Bilinear interpolation
In mathematics, bilinear interpolation is an extension of linear interpolation for interpolating functions of two variables (e.g., x and y) on a rectilinear 2D grid.
The key idea is to perform linear interpolation first in one direction, and then again in the other direction. Although each step is linear in the sampled values and in the position, the interpolation as a whole is not linear but rather quadratic in the sample location.
Suppose that we want to find the value of the unknown function f at the point (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
We proceed by interpolating in the y-direction to obtain the desired estimate:
Note that we will arrive at the same result if the interpolation is done first along the y-direction and then along the x-direction.
An alternative way to write the solution to the interpolation problem is
Where the coefficients are found by solving the linear system
If a solution is preferred in terms of f(Q) then we can write
Where the coefficients are found by solving
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
Or equivalently, in matrix operations:
Contrary to what the name suggests, the bilinear interpolant is not linear; but it is the product of two linear functions.
Alternatively, the interpolant can be written as
where
In both cases, the number of constants (four) correspond to the number of data points where f is given. The interpolant is linear along lines parallel to either the x or the y direction, equivalently if x or y is set constant. Along any other straight line, the interpolant is quadratic. However, even if the interpolation is not linear in the position (x and y), it is linear in the amplitude, as it is apparent from the equations above: all the coefficient bj, j=1..4, are proportional to the value of the function f(,).
...
Wikipedia