Classical Interpolation Methods Research Articles

Approximate reconstruction of randomly sampled signals

We study the use of polynomial interpolation to approximate a function specified by samples taken at random moments satisfying a Poisson distribution with uniform mean sampling rate. Two different selection schemes are considered to determine which samples should be used in the construction of the polynomials, and detailed error estimates are derived for each case. The results are compared with the classical interpolation methods of convolution with a smoothing window. It is concluded that only low order polynomials are useful for interpolation in the presence of noise, but that they are comparable or superior to nonadaptive convolution in most cases, as well as computationally more efficient. Some simulation experiments are presented to support the theoretical estimates.

Completing missing groundwater observations by interpolation

This paper considers the problem of completing the missing records of an observation well. It is assumed that the water level records were taken once a month during a sufficiently long period in the past such that the existing records enable us to constitute an adequate model. It is also assumed that each year is composed of two seasons, a rainy one and a dry one, the water levels are monotonically increasing during the rainy season and decreasing during the dry season. Hence it is desired that the computed water levels will follow this seasonal pattern. The paper presents a simple interpolation method for automatic reconstruction of the missing observations. The only information which is needed by the proposed algorithm is the series of the recorded water levels and a suitable model for this series. The method can be viewed as a modified version of the classical interpolation method described earlier by Friedman. The main difference is that here the related series is generated by the enclosed model. A further difference is that the interpolation rule is modified to ensure that the seasonal monotony of the water levels is not violated.

Triangle based interpolation

Triangle based interpolation is introduced by an outline of two classical planar interpolation methods, viz. linear triangular facets and proximal polygons. These are shown to have opposite local bias. By applying cross products of triangles to obtain local gradients, a method designated “slant-top proximal polygon interpolation” is introduced that is intermediate between linear facets and polygonal interpolation in its local bias. This surface is not continuous, but, by extending and weighting the gradient planes, a C1 surface can be obtained. The gradients also allow a roughness index to be calculated for each data point in the set. This index is used to control the shape of a blending function that provides a weighted combination of the gradient planes and linear interpolation. This results in a curvilinear, C1,interpolation of the data set that is bounded by the linear interpolation and the weighted gradient planes and is tangent to the slant-top interpolation at the data points. These procedures may be applied to data with two, three, or four independent variables.

Polygonal interpolation, a simple, rapid, and versatile approximation method requiring minimal computing facilities

A method is described for interpolation of data points by a polygon of arbitrarily high subdivision, the breakpoints of which are obtained by a simple and rapid process, which can be performed on programmable desk calculators. When compared with classical interpolation methods in quality of approximation and computational expense, polygonal interpolation proves as a useful and versatile tool for approximation of general functions.

A digital signal processing approach to interpolation

In many digital signal precessing systems, e.g., vacoders, modulation systems, and digital waveform coding systems, it is necessary to alter the sampling rate of a digital signal Thus it is of considerable interest to examine the problem of interpolation of bandlimited signals from the viewpoint of digital signal processing. A frequency dmnain interpretation of the interpolation process, through which it is clear that interpolation is fundamentally a linear filtering process, is presented, An examination of the relative merits of finite duration impulse response (FIR) and infinite duration impulse response (IIR) digital filters as interpolation filters indicates that FIR filters are generally to be preferred for interpolation. It is shown that linear interpolation and classical polynomial interpolation correspond to the use of the FIR interpolation filter. The use of classical interpolation methods in signal processing applications is illustrated by a discussion of FIR interpolation filters derived from the Lagrange interpolation formula. The limitations of these filters lead us to a consideration of optimum FIR filters for interpolation that can be designed using linear programming techniques. Examples are presented to illustrate the significant improvements that are obtained using the optimum filters.