Convolutional kernel function algebra

Abstract

Many systems for image manipulation, signal analysis, machine learning, and scientific computing make use of discrete convolutional filters that are known before computation begins. These contexts benefit from common sub-expression elimination to reduce the number of calculations required, both multiplications and additions. We present an algebra for describing convolutional kernels and filters at a sufficient level of abstraction to enable intuitive common sub-expression based optimizations through decomposing filters into smaller, repeated, kernels. This enables the creation of an enormous search space of potential implementations of filters via algebraic manipulation. We demonstrate how integral image and sliding window optimizations can be expressed in the context of common sub-expression elimination as well as show the direct use case for this algebra in massively SIMD multiply-free contexts such as in cellular processor arrays. We then show that this algebra is general enough to express and optimize kernels that use non-standard semi-rings to enable shortest path algorithms.