In mathematics, bilinear interpolation is an extension of linear interpolation for interpolating functions of two variables. The key idea is to perform linear interpolation first in one direction, and then in the other direction.

Suppose that we want to find the value of the unknown function f at the point P = (x, y). It is assumed that we know the value of f at the four points Q₁₁ = (x₁, y₁), Q₁₂ = (x₁, y₂), Q₂₁ = (x₂, y₁), and Q₂₂ = (x₂, y₂).

We first do linear interpolation in the x-direction. This yields

$f(R\_1)\; \backslash approx\; \backslash frac\{x\_2-x\}\{x\_2-x\_1\}\; f(Q\_\{11\})\; +\; \backslash frac\{x-x\_1\}\{x\_2-x\_1\}\; f(Q\_\{21\})\; \backslash quad\backslash mbox\{where\}\backslash quad\; R\_1\; =\; (x,y\_1),$

$f(R\_2)\; \backslash approx\; \backslash frac\{x\_2-x\}\{x\_2-x\_1\}\; f(Q\_\{12\})\; +\; \backslash frac\{x-x\_1\}\{x\_2-x\_1\}\; f(Q\_\{22\})\; \backslash quad\backslash mbox\{where\}\backslash quad\; R\_2\; =\; (x,y\_2).$

We proceed by interpolating in the y-direction.

$f(P)\; \backslash approx\; \backslash frac\{y\_2-y\}\{y\_2-y\_1\}\; f(R\_1)\; +\; \backslash frac\{y-y\_1\}\{y\_2-y\_1\}\; f(R\_2).$

This gives us the desired estimate of f(x, y).

$f(x,y)\; \backslash approx\; \backslash frac\{f(Q\_\{11\})\}\{(x\_2-x\_1)(y\_2-y\_1)\}\; (x\_2-x)(y\_2-y)\; +\; \backslash frac\{f(Q\_\{21\})\}\{(x\_2-x\_1)(y\_2-y\_1)\}\; (x-x\_1)(y\_2-y)$

$\{\}\; +\; \backslash frac\{f(Q\_\{12\})\}\{(x\_2-x\_1)(y\_2-y\_1)\}\; (x\_2-x)(y-y\_1)\; +\; \backslash frac\{f(Q\_\{22\})\}\{(x\_2-x\_1)(y\_2-y\_1)\}\; (x-x\_1)(y-y\_1).$

If we choose a coordinate system in which the four points where f is known are (0, 0), (0, 1), (1, 0), and (1, 1), then the interpolation formula simplifies to

$f(x,y)\; \backslash approx\; f(0,0)\; \backslash ,\; (1-x)(1-y)\; +\; f(1,0)\; \backslash ,\; x(1-y)\; +\; f(0,1)\; \backslash ,\; (1-x)y\; +\; f(1,1)\; xy.$

Or equivalently, in matrix operations:

$f(x,y)\; \backslash approx\; \backslash begin\{bmatrix\}$

1-x & x \end{bmatrix} \begin{bmatrix} f(0,0) & f(0,1) \\ f(1,0) & f(1,1) \end{bmatrix} \begin{bmatrix} 1-y \\ y \end{bmatrix} Contrary to what the name suggests, the interpolant is not linear. Instead, it is of the form

$(a\_1\; x\; +\; a\_2)(a\_3\; y\; +\; a\_4),\; \backslash ,$

so it is a product of two linear function. Alternatively, the interpolant can be written as

$b\_1\; +\; b\_2\; x\; +\; b\_3\; y\; +\; b\_4\; x\; y.\; \backslash ,$

In both cases, the number of constants (four) correspond to the number of data points where f is given.

The result of bilinear interpolation is independent of the order of interpolation. If we had first performed the linear interpolation in the y-direction and then in the x-direction, the resulting approximation would be the same.

The obvious extension of bilinear interpolation to three dimensions is called trilinear interpolation.

See also

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• Bicubic interpolation
• Spline interpolation
• Lanczos resampling

Interpolation