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Kinzel, Meike: Nuclear norms in the context of dynamic MRI : regularization techniques and asymptotic analysis. 2021
Inhalt
Introduction
Mathematical preliminaries
Function spaces
Integral operators
Singular value decomposition
-convergence
Inverse problems and variational methods
Inverse problems
Variational methods
Bayesian modeling
Transition to the infinite-dimensional case
Example: Sparse signal reconstruction
Existence of solutions
Variational methods for dynamic MRI
Dynamic MRI: Physics and methodical limits
The measurement process
The (ill-posed) inverse problem to dynamic MRI
Matrix norms in the context of dynamic MRI
Mixed norms
Nuclear norm
Generalized nuclear norm
Variational methods for the reconstruction of dynamics
Linear dependence among frames
Superposition of background and dynamics
Linear dependence favoring smooth dynamics
Numerical implementation and results
Algorithmic solution
Computational experiments
Undersampling imposing a superposition of dynamics
Tracking of dynamics
Undersampling imposing the occurrence of few smooth dynamics?
Asymptotic behavior
Analytical study of general energy functionals
Formal setting
Coercivity
Continuity of data fidelity terms
-convergence of regularizing norms
Mixed norms
Nuclear norm
Generalized nuclear norm
Conclusion and outlook
Bibliography
List of figures and tables