  Geometric alignment appears in a variety of applications, ranging from domain
adaptation, optimal transport, and normalizing flows in machine learning;
optical flow and learned augmentation in computer vision and deformable
registration within biomedical imaging. A recurring challenge is the alignment
of domains whose topology is not the same; a problem that is routinely ignored,
potentially introducing bias in downstream analysis. As a first step towards
solving such alignment problems, we propose an unsupervised algorithm for the
detection of changes in image topology. The model is based on a conditional
variational auto-encoder and detects topological changes between two images
during the registration step. We account for both topological changes in the
image under spatial variation and unexpected transformations. Our approach is
validated on two tasks and datasets: detection of topological changes in
microscopy images of cells, and unsupervised anomaly detection brain imaging.
