Fast 2D Convolutions and Cross-Correlations Using Scalable Architectures

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 for general kernels and the use of singular value decomposition -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 <inline-formula> <tex-math notation="LaTeX">$P\times P$ </tex-math></inline-formula> blocks, 2D convolutions and cross-correlations can be computed in just <inline-formula> <tex-math notation="LaTeX">$O(P)$ </tex-math></inline-formula> clock cycles up to <inline-formula> <tex-math notation="LaTeX">$O(P^{2})$ </tex-math></inline-formula> 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.