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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 505 (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.
Nonrigid point set registration: Coherent Point Drift (CPD)
 IN ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS 19
, 2006
"... We introduce Coherent Point Drift (CPD), a novel probabilistic method for nonrigid registration of point sets. The registration is treated as a Maximum Likelihood (ML) estimation problem with motion coherence constraint over the velocity field such that one point set moves coherently to align with ..."
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Cited by 141 (0 self)
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We introduce Coherent Point Drift (CPD), a novel probabilistic method for nonrigid registration of point sets. The registration is treated as a Maximum Likelihood (ML) estimation problem with motion coherence constraint over the velocity field such that one point set moves coherently to align with the second set. We formulate the motion coherence constraint and derive a solution of regularized ML estimation through the variational approach, which leads to an elegant kernel form. We also derive the EM algorithm for the penalized ML optimization with deterministic annealing. The CPD method simultaneously finds both the nonrigid transformation and the correspondence between two point sets without making any prior assumption of the transformation model except that of motion coherence. This method can estimate complex nonlinear nonrigid transformations, and is shown to be accurate on 2D and 3D examples and robust in the presence of outliers and missing points.
Fast solution of the radial basis function interpolation equations: Domain decomposition methods
 SIAM Journal of Scientific Computing
, 2000
"... Abstract. In this paper we consider domain decomposition methods for solving the radial basis function interpolation equations. There are three interwoven threads to the paper. The first thread provides good ways of setting up and solving small to mediumsized radial basis function interpolation pr ..."
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Cited by 76 (3 self)
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Abstract. In this paper we consider domain decomposition methods for solving the radial basis function interpolation equations. There are three interwoven threads to the paper. The first thread provides good ways of setting up and solving small to mediumsized radial basis function interpolation problems. These may occur as subproblems in a domain decomposition solution of a larger interpolation problem. The usual formulation of such a problem can suffer from an unfortunate scale dependence not intrinsic in the problem itself. This scale dependence occurs, for instance, when fitting polyharmonic splines in even dimensions. We present and analyze an alternative formulation, available for all strictly conditionally positive definite basic functions, which does not suffer from this drawback, at least for the very important example previously mentioned. This formulation changes the problem into one involving a strictly positive definite symmetric system, which can be easily and efficiently solved by Cholesky factorization. The second section considers a natural domain decomposition method for the interpolation equations and views it as an instance of von Neumann’s alternating projection algorithm. Here the underlying Hilbert space is the reproducing kernel Hilbert space induced by the strictly conditionally positive definite basic function. We show that the domain decomposition method presented converges linearly under very weak nondegeneracy conditions on the possibly overlapping subdomains. The last section presents some algorithmic details and numerical results of a domain decomposition interpolatory code for polyharmonic splines in 2 and 3 dimensions. This code has solved problems with 5 million centers and can fit splines with 10,000 centers in approximately 7 seconds on very modest hardware.
The Uniform Convergence of Thin Plate Spline Interpolation in Two Dimensions
, 1994
"... that is not a subset of a single straight line, then we prove that a sequence of thin plate spline interpolants converges to f uniformly on D. Specifically, we require h!0, where h is now the least upper bound on the numbers fd(x; V) : x 2 Dg and where d(x; V), x2R 2 , is the least Euclidean di ..."
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Cited by 26 (0 self)
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that is not a subset of a single straight line, then we prove that a sequence of thin plate spline interpolants converges to f uniformly on D. Specifically, we require h!0, where h is now the least upper bound on the numbers fd(x; V) : x 2 Dg and where d(x; V), x2R 2 , is the least Euclidean distance from x to an interpolation point. Our method of analysis applies integration by parts and the CauchySchwarz inequality to the scalar product between second derivatives that occurs in the variational calculation of thin plate spline interpolation. Mathematics Subject Classification (1991): 65D07 1. Introduction Let f be a function from R 2 to R that has square integr
Temporally Coherent Completion of Dynamic Shapes
"... We present a novel shape completion technique for creating temporally coherent watertight surfaces from realtime captured dynamic performances. Because of occlusions and low surface albedo, scanned mesh sequences typically exhibit large holes that persist over extended periods of time. Most convent ..."
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Cited by 20 (4 self)
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We present a novel shape completion technique for creating temporally coherent watertight surfaces from realtime captured dynamic performances. Because of occlusions and low surface albedo, scanned mesh sequences typically exhibit large holes that persist over extended periods of time. Most conventional dynamic shape reconstruction techniques rely on template models or assume slow deformations in the input data. Our framework sidesteps these requirements and directly initializes shape completion with topology derived from the visual hull. To seal the holes with patches that are consistent with the subject’s motion, we first minimize surface bending energies in each frame to ensure smooth transitions across hole boundaries. Temporally coherent dynamics of surface patches are obtained by unwarping all frames within a time window using accurate interframe correspondences. Aggregated surface samples are then filtered with a temporal visibility kernel that maximizes the use of nonoccluded surfaces. A key benefit of our shape completion strategy is that it does not rely on longrange correspondences or a template model. Consequently, our method does not suffer from error accumulation typically introduced by noise, large deformations, and drastic topological changes. We illustrate the effectiveness of our method on several highresolution scans of human performances captured with a stateoftheart multiview 3D acquisition system.
Finite Element Thin Plate Splines for Data Mining Applications
, 1998
"... Thin plate splines have been used successfully to model curves and surfaces. A new application is in data mining where they are used to model interaction terms. These interaction splines break the "curse of dimensionality" by reducing the highdimensional nonparametric regression problem t ..."
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Cited by 13 (8 self)
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Thin plate splines have been used successfully to model curves and surfaces. A new application is in data mining where they are used to model interaction terms. These interaction splines break the "curse of dimensionality" by reducing the highdimensional nonparametric regression problem to the determination of a set of interdependent surfaces. However, the determination of the corresponding thin plate splines requires the solution of a dense linear system of equations of order n where n is the number of observations. For data mining applications n can be in the millions, and so standard thin plate splines, even using fast algorithms may not be practical. A finite element approximation of the thin plate splines will be described. The method uses H¹ elements in a formulation which only needs first order derivatives. The resolution of the method is chosen independently of the number of observations which only need to be read from secondary storage once and do not need to be stored ...
Meshfree methods
 Handbook of Theoretical and Computational Nanotechnology. American Scientific Publishers
, 2005
"... Meshfree methods are the topic of recent research in many areas of computational science and approximation theory. These methods come in various flavors, most of which can be explained either by what is known in the literature as radial basis functions (RBFs), or in terms of the moving least squares ..."
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Meshfree methods are the topic of recent research in many areas of computational science and approximation theory. These methods come in various flavors, most of which can be explained either by what is known in the literature as radial basis functions (RBFs), or in terms of the moving least squares (MLS) method. Over the past several years meshfree approximation methods have found their way into many different application areas ranging from artificial intelligence, computer graphics, image processing and optimization to the numerical solution of all kinds of (partial) differential equations problems. Applications in computational nanotechnology are still somewhat rare, but do exist in the literature. In this chapter we will focus on the mathematical foundation of meshfree methods, and the discussion of various computational techniques presently available for a successful implementation of meshfree methods. At the end of this review we mention some initial applications of meshfree methods to problems in computational nanotechnology, and hope that this introduction will serve as a motivation for others to apply meshfree methods to many other challenging problems in computational nanotechnology.