Parallel Algorithms via Scaled Paraboloid Structuring Functions for Spatially-Variant and Label-Set Dilations and Erosions

Although most greyscale morphology is performed with ``flat'' structuring functions because these are widely available, the use of scaled paraboloid (or quadratic) structuring functions offers a far wider range of applicability, better theoretical properties, and can also be computed efficiently. We demonstrate the novel application of scaled paraboloid structuring functions to parallel algorithms for two important classes of morphology -- binary dilations and erosions using spatially variant structuring elements, and dilations and erosions on label sets. These algorithms exploit the dimensional-separability properties of parabolic structuring functions to process each scan line of an image independently, leading to highly efficient parallel implementations particularly in higher dimensions.