A Convolutional Persistence Transform

We consider a new topological feauturization of $d$-dimensional images, obtained by convolving images with various filters before computing persistence. Viewing a convolution filter as a motif within an image, the persistence diagram of the resulting convolution describes the way the motif is distributed throughout that image. This pipeline, which we call convolutional persistence, extends the capacity of topology to observe patterns in image data. Indeed, we prove that (generically speaking) for any two images one can find some filter for which they produce different persistence diagrams, so that the collection of all possible convolutional persistence diagrams for a given image is an injective invariant. This is proven by showing convolutional persistence to be a special case of another topological invariant, the Persistent Homology Transform. Other advantages of convolutional persistence are improved stability and robustness to noise, greater flexibility for data-dependent vectorizations, and reduced computational complexity for convolutions with large stride vectors. Additionally, we have a suite of experiments showing that convolutions greatly improve the predictive power of persistence on a host of classification tasks, even if one uses random filters and vectorizes the resulting diagrams by recording only their total persistences.