Separable 2D Transform

We consider a separable two dimensional transform by representing the 2D basis function as the product of two 1D basis functions.

Here f(n₁,n₂) is a 2D signal, F_m1,m2 is a 2D array of coefficients, and the g_m are the same basis functions as in the 1D case.
This expression for representing a signal, f(n₁,n₂), as a linear combination of separable basis functions can be written in matrix form and the MMSE estimate of the coefficient array can be found.
Again, when we have the same number of basis functions as original data points the estimate of F is exact and the basis is complete.
G and A are the 1D, image independent synthesis and analysis matrices and the forward and inverse transforms are simply matrix multiplications.