  The persistent homology transform (PHT) represents a shape with a multiset of
persistence diagrams parameterized by the sphere of directions in the ambient
space. In this work, we describe a finite set of diagrams that discretize the
PHT such that it faithfully represents the underlying shape. We provide a
discretization that is exponential in the dimension of the shape. Moreover, we
show that this discretization is stable with respect to various perturbations
and we provide an algorithm for computing the discretization. Our approach
relies only on knowing the heights and dimensions of topological events, which
means that it can be adapted to provide discretizations of other
dimension-returning topological transforms, including the Betti function
transform. With mild alterations, we also adapt our methods to faithfully
discretize the Euler characteristic function transform.
