Morphological Simplification

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Williams, Jason Daniel
Rossignac, Jarek
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Morphological filters, such as closuer, opening, and their combinations, may be used for cleaning and analyzing images and shapes. We focus on the most popular special cases of these operators: the rounding R(S) and filleting F(S) of an arbitrary set S and the combinations R(F(S)) and F(R(S)). These operators may be obtained by combining growing and shrinking operators, which are Minkowski sums and differences with a ball of a given radius r. We define the mortar M(S) as F(S)-R(S). Note that the mortar occupies the thin cracks, protrusions, constrictions, and areas near the high-curvature portions of the boundary of S. Thus, we argue that confining the effect of shape simplification to the mortar has advantages over previously proposed tolerance zones and error metrics, which fail to differentiate between the irregular regions contained in the mortar and the regular (low-curvature) regions of S. We point out that R(F(S)) and F(R(S)) are suitable filters in this context, because their effects are confined to M(S) and leave the core R(S) and the anticore, which is the complement of F(S), unchanged. Furthermore, they tend to replace the high-curvature portions of the boundary of S with with regular portions where the radius of curvature exceeds r. Unfortunately, these operators have a bias, which may result in a large total volume of the symmetric difference between S and its simplified version S'. In order to minimize this volume, we propose to select the filter locally, for each connected component of the mortar. Thus, some portions of the mortar will be simplified using F(R(S)) and some using R(F(S)). This approach, which we call the Mason filter, can be used for the simplification of shapes regardless of their representation or dimensionality. We demonstrate its application to discrete two-dimensional binary sets (i.e. black and white images) and discuss implementation details.
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