On generalized separation and the speedup of local operators on multi-dimensional signals

Along with the boom of computer vision- and other signal processing-based applications, the question of fast local operator application has become increasingly important over the past decade. For two-dimensional signals, such as images, local operator separation has proven to ameliorate the computational effort incurred by prominent operators such as mean filter, binomial filter or Sobel filter. However, many modern days´ applications involve higher-dimensional input signals including tomography, functional magnetic resonance imaging (fMRI), selective plane illumination microscopy (SPIM), confocal laser scanning microscopy (CLSM), special image fusion. In this work, we generalize previous separation methods in order to make them applicable to input spaces of arbitrary (but finite) dimensionality. We show how this approach renders the computational effort of the dominant multiplicative part of local filter application linear both in size and dimension. Thus, filtering can become feasible for input signals whose dimensionality and size complexity would otherwise be prohibitively high. Finally, we present experiments that demonstrate our approach to be highly beneficial not only in theory but also in practice.