  The manuscript describes fast and scalable architectures and associated
algorithms for computing convolutions and cross-correlations. The basic idea is
to map 2D convolutions and cross-correlations to a collection of 1D
convolutions and cross-correlations in the transform domain. This is
accomplished through the use of the Discrete Periodic Radon Transform (DPRT)
for general kernels and the use of SVD-LU decompositions for low-rank kernels.
The approach uses scalable architectures that can be fitted into modern FPGA
and Zynq-SOC devices. Based on different types of available resources, for
$P\times P$ blocks, 2D convolutions and cross-correlations can be computed in
just $O(P)$ clock cycles up to $O(P^2)$ clock cycles. Thus, there is a
trade-off between performance and required numbers and types of resources. We
provide implementations of the proposed architectures using modern programmable
devices (Virtex-7 and Zynq-SOC). Based on the amounts and types of required
resources, we show that the proposed approaches significantly outperform
current methods.
