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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 123 (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.
A Multiscale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis Functions
, 2003
"... In this paper, we propose a hierarchical approach to 3D scattered data interpolation with compactly supported basis functions. Our numerical experiments suggest that the approach integrates the best aspects of scattered data fitting with locally and globally supported basis functions. Employing loca ..."
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Cited by 61 (3 self)
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In this paper, we propose a hierarchical approach to 3D scattered data interpolation with compactly supported basis functions. Our numerical experiments suggest that the approach integrates the best aspects of scattered data fitting with locally and globally supported basis functions. Employing locally supported functions leads to an efficient computational procedure, while a coarsetofine hierarchy makes our method insensitive to the density of scattered data and allows us to restore large parts of missed data. Given a point
Scattered Data Fitting on the Sphere
 in Mathematical Methods for Curves and Surfaces II
, 1998
"... . We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulat ..."
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Cited by 50 (5 self)
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. We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulations, spherical splines, spherical radial basis functions, and some associated multiresolution methods. In addition, we briefly discuss spherelike surfaces, visualization, and methods for more general surfaces. The paper includes a total of 206 references. x1. Introduction Let S be the unit sphere in IR 3 , and suppose that fv i g n i=1 is a set of scattered points lying on S. In this paper we are interested in the following problem: Problem 1. Given real numbers fr i g n i=1 , find a (smooth) function s defined on S which interpolates the data in the sense that s(v i ) = r i ; i = 1; : : : ; n; (1) or approximates it in the sense that s(v i ) ß r i ; i = 1; : : : ; n: (2) Data f...
3D Scattered Data Approximation with Adaptive Compactly Supported Radial Basis Functions
"... In this paper, we develop an adaptive RBF fitting procedure for a high quality approximation of a set of points scattered over a piecewise smooth surface. We use compactly supported RBFs whose centers are randomly chosen from the points. The randomness is controlled by the point density and surface ..."
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Cited by 38 (2 self)
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In this paper, we develop an adaptive RBF fitting procedure for a high quality approximation of a set of points scattered over a piecewise smooth surface. We use compactly supported RBFs whose centers are randomly chosen from the points. The randomness is controlled by the point density and surface geometry. For each RBF, its support size is chosen adaptively according to surface geometry at a vicinity of the RBF center. All these lead to a noiserobust high quality approximation of the set. We also adapt our basic technique for shape reconstruction from registered range scans by taking into account measurement confidences. Finally, an interesting link between our RBF fitting procedure and partition of unity approximations is established and discussed.
Kernel Techniques: From Machine Learning to Meshless Methods
, 2006
"... Kernels are valuable tools in various fields of Numerical Analysis, including approximation, interpolation, meshless methods for solving partial differential equations, neural networks, and Machine Learning. This contribution explains why and how kernels are applied in these disciplines. It uncovers ..."
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Cited by 35 (11 self)
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Kernels are valuable tools in various fields of Numerical Analysis, including approximation, interpolation, meshless methods for solving partial differential equations, neural networks, and Machine Learning. This contribution explains why and how kernels are applied in these disciplines. It uncovers the links between them, as far as they are related to kernel techniques. It addresses nonexpert readers and focuses on practical guidelines for using kernels in applications.
Solving Differential Equations with Radial Basis Functions: Multilevel Methods and Smoothing
 Advances in Comp. Math
"... . Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a ..."
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Cited by 35 (7 self)
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. Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a multilevel framework is demonstrated. A possible connection between multigrid finite elements and multilevel radial basis function methods with smoothing is explored. Various numerical examples are also provided throughout the paper. 1. Introduction During the past few years the idea of using socalled meshless methods for the numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community. As a few representative examples we mention Belytschko and coworker's results [3] using the socalled elementfree Galerkin method, Duarte and Oden's work [11] using hp clouds, Babuska and Melenk 's work [1] on the partition of unity method, ...
On the Efficiency of Interpolation by Radial Basis Functions
, 1997
"... . We study the computational complexity, the error behavior, and the numerical stability of interpolation by radial basis functions. It turns out that these issues are intimately connected. For the case of compactly supported radial basis functions, we consider the possibility of getting reasonably ..."
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Cited by 30 (8 self)
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. We study the computational complexity, the error behavior, and the numerical stability of interpolation by radial basis functions. It turns out that these issues are intimately connected. For the case of compactly supported radial basis functions, we consider the possibility of getting reasonably good reconstructions of dvariate functions from N data at O(Nd) computational cost and give some supporting theoretical results and numerical examples. x1. Optimal Recovery Given function values f(x 1 ); : : : ; f(xN ) on a discrete set X = fx 1 ; : : : ; xN g of scattered locations x j 2 IR d , we want to recover a function f on some given domain\Omega ` IR d that contains X. Under certain assumptions to be stated below, an optimal reconstruction takes the form of interpolation by another function s 2 S ae C(IR d ) with s(x k ) = f(x k ); 1 k N . Due to the MairhuberCurtis [3] theorem, the space S of interpolants must depend on X, as is the case for classical splines and fini...
Multistep Approximation Algorithms: Improved Convergence Rates through Postconditioning with Smoothing Kernels
 Advances in Comp. Math. 10
, 1999
"... . We show how certain widely used multistep approximation algorithms can be interpreted as instances of an approximate Newton method. It was shown in an earlier paper by the second author that the convergence rates of approximate Newton methods (in the context of the numerical solution of PDEs) suff ..."
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Cited by 28 (11 self)
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. We show how certain widely used multistep approximation algorithms can be interpreted as instances of an approximate Newton method. It was shown in an earlier paper by the second author that the convergence rates of approximate Newton methods (in the context of the numerical solution of PDEs) suffer from a "loss of derivatives", and that the subsequent linear rate of convergence can be improved to be superlinear using an adaptation of NashMoser iteration for numerical analysis purposes; the essence of the adaptation being a splitting of the inversion and the smoothing into two separate steps. We show how these ideas apply to scattered data approximation as well as the numerical solution of partial differential equations. We investigate the use of several radial kernels for the smoothing operation. In our numerical examples we use radial basis functions also in the inversion step. 1. Introduction It has been only very recently that the idea of multistep (or multilevel) interpolation ...
A Meshless Hierarchical Representation for Light Transport
"... Figure 1: Left: Realtime global illumination on a static 2.3M triangle scene. Both the light and the viewpoint can be moved freely at 721 frames per second after a little less than half an hour of precomputation on a single PC. Right: The indirect illumination expressed in our meshless hierarchica ..."
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Cited by 23 (0 self)
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Figure 1: Left: Realtime global illumination on a static 2.3M triangle scene. Both the light and the viewpoint can be moved freely at 721 frames per second after a little less than half an hour of precomputation on a single PC. Right: The indirect illumination expressed in our meshless hierarchical basis (emphasized for visualization). Green dots represent nonzero coefficients. We introduce a meshless hierarchical representation for solving light transport problems. Precomputed radiance transfer (PRT) and finite elements require a discrete representation of illumination over the scene. Nonhierarchical approaches such as pervertex values are simple to implement, but lead to long precomputation. Hierarchical bases like wavelets lead to dramatic acceleration, but in their basic form they work well only on flat or smooth surfaces. We introduce a hierarchical function basis induced by scattered data approximation. It is decoupled from the geometric representation, allowing the hierarchical representation of illumination on complex objects. We present simple data structures and algorithms for constructing and evaluating the basis functions. Due to its hierarchical nature, our representation adapts to the complexity of the illumination, and can be queried at different scales. We demonstrate the power of the new basis in a novel precomputed directtoindirect light transport algorithm that greatly increases the complexity of scenes that can be handled by PRT approaches.