Before revision:  The theoretical justification for their use is based on the fact that shapes can be fully reconstructed using just the descriptors .
After revision:  One common summary tool is the Persistent Homology Transform (PHT), which represents a shape with a multiset of persistence diagrams parameterized by directions in the ambient space. For practical applicability, we must bound the number of directions needed in order to ensure that the PHT is a faithful representation of a shape .