Diabetic retinopathy is a complication of diabetes that produces changes in the blood vessel structure in the retina, which can cause severe vision problems and even blindness. In this paper, we demonstrate that by identifying topological features in very high resolution retinal images, we can construct a classifier that discriminates between healthy patients and those with diabetic retinopathy using summary statistics of these features. Topological data analysis identifies the features as connected components and holes in the images and describes the extent to which they persist across the image. These features are encoded in persistence diagrams, summaries of which can be used to discrimate between diabetic and healthy patients. The method has the potential to be an effective automated screening tool, with high sensitivity and specificity.

The interstellar medium (ISM) is a magnetised system in which transonic or supersonic turbulence is driven by supernova explosions. This leads to the production of intermittent, filamentary structures in the ISM gas density, whilst the associated dynamo action also produces intermittent magnetic fields. The traditional theory of random functions, restricted to second-order statistical moments (or power spectra), does not adequately describe such systems. We apply topological data analysis (TDA), sensitive to all statistical moments and independent of the assumption of Gaussian statistics, to the gas density fluctuations in a magnetohydrodynamic (MHD) simulation of the multi-phase ISM. This simulation admits dynamo action, so produces physically realistic magnetic fields. The topology of the gas distribution, with and without magnetic fields, is quantified in terms of Betti numbers and persistence diagrams. Like the more standard correlation analysis, TDA shows that the ISM gas density is sensitive to the presence of magnetic fields. However, TDA gives us important additional information that cannot be obtained from correlation functions. In particular, the Betti numbers per correlation cell are shown to be physically informative. Magnetic fields make the ISM more homogeneous, reducing the abundance of both isolated gas clouds and cavities, with a stronger effect on the cavities. Remarkably, the modification of the gas distribution by magnetic fields is captured by the Betti numbers even in regions more than 300 pc from the midplane, where the magnetic field is weaker and correlation analysis fails to detect any signatures of magnetic effects.

The two aspect ratios of randomly oriented triaxial ellipsoids (representing isosurfaces of an isotropic 3D random field) can be determined from a single 2D cross-section of their sample using the probability density function (PDF) of the filamentarity F of individual structures seen in cross-section (F = 0 for a circle and F = 1 for a line). The PDF of F has a robust form with a sharp maximum and truncation at larger F , and the most probable and maximum values of F are uniquely and simply related to the two aspect ratios of the triaxial ellipsoids. The parameters of triaxial ellipsoids of randomly distributed sizes can still be recovered from the PDF of F . This method is applicable to many shape recognition problems, here illustrated by the neutral hydrogen density in the turbulent interstellar medium of the Milky Way. The gas distribution is shown to be filamentary with axis ratios of about 1:2:20.

Persistent homology is used to track the appearance and disappearance of features as we move through a nested sequence of topological spaces. Equating the nested sequence to a filtration and the appearance and disappearance of features to events, we show that simple event history methods can be used for the analysis of topological data. We propose a version of the well known Nelson-Aalen cumulative hazard estimator for the comparison of topological features of random fields and for testing parametric assumptions. We suggest a Cox proportional hazards approach for the analysis of embedded metric trees. The Nelson-Aalen method is illustrated on globally distributed climate data and on neutral hydrogen distribution in the Milky Way. The Cox method is use to compare vascular patterns in fundus images of the eyes of healthy and diabetic retinopathy patients.

Current astrophysical models of the interstellar medium assume that small scale variation and noise can be modelled as Gaussian random fields or simple transformations thereof, such as lognormal. We use topological methods to investigate this assumption for three regions of the southern sky. We consider Gaussian random fields on two-dimensional lattices and investigate the expected distribution of topological structures quantified through Betti numbers. We demonstrate that there are circumstances where differences in topology can identify differences in distributions when conventional marginal or correlation analyses may not. We propose a non-parametric method for comparing two fields based on the counts of topological features and the geometry of the associated persistence diagrams. When we apply the methods to the astrophysical data, we find strong evidence against a Gaussian random field model for each of the three regions of the interstellar medium that we consider. Further, we show that there are topological differences at a local scale between these different regions.