Deformation and transport of image data
Sebastian Neumayer · DepositOnce · 2021
Image and data processing have become increasingly important over recent years. In particular, robust distance measures and transformation methods on large and perturbed data sets are of growing interest. These tasks become even more challenging if the measurement locations are distributed on some manifold or if the data itself has a manifold structure. In this thesis, we start by examining a time discrete manifold-valued morphing model together with a time continuous counterpart which includes a variational inequality and is inspired by the classical metamorphosis approach. For both models, existence of minimizers is shown under the assumption that the data lies in a finite-dimensional Hadamard manifold. Further, we prove Mosco convergence of the time discrete model to the time continuous one. Then, we propose to use the real-valued version of this model as variational regularizer for inverse problems if some template image is available. As this incorporates additional information into the reconstruction process, such an approach is particularly useful if the data is sparse. In cases where an even stronger regularization is necessary, we propose to use a more restrictive flow of diffeomorphism based model instead. Additionally, increased robustness to intensity differences between template and ground-truth is achieved by using a distance measure based on normalized cross-correlation. These modifications enable us to obtain promising reconstructions for computed tomography with a very sparse number of angles. For both models existence, stability and convergence for vanishing noise are proven. Besides metamorphosis and flow of diffeomorphism, we are interested in optimal transport between images. More precisely, we investigate Sinkhorn divergences, which interpolate between optimal transport and discrepancies. Here, we focus on the behavior of the corresponding optimal dual potentials and establish a relation between their limits as the regularization parameter goes to infinity and the optimal potential of the associated discrepancy. Additionally, we compare these different distance measures for a process called dithering. Finally, we investigate the approximation of probability measures on compact metric spaces and in particular on Riemannian manifolds by measures supported on Lipschitz curves. For this purpose, the approximation quality is assessed in terms of discrepancies as these do not suffer from the so-called curse of dimensionality. % Potential applications are 3D printing and magnetic resonance imaging. While we also investigate the general case via the traveling salesman approach, a strong focus lies on the push-forward of the Lebesgue measure on the unit interval by Lipschitz continuous curves. Here, we utilize a result on the quadrature error achievable by integration with respect to a measure that exactly integrates all eigenfunctions of the Laplace--Beltrami operator with eigenvalues smaller than a fixed number. In particular, this enables us to provide optimal approximation rates in terms of the curve's length and Lipschitz constant for measures fulfilling smoothness requirements. All results established in this thesis are illustrated by numerical proof-of-concept examples and comparisons. These include real-world data experiments in particular for our flow of diffeomorphism based reconstruction model. For the necessary discrepancy evaluations, recent fast Fourier transform techniques on certain manifolds are exploited.