Feb 07. 2025, 14:55 — 15:20
Medical data often involve signals with values in nonlinear spaces. For instance, in medical imaging, pixel values may belong to more complex spaces than the real line such as SPD matrices (e.g. diffusion tensor imaging), the probability simplex (e.g. soft segmentation maps to label different tissues while accounting for uncertainty), or probability measures, say, supported on the 2-sphere (e.g. orientation density functions). Furthermore, such signals cannot usually be assumed to be continuous due to the multiphase nature of anatomy. Hence, studying merely measurable mappings with values in metric spaces is of importance.
I will thus first describe a generalization of Lebesgue spaces, where elements take values in arbitrary metric spaces and show that many properties from the vector-valued case also hold in this general setting. In these particular spaces, I will introduce the class of pointwise absolutely continuous curves, which is stronger than the usual absolute continuity in metric spaces. This distinction only exists outside the vector-valued case and allows the study of the metric geometry of these general Lebesgue spaces at the pointwise scale even in this general setting. Finally, following ideas from metamorphoses, I will show that these objects can be useful in deriving properties of a metric accounting for the action of the group of Sobolev diffeomorphisms on general Lebesgue spaces. Depending on time, I will present some experiments illustrating matching results obtained using the latter metric in the case of manifold-valued images.