Scattered Data Approximation Research Articles

A dynamic meshless method for the least squares problem with some noisy subdomains

The interpolation method by radial basis functions is used widely for solving scattered data approximation. However, sometimes it makes more sense to approximate the solution by least squares fit. This is especially true when the data are contaminated with noise. A meshfree method namely, meshless dynamic weighted least squares (MDWLS) method, is presented in this paper to solve least squares problems with noise. The MDWLS method by Gaussian radial basis function is proposed to fit scattered data with some noisy areas in the problem’s domain. Existence and uniqueness of a solution is proved. This method has one parameter which can adjusts the accuracy according to the size of noises. Another advantage of the developed method is that it can be applied to problems with nonregular geometrical domains. The new approach is applied for some problems in two dimensions and the obtained results confirm the accuracy and efficiency of the proposed method. The numerical experiments illustrate that our MDWLS method has better performance than the traditional least squares method in case of noisy data.

Orientational Interpolation of the Optical Spectra of Nonspherical Nanoparticles

A general method called orientational interpolation (OI) is introduced to compute arbitrary-orientation and orientation averaged optical spectra of nonspherical plasmonic nanoparticles starting from a small number (4–7) of simulated spectra at fixed orientations. OI is a postprocessing method independent of the choice of simulation technique. It uses scattered data approximation on the configuration manifold of the particle and exploits electrodynamic and point-group symmetries to make efficient use of simulation results. OI is shown to agree closely with the exact solution for a silver ellipsoid in the dipole limit and with the more complicated orientation dependence of a larger ellipsoid. The method is then applied successfully to describe the ensemble UV–visible spectra of colloidal silver decahedra and pentatwinned nanorods. OI provides insight into the orientation-dependent plasmon lineshapes of nonspherical nanoparticles and will be of practical use as the computational counterpart of the spectroscopy of single particles and randomly oriented ensembles.

A unified approach to scattered data approximation on $\mathbb{S}^{\bf 3}$ and SO(3)

In this paper we use the connection between the rotation group SO(3) and the three-dimensional Euclidean sphere $\mathbb{S}^{3}$ in order to carry over results on the sphere $\mathbb{S}^{3}$ directly to the rotation group SO(3) and vice versa. More precisely, these results connect properties of sampling sets and quadrature formulae on SO(3) and $\mathbb{S}^{3}$ respectively. Furthermore we relate Marcinkiewicz---Zygmund inequalities and conditions for the existence of positive quadrature formulae on the rotation group SO(3) to those on the sphere $\mathbb{S}^{3}$ , respectively.

Integration of Moving Least Squares Method and Multilevel B-Spline Method for Scattered Data Approximation

For scattered data approximation with multilevel B-spline(MBS) method, accuracy could be enhanced by densifying control lattice. Nevertheless, when control lattice density reaches to some extent, approximation accuracy could not be enhanced further. A strategy based on integration of moving least squares(MLS) and multilevel B-spline(MBS) is presented. Experimental results demonstrate that the presented strategy has higher approximation accuracy.

An Adaptive Method for Choosing Center Sets of RBF Interpolation

Radial basis functions (RBF) provide powerful meshfree methods for multivariate interpolation for scattered data. RBF methods have been praised for their simplicity and ease of implementation in multivariate scattered data approximation. But both the approximation quality and stability depend on the distribution of the center set. It leads immediately to the problem of finding good or even optimal point sets for the reconstruction process. Many methods are constructed for center choosing. In this paper, we give a short overview of these algorithms including thinning algorithm, greedy algorithm, arclength equipartition like algorithm and k-means clustering algorithm. A new adaptive data-dependent method is provided at the end with some numerical examples to show its effectiveness.

Choosing Basis Functions and Shape Parameters for Radial Basis Function Methods

Radial basis function (RBF) methods have broad applications in numerical analysis and statistics. They have found uses in the numerical solution of PDEs, data mining, machine learning, and kriging methods in statistics. This work examines the use of radial basis functions in scattered data approximation. In particular, the experiments in this paper test the properties of shape parameters in RBF methods, as well as methods for finding an optimal shape parameter. Locating an optimal shape parameter is a difficult problem and a topic of current research. Some experiments also consider whether the same methods can be applied to the more general problem of selecting basis functions.

Fast high-dimensional approximation with sparse occupancy trees

This paper is concerned with scattered data approximation in high dimensions: Given a data set X ⊂ R d of N data points x i along with values y i ∈ R d ′ , i = 1 , … , N , and viewing the y i as values y i = f ( x i ) of some unknown function f , we wish to return for any query point x ∈ R d an approximation f ̃ ( x ) to y = f ( x ) . Here the spatial dimension d should be thought of as large. We emphasize that we do not seek a representation of f ̃ in terms of a fixed set of trial functions but define f ̃ through recovery schemes which are primarily designed to be fast and to deal efficiently with large data sets. For this purpose we propose new methods based on what we call sparse occupancy trees and piecewise linear schemes based on simplex subdivisions.

Surface Reconstruction Based on Hierarchical Floating Radial Basis Functions

Abstract In this paper we address the problem of optimal centre placement for scattered data approximation using radial basis functions (RBFs) by introducing the concept of floating centres. Given an initial least‐squares solution, we optimize the positions and the weights of the RBF centres by minimizing a non‐linear error function. By optimizing the centre positions, we obtain better approximations with a lower number of centres, which improves the numerical stability of the fitting procedure. We combine the non‐linear RBF fitting with a hierarchical domain decomposition technique. This provides a powerful tool for surface reconstruction from oriented point samples. By directly incorporating point normal vectors into the optimization process, we avoid the use of off‐surface points which results in less computational overhead and reduces undesired surface artefacts. We demonstrate that the proposed surface reconstruction technique is as robust as recent methods, which compute the indicator function of the solid described by the point samples. In contrast to indicator function based methods, our method computes a global distance field that can directly be used for shape registration.

A hybrid approach to interactive global illumination and soft shadows

We present a hybrid approach to simulate global illumination and soft shadows at interactive frame rates. The strengths of hardware-accelerated GPU techniques are combined with CPU methods to achieve physically consistent results while maintaining reasonable performance. The process of image synthesis is subdivided into multiple passes accounting for the different illumination effects. While direct lighting is rendered efficiently by rasterization, soft shadows are simulated using a novel approach combining the speed of shadow mapping and the accuracy of visibility ray tracing. A shadow refinement mask is derived from the result of the direct lighting pass and from a small number of shadow maps to identify the penumbral region of an area light source. This region is accurately rendered by ray tracing. For diffuse indirect illumination, we introduce radiosity photons to profit from the flexibility of a point-based sampling while maintaining the benefits of interpolation over scattered data approximation or density estimation. A sparse sampling of the scene is generated by particle tracing. An area is approximated for each point sample to compute the radiosity solution using a relaxation approach. The indirect illumination is interpolated between neighboring radiosity photons, stored in a multidimensional search tree. We compare different neighborhood search algorithms in terms of image quality and performance. Our method yields interactive frame rates and results consistent with path tracing reference solutions.

A scattered data approximation tool to map single-walled carbon nanotube dispersion to the processing parameters in polymer nanocomposites

The relationship of nanocomposite dispersion to dispersion techniques and other processing parameters was studied. Examining all permutations of the various factors in the laboratory is a challenging task. In this paper, we propose to map a correlation between inputs and output via a self-adaptive scattered data approximation method. The proposed greedy algorithm, sequential function approximation (SFA), reveals the multidimensional behaviour of the system, provides the sensitivity of each input and presents the combination of inputs that is most suitable for a specific output. In this research, we have collected data from various research institutions and applied it to SFA. The results show that SWNT weight percent, sonication time, SWNT modification and high shear mixing time are key factors that affect the dispersion. This text discusses SFA, the data and the results in detail. This work serves as a proof of concept for functional mapping to be applied to polymer processing.

Stable Moving Least-Squares

It is a common procedure for scattered data approximation to use local polynomial fitting in the least-squares sense. An important instance is the Moving Least-Squares where the corresponding weights of the data site vary smoothly, resulting in a smooth approximation. In this paper we build upon the techniques presented by Wendland and present a somewhat simpler error analysis of the MLS approximation. Then, we show by example that the N factor, which appears in the bound on the Lebesgue constant in [Holger Wendland, Local polynomial reproduction and moving least squares approximation, IMA J. Numer. Anal. 21 (1) (2001) 285–300], where N is the number of points used in the approximation, can be realized. Hence, we devise a method for choosing the weights smoothly so that the corresponding Lebesgue constant can be bounded independently of N . This is done by employing Voronoi weights. We conclude with some numerical examples exhibiting the effectiveness of the suggested method for highly irregular data sites.

Numerical solutions to a boundary-integral equation with spherical radial basis functions

The Laplace equation in the exterior of the unit sphere with a Dirichlet boundary condition arises from geodesy, oceanography and meteorology. This problem is reformulated into a weakly singular integral equation on the sphere. We study the use of spherical radial basis functions to find approximate solutions to this integral equation using collocation methods. Experiments with data collected by a NASA satellite are performed to clarify the method. Our results illustrate how scattered data can be handled when solving boundary value problems in the exterior of the sphere. References W. Freeden, T. Gervens, and M. Schreiner. Constructive approximation on the sphere with applications to geomathematics. Oxford: Clarendon Press, New York, 1998. T. M. Morton and M. Neamtu. Error bounds for solving pseudodifferential equations on spheres. J. Approx. Theory, 114 (2002), 242--268. doi:10.1006/jath.2001.3642 C. Mueller. Spherical Harmonics, volume 17 of Lecture Notes in Mathematics. Springer--Verlag, Berlin, 1966. F. J. Narcowich and J. D. Ward. Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math. Anal., 33 (2002), 1393--1410. doi:10.1137/S0036141001395054 J.-C. Nedelec. Acoustic and Electromagnetic Equations. Springer--Verlag, New York, 2000. T. D. Pham and T. Tran. Collocation solutions to pseudodifferential equations of negative orders on the sphere using spherical radial basis functions. In preparation. I. J. Schoenberg. Positive definite function on spheres. Duke Math. J., 9 (1942), 96--108. doi:10.1215/S0012-7094-42-00908-6 S. Svensson. Pseudodifferential operators -- a new approach to the boundary problems of physical geodesy. Manuscr. Geod., 8 (1983), 1--40. H. Wendland. Meshless Galerkin methods using radial basis functions. Math. Comp., 68 (1999), 1521--1531. doi:10.1090/S0025-5718-99-01102-3 H. Wendland. Scattered Data Approximation. Cambridge University Press, Cambridge, 2005. Y. Xu and E. W. Cheney. Strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 116 (1992), 977--981. doi:10.2307/2159477

Marcinkiewicz–Zygmund Inequalities and Polynomial Approximation from Scattered Data on SO(3)

We consider scattered data approximation problems on SO(3). To this end, we construct a new operator for polynomial approximation on the rotation group. This operator reproduces Wigner-D functions up to a given degree and has uniformly bounded L p -operator norm for all 1 ≤ p ≤ ∞. The operator provides a polynomial approximation with the same approximation degree of the best polynomial approximation. Moreover, the operator together with a Markov type inequality for Wigner-D functions enables us to derive scattered data L p -Marcinkiewicz–Zygmund inequalities for these functions for all 1 ≤ p ≤ ∞. As a major application of such inequalities, we consider the stability of the weighted least squares approximation problem on SO(3).

A meshless hierarchical representation for light transport

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. Non-hierarchical approaches such as per-vertex 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 direct-to-indirect light transport algorithm that greatly increases the complexity of scenes that can be handled by PRT approaches.

Improved Adaptive–Reinforcement Learning Control for Morphing Unmanned Air Vehicles

This paper presents an improved Adaptive-Reinforcement Learning Control methodology for the problem of unmanned air vehicle morphing control. The reinforcement learning morphing control function that learns the optimal shape change policy is integrated with an adaptive dynamic inversion control trajectory tracking function. An episodic unsupervised learning simulation using the Q-learning method is developed to replace an earlier and less accurate Actor-Critic algorithm. Sequential Function Approximation, a Galerkin-based scattered data approximation scheme, replaces a K-Nearest Neighbors (KNN) method and is used to generalize the learning from previously experienced quantized states and actions to the continuous state-action space, all of which may not have been experienced before. The improved method showed smaller errors and improved learning of the optimal shape compared to the KNN.

On Spherical Averages of Radial Basis Functions

A radial basis function (RBF) has the general form $$s(x)=\sum_{k=1}^{n}a_{k}\phi(x-b_{k}),\quad x\in\mathbb{R}^{d},$$ where the coefficients a 1,…,a n are real numbers, the points, or centres, b 1,…,b n lie in ℝ d , and φ:ℝ d →ℝ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when φ is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm φ(x)=‖x‖ when d is an odd positive integer, the thin plate spline φ(x)=‖x‖2log ‖x‖ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243–264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the Paley–Wiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserles’s study of geometric integration.

Smoothing Lipschitz functions

This paper describes a new approach to multivariate scattered data smoothing. It is assumed that the data are generated by a Lipschitz continuous function f, and include random noise to be filtered out. The proposed approach uses known, or estimated value of the Lipschitz constant of f, and forces the data to be consistent with the Lipschitz properties of f. Depending on the assumptions about the distribution of the random noise, smoothing is reduced to a standard quadratic or a linear programming problem. We discuss an efficient algorithm which eliminates the redundant inequality constraints. Numerical experiments illustrate applicability and efficiency of the method. This approach provides an efficient new tool of multivariate scattered data approximation.

Robust mosaicing with correction of motion distortions and tissue deformations for in vivo fibered microscopy

Real-time in vivo and in situ imaging at the cellular level can be achieved with fibered confocal microscopy. As interesting as dynamic sequences may be, there is a need for the biologist or physician to get an efficient and complete representation of the entire imaged region. For this demand, the potential of this imaging modality is enhanced by using video mosaicing techniques. Classical mosaicing algorithms do not take into account the characteristics of fibered confocal microscopy, namely motion distortions, irregularly sampled frames and non-rigid deformations of the imaged tissue. Our approach is based on a hierarchical framework that is able to recover a globally consistent alignment of the input frames, to compensate for the motion distortions and to capture the non-rigid deformations. The proposed global alignment scheme is seen as an estimation problem on a Lie group. We model the relationship between the motion and the motion distortions to correct for these distortions. An efficient scattered data approximation scheme is proposed both for the construction of the mosaic and to adapt the demons registration algorithm to our irregularly sampled inputs. Controlled experiments have been conducted to evaluate the performance of our algorithm. Results on several sequences acquired in vivo on both human and mouse tissue also demonstrate the relevance of our approach.

N-D $C^k$ B-Spline Scattered Data Approximation

Since the 1970’s B-splines have evolved to become the {} standard for curve and surface representation due to many of their salient properties. Conventional least-squares scattered data fitting techniques for B-splines require the inversion of potentially large matrices. This is time-consuming as well as susceptible to ill-conditioning which leads to undesired results. Lee {} proposed a novel B-spline algorithm for fitting a 2-D cubic B-spline surface to scattered data in . The proposed algorithm utilizes an optional multilevel approach for better fitting results. We generalize this technique to support N-dimensional data fitting as well as arbitrary degree of B-spline. In addition, we generalize the B-spline kernel function class to accommodate this new image filter.

Multilevel scattered data approximation by adaptive domain decomposition

A new multilevel approximation scheme for scattered data is proposed. The scheme relies on an adaptive domain decomposition strategy using quadtree techniques (and their higher-dimensional generalizations). It is shown in the numerical examples that the new method achieves an improvement on the approximation quality of previous well-established multilevel interpolation schemes.