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 $M$ of $\mathbb{R}^d$, and associates to each direction $v\in S^{d-1}$ a shape summary obtained by scanning $M$ in the direction $v$. 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 $O(d)$ 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.