Segment-based simple-connectivity measure design and implementation

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In developing different measures for the description of a segment’s shape, we noted that it would be useful to include a measure capable of quantifying the presence of holes. This was motivated by the following scenario. The measures we use to characterize a segment’s shape, such as RoundnessAndNoHole (also known as compactness), ConvexityAndNoHole and RectangularityAndNoHole are monotonically decreasing with the presence of holes, namely: • RoundnessAndNoHole is high if Roundness is high and condition NoHole is true, • ConvexityAndNoHole is high if Convexity is high and condition NoHole is true and, finally, • RectangularityAndNoHole is high if Rectangularity is high and condition NoHole is true. For example, a region with a perfectly round external boundary, but containing several holes, will present a low RoundnessAndNoHole measure. Were the holes not present in the region, it would instead feature a very high RoundnessAndNoHole measure. Besides these measures, our newly introduced version of a measure of elongatedness is also affected by the presence of holes, increasing as the number of holes increases.

In our study of satellite images, it is very common to find segments that contain holes, whether due to the underlying holes in the original observed structure or whether due to segmentation errors. In order to reason about these types of situations without having to change the definitions of the shape measures already in use (which are quite natural and intuitive), we introduce a new measure to quantify the presence of holes, which we call simple-connectivity. The simple-connectivity measure quantifies the extent to which a region is simply-connected, i.e., the measure should be monotonically decreasing with holes whose cardinality increases or whose size increases (at fixed cardinality).