Article

Zur Frage nach der Rekonstruktion einer ganzen Funktion aus den Werten in den Punkten {λ n } und {±λ n }

January 1978
Sibirskii matematicheskii zhurnal 19

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The Simplest Way to Interpolate in a Spreadsheet

January 2019

Albert Loh

Given a table of data points x, y, values in between data points are usually derived using the following interpolation formula:- y = y0 + (y1 - y0) / (x1 - x0) * (x - x0) (1) This formula is quite difficult to program using a spreadsheet. A far simpler way to interpolate data in a spreadsheet is shown here, using the formula:- y = slope * x + intercept (2) instead of the original formula ... [Show full abstract] (1). This formula (2) is much easier to program in a spreadsheet.

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National Aeronautics and

May 2000

The generalized hydrodynamics #the wavevector dependence of the transport coe#cients# of a generalized lattice Boltzmann equation #LBE# is studied in detail. The generalized lattice Boltzmann equation is constructed in moment space rather than in discrete velocity space. The generalized hydrodynamics of the model is obtained by solving the dispersion equation of the linearized LBE either ... [Show full abstract] analytically by using perturbation technique or numerically. The proposed LBE model has a maximum number of adjustable parameters for the given set of discrete velocities. Generalized hydrodynamics characterizes dispersion, dissipation #hyper-viscosities#, anisotropy, and lack of Galilean invariance of the model, and can be applied to select the values of the adjustable parameters which optimize the properties of the model. The proposed generalized hydrodynamic analysis also provides some insights into stability and proper initial conditions for LBE simulations. The stability properties of some 2D LBE models are analyzed and compared with each other in the parameter space of the mean streaming velocity and the viscous relaxation time. The procedure described in this work can be applied to analyze other LBE models. As examples, LBE models with various interpolation schemes are analyzed. Numerical results on shear #ow with an initially discontinuous velocity pro#le #shock# with or without a constant streaming velocity are shown to demonstrate the dispersion e#ects in the LBE model; the results compare favorably with our theoretical analysis. We also show that whereas linear analysis of the LBE evolution operator is equivalent to Chapman-Enskog analysis in the long wave-length limit #wave vector k = 0#, it can also provide results for large values of k. Such results are important for the...

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Light Field Rendering

September 2002 · ACM SIGGRAPH Computer Graphics

A number of techniques have been proposed for flying through scenes by redisplaying previously rendered or digitized views. Techniques have also been proposed for interpolating between views by warping input images, using depth information or correspondences between multiple images. In this paper, we describe a simple and robust method for generating new views from arbitrary camera positions ... [Show full abstract] without depth information or feature matching, simply by combining and resampling the available images. The key to this technique lies in interpreting the input images as 2D slices of a 4D function - the light field. This function completely characterizes the flow of light through unobstructed space in a static scene with fixed illumination.

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Numerical integration of functions using the interpolation of Chebyshev-type polynomials

January 1986

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Blind Authentication Using Periodic Properties of Interpolation

October 2008 · IEEE Transactions on Information Forensics and Security

In this paper, we analyze and analytically describe the specific statistical changes brought into the covariance structure of signal by the interpolation process. We show that interpolated signals and their derivatives contain specific detectable periodic properties. Based on this, we propose a blind, efficient, and automatic method capable of finding traces of resampling and interpolation. The ... [Show full abstract] proposed method can be very useful in many areas, especially in image security and authentication. For instance, when two or more images are spliced together, to create high quality and consistent image forgeries, almost always geometric transformations, such as scaling, rotation, or skewing are needed. These procedures are typically based on a resampling and interpolation step. By having a method capable of detecting the traces of resampling, we can significantly reduce the successful usage of such forgeries. Among other points, the presented method is also very useful in estimation of the geometric transformations factors.

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Monatsberichte aus dem Meßnetz; Gleitende Mittelwerte mit inverser Distanzgewichtung zur Glättung un...

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Ludwig Christian Ries

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Geometric Motion Design

August 1996

Michael G Wagner

We describe the use of rational B--spline motions for the solution of motion design problems. It is shown that such motions possess two kinds of control structures suitable for interactive design or geometry processing, respectively. A main advantage of the proposed technique over previous approaches is that it is built directly onto standard CAD algorithms and thus it inherits their advantages. ... [Show full abstract] As examples, we briefly examine interpolation problems and relations to rational curves and surfaces with rational offsets. 1. Introduction This paper reports on recent progress in a research direction that tries to combine methods from kinematics and computer aided geometric design in order to solve motion design problems. This has applications in several areas. For example, in computer animation one often wants to design a smooth motion of a camera that interpolates or approximates some given positions (key frames). Other applications are in mechanical systems animation or motion planning fo...

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Experiments with Local Search on Random 2+p-SAT

April 2002

Random-2+p-SAT interpolates between the polynomial-time problem Random-2-SAT when p = 0 and the NP-complete problem Random3 -SAT when p = 1. At some value p = p 0 0:41, a dramatic change in the structural nature of instances is predicted by statistical mechanics methods. This is reected by a change in the typical cost scaling for a complete search method Tableau, seen experimentally. We show ... [Show full abstract] empirically the same change of of behaviour in the local search algorithm WSat/Novelty , a recent variant of WSat. Between p = 0:3 and p = 0:5 we see typical cost scaling of WSat/Novelty at the 50% satis ability point apparently change from slow polynomial growth to superpolynomial.

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Application of UV laser fabrication to organ models interpolated from computed tomography and magnet...

September 1992 · Applied Optics

We have developed a unique laser fabrication system that uses an ultraviolet laser beam and liquid photopolymer. The system can easily be used to fabricate physical models without milling tools in only one process by using digital data obtained from medical computed tomography (CT) scanners or computer-aided design systems. We describe the fabrication of a smooth physical model such as a ... [Show full abstract] cerebrum, using the laser fabrication system, with the help of CT and magnetic resonance images that are made with coarse slices. Each sandwiched area between adjoining images is interpolated by using third-order spline curves in the cylindrical coordinate system. This modeling technique can play a major role in personal prosthesis, surgical planning, and implant design.

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Denoising and Signal-to-Noise Ratio Enhancement: Derivatives

December 2009

Veli-Matti Taavitsainen

This chapter presents the most common types of splines used in chemometric applications such as interpolation, smoothing, baseline correction, differentiation, and data compression. It does not go deep into the theory, but presents the basic ideas and algorithms. The scope is limited to polynomial splines, which are the most common ones in chemometric applications. Both piecewise and basis ... [Show full abstract] function presentations (i.e., B-splines) of splines are given. The emphasis is on smoothing splines, which are typically needed in processing spectral data. The methods are illustrated using infrared (IR) spectra.

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An innovative method for coordinate measuring machine one-dimensional self-calibration with simplifi...

May 2013 · Review of Scientific Instruments

In this paper, an innovative method for CMM (Coordinate Measuring Machine) self-calibration is proposed. In contrast to conventional CMM calibration that relies heavily on a high precision reference standard such as a laser interferometer, the proposed calibration method is based on a low-cost artefact which is fabricated with commercially available precision ball bearings. By optimizing the ... [Show full abstract] mathematical model and rearranging the data sampling positions, the experimental process and data analysis can be simplified. In mathematical expression, the samples can be minimized by eliminating the redundant equations among those configured by the experimental data array. The section lengths of the artefact are measured at arranged positions, with which an equation set can be configured to determine the measurement errors at the corresponding positions. With the proposed method, the equation set is short of one equation, which can be supplemented by either measuring the total length of the artefact with a higher-precision CMM or calibrating the single point error at the extreme position with a laser interferometer. In this paper, the latter is selected. With spline interpolation, the error compensation curve can be determined. To verify the proposed method, a simple calibration system was set up on a commercial CMM. Experimental results showed that with the error compensation curve uncertainty of the measurement can be reduced to 50%.

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Two Side Bounds on the Inverses of Diagonally Dominant Tridiagonal Matrices

March 1998

Reinhard Nabben

We establish upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices. These bounds improve the bounds recently given by Shivakumar and Ji. Moreover, we show how to improve our bounds iteratively. For an n Theta n M--matrix this iterative refinement yields the exact inverse after n Gamma 1 steps. 1 Introduction Tridiagonal matrices arise in many ... [Show full abstract] applications, e.g. the numerical solution of PDE's and cubic spline interpolation. Therefore this class of matrices is studied very well and a lot of theory is known about tridiagonal matrices and their inverses. One of the most beautiful result is due to Gantmacher and Krein [GK1], [GK2] who proved that the inverse of an irreducible symmetric tridiagonal matrix is given by two vectors. A similar result is also proved by Barrett in [Ba]. This result is generalized in several directions. For a review on symmetric tridiagonal matrices and their inverses see Meurant [M]. For many problems it is very hel...

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Last Updated: 22 Oct 2024

Zur Frage nach der Rekonstruktion einer ganzen Funktion aus den Werten in den Punkten {λ n } und {±λ n }

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