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Monday, June 4, 2018

Integral transforms

Let X,Y be topological spaces.

Definition: A set UX is constructible if it is a finite union of locally closed sets. A function f:XY is constructible if f1(y)X is constructible for all yY.

Write CF(X) for the set of constructible functions f:XZ. Recall if UX is constructible, it is triangulable.

Definition: Let XRN be constructible and {Xr}rR a filtration of X by constructible sets Xr. The kth persistence diagram of X is the set PD(Xr,k)={(a,b)(R{±})2:a<b}, where each element represents the longest sequence of identity morphisms in the decomposition of the image of the kth 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).

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