# 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.

## interpolation

see MEASURES OF CENTRAL TENDENCY.

## 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, it 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).
References in periodicals archive ?
Ali Shahnawaz Khan, Executive Director of the Kashmiri Scandinavian Council, Norway and leader of the Christian Democratic Party, lauded Norwegian Government for Kashmir interpolation move in the parliament.
Survey: Interpolation methods in medical image processing," IEEE Trans.
1982) evaluated three interpolation methods: linear interpolation with four data points found in the grid axis directions (LIXY), linear interpolation between two data points found in the approximate direction of steepest slope (LISS), and cubic interpolation using four data points found in the approximate direction of steepest slope (CISS).
j], namely, the smooth condition on the interpolation nodes: [S'.
The basic idea of Edge-Directed Interpolation techniques is to analyze edge information in the LR image in order to aid in the interpolation step [1, 7-22].
The latter indication (no ceiling) is used in CVA to limit the interpolation of observed ceilings across the regions between METAR sites.
Function for MHR calculations is chosen individually at each interpolation and it represents probability distribution function of parameter [alpha] [member of] [0,1] for every point situated between two interpolation knots.
We recall that the Hermite interpolation problem on the unit circle consists in obtaining a Laurent polynomial [H.
IDW is the most appropriate interpolation technique that helps in giving an impression that the variables being presented share common characteristics and are close to one another (Patel and Chopra 2007).
There are many different interpolation techniques, and a comprehensive review is beyond the scope of this work.
Previous researches have been conducted by researchers around the world relating to spatial interpolation (Anderson, 2002; Caruso and Quarta, 1998).
Jung and Yoo [19] proposed the neighbor mean interpolation (NMI), which is a novel image interpolation method.

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