Scattered Data Interpolation Methods Research Articles

Weather daily data approximation using point adaptive ellipsoidal neighborhood in scattered data interpolation methods

Many papers have applied the meshless method to approximate a function by using a set of scattered data. To use a meshless method, we need to predefine a positive real number as a radius of the local sphere or a positive integer as the number of interior points. This is while the effect of a fixed number as the radius of a local sphere or as the number of interior points could be different for different parts of a complex domain. This paper contains the construction of ellipsoidal neighborhoods for the meshless interpolation and approximation methods with local support functions and local behavior. By using these new neighborhoods, the trend of local data could be found easily. The advantage of this method over the current methods is the use of ellipsoidal local neighborhoods which include points with the most impact on the approximation of the function. We applied these methods to the data of the daily temperature and humidity of Alberta, Canada.

Local multilevel scattered data interpolation

Radial basis functions play an increasingly prominent role in modern approximation. They are widely used in scattered data fitting, numerical solution of partial differential equations, machine learning and others. Although radial basis functions have excellent approximation properties, they often produce highly ill-conditioned discrete algebraic system and lead to a high computational cost. The paper introduces local multilevel scattered data interpolation method, which employ nested scattered data sets and scaled compactly supported radial basis functions with varying support radii. We will provide convergence theory for Sobolev target functions. And several numerical experiments will be provided to conform the efficiency of new method.

Interpolating sparse scattered data using flow information

Scattered data interpolation and approximation techniques allow for the reconstruction of a scalar field based upon a finite number of scattered samples of the field. In general, the fidelity of the reconstruction with respect to the original scalar field tends to deteriorate as the number of samples decreases. For the situation of very sparse sampling, the results may not be acceptable at all. However, if it is known that the scalar field of interest is correlated with a known flow field – as is the case when the scalar field represents the value of an oceanographic tracer that propagates under the influence of the ocean's flow – then this knowledge can be exploited to enhance the scattered data reconstruction method. One way to exploit flow field information is to use it to construct a modified notion of distance between points. Replacing the standard Euclidean distance metric with a flow-field-aware notion of distance provides a method for extending standard scattered data interpolation methods into flow-based methods that produce superior results for very sparse data. The resulting reconstructions typically have lower root-mean-square errors than reconstructions that do not use the flow information, and qualitatively they often appear physically more realistic.

Regularized Minimum Length Method in Scattered Data Interpolation

In an attempt of accumulating more experiences of interpolating scattered data using the minimum length method, this study chooses new kernel functions from the machine learning technique to implementing this minimum length method. But, consulting with the regularization theory, a regularized minimum length method is created by solving coefficient of it in a penalized least squares approximation problem. The purpose of creating this regularized minimum length method is responding to a pilot observation finding the instability of original minimum length method under dense interpolation points. Testing the regularized minimum length method finds that applying it is time-saving but its performance is comparable to the radial point interpolation with polynomial reproduction. Inverse multiquadric and rational quadric kernel functions are two preferred kernel function to perform the regularized minimum length method. In conclusion, the proposed regularized minimum length method can be a useful scattered data interpolation method.

Pose Space Image Based Rendering

AbstractThis paper introduces a new image‐based rendering approach for articulated objects with complex pose‐dependent appearance, such as clothes. Our approach combines body‐pose‐dependent appearance and geometry to synthesize images of new poses from a database of examples. A geometric model allows animation and view interpolation, while small details as well as complex shading and reflection properties are modeled by pose‐dependent appearance examples in a database. Correspondences between the images are represented as mesh‐based warps, both in the spatial and intensity domain. For rendering, these warps are interpolated in pose space, i.e. the space of body poses, using scattered data interpolation methods. Warp estimation as well as geometry reconstruction is performed in an offline procedure, thus shifting computational complexity to an a‐priori training phase.

Hardy's Multi-Quadric method of calibrating a wireless electromagnetic localization system for intestinal micro-devices

A wireless electromagnetic localization method for tracking a medical micro-device in the gastrointestinal tract has been proposed and a prototype for the localization system has been developed. In the electromagnetic localization system, several excitation coils were mounted above the abdomen of human body to generate a varying electromagnetic field one by one. Then a wireless magnetic sensor was used to measure the magnetic field strength at the location of the micro-device. Thus the position and orientation of the micro-device could be obtained by solving the inverse problem of magnetic fields. In order to improve the localization precision, a global error correction technique based on the Hardy's Multi-Quadric scattered data interpolation method was investigated to correct the static errors. In the calibration experiment, dependencies between true locations of the micro-device and the corresponding tracked locations within the operational space were established. Based on these data sets, the Hardy's Multi-Quadric correction function was derived. The experimental results show that the localization error can be greatly reduced by using the Hardy's Multi-Quadric correction method. The advantage of the approach is that it allows for smooth continuous mapping from the tracked space to the true space within the entire tracked space.

A Multiscale Framework for EHL and Micro-EHL

In this article, a heterogeneous multiscale method is introduced to analyze the microelastohydrodynamic lubrication (micro-EHL) of bearings with topological features. Two scales are adopted in the analysis: the large-scale simulations describe the entire bearing domain, and the small-scale simulations describe the fluid–structure interaction (FSI) at the small-scale features. Conservation of mass and momentum of the lubricant and the bearing's elastic deformation are solved for. The relationship between the pressure gradient and mass flow is obtained from homogenized small-scale FSI simulations and applied on a global scale via a scattered data interpolation method. When the micro structure is periodic the exact model at micro scale is replaced by an effective derived equation, i.e., homogenized model. The elastic deformation of the textured bearing surface is addressed at both the large and small scales, by decomposing the displacement influence matrix into the diagonal terms and nondiagonal terms (sorted at the small scale and large scale, respectively). The multiscale method was demonstrated as being capable of modeling the global pressure and film thickness for a bearing with surface texture while maintaining the accuracy of the small-scale modeling features. The illustrative geometry was that of a linear converging pad bearing in two dimensions. The solutions were compared with those obtained using lubrication theory for the smooth surface case, and good agreement was obtained. The method was then demonstrated for geometries incorporating topographical features.

A symmetric collocation method with fast evaluation

Symmetric collocation, which can be used to numerically solve linear partial differential equations, is a natural generalization of the well-established scattered data interpolation method known as radial basis function interpolation. As with radial basis function interpolation, a major shortcoming of symmetric collocation is the high cost, in terms of floating-point operations, of evaluating the obtained function. When solving a linear partial differential equation, one usually has some freedom in choosing the collocation points. We explain how this freedom can be exploited to allow the fast evaluation of the obtained function provided the basic function is chosen as a tensor product of compactly supported piecewise polynomials. Our proposed fast evaluation method, which is exact in exact arithmetic, is initially designed and analysed in the univariate case. The multivariate case is then reduced, recursively, to multiple univariate evaluations. Along with the theoretical development of the method, we report the results of selected numerical experiments which help to clarify expectations.

STABLE COMPUTATION OF NATURAL NEIGHBOR INTERPOLATION

Natural neighbor interpolation is a scattered data interpolation method based on Voronoi diagrams. A class of natural neighbor interpolants are defined in an integral form which is not suitable for computation. This paper demonstrates that these interpolants can be computed solely with algebraic operations and conditional branches. In addition, it is noted that special care is required to achieve stable computation, and a procedure is proposed for stably calculating natural neighbor interpolation with computers. Its usability is ascertained by surface drawings of natural neighbor interpolation.

Fast and Adaptive Bidimensional Empirical Mode Decomposition Using Order-Statistics Filter Based Envelope Estimation

A novel approach for bidimensional empirical mode decomposition (BEMD) is proposed in this paper. BEMD decomposes an image into multiple hierarchical components known as bidimensional intrinsic mode functions (BIMFs). In each iteration of the process, two-dimensional (2D) interpolation is applied to a set of local maxima (minima) points to form the upper (lower) envelope. But, 2D scattered data interpolation methods cause huge computation time and other artifacts in the decomposition. This paper suggests a simple, but effective, method of envelope estimation that replaces the surface interpolation. In this method, order statistics filters are used to get the upper and lower envelopes, where filter size is derived from the data. Based on the properties of the proposed approach, it is considered as fast and adaptive BEMD (FABEMD). Simulation results demonstrate that FABEMD is not only faster and adaptive, but also outperforms the original BEMD in terms of the quality of the BIMFs.

Interpolating scattered data using 2D self-organizing feature maps

Many computer-aided design, computer graphics, and data visualization applications require freeform surfaces to be created from irregularly spaced and unorganized digitized data. Most surface interpolation and approximation techniques require information about the connectivity between these measured points. In contrast, the scattered data interpolation method described in this paper exploits the topological structure and unsupervised learning algorithm of a 2D self-organizing feature map (SOFM) to iteratively create a polygonal surface mesh that takes the general shape of the underlying object. The mesh representation, with quadrilateral elements, can be used to produce a facetted surface model for direct visualization or provide the means to “parametrize” the scattered data prior to generating a smooth continuous surface. Several illustrative examples using scattered range data are provided to demonstrate the data interpolation and surface reconstruction capability of the proposed 2D SOFM.

Scattered data interpolation methods for electronic imaging systems: a survey

Numerous problems in electronic imaging systems in- volve the need to interpolate from irregularly spaced data. One ex- ample is the calibration of color input/output devices with respect to a common intermediate objective color space, such as XYZ or L*a*b*. In the present report we survey some of the most impor- tant methods of scattered data interpolation in two-dimensional and in three-dimensional spaces. We review both single-valued cases, where the underlying function has the form f:R 2 !R or f:R 3 !R, and multivalued cases, where the underlying function is f:R 2 !R 2 or f:R 3 !R 3 . The main methods we review include linear triangular (or tetrahedral) interpolation, cubic triangular (Clough-Tocher) interpo- lation, triangle based blending interpolation, inverse distance weighted methods, radial basis function methods, and natural neigh- bor interpolation methods. We also review one method of scattered data fitting, as an illustration to the basic differences between scat- tered data interpolation and scattered data fitting. © 2002 SPIE and

A New Approach in Offline-Optimization of the Injection Molding Process

Abstract For the purpose of process optimization of the injection-molding process a robust processing point is determined, at which specified characteristics of the parts are met and the process is insensitive to disturbances. The idea of the new optimization concept is to ensure that also around the robust processing point, within the so-called robust processing window, the part geometry is in tolerance and a low standard deviation respectively a high SN-ratio is achieved. Instead of reducing the optimization task to a one-dimensional problem by combining the target functions, in this approach one defines lower and upper bounds for the characteristics to be optimized. These bounds must be fulfilled in the range of the robust processing window and, if necessary, might be sharpened or relaxed during the optimization procedure. The corresponding response surfaces are generated by means of new Scattered Data Interpolation-methods.

Image warping with scattered data interpolation

Discusses a new approach to image warping based on scattered data interpolation methods which provides smooth deformations with easily controllable behavior. A new, efficient deformation algorithm underlies the method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Spatial free-form deformation with scattered data interpolation methods

The problem of deforming a given spatial shape is treated. There are many examples of applications in visual computing: fitting surfaces to sampled data points in space, correction of distortions in tomographic imaging, modeling of free-form geometric shapes, and animating metamorphoses of geometric objects. Our solution warps the space surrounding the given shape with the effect of deforming the embedded shape, too, with a function derived with scattered data interpolation methods from the displacements of a finite set of control points that can be placed arbitrarily and adaptively. We present algorithms implementing this idea on parametric surfaces and rasterized volume data.

Deformed cross‐dissolves for image interpolation in scientific visualization

AbstractDeformed cross‐dissolves are methods for inconspicuous interpolation between images. We describe methods for deformation based on scattered data interpolation methods using correspondence points in the images to be interpolated and an algorithm for automatic establishment of these correspondence points. We also describe efficient cross‐dissolve algorithms for the computation of intermediate images. Results for interpolation in the field of medical visualization are presented.

Interpolating scattered multivariate data as a function of time

Two classes of methods are presented for interpolating scattered data sampled in a spatial domain at different times. Instead of treating time as another Euclidean variable, time is treated as a special variable in our two approaches. These methods make use of scattered data interpolants over the spatial domain and univariate interpolants over the time domain. When compared with existing scattered data interpolation methods, the new methods are more effective.

Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation

A class of multivariate scattered data interpolation methods which includes the so-called multiquadrics is considered. Pointwise error bounds are given in terms of several parameters including a parameter d which, roughly speaking, measures the spacing of the points at which interpolation occurs. In the multiquadric case these estimates are O(λ 1 d ) as d → 0, where λ is a constant which satisfies 0 < λ < 1. An essential ingredient in this development which may be of independent interest is a bound on the size of a polynomial over a cube in R n in terms of its values on a discrete subset which is scattered in a sufficiently uniform manner.