Data Interpolation Method Research Articles

Interpolation of principal stress directions by nonparametric statistics: Global maps with confidence limits

We suggest a simple method for interpolation of data on the horizontal most compressive principal stress direction ( ) which makes no assumptions about their distribution, except that the probability of a given discrepancy is primarily a function of distance. Using the 6000 data of the World Stress Map, we obtain empirical two‐point distributions of direction differences at a number of distances. Then, to find the probability distribution at an interpolation point, we form the product of the distributions implied by all the available data within the correlation horizon (with or without preaveraging of clustered data). It is then simple to select the most likely direction and 90% (or other) confidence bounds. The resulting global maps of directions have 90% confidence limits of less than ±45° over 20–65% of the globe, and small uncertainties of less than ±10° over 5–25% of the globe (depending on the method). Both maps illustrate how radial directions encircle high topography (or unusually shallow bathymetry), suggesting a generally low level of deviatoric stress in the lithosphere.

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.

The equivalent data concept applied to the interpolation of potential field data

The equivalent layer calculation becomes more efficient by first converting the observed potential data set to a much smaller equivalent data set, thus saving considerable CPU time. This makes the equivalent‐source method of data interpolation very competitive with other traditional gridding techniques that ignore the fact that potential anomalies are harmonic functions. The equivalent data set is obtained by using a least‐squares iterative algorithm at each iteration that solves an underdetermined system fitting all observations selected from previous iterations and the observation with the greatest residual in the preceding iteration. The residuals are obtained by computing a set of “predicted observations” using the estimated parameters at the current iteration and subtracting them from the observations. The use of Cholesky’s decomposition to implement the algorithm leads to an efficient solution update everytime a new datum is processed. In addition, when applied to interpolation problems using equivalent layers, the method is optimized by approximating dot products by the discrete form of an analytic integration that can be evaluated with much less computational effort. Finally, the technique is applied to gravity data in a 2 × 2 degrees area containing 3137 observations, from Equant‐2 marine gravity survey offshore northern Brazil. Only 294 equivalent data are selected and used to interpolate the anomalies, creating a regular grid by using the equivalent‐layer technique. For comparison, the interpolation using the minimum‐curvature method was also obtained, producing equivalent results. The number of equivalent observations is usually one order of magnitude smaller than the total number of observations. As a result, the saving in computer time and memory is at least two orders of magnitude as compared to interpolation by equivalent layer using all observations.

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.

Oceanographic data interpolation: Objective analysis and splines

Two data interpolation methods are reviewed and compared. They are objective analysis (also known as objective mapping, statistical interpolation or Gauss‐Markov interpolation), and spline interpolation. The former is a statistical method based on minimizing the interpolation error variance; the latter is a deterministic method which seeks to obtain the smoothest interpolated field consistent with the data. The two methods are formally equivalent. This means that a single interpolated field may be viewed as deriving from the extremization of either a statistical or a deterministic quantity. This helps to explain why objective analysis remains a useful tool even when the statistics are poorly known. Two types of spline interpolation are considered: “norm splines”, for which smoothness is defined in terms of the field amplitude and curvature, and “semi‐norm splines”, which consider only the field curvature. The objective analysis algorithm may be used to perform norm spline interpolation simply by a suitable choice of covariance function. An example of such a choice is given. The statistical, norm spline, and semi‐norm spline methods are tested on 100 realizations of a random field with known covariance structure. It is shown that the interpolation error is a weak function of the length scale needed by objective analysis and norm splines, but that the statistical error estimate depends strongly on the length scale, and can be quite misleading. The data spacing is shown to have more influence on the interpolation error than does the length scale. Semi‐norm splines are shown to have poor performance near the boundaries of the data set, and do not provide an error estimate. However, they have the advantage of not requiring the specification of a length scale, and provide a more accurate interpolation if a good estimate of the length scale is not available. The objective analysis error estimate does not reflect any error in the estimate of the true mean of the field, a statistic which is required by the method. Fortunately, all the methods appear to be insensitive to errors in estimating the mean, with the exception of objective analysis with an oscillatory covariance function. This result is all the more surprising because the data sets were generated using exactly this covariance function.

Rangeland Grasshopper (Orthoptera: Acrididae) Spatial Variability: Macroscale Population Assessment

Variability of adult grasshoppers counts was studied for statewide collections made in Montana during 1987. Geostatistical techniques were used to compare the spatial relationships among counts from three recognized environmental regions. Related geostatistical methods also were used to develop macroscale (regional) grasshopper hazard maps based on spatial variability. Development of semivariograms and kriging (an interpolation method) for grasshopper count data are discussed, as well as potential for the application of such methods for use in pest management. Results and methods will be useful to insect ecologists and pest managers interested in developing hazard maps and understanding the spatial relatedness of rangeland grasshopper populations.

A new method for image segmentation

In applications involving visual inspection, it is often required to separate objects from background, in conditions of poor and nonuniform illumination. In such cases one has to rely on adaptive methods that learn the illumination from the given images and base the object/background decision on this information. We here present a new method for image segmentation via adaptive thresholding. The threshold surface is determined by interpolating the image gray levels at points where the gradient is high, indicating probable object edges. Several methods of data interpolation to levels given at scattered points in the image plane are discussed. One method is tested on several examples and the segmentation results are compared to previously proposed adaptive thresholding algorithms.

Experimental verification study for system for prediction of environmental emergency dose information; SPEEDI. (I). Three-dimensional interpolation method for surface wind observations in complex terrain to produce gridded wind field.

In order to calculate airflow around a nuclear site, a new weighted interpolation method for sparse wind data at various terrain heights is developed. This method employs weighting functions for vertical distance and the topographic barrier between the station and a grid point, in addition to a weighting function for horizontal distance which is employed in usual methods. The new weighting functions are developed from the analysis of field measurements taken in complex terrain. This method can represent the wind-fields from the bottom to the upper boundary better than usual methods which use only the horizontal weighting factor, even when the upper wind data are not available.

A comparative test of photogrammetrically sampled digital elevation models

A working group within the ISPRS has conducted an international comparative test of digital elevation models (DEM) based on a resolution from the ISPRS Congress in Hamburg. The object of the test was to study the relations among methods for data acquisition, interpolation, resulting accuracy and type of terrain. Six test areas were selected, having topographic structures varying from smooth rolling farmland to very steep mountainous terrain with forested valleys. Fifteen organisations have produced DEM's from aerial photographs in scales 1:4,000–1:30,000. The DEM's have then been used for the derivation of elevations in a set of check points, the location of which were unknown when the DEM was measured in the stereo-instrument. The elevations of these check points were then compared with their “true” values. The “true” errors found in the check points are presented and analysed. After elimination of blunders the remaining errors are composed of systematic and random parts. The systematic parts can originate from the reconstruction of the stereomodel, from the interpolation of the DEM, and from effects of the vegetation height. The size and distribution of the “true” errors are presented in Diagrammes and Illustrations. Errors of functions of the DEM (e.g. slope and curvature) are also studied.

Numerical Variational Methods for Data Filtering and Interpolation

Abstract An objective representation of an observed meteorological field is obtained by minimizing a quadratic functional that measures both the smoothness (and regularity) of the objective field and the closeness to the observed data. This is a particular form of the general model for numerical variational analysis suggested by Wahba and Wendelberger to generalize the idea introduced by Sasaki. The solution of this minimization problem can be obtained by using homogeneous splines and cross validation or by using the finite element method to determine an approximate solution. Finite elements can provide data compression for large N. Testud and Chong proposed such a method, based on bilinear finite elements for 3-dimensional wind field analysis. We go further with the same principle and study a more regular approximation, which can be analytically differentiated, obtained by bicubic splines. The numerical simulations proposed by Chong and Testud are used to compare the capability of all these methods (spli...

Neutron Fluence-to-Dose Equivalent Conversion Factors: A Comparison of Data Sets and Interpolation Methods

Neutron Fluence-to-Dose Equivalent Conversion Factors: A Comparison of Data Sets and Interpolation Methods

An interpolation method of randomly distributed atmospheric data in the height-time domain

A numerical method for objective interpolation of boundary-layer data in the height-time domain is presented. The method is based on a diffusive transport hypothesis and allows for the determination of the optimal ratio between the height and time scales to be used for non-dimensionalising the independent variables z and t. This ratio, with dimensions of a velocity, may be related to the vertical transfer properties of the atmosphere. A few cases during springtime with different stability conditions are discussed, showing that this ratio varies by at least one order of magnitude between day and night.

Method for smooth approximation of data

A new method for smooth interpolation and approximation of data is introduced. The method is based on choosing the smoothest analytical curve that approximates the data points. The method allows for the inclusion of additional information, such as uncertainties in the data and other constraints given by the general phenomenology of the experiment or theory. The novelty in the method, as well as its most efficient feature, is in its ability to obtain a good approximation, not only of the data points, but also of their first and higher derivatives. The method is applied to several examples. Among them we include function approximations and several physical experiments, such as heat capacity measurement and Mössbauer spectrum. Another interesting example that is treated involves a calibration of GaAs resistance thermometer. The method is extended to multidimensional spaces, and can also be applied to the calculation of numerical derivatives and integrals of data points.