Bilinear Interpolation

Used often in image processing (2D raster grid) to find the value of an intermediate point in the 2D pixel space.

Bilinear interpolation is a simple procedure, representable as the interpolation from four known points to another point (I guess in the hull of that point) via sequential interpolation in the $x$ then $y$ axes, and this is equivalent to performing the interpolation in the opposite order ($y$ then $x$).

See Bilinear interpolation - Wikipedia for the various forms for computing the bilinear interpolation including:

• repeated linear interpolation
• weighted mean
• matrix formulation
• polynomial fit

Bilinearity $⟹$ non-linear

Bilinear is nonlinear. It’s linear in both main variables, but not in any superposition. Naively speaking, it’s linear if you cut along 𝑥x or 𝑦y axis, but you’re not allowed to rotate the frame (which is what a proper linear function allows, even requires, as linearity is independent of choice of coordinates). A typical example is bilinear interpolation (splines & such), which puts a smooth interpolation over a rectangular frame (a classic problem that a 3D quadrilateral is almost never planar).

— https://math.stackexchange.com/a/1650733/556438