Fast implementation for symmetric non-separable transforms based on grids

When a line graph is symmetric, the associated graph Fourier transform has a fast implementation. In this paper, we extend this idea to the 2D non-separable case, where the graph of interest is a square-shaped grid. We investigate a number of symmetry types for 2D grids. Then, for each type of symmetry we derive a block-diagonalization form of the graph Laplacian matrix, based on which fast implementations with reduced number of multiplications can be obtained. We show that for moderate block sizes, certain types of grid symmetry enable us to design non-separable block transforms that have computational complexities comparable to those of separable ones.