Persistent homology distinguishes image features from noise

In an image, topological features that persist over a wide range of scales provide quality measures of the overall shape of the data. In contrast, features sensitive to small changes in scale can be attributed to noise. Persistent homology distinguishes between the two by computing the shape of complex datasets over many scales.

By essentially reducing data to a smaller number of robust topological features, persistent homology can make the training and computing of machine learning models easier and faster. Leykam et al. tested the technique on images of dark solitons in Bose-Einstein condensates.

Solitons are self-localizing wave pulses that emerge in a variety of nonlinear wave media, including waves in the ocean. Dark solitons emerge when repulsive interactions are present, forming localized dips in wave amplitude and stripes of low intensity in 2D systems.

To identify the dark solitons, the researchers converted grayscale images to binary images based on an adjustable threshold intensity.

“We then construct a graph by taking each ‘0’ pixel as a vertex, and connecting neighboring ‘0’ pixels by edges,” said author Daniel Leykam. “Once a graph is constructed, its topological features — distinct clusters and cycles formed by the low-intensity parts of the image — can be readily computed using simple linear algebra.”

The robust features are those that persisted through a wide range of threshold intensities, separate from the small-scale density fluctuations of the Bose-Einstein condensate.

The team hopes other physicists will use persistent homology to analyze experimental data. They are working to apply similar techniques to optical waveguide devices, where image data could point to fabrication defects.

Source: “Dark soliton detection using persistent homology,” by Daniel Leykam, Irving Rondón, and Dimitris G. Angelakis, Chaos (2022). The article can be accessed at https://doi.org/10.1063/5.0097053 .