Discrete Deep Structure

Abstract

The discrete scale space representation L of f is continuous in scale t. A computational investigation of L however must rely on a finite number of sampled scales. There are multiple approaches to sampling L differing in accuracy, runtime complexity and memory usage. One apparent approach is given by the definition of L via discrete convolution with a scale space kernel. The scale space kernel is of infinite domain and must be truncated in order to compute an individual scale, thus introducing truncation errors. A periodic boundary condition for f further complicates the computation. In this case, circular convolution with a Laplacian kernel provides for an elegant but still computationally complex solution. Applied in its eigenspace however, the circular convolution operator reduces to a simple and much less complex scaling transformation. This paper details how to efficiently decompose a scale of L and its derivative ∂  _t L into a sum of eigenimages of the Laplacian circular convolution operator and provides a simple solution of the discretized diffusion equation, enabling for fast and accurate sampling of L.