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Reconstruction and Representation of 3D Objects with Radial Basis Functions
 Computer Graphics (SIGGRAPH ’01 Conf. Proc.), pages 67–76. ACM SIGGRAPH
, 2001
"... We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs al ..."
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Cited by 500 (1 self)
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We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow us to model large data sets, consisting of millions of surface points, by a single RBFpreviously an impossible task. A greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages. The energyminimisation characterisation of polyharmonic splines result in a "smoothest" interpolant. This scaleindependent characterisation is wellsuited to reconstructing surfaces from nonuniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a noninterpolating approximation when the data is noisy. The functional representation is in effect a solid model, which means that gradients and surface normals can be determined analytically. This helps generate uniform meshes and we show that the RBF representation has advantages for mesh simplification and remeshing applications. Results are presented for realworld rangefinder data.
Surface Interpolation With Radial Basis Functions for Medical Imaging
 IEEE Transactions on Medical Imaging
, 1997
"... Radial basis functions are presented as a practical solution to the problem of interpolating incomplete surfaces derived from threedimensional (3D) medical graphics. The specific application considered is the design of cranial implants for the repair of defects, usually holes, in the skull. Radial ..."
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Cited by 92 (2 self)
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Radial basis functions are presented as a practical solution to the problem of interpolating incomplete surfaces derived from threedimensional (3D) medical graphics. The specific application considered is the design of cranial implants for the repair of defects, usually holes, in the skull. Radial basis functions impose few restrictions on the geometry of the interpolation centers and are suited to problems where the interpolation centers do not form a regular grid. However, their high computational requirements have previously limited their use to problems where the number of interpolation centers is small (! 300). Recently developed fast evaluation techniques have overcome these limitations and made radial basis interpolation a practical approach for larger data sets. In this paper radial basis functions are fitted to depthmaps of the skull's surface, obtained from Xray CT data using raytracing techniques. They are used to smoothly interpolate the surface of the skull across defe...
Approximating and Intersecting Surfaces from Points
, 2003
"... Point sets become an increasingly popular shape representation. Most shape processing and rendering tasks require the approximation of a continuous surface from the point data. We present a surface approximation that is motivated by an efficient iterative ray intersection computation. On each poin ..."
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Cited by 74 (3 self)
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Point sets become an increasingly popular shape representation. Most shape processing and rendering tasks require the approximation of a continuous surface from the point data. We present a surface approximation that is motivated by an efficient iterative ray intersection computation. On each point on a ray, a local normal direction is estimated as the direction of smallest weighted covariances of the points. The normal direction is used to build a local polynomial approximation to the surface, which is then intersected with the ray. The distance to the polynomials essentially defines a distance field, whose zeroset is computed by repeated ray intersection. Requiring the distance field to be smooth leads to an intuitive and natural sampling criterion, namely, that normals derived from the weighted covariances are well defined in a tubular neighborhood of the surface. For certain, wellchosen weight functions we can show that wellsampled surfaces lead to smooth distance fields with nonzero gradients and, thus, the surface is a continuously differentiable manifold. We detail spatial data structures and efficient algorithms to compute raysurface intersections for fast ray casting and ray tracing of the surface.
A short course on fast multipole methods
 Wavelets, Multilevel Methods and Elliptic PDEs
, 1997
"... In this series of lectures, we describe the analytic and computational foundations of fast multipole methods, as well as some of their applications. They are most easily understood, perhaps, in the case of particle simulations, where they reduce the cost of computing all pairwise interactions in a s ..."
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Cited by 63 (5 self)
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In this series of lectures, we describe the analytic and computational foundations of fast multipole methods, as well as some of their applications. They are most easily understood, perhaps, in the case of particle simulations, where they reduce the cost of computing all pairwise interactions in a system of N particles from O(N 2)toO(N)orO(N log N) operations. They are equally useful, however, in solving certain partial differential equations by first recasting them as integral equations. We will draw heavily from the existing literature, especially Greengard [23, 24, 25]; Greengard and Rokhlin [29, 32]; Greengard and Strain [34].
Variational Implicit Surfaces
, 1999
"... We introduce a new method of creating smooth implicit surfaces of arbitrary manifold topology. These surfaces are described by specifying locations in 3D through which the surface should pass, and also identifying locations that are interior or exterior to the surface. A 3D implicit function is crea ..."
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Cited by 62 (2 self)
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We introduce a new method of creating smooth implicit surfaces of arbitrary manifold topology. These surfaces are described by specifying locations in 3D through which the surface should pass, and also identifying locations that are interior or exterior to the surface. A 3D implicit function is created from these constraints using a variational scattered data interpolation approach. We call the isosurface of this function a variational implicit surface. Like other implicit surface descriptions, these surfaces can be used for CSG and interference detection, may be interactively manipulated, are readily approximated by polygonal tilings, and are easy to ray trace. A key strength is that variational implicit surfaces allow the direct specification of both the location of points on the surface and surface normals. These are two important manipulation techniques that are difficult to achieve using other implicit surface representations such as sums of spherical or ellipsoidal Gaussian functions ("blobbies"). We show that these properties make variational implicit surfaces particularly attractive for interactive sculpting using the particle sampling technique introduced by Witkin and Heckbert in [30]. Our formulation also yields a simple method for converting a polygonal model to a smooth implicit model.
Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions
, 2003
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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, ...
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 (10 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.
Transport schemes on a sphere using radial basis functions
 J. Comp. Phys
"... The aim of this work is to introduce the physics community to the high performance of radial basis functions (RBFs) compared to other spectral methods for modeling transport (pure advection) and to provide the first known application of the RBF methodology to hyperbolic partial differential equation ..."
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Cited by 34 (7 self)
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The aim of this work is to introduce the physics community to the high performance of radial basis functions (RBFs) compared to other spectral methods for modeling transport (pure advection) and to provide the first known application of the RBF methodology to hyperbolic partial differential equations on a sphere. First, it is shown that even when the advective operator is posed in spherical coordinates (thus having singularities at the poles), the RBF formulation of it is completely singularityfree. Then, two classical test cases are conducted: 1) linear advection, where the initial condition is simply transported around the sphere and 2) deformational flow (idealized cyclogenesis), where an angular velocity is applied to the initial condition, spinning it up around an axis of rotation. The results show that RBFs allow for a much lower spatial resolution (i.e. lower number of nodes) while being able to take unusually large timesteps to achieve the same accuracy as compared to other commonly used spectral methods on a sphere such as spherical harmonics, double Fourier series, and spectral element methods. Furthermore, RBFs are algorithmically much simpler to program.
Efficient Reconstruction of Large Scattered Geometric Datasets Using the Partition of Unity and Radial Basis Functions
, 2004
"... We present a new scheme for the reconstruction of large geometric data. It is based on the wellknown radial basis function model combined with an adaptive spatial and functional subdivision associated with a family of functions forming a partition of unity. This combination offers robust and effi ..."
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Cited by 22 (1 self)
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We present a new scheme for the reconstruction of large geometric data. It is based on the wellknown radial basis function model combined with an adaptive spatial and functional subdivision associated with a family of functions forming a partition of unity. This combination offers robust and efficient solution to a great variety of 2D and 3D reconstruction problems, such as the reconstruction of implicit curves or surfaces with attributes starting from unorganized point sets, image or mesh repairing, shape morphing or shape deformation, etc. After having presented the theoretical background, the paper mainly focuses on implementation details and issues, as well as on applications and experimental results.