In this talk, I will present a motion-aware variational approach, based on a new multiscale directional system of functions called cylindrical shearlets, to reconstruct moving objects from sparse dynamic data. Compared to conventional separable representations, cylindrical shearlets are very efficient in representing spatio-temporal data, since they are better suited to handle the geometry of these data. We test our approach on both simulated and measured data. Numerical results demonstrate the advantages of our novel approach with respect to conventional multiscale methods.

A novel Bayesian prior distribution family is introduced, based on wavelet transforms.The priors correspond to well-defined infinite-dimensional random variables and can be approximated by finite-dimensional models. The non-zero wavelet coefficients are chosen in a systematic way so that prior draws have a specific fractal behaviour. This paves the way for new types of signal and image processing methods that can either promote certain fractal properties in the underlying data, or serve as smart "fingerprints" for measured object types. Realisations of the new priors take values in Besov spaces and have singularities only on a small set with a certain Hausdorff dimension. We also introduce an efficient algorithm for calculating the MAP estimator in the denoising problem.

Besov priors for Bayesian inverse problems are interesting since they promote various types of regularity on the unknown, especially non-smooth regularity, depending on the choice of basis and parameters of the prior. Besov priors introduces a $p$-norm into the posterior, which makes effective Gaussian samplers inapplicable. Randomize-Then-Optimize (RTO) is an optimization-based sampling algorithm, that computes exact independent samples from a posterior with Gaussian prior and a linear forward operator. We introduce a prior transformation that transforms a Besov prior into a Gaussian prior, which makes Gaussian samplers like RTO applicable. The caveat of the transformation is that the forward operator becomes non-linear even though it originally was linear. To sample from the transformed posterior we use RTO samples as proposals for the Metropolis-Hastings algorithm. We apply this sampling method to a deconvolution problem where the type of Besov prior is varied, to discover the quality of the method and the posterior dependencies on the choice of Besov prior. Our results validate that the computed samples come from the original posterior with Besov prior and shows that the choice of prior basis and parameters has a significant impact on the posterior.

We consider the challenge of solving the ill-posed reconstruction problem in computed tomography using a translation-invariant diagonal frame decomposition (TI-DFD). First, we review the concept of diagonal frame decompositions (DFD) and their translation-invariant counterparts for general linear operators. Subsequently, we explain how the filter-based regularization methods can be defined using these frame decompositions. Finally, as an example, we introduce the TI-DFD for the Radon transform on $L^2 (\mathbb{R}^2)$ and provide an exemplary construction using the TI wavelet transform. In numerical results, we demonstrate the advantages of our approach over non-translation invariant counterparts.