Interpolation

What Is Interpolation?

Interpolation is a mathematical method for estimating unknown values that fall between known data points by constructing a function that passes exactly through those points. Given a set of sample values at specified locations or times, an interpolating function can be evaluated at any intermediate position to recover an approximation of the underlying continuous quantity. The technique is fundamental in numerical analysis, signal processing, computer graphics, and scientific computing, where physical measurements or digital samples are inherently discrete but applications require continuous or higher-resolution representations.

Interpolation is distinct from approximation or curve fitting, which seeks a function that fits data within a tolerance rather than passing exactly through each point. When data are noisy, fitting methods such as least-squares regression are preferred; when data are exact or nearly so, interpolation is appropriate. The choice of interpolating function determines the accuracy, computational cost, and smoothness of the result.

Polynomial and Spline Methods

The simplest interpolating functions are polynomials. Given n+1 data points, a unique polynomial of degree n exists that passes through all of them. The Lagrange and Newton divided-difference forms both construct this polynomial, though direct high-degree polynomials can oscillate wildly between data points, a phenomenon known as Runge's phenomenon. Piecewise polynomial methods, particularly cubic splines, address this by fitting low-degree polynomials in adjacent intervals while enforcing continuity of the function and its first and second derivatives at each internal node. As detailed in MIT OpenCourseWare materials on numerical analysis, cubic splines provide smooth interpolants that avoid oscillation and are widely used in engineering simulations and computer-aided design.

Linear interpolation is the simplest and most computationally efficient form: it connects successive data points with straight-line segments and is the default in many real-time applications where processing budget is limited. Bilinear and trilinear interpolation extend the concept to two and three spatial dimensions, as used in image resizing and volumetric rendering.

Radial Basis Function Networks

For interpolation over scattered data in multiple dimensions, where sample points are not arranged on a regular grid, radial basis function (RBF) methods provide a general framework. An RBF network represents the interpolating surface as a weighted sum of basis functions, each centered on one data point and depending only on the Euclidean distance from that center. The weights are determined by solving a linear system. RBF interpolation has been applied to terrain modeling, meteorological field reconstruction, and mesh-free simulation of physical systems. The approach is related to Gaussian process regression and to the training of certain neural network architectures, as the RBF network is a single hidden layer network with fixed basis functions.

Signal Processing and Digital-to-Analog Conversion

In digital signal processing, interpolation refers specifically to increasing the sample rate of a discrete signal. A digital-to-analog converter (DAC) must reconstruct a continuous waveform from digital samples; the ideal reconstruction filter implements sinc interpolation, which recovers the original band-limited signal exactly when the Nyquist criterion is satisfied. Practical implementations use finite-length filters that approximate the ideal. A DSP-focused treatment of interpolation by Robertson describes how polynomial and linear-phase FIR filters achieve efficient interpolation with controllable alias rejection in real hardware. The related IEEE paper on digital signal processing approaches to interpolation demonstrates that well-designed FIR interpolators can match or exceed the accuracy of classical polynomial methods for sampled data.

Applications

Interpolation has applications in a wide range of fields, including: