# interpolation

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## interpolation

[in‚tər·pə′lā·shən]
(mathematics)
A process used to estimate an intermediate value of one (dependent) variable which is a function of a second (independent) variable when values of the dependent variable corresponding to several discrete values of the independent variable are known.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## interpolation

see MEASURES OF CENTRAL TENDENCY.
Collins Dictionary of Sociology, 3rd ed. © HarperCollins Publishers 2000
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Interpolation

an insertion or correction in an original text made by someone other than the author.

Interpolations played a pivotal role in the texts by Roman jurists that are compiled in the Digest. They were made to eliminate contradictions in the texts as well as the statutes and attitudes that were inappropriate for the Justinian era. Various kinds of interpolations were made, including specification and substitutions of the rule of law, substitution or elimination of terms, and lexical changes. The medieval humanists were the first to discover the interpolations in the Digest.

## Interpolation

in mathematics and statistics, the process of finding values of a quantity between some of its known values. An example is finding values of the function f(x) at points x lying between the points (nodes of interpolation) x0 < x1 < … < xn by means of the known values yi = f(x1), where i = 0, 1, …, n. In the case when x lies outside the interval included between x0 and xn, the analogous problem is called an extrapolation problem.

In the simplest case, linear interpolation, the value of f(x) at a point x satisfying the inequality x0 < x1, is taken to be equal to the value of the linear function coinciding with f(x) at the points x = x0 and x = x1. The interpolation problem is undefined from a strict mathematical viewpoint: if nothing is known about the function f(x) except its values at the points x0, x1, …, xn, then its value at a point x, which is different from all these points, remains completely arbitrary. The interpolation problem acquires a definite meaning if the function f(x) and its derivatives are subject to certain inequalities. If, for example, the values f(x0) and f(x1) are given and it is known that for xo < x < x1 the inequality | f”(x) | ≤ M is fulfilled, then the error of the formula (*) may be estimated with the aid of the inequality It makes sense to use more complex interpolation formulas only in the case when it is certain that the function is sufficiently “smooth,” that is, when it has a sufficient number of derivatives that do not increase rapidly.

In addition to the computation of values of functions, interpolation has numerous other applications (for example, approximate integration, approximate solution of equations, and, in statistics, the smoothing of distribution series with the aim of eliminating random distortions).

### REFERENCES

Goncharov, V. L. Teoriia interpolirovaniia i priblizheniia funktsii, 2nd ed. Moscow, 1954.
Krylov, A. N. Lektsii o pribiizhennykh vychisleniiakh, 6th ed. Moscow, 1954.
Yule, G. U., and M. G. Kendall. Teoriia statistiki, 14th ed. Moscow, 1960. (Translated from English.)

## interpolation

In computer graphics, interpolation is the creation of new values that lie between known values. For example, when objects are rasterized into two-dimensional images from their corner points (vertices), all the pixels between those points are filled in by an interpolation algorithm, which determines their color and other attributes (see graphics pipeline).

Another example is when a video image in a low resolution is upscaled to display on a monitor with a higher resolution, the missing lines are created by interpolation. In a digital camera, the optical zoom is based on the physical lenses, but the digital zoom is accomplished by algorithms (see interpolated resolution).
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References in periodicals archive ?
According to the interpolation reminder of cubic Hermite spline, when interpolation interval approaches zero, the cubic Hermite spline curve can approximate well to the function being interpolated. However, for the EH interpolation spline we constructed, it can approximate well to the function being interpolated without interpolation interval approaching zero, and it can approximate to the interpolated functions better than the standard cubic Hermite interpolation spline.
Compared with other methods to generate a PSD curve, the log-cubic method is superior to the freehand method in its objectivity, superior to the log-linear method in the smoothness and accuracy of the interpolated PSD curve, and superior to the log-spline method in the monotonicity of the interpolated PSD curve.
These algorithms also generate interpolated images with blurred edges and annoying artifacts near areas that contains sharp edges.
Chiang, "Analysis of the exponential signal by the interpolated DFT algorithm", IEEE Trans.
Caption: Figure 2: Comparison of analytical and interpolated right boundary conditions c(1,t) (kg/[m.sup.3]) ([increment of x] = 0.0500, [DELTA]t = 0.0100, Pe = 5.0).
To ensure that the interpolated results can be constrained by the values of the contour lines, virtual control points were added to control the precision of interpolation.
Last, we report that the features extracted from interpolated arrays were significantly less deviated from the features extracted from the raw data than were those following low-pass filter (hypothesis 4).
A general form of predicting an interpolated value Z at a given point x based on samples [Z.sub.i] = Z([x.sub.i]) for i = 1,2, ..., n using IDW is an interpolating function is
The same procedure was used in the data analysis to ensure the comparability of results between the original images with a 64 x 64 matrix and the interpolated images.
Suppose [f.sub.t] is the interpolated frame to be estimated, and the previous and the following neighboring frames of [f.sub.t] refer to [f.sub.t-1] and [f.sub.t-1] respectively.
The Western Roscoelite Zone (WRZ): predominantly quartz-roscoelite mineralisation, located along a major north striking fault, crosscut by north-west striking thrusts, which contains 52% of the interpolated mineralisation.
The main topic of this paper is to present a 3-D imaging algorithm for tomography SAR by introducing the projection technique into the interpolated array transform.

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