Manifold-Valued Data in Medical Imaging Applications

Maximilian Baust () and Andreas Weinmann ()
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Maximilian Baust: Chair of Computer Aided Medical Procedures, TU Munich
Andreas Weinmann: Hochschule Darmstadt, Department of Mathematics and Natural Sciences

Chapter Chapter 22 in Handbook of Variational Methods for Nonlinear Geometric Data, 2020, pp 613-647 from Springer

Abstract: Abstract The last decade has witnessed a considerable amount of research being devoted on how to process big and often unstructured data. However, one often neglects the fact that a considerable portion of today’s data deluge is actually structured, particularly when we consider measured data. The reason for this fact is that sensors either directly record structured data, e.g., object poses, or are used to record data of a certain type, i.e., magnetic resonance images of the human heart. In both cases, the measured data is subject to physical or physiological constraints causing the measured data to enjoy a manifold structure. In this chapter, we make extensive use of this observation and discuss several applications within the realm of medical imaging. We briefly discuss the general mathematical structure of these problems and elaborate on why recently proposed formulations of manifold-valued regularizers are applicable to them. We describe the employed numerical schemes to provide application-focused readers with a guide to manifold-valued regularization techniques. Further, we discuss three entirely disjoint applications: regularization of pose signals for 3D freehand ultrasound compounding, estimation and regularization of diffusion tensors measured by magnetic resonance imaging, and estimation and regularization of shape signals. Finally, we discuss possible extensions.

Date: 2020
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DOI: 10.1007/978-3-030-31351-7_22

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