  Topological transforms are parametrized families of topological invariants,
which, by analogy with transforms in signal processing, are much more
discriminative than single measurements. The first two topological transforms
to be defined were the Persistent Homology Transform and Euler Characteristic
Transform, both of which apply to shapes embedded in Euclidean space. The
contribution of this paper is to define topological transforms that depend only
on the intrinsic geometry of a shape, and hence are invariant to the choice of
embedding. To that end, given an abstract metric measure space, we define an
integral operator whose eigenfunctions are used to compute sublevel set
persistent homology. We demonstrate that this operator, which we call the
distance kernel operator, enjoys desirable stability properties, and that its
spectrum and eigenfunctions concisely encode the large-scale geometry of our
metric measure space. We then define a number of topological transforms using
the eigenfunctions of this operator, and observe that these transforms inherit
many of the stability and injectivity properties of the distance kernel
operator.
