What is Interpolation?

One thing that is rightly said about the term interpolation is that without interpolation all science would be impossible. And after such a strong statement it is very important to know what is interpolation and its basic working.

Table of Content

• What is Interpolation?
• Interpolation Formula
• Interpolation and Extrapolation
• Types of Interpolation
• Explaining Each Type in Detail
• Summing Up

What is Interpolation?

Mathematical interpolation is a way of estimating the value of a function at those places where the function is implicitly defined within its domain. For predicting the value of that function at other places, one must first identify another curve or surface that passes through certain known data points.

In the above-discussed disciplines, interpolation has wide-ranging applications, numerical analysis and data engineering, computer graphics, and mathematics. Mathematics employs this phase extensively to analyze pictures and videos, to estimate values among known data points, and to decrease the amount of data needed for defining curves and surfaces.

There are several ways of interpolating, which include linear interpolation, polynomial interpolation, and spline interpolation, among others. However, the choice of a particular interpolation scheme depends on certain criteria pertaining to the use of that scheme and on the particular characteristics of data for the specific kind of interpolation.

The following are some benefits of interpolation:

1. Value estimation: Smooth curves and surfaces can be produced using interpolation by using it to estimate values between known data points.
2. Data Compression: Interpolation can be utilized to minimize the quantity of data that must be kept or sent by decreasing the number of data points required to depict a curve or surface.
3. Image and Video Processing: Interpolation is a technique that is often used in picture and video processing to boost image resolution or smooth out pixelated images.
4. Numerical Analysis: Interpolation is frequently used in numerical analysis to resolve differential equations because it allows an equation’s solution to be approximated at intermediate places.

Interpolation is a helpful approach in general for estimating values between known data points, reducing the amount of data required to depict a curve or surface, or processing photos or videos.

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Interpolation Formula

The Interpolation formula is as follows-

y- y1= ((y2-y1)/ (x2- x1))* (x2- x1)

where

• Linear interpolation value is denoted by y.
• The independent variable is denoted by x.
• Values of the function at one point are denoted by x1,y1.
• Values of the function at another point are denoted by x2,y2.

Interpolation and Extrapolation

Interpolation and Extrapolation are both methods used to estimate the value of a function at a point based on known values at surrounding points. Here is a table showing the difference between Interpolation and Extrapolation.

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Types of Interpolation

Interpolation can be calculated in a variety of ways. A few methods of Interpolation are the following:

1. Linear Interpolation: A straightforward approach for predicting the value of a function at a position between two known data points by determining the equation of the line that connects these two points.
2. Polynomial Interpolation: A technique for predicting the value of a function at a position between two known data points by locating a polynomial that passes between the data points.
3. Spline Interpolation:  A method for predicting the value of a function at a position between two known data points by locating a smooth curve that passes between the data points and has continuous derivatives up to a particular order.
4. Cubic Interpolation: Determining a cubic polynomial that crosses through the data points and has constant first derivatives to estimate the value of a function at a position between two known data points.
5. B-spline Interpolation: A technique for predicting the value of a function at a position between two known data points by locating a spline that passes through the data points and is depicted as a linear combination of basis functions.
6. Multivariate Interpolation: A method for predicting the value of a function at a location in several dimensions between two known data points.
7. Radial Basis Function (RBF) Interpolation: A method of estimating the value of a function at a point between two known data points by finding a radial basis function that passes through the data points and is represented as a linear combination of basis functions.

The method of interpolation used is determined by the application’s unique needs and the type of data being interpolated. Simpler approaches, such as linear interpolation, are beneficial for basic estimations. And complicated methods, such as spline or polynomial interpolation, are useful for more accurate guesses of more complex functions.

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Explaining Each Type in Detail

It is important to know about the above-discussed types or methods of interpolation and discuss a few in detail.

Polynomial Interpolation

Polynomial Interpolation is an approximation in which the value of a function at a position between two known data points is determined by determining a polynomial that passes between these points in an elegant, smooth manner. In that approach, the data positions are represented as coefficients on a polynomial equation that ultimately is used to estimate what the function value would have been at the unknown place

Here’s the general form of the polynomial equation:

P(x) = a0 + a1x + a2x^2 +…+ anx^n where x is the independent variable, ai are the coefficients of the polynomial and n is the degree of the polynomial which is equal to the number of data points minus one.

The degree of the polynomial depends on the number of data points; a polynomial of a higher degree tends to give a better estimate. However, caution is required because the polynomials of a higher degree have a tendency to overfit the data; the resulting polynomial will not generalize well to new data points. Therefore, it is very application dependent and depends upon the nature of the data that is being interpolated.

Spline Interpolation

Spline interpolation is a technique that uses piecewise polynomial functions known as splines for approximating the value of a function between two known data points. This method is very versatile and ensures a smooth and continuous transition between the data points.

The spline function is described to be a continuous and smooth polynomial function that beautifully crosses all the data points while maintaining specific conditions, such as the continuity of the first and second derivatives. In this regard, the most prominent type of splines is known as the cubic spline, whereby each segment uses a third-degree polynomial.

The spline function is defined as follows:

S(x) = a0 + a1x + a2x^2 + a3x^3, where x is the independent variable, ai are the coefficients of the polynomial, and n is the degree of the polynomial.

Cubic Interpolation

Cubic interpolation is a procedure used for estimating the function value of a point, lying somewhere between two already known data points. The methodology involves fitting of a cubic polynomial to that dataset with the choice being such that it passes not only through these two known points but also attains the same first and second derivatives at the points. Thus this technique offers a smooth as well as a continuous estimation for the function over the interval bounded by those data points.

The cubic polynomial function is defined as:

P(x) = a0 + a1x + a2x^2 + a3x^3, where x is the independent variable, ai are the coefficients of the polynomial, and n is the degree of the polynomial.

Multivariate Interpolation

Multivariate interpolation refers to estimating the value of the multivariate function at any point in the domain based on the values known in the neighborhood. The purpose is to use the existing known values in the neighborhood to approximate the unknown value of the function at that particular point.

There are various methods of multivariate interpolation, including polynomial interpolation, spline interpolation, and kriging. These methods differ in the mathematical models used to estimate the unknown values and the computational methods used to solve the problem.

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Summing Up

With the ever-increasing amount of data generated, interpolation techniques shall prove necessary forever in the efficient analysis and processing of this data. Moreover, the rapid development of new technologies would, in fact, advance them as even more sophisticated and exact techniques that would further enhance the accuracy of predictions and estimates.

To sum up, interpolation is one of the highly significant subjects that continues to have innumerable applications, perhaps even for a long time to come.