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How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms
In this paper we consider two topological transforms based on Euler calculus:
the Persistent Homology Transform (PHT) and the Euler Characteristic Transform
(ECT). Both of these transforms are of interest for their mathematical
properties as well as their applications to science and engineering, because
they provide a way of summarizing shapes in a topological, yet quantitative,
way. Both transforms take a shape, viewed as a tame subset of
, and associates to each direction a shape summary
obtained by scanning in the direction . These shape summaries are either
persistence diagrams or piecewise constant integer valued functions called
Euler curves. By using an inversion theorem of Schapira, we show that both
transforms are injective on the space of shapes---each shape has a unique
transform. We also introduce a notion of a "generic shape", which we prove can
be uniquely identified up to an element of by using the pushforward of
the Lebesgue measure from the sphere to the space of Euler curves. Finally, our
main result proves that any shape in a certain uncountable set of non-axis
aligned shapes can be specified using only finitely many Euler curves.Comment: October 10, 2019 version is 30 pages with several additional details
for modified proposition and theorem statements. Formula for the bound on the
number of directions has changed slightl