## Bilinear Interpolation

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An image (or cross section) is a 2D matrix of picture elements (pixels). The image is defined by the function values (resp. color triples) $C_{ij}$ at the pixel’s center points $P_{ij}$.

The latter grid layout is called **cell-centric** representation. If function values are given directly at the grid points, the representation is called **grid-centric**.

An image (or cross section) is interpolated with **bi-linear interpolation**, which is short for 2 consecutive linear interpolations in the x- and y-direction.

A single **linear interpolation** has the form

with u being the normalized interpolation coefficient in the range [0,1] and a and b being the two values to be interpolated. A linear interpolation has the following characteristics:

Bi-linear interpolation creates a single interpolated value (blue) from **4** neighbouring pixels (red) by two subsequent linear interpolations along the two axis with respective normalized interpolation weights $u$ and $v$.

Given: 4 discrete color values $C_{00},C_{10},C_{01},C_{1}$ at the centers $P_{00},P_{10},P_{01},P_{1}$ of 4 neighboring pixels.

Wanted: continuous interpolated color C(u,v) within the interpolation area $(u,v)\in[0,1]x[0,1]$

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