Bicubic interpolation

In mathematics, bicubic interpolation is an extension of cubic interpolation for interpolating data points on a two dimensional regular grid. The interpolated surface is smoother than corresponding surfaces obtained by bilinear interpolation or nearest-neighbor interpolation. Bicubic interpolation can be accomplished using either Lagrange polynomials, cubic splines, or cubic convolution algorithm.

In image processing, bicubic interpolation is often chosen over bilinear interpolation or nearest neighbor in image resampling, when speed is not an issue. In contrast to bilinear interpolation, which only takes 4 pixels (2×2) into account, bicubic interpolation considers 16 pixels (4×4). Images resampled with bicubic interpolation are smoother and have fewer interpolation artifacts.

Suppose the function values ${\displaystyle f}$ and the derivatives ${\displaystyle f_{x}}$, ${\displaystyle f_{y}}$ and ${\displaystyle f_{xy}}$ are known at the four corners ${\displaystyle (0,0)}$, ${\displaystyle (1,0)}$, ${\displaystyle (0,1)}$, and ${\displaystyle (1,1)}$ of the unit square. The interpolated surface can then be written

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