On a novel matrix approach for linear space-invariant 2-D deconvolutions

The linear and space-invariant convolution
Y = H \bigotimes X
of a matrix
X
with the kernel (impulse response, point spread function) matrix
H
is modeled as a linear matrix equation of the form
Y=\sum_{i}A_{i}XB_{i}
. Under periodicity assumptions on matrix
X
analytical and numerical solution methods of the 2-D inverse filtering problem are presented jointly with an analysis of the computational complexity and convergence conditions. As a consequence of this new matrix approach, the discrete Fourier transform (DFT) method is derived as a particular case of more powerful algebraic operators to solve general matrix equations.