  In this paper we consider two topological transforms that are popular in
applied topology: 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, i.e.~each shape has a
unique transform. Moreover, we prove that these transforms determine continuous
maps from the sphere to the space of persistence diagrams, equipped with any
Wasserstein $p$-distance, or the space of Euler curves, equipped with certain
$L^p$ norms. By making use of a stratified space structure on the sphere,
induced by hyperplane divisions, we prove additional uniqueness results in
terms of distributions on the space of Euler curves. Finally, our main result
proves that any shape in a certain uncountable space of PL embedded shapes with
plausible geometric bounds can be uniquely determined using only finitely many
directions.
