Efficient neighborhood-based computations on regions using scans

Many basic computations can be done by means of iterative neighborhood-based calculations, including threshold, optimum, distance transform, contour closing, mathematical morphology, etc. Some of them can be performed using rows-per-rows scans (A. Rosenfeld and J.-L. Pfaltz, 1966) (G. Borgefors, 1986). Such regular computations allow to optimize the use of caches on standard architecture, and to achieve computations in good times. However, these basic computations are also useful inside the regions of the images. When applied on some regions instead of the whole image, more scans could be necessary because of the irregular shape of the regions. In this paper, we show that row-per-row scans can be used for a large class of operators, so-called idempotent r-operators (including the previously cited computations). Moreover, we give conditions on the use of scans to perform neighborhood-based computations inside any regions. Among other results, we show that only two scans allow to compute a distance transform in every regions used in classical split and merge algorithms (S.L. Horowitz and T. Pavlidis, 1976), and only three scans are sufficient on Voronoi regions (M. Tuceryan and A.K. Jain, 1990). These results extend the use of row per row scans to more cases, and improve many algorithms that rely on some neighborhood-based computations inside regions.