In numerical analysis, a branch of mathematics, bicubic interpolation is one of the most common interpolation methods in two dimensions. With this method, the value f(x, y) of a function f at a point (x, y) is computed as a weighted average of the nearest sixteen samples in a rectangular grid (a 4x4 array). Here, two cubic interpolation polynomials, one for each plane direction, are used. Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ... :For other senses of this word, see dimension (disambiguation). ... Partial plot of a function f. ... In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. ... GRID can refer to : GRID computing short for gay-related immune deficiency, a former name for AIDS. See also homosexuality and medical science General Repository for Interaction Datasets, a database of biological interactions hosted at Mount Sinai Hospital in Toronto, Canada This is a disambiguation page — a navigational aid... Polynomial of degree 3 In mathematics, a cubic function is a function of the form where a is nonzero; or in other words, a polynomial of degree three. ... In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial. ...

Properties

Bicubic interpolation results in an interpolating function which is continuous, has continuous first partial derivatives, and has continuous cross derivatives everywhere.

Formula

The bicubic interpolation is calculated as follows:

a₀₀ + a₁₀x + a₀₁y + a₂₀x² + a₁₁xy + a₀₂y² + a₂₁x²y + a₁₂xy² + a₂₂x²y² + a₃₀x³ + a₀₃y³ + a₃₁x³y + a₁₃xy³ + a₃₂x³y² + a₂₃x²y³ + a₃₃x³y³

Or, in more compact form:

The procedure used to calculate the coefficients a_ij depends on the interpolated data source properties. If we are interpolating a function with known derivatives, the common approach is to use the height at the four vertices, together with three derivatives at each vertex. The first derivatives h'x and h'y express the slope of the surface in the x and y directions, while the second (cross) derivative h''xy represents the slope in both x and y. In terms of solving the coefficients of the formula, these values can be expressed by differentiating the x and y vectors independently, and then consecutively. For each of the vertices of the grid cell, the local coordinates (at 0,0, 1,0, 0,1 and 1,1) can be input into these equations to generate the 16 equations to solve.

Use in computer graphics

The bicubic algorithm is frequently used for scaling images and video for display. It preserves fine detail better than the predominant bilinear algorithm. A zoomed small portion of a bitmap of a cat, using nearest neighbor filtering (left) and bicubic filtering (right). ...

See also

• Cubic Hermite spline, the one-dimensional analogue
• Bilinear interpolation
• Spline interpolation
• Anti-aliasing
• Sinc filter
• Lanczos resampling

In the mathematical subfield of numerical analysis a cubic Hermite spline, named in honor of Charles Hermite (Hermite is pronounced air MIT), is a third-degree spline with each polynomial of the spline in Hermite form. ... In mathematics, bilinear interpolation is an extension of linear interpolation for interpolating functions of two variables. ... In the mathematical subfield of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. ... In digital signal processing, anti-aliasing is the technique of minimizing aliasing (jagged or blocky patterns) when representing a high-resolution signal at a lower resolution. ... In signal processing, the sinc filter strips high-frequency data from a signal. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

External links

• Application of interpolation to elevation samples
• Comparison of interpolation functions