a comparison with results obtained by normal cycles and local fitting; whereas the former method somewhat lacks in robustness, the latter does not achieve a consistent behavior at features on coarse scales. As to applications, we address computing principal curves and feature extraction on multiple scales. 1.