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We study an inverse problem of determining a time-dependent potential appearing in the wave equation on conformally transversally anisotropic manifolds of dimension three or higher. These are compact Riemannian manifolds with boundary that are conformally embedded in a product of the real line and a transversal manifold. Under the assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove the unique determination of time-dependent potentials from the knowledge of a certain partial Cauchy data set. 

We provide new proofs based on the Myers–Steenrod theorem to confirm that travel time data, travel time difference data and the broken scattering relations determine a simple Riemannian metric on a disc up to the natural gauge of a boundary fixing diffeomorphism. Our method of the proof leads to a Lipschitz-type stability estimate for the first two data sets in the class of simple metrics. 

We study an inverse problem of determining a time-dependent damping coefficient and potential appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines the time-dependent damping coefficient and potential uniquely. 

Abstract Alzheimer's disease is the leading cause of dementia. The long progression period in Alzheimer's disease provides a possibility for patients to get early treatment by having routine screenings. However, current clinical diagnostic imaging tools do not meet the specific requirements for screening procedures due to high cost and limited availability. In this work, we took the initiative to evaluate the retina, especially the retinal vasculature, as an alternative for conducting screenings for dementia patients caused by Alzheimer's disease. Highly modular machine learning techniques were employed throughout the whole pipeline. Utilizing data from the UK Biobank, the pipeline achieved an average classification accuracy of 82.44%. Besides the high classification accuracy, we also added a saliency analysis to strengthen this pipeline's interpretability. The saliency analysis indicated that within retinal images, small vessels carry more information for diagnosing Alzheimer's diseases, which aligns with related studies.