Bilinear interpolation is a method for estimating the value of a function at a point $(x, y)$ using the known values at four surrounding rectangular points: $Q_{11}=(x_1, y_1)$, $Q_{12}=(x_1, y_2)$, $Q_{21}=(x_2, y_1)$, and $Q_{22}=(x_2, y_2)$.

It works in two steps:

1. Perform two linear interpolations in the x-direction to find intermediate values at $(x, y_1)$ and $(x, y_2)$.
2. Perform one linear interpolation in the y-direction using the two intermediate values to find the final value at $(x, y)$.

Formula for Step 1:

$$ f(x, y_1) \approx \frac{x_2-x}{x_2-x_1}f(Q_{11}) + \frac{x-x_1}{x_2-x_1}f(Q_{21}) $$

Formula for Step 2:

$$ f(x, y) \approx \frac{y_2-y}{y_2-y_1}f(x, y_1) + \frac{y-y_1}{y_2-y_1}f(x, y_2) $$