Topological data analysis in 3D ultrasound tomography

Topological data/image analysis has grown into an independent area of practical research in the past 8 years due to the increased availability of software to carry out persistent homology, etc. Other aspects include cohomology, graph theory, Morse theory and homotopy groups. We focus here on homotopy groups. We introduce a specific map from S1 to the N-torus, where N is the number of elements in the receiver array in the QT Imaging system. We show the homotopic equivalence classes of these maps (the elements of the (known) first homotopy group of the N-Torus) have direct relevance to phase wrapping in frequency domain 3D ultrasound tomography in a manner somewhat different from the normal understanding of phase wrapping/cycle-skipping. The maps are related to the Lie symmetry group associated with full view tomography: SO(2)xR which arise in the equivalence of our method to training a neural net previously shown. This characterization as a known homotopy group gives constraints on frequency stepping in reconstruction and/or vertical differences between data levels in data acquisition scenarios, depending on the interpretation of the R coordinate in the Lie symmetry group.