MORPHOLOGICAL SAMPLING OF CLOSED SETS - Christian Ronse; Mohamed Tajine (Image Analysis and Stereology, 2011)
We briefiy survey the standard morphological approach (Heijmans, 1994) to the sampling (or discretization) of sets. Then we summarize the main results of our metric theory of sampling (Ronse and Tajine, 2000; 2001; 2002; Tajine and Ronse, 2002), which can be used to analyse several sampling schemes, in particular the morphological one. We extend it to the sampling of closed sets (instead of compact ones), and to the case where the sampling subspace is boundedly compact (instead of boundedly finite), and obtain new results on morphological sampling. In the original morphological theory for sampling (Heijmans, 1994), some reconstruction of the sampling was shown to converge, relative to the Fell topology (or Hausdorff-Busemann metric), to the original closed set when the resolution of the sampling space tends to zero. As we explain here, this reconstruction step is artificial and unnecessary, the sampling itself converges to the original closed set under the Hausdorff metric, which is a stronger convergence than for the Hausdorff-Busemann metric.