Integral transforms

Let $X,Y$ be topological spaces.

Definition: A set $U\subseteq X$ is constructible if it is a finite union of locally closed sets. A function $f\colon X\to Y$ is constructible if $f^{-1}(y)\subseteq X$ is constructible for all $y\in Y$.

Write $CF(X)$ for the set of constructible functions $f\colon X\to \Z$. Recall if $U\subseteq X$ is constructible, it is triangulable.

Definition: Let $X\subseteq \R^N$ be constructible and $\{X_r\}_{r\in \R}$ a filtration of $X$ by constructible sets $X_r$. The $k$th persistence diagram of $X$ is the set $PD(X_r,k)= \{(a,b)\subseteq (\R\cup \{\pm\infty\})^2 : a<b \}$, where each element represents the longest sequence of identity morphisms in the decomposition of the image of the $k$th persistent homology functor $PH(X_r,k)\colon (\R,\leqslant )\to Vect$ to each component.

Write $D$ for the set of all persistence diagrams.

Definition: Let $X,Y\subseteq \R^N$ be constructible, $S\subseteq X\times Y$ also constructible with $\pi_1,\pi_2$ the natural projections, and $\sigma$ a simplex in a triangulation of $X$. The Euler integral of elements of $CF(X)$ is the assignment $$\begin{array}{r c l} \displaystyle \int_X\ \cdot\ d\chi\colon CF(X) & \to & \Z, \\ \mathbf{1}_\sigma & \mapsto & (-1)^{\dim(\sigma)}. \end{array}$$ The Radon transform of elements of $CF(X)$ is the assignment $$\begin{array}{r c l} \mathcal R_S \colon CF(X) & \to & CF(Y), \\ (x\mapsto h(x)) & \mapsto & \left(y\mapsto \displaystyle \int_{\pi_2^{-1}(y)} \pi_1^*h\ d\chi\right). \end{array}$$ The persistent homology transform of $X$ is the assignment $$\begin{array}{r c l} PHT_X \colon S^{N-1} & \to & D^N, \\ v & \mapsto & \left\{PD(\{x\in X:x\cdot v\leqslant r\},0),\dots,PD(\{x\in X : x\cdot v\leqslant r\},N-1)\right\} \end{array}$$

The Euler integral is also called the Euler transform, or the Euler charateristic transform. The Radon transform has a weighted version, where every simplex in $S$ is assigned a weight.

References: Schapira (Tomography of constructible functions), Baryhsnikov, Ghrist, Lipsky (Inversion of Euler integral transforms), Turner, Mukherjee, Boyer (Persistent homology transform).