The discrete triangle transform

We introduce the discrete triangle transform (DTT), a non-separable transform for signal processing on a two-dimensional equispaced triangular grid. The DTT is, in a strict mathematical sense, a generalization of the DCT, type III, to two dimensions, since the DTT is built from Chebyshev polynomials in two variables in the same way as the DCT, type III, is built from Chebyshev polynomials in one variable. We provide boundary conditions, signal extension, and diagonalization properties for the DTT. Finally, we give evidence that the DTT has Cooley-Tukey FFT like algorithms that enable its efficient computation.