Program within Applied Differential Geometry

Cardiovascular geometry

Analysing vascular images from various medical imaging modalities, based on the generic prior that vessels are spatially elongated structures.

Complicating factors, such as the occurrence of bifurcations and inevitable imaging imperfections, are effectively handled by a geometrical paradigm that (i) `lifts’ spatial evidence to an abstract manifold in which space and orientation are pried apart, and (ii) exploits geometric deep learning.

Details: Having a healthy cardiovascular system is vital to our health and well-being. A proper assessment of the circulatory system from a medical scan is not merely a matter of depicting arteries and veins, but requires a topologically correct mapping of full vessel trees. Data noise and vessel crossings and bifurcations often lead to confusion due to ambiguities or missing pieces of information. To properly deal with these issues we consider ‘liftings’ of spatial data into higher-dimensional homogeneous spaces, a trick of the trade from differential geometry that allows one to pry apart position and orientation. After lifting, empirical data are more sparsely distributed over a higherdimensional domain, but, in a precise sense, still carry exactly the same information. Subject to geometric partial (linear, morphological) differential equations of parabolic type, the higher-dimensional images can be processed so as to enhance, complete, and delineate the line-like structures of interest, with much less chance of confusion as compared to traditional ‘direct’ routes. After processing, the lifted data can be back-projected into 3-space for human inspection. We furthermore propose ‘geometric deep learning’ as a natural augmentation to learn from extrinsic expert annotations, thus complementing the data intrinsic analysis sketched above.