Automatic selection of interpolation segments for a machine tool with linear interpolator
Automatic selection of interpolation segments for machine tool with linear interpolator
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Nearest neighbor, bilinear interpolation and bicubic interpolation geographic correction effects on LANDSAT imagery
Geographical correction effects on LANDSAT image data are identified, using the nearest neighbor, bilinear interpolation and bicubic interpolation techniques. Potential impacts of registration on image compression and classification are explored.
Jayroe, R. R., Jr.↗
Efficient Implementation of an Optimal Interpolator for Large Spatial Data Sets
Scattered data interpolation is a problem of interest in numerous areas such as electronic imaging, smooth surface modeling, and computational geometry. Our motivation arises from applications in geology and mining, which often involve large scattered data sets and a demand for high accuracy. The method of choice is ordinary kriging. This is because it is a best unbiased estimator. Unfortunately, this interpolant is computationally very expensive to compute exactly. For n scattered data points, computing the value of a single interpolant involves solving a dense linear system of size roughly n x n. This is infeasible for large n. In practice, kriging is solved approximately by local approaches that are based on considering only a relatively small'number of points that lie close to the query point. There are many problems with this local approach, however. The first is that determining the proper neighborhood size is tricky, and is usually solved by ad hoc methods such as selecting a fixed number of nearest neighbors or all the points lying within a fixed radius. Such fixed neighborhood sizes may not work well for all query points, depending on local density of the point distribution. Local methods also suffer from the problem that the resulting interpolant is not continuous. Meyer showed that while kriging produces smooth continues surfaces, it has zero order continuity along its borders. Thus, at interface boundaries where the neighborhood changes, the interpolant behaves discontinuously. Therefore, it is important to consider and solve the global system for each interpolant. However, solving such large dense systems for each query point is impractical. Recently a more principled approach to approximating kriging has been proposed based on a technique called covariance tapering. The problems arise from the fact that the covariance functions that are used in kriging have global support. Our implementations combine, utilize, and enhance a number of different approaches that have been introduced in literature for solving large linear systems for interpolation of scattered data points. For very large systems, exact methods such as Gaussian elimination are impractical since they require 0(n(exp 3)) time and 0(n(exp 2)) storage. As Billings et al. suggested, we use an iterative approach. In particular, we use the SYMMLQ method, for solving the large but sparse ordinary kriging systems that result from tapering. The main technical issue that need to be overcome in our algorithmic solution is that the points' covariance matrix for kriging should be symmetric positive definite. The goal of tapering is to obtain a sparse approximate representation of the covariance matrix while maintaining its positive definiteness. Furrer et al. used tapering to obtain a sparse linear system of the form Ax = b, where A is the tapered symmetric positive definite covariance matrix. Thus, Cholesky factorization could be used to solve their linear systems. They implemented an efficient sparse Cholesky decomposition method. They also showed if these tapers are used for a limited class of covariance models, the solution of the system converges to the solution of the original system. Matrix A in the ordinary kriging system, while symmetric, is not positive definite. Thus, their approach is not applicable to the ordinary kriging system. Therefore, we use tapering only to obtain a sparse linear system. Then, we use SYMMLQ to solve the ordinary kriging system. We show that solving large kriging systems becomes practical via tapering and iterative methods, and results in lower estimation errors compared to traditional local approaches, and significant memory savings compared to the original global system. We also developed a more efficient variant of the sparse SYMMLQ method for large ordinary kriging systems. This approach adaptively finds the correct local neighborhood for each query point in the interpolation process.
Memarsadeghi, Nargess↗
Minimal norm constrained interpolation
In computational fluid dynamics and in CAD/CAM, a physical boundary is usually known only discreetly and most often must be approximated. An acceptable approximation preserves the salient features of the data such as convexity and concavity. In this dissertation, a smooth interpolant which is locally concave where the data are concave and is locally convex where the data are convex is described. The interpolant is found by posing and solving a minimization problem whose solution is a piecewise cubic polynomial. The problem is solved indirectly by using the Peano Kernal theorem to recast it into an equivalent minimization problem having the second derivative of the interpolant as the solution. This approach leads to the solution of a nonlinear system of equations. It is shown that Newton's method is an exceptionally attractive and efficient method for solving the nonlinear system of equations. Examples of shape-preserving interpolants, as well as convergence results obtained by using Newton's method are also shown. A FORTRAN program to compute these interpolants is listed. The problem of computing the interpolant of minimal norm from a convex cone in a normal dual space is also discussed. An extension of de Boor's work on minimal norm unconstrained interpolation is presented.
Irvine, L. D.↗
Filtering Interpolators for Image Comparison Algorithms
Comparing two or more images, either by differencing or ratioing, is important to many remote sensing problems. Because the pixel sample points for the images are (almost) always separated by some nonzero shift, a resampling, or interpolation, process must be performed if one image is to be accurately compared to another. Considered in Fourier space, an interpolator acts as a filter that attenuates some frequencies (usually high) of the image. Thus, when the shifted and unshifted images are compared, the former has been filtered, while the latter has not; the effect of this difference is called interpolation error. The key idea of this paper is to apply a filter to the unshifted image that matches the filtering effect of applying the interpolator to the shifted image, thereby drastically reducing interpolation error. The resulting interpolators, called filtering interpolators, are derived and discussed in detail elsewhere. Basic results will be given in this presentation.
Lucke, R. L.↗
Effective Interpolation of Incomplete Satellite-Derived Leaf-Area Index Time Series for the Continental United States
Many earth science modeling applications employ continuous input data fields derived from satellite data. Environmental factors, sensor limitations and algorithmic constraints lead to data products of inherently variable quality. This necessitates interpolation of one form or another in order to produce high quality input fields free of missing data. The present research tests several interpolation techniques as applied to satellite-derived leaf area index, an important quantity in many global climate and ecological models. The study evaluates and applies a variety of interpolation techniques for the Moderate Resolution Imaging Spectroradiometer (MODIS) Leaf-Area Index Product over the time period 2001-2006 for a region containing the conterminous United States. Results indicate that the accuracy of an individual interpolation technique depends upon the underlying land cover. Spatial interpolation provides better results in forested areas, while temporal interpolation performs more effectively over non-forest cover types. Combination of spatial and temporal approaches offers superior interpolative capabilities to any single method, and in fact, generation of continuous data fields requires a hybrid approach such as this.
Jasinski, Michael F.↗
Supersonic inlet contour interpolation
A method for designing supersonic inlet contours is described which consists in the interpolation of the contours of two known inlets designed for different Mach numbers, thereby determining the contours for a third inlet at an intermediate design Mach number. Several similar axisymmetric inlet contours were interpolated from known inlets with design Mach numbers ranging from 2.16 to 4.0 and with design Mach numbers differing by as much as 1.0. The flowfields were calculated according to Sorensen's (1965) computer program. Shockwave structure and pressure distribution characteristics are shown for the interpolated inlets. The validity of the interpolation is demonstrated by comparing the plots of the flowfield properties across the throat station of the interpolated inlet with the known inlets which were designed iteratively. It seems possible to write a computer program so that a matrix of known inlet contours can be interpolated.
Sorensen, N. E.↗
Efficient Implementation of an Optimal Interpolator for Large Spatial Data Sets
Interpolating scattered data points is a problem of wide ranging interest. A number of approaches for interpolation have been proposed both from theoretical domains such as computational geometry and in applications' fields such as geostatistics. Our motivation arises from geological and mining applications. In many instances data can be costly to compute and are available only at nonuniformly scattered positions. Because of the high cost of collecting measurements, high accuracy is required in the interpolants. One of the most popular interpolation methods in this field is called ordinary kriging. It is popular because it is a best linear unbiased estimator. The price for its statistical optimality is that the estimator is computationally very expensive. This is because the value of each interpolant is given by the solution of a large dense linear system. In practice, kriging problems have been solved approximately by restricting the domain to a small local neighborhood of points that lie near the query point. Determining the proper size for this neighborhood is a solved by ad hoc methods, and it has been shown that this approach leads to undesirable discontinuities in the interpolant. Recently a more principled approach to approximating kriging has been proposed based on a technique called covariance tapering. This process achieves its efficiency by replacing the large dense kriging system with a much sparser linear system. This technique has been applied to a restriction of our problem, called simple kriging, which is not unbiased for general data sets. In this paper we generalize these results by showing how to apply covariance tapering to the more general problem of ordinary kriging. Through experimentation we demonstrate the space and time efficiency and accuracy of approximating ordinary kriging through the use of covariance tapering combined with iterative methods for solving large sparse systems. We demonstrate our approach on large data sizes arising both from synthetic sources and from real applications.
Memarsadeghi, Nargess↗
Interpolation of a surface from sets of discrete height data of different statistical characteristics
This paper presents and analyzes a method for the interpolation of a unique surface from two sets of independent digital height data of differing statistical characteristics. This method is based on linear prediction and thus relies on the concepts of auto- and cross-covariance functions. The linear prediction algorithm for two sets of digital height measurements is first derived and then evaluated using the method of moving averages and bilinear interpolation for comparison. It is found that the overall root mean square interpolation errors of linear prediction are similar to those from moving averages and bilinear interpolation. This accuracy performance, together with the well known potential for controlled filtering of measuring errors and good-behavior in areas of poor control, makes linear prediction a versatile and general method for interpolating a unique surface from two sets of digital height data, with applications in photogrammetric mapping, remote sensing, and other fields.
Leberl, F.↗
Modal interpolation program, L215 (INTERP). Volume 1: Engineering and usage
The usage of the Modal Interpolation Program L215 (INTERP) is described. The program uses modal data to form sets of arrays containing interpolation coefficients. The interpolation arrays can then be used to determine displacements at various aerodynamic surface and surface slopes that are parallel and perpendicular to the freestream direction. Five different interpolation methods are available. A description of the data manipulation and the interpolation methods is presented.
Kroll, R. I.↗
Use of shape-preserving interpolation methods in surface modeling
In many large-scale scientific computations, it is necessary to use surface models based on information provided at only a finite number of points (rather than determined everywhere via an analytic formula). As an example, an equation of state (EOS) table may provide values of pressure as a function of temperature and density for a particular material. These values, while known quite accurately, are typically known only on a rectangular (but generally quite nonuniform) mesh in (T,d)-space. Thus interpolation methods are necessary to completely determine the EOS surface. The most primitive EOS interpolation scheme is bilinear interpolation. This has the advantages of depending only on local information, so that changes in data remote from a mesh element have no effect on the surface over the element, and of preserving shape information, such as monotonicity. Most scientific calculations, however, require greater smoothness. Standard higher-order interpolation schemes, such as Coons patches or bicubic splines, while providing the requisite smoothness, tend to produce surfaces that are not physically reasonable. This means that the interpolant may have bumps or wiggles that are not supported by the data. The mathematical quantification of ideas such as physically reasonable and visually pleasing is examined.
Ftitsch, F. N.↗
Spectral analysis of linear, shift-invariant interpolants
The use of a spectral analysis technique to evaluate the reconstruction/interpolation performance of linear, shift-invariant interpolants is examined. The technique was utilized to measure the performance of cubic hermite, quintic hermite, exponential, cubic, spline, Nu, PCC, Keys cubic, and BAWA cubic interpolants. The performance criterion is based upon the mean square error of the difference between the sampled and reconstructed functions. The reconstruction properties, interpolation functions, and reconstruction filters for the interpolants are studied and compared. It is noted that the spectral analysis technique is applicable to reconstruction algorithms used in signal and image processes, and interpolants used in numerical analysis, computer-aided design, and computer graphics.
Lansing, Donald L.↗
Transition elements based on transfinite interpolation
In this study the transfinite interpolation methodology, a 'blending-function' method in particular, is utilized for the formulation of transition elements. The method offers a formal way of meeting continuity requirements in a transition element. Element shape functions are derived by blending the continuity requirements of individual boundary segments. The blending directions are naturally orthogonal in rectangular domains therefore interpolation of the boundaries over rectangular 2D and 3D elements can be performed with minimal effort. In triangular domains, however, the choice of blending directions and interpolants is not straightforward. For that reason, two interpolation techniques are proposed for blending of the boundaries of triangular domains. A series of transition elements of various classes compatible with elements of different orders and dimensions is developed and the full potential of the transfinite interpolation, as it applies to element formulation, is explored.
Odabas, Onur R.↗
Interpolant Improvements and Lessons Learned
This presentation is for the OpenMDAO workshop 2022 and updates users on recent improvements to interpolant methods. Specifically, we discuss computational improvements, visualization capabilities, and suggested best practices for using interpolants. The term interpolants is used synonymously with metamodels and surrogate models. The goal of the presentation is to increase adoption of efficient interpolant methods and increase users’ awareness to built-in features within OpenMDAO.
Multidisciplinary Design Optimization↗
INPUTB: A thermal/structural data interface program for 2-dimensional and 3-dimensional interpolation
A computer program (INPUTB) for interpolation in both space and time, and based on a linear interpolation scheme using simplex spatial regions is described. The program was developed to provide data interfacing between the output from thermal analyzers and the input to the BOPACE 3-D program. The INPUTB interpolator is of a general nature and could be used for other tasks. The INPUTB program utilizes temperature values which are given at some sequence of time for a list of strategically located thermal nodes. It operates on these values by performing a double interpolation to provide temperature values at another desired sequence of times for a list of structural nodes.
Vos, R. G.↗
Interpolation schemes for orbit determination with the global positioning system
This paper demonstrates that the Global Positioning System (GPS) and simple interpolation schemes can be utilized to satisfy typical orbit determination demands of applications satellites. The complete GPS consists of 24 satellites and permits a position fix of a user satellite at any arbitrary instant. The interpolation formulae used in this report fit generalized Keplerian orbits through a number of position fixes. For a given accuracy level the telemetry requirements as measured by the time intervals between position fixes and the computational load of an interpolation as measured by the number of points through which a generalized Keplerian orbit is fitted vary inversely. A set of possible compromises between these two factors is presented. The Phase I GPS consists of 6 satellites and permits a position fix of a user satellite just 36% of the time. It is shown that with this system more sophisticated interpolation schemes which model atmospheric drag and higher degree terms of the earth's gravity field must be employed to obtain accurate orbits.
Argentiero, P.↗
Modal Interpolation Program, L 215 (INTERP): Supplemental System Design and Maintenance Document - Volume 2
The design, structure, and usage of the modal interpolation program L215 are presented. The program uses modal data sets of arrays containing interpolation coefficients. The interpolation arrays are used to determine displacements at various aerodynamic control points. The displacements consist of translations normal to the aerodynamic surface and surface slopes that are parallel and perpendicular to the free stream direction. Five different interpolation methods are available.
Hirayama, M. Y.↗
Quantitative analysis of the reconstruction performance of interpolants
The analysis presented provides a quantitative measure of the reconstruction or interpolation performance of linear, shift-invariant interpolants. The performance criterion is the mean square error of the difference between the sampled and reconstructed functions. The analysis is applicable to reconstruction algorithms used in image processing and to many types of splines used in numerical analysis and computer graphics. When formulated in the frequency domain, the mean square error clearly separates the contribution of the interpolation method from the contribution of the sampled data. The equations provide a rational basis for selecting an optimal interpolant; that is, one which minimizes the mean square error. The analysis has been applied to a selection of frequently used data splines and reconstruction algorithms: parametric cubic and quintic Hermite splines, exponential and nu splines (including the special case of the cubic spline), parametric cubic convolution, Keys' fourth-order cubic, and a cubic with a discontinuous first derivative. The emphasis in this paper is on the image-dependent case in which no a priori knowledge of the frequency spectrum of the sampled function is assumed.
Lansing, Donald L.↗