Editing Bilinear interpolation (section)

=== Weighted mean ===
[[File:Bilinear interpolation visualisation.svg|thumb|upright|A geometric visualisation of bilinear interpolation. The product of the value at the desired point (black) and the entire area is equal to the sum of the products of the value at each corner and the partial area diagonally opposite the corner (corresponding colours).]]
The solution can also be written as a [[Weighted Mean|weighted mean]] of the ''f''(''Q''):
:<math>f(x, y) \approx w_{11} f(Q_{11}) + w_{12} f(Q_{12}) + w_{21} f(Q_{21}) + w_{22} f(Q_{22}),</math>

where the weights sum to 1 and satisfy the transposed linear system
:<math>
  \begin{bmatrix}
    1       & 1         & 1         & 1         \\
    x_1     & x_1       & x_2       & x_2       \\
    y_1     & y_2       & y_1       & y_2       \\
    x_1y_1  & x_1y_2    & x_2y_1    & x_2y_2
\end{bmatrix}
\begin{bmatrix}
    w_{11} \\ w_{12} \\ w_{21} \\ w_{22}
\end{bmatrix} =
\begin{bmatrix}
    1 \\ x \\ y \\ xy
\end{bmatrix},
</math>

yielding the result
:<math>\begin{align}
\begin{bmatrix}
w_{11}\\w_{21}\\w_{12}\\w_{22}
\end{bmatrix} = \frac{1}{(x_2-x_1)(y_2-y_1)}\begin{bmatrix}
x_2y_2 & -y_2 & -x_2 & 1 \\
-x_2y_1 & y_1 & x_2 & -1 \\
-x_1y_2 & y_2 & x_1 & -1 \\
x_1y_1 & -y_1 & -x_1 & 1 
\end{bmatrix}\begin{bmatrix}
1\\x\\y\\xy
\end{bmatrix},
\end{align}</math>
which simplifies to
:<math>\begin{align}
  w_{11} &= \frac{(x_2 - x  )(y_2 - y  )}{(x_2 - x_1)(y_2 - y_1)}, \\
  w_{12} &= \frac{(x_2 - x  )(y   - y_1)}{(x_2 - x_1)(y_2 - y_1)}, \\
  w_{21} &= \frac{(x   - x_1)(y_2 - y  )}{(x_2 - x_1)(y_2 - y_1)}, \\
  w_{22} &= \frac{(x   - x_1)(y   - y_1)}{(x_2 - x_1)(y_2 - y_1)},
\end{align}</math>
in agreement with the result obtained by repeated linear interpolation. The set of weights can also be interpreted as a set of [[generalized barycentric coordinates]] for a rectangle.