Magnetic resonance imaging (MRI) underwent a rapid development during the last decades. Nowadays, not only static morphological images can be obtained via different MRI acquisition modalities, but also quantitative and dynamic data may be reconstructed. In this context, variational models are effective for solving the underlying inverse problems with high accuracy and visual quality. In the talk, we discuss variational approaches for two recent developments in MRI: quantitative susceptibility mapping (QSM) and magnetic-resonance positron emission tomography (MR-PET). For both techniques, variational modelling and optimization provides flexible and robust solution methods. In particular, they allow for a novel one-step reconstruction technique for QSM as well as for joint edge-aligned reconstruction in MR-PET. The underlying principles of these approaches such as forward problem, regularization and numerical optimization will be shown and examples will be discussed.

In this talk we provide an overview of the history of l1-norm minimization applied to underdetermined inverse problems. In the 70s and 80s geophysicists proposed using l1-norm minimization for deconvolution from bandpass data in reflection seismography. In the 2000s, inspired by this approach and by magnetic resonance imaging, a method to provably recover sparse signals from random projections, known as compressed sensing, was developed. Theoretical insights used to analyze compressed sensing have recently been adapted to understand the potential and limitations of l1-norm minimization for deterministic problems. These include super-resolution from low-pass data and the deconvolution problem that originally motivated the geophysicists.

The iterative reconstruction method known as ART (Algebraic Reconstruction Technique) and its many variants are surprisingly simple and efficient methods with many applications in computed tomography. On the theoretical side we are interested in explaining why it is so successful, and on the practical side how to implement it efficiently, how to select the relaxation parameter, and how to stop the iterations. This talk presents new theoretical and experimental results about the semi-convergence of ART, and we also discuss how to implement block-versions on multi-core computers.

In the past decade revolutionary advances have been made in the field of microscopy imaging, some of which have been honored by the Nobel prize in Chemistry 2014. One high-resolution method is based on transforming conventionally unresolvable details into measurable patterns with the help of an effect most people have already personally experienced: the Moiré effect. If two fine periodic patterns overlap, coarse patterns emerge. This is typically seen on a finely weaved curtain folding back onto itself. Another example is fast moving coarse patterns on both fences of a bridge above a motorway, when approaching it with the car. The microscopy method of structured illumination utilizes this effect by projecting a fine grating onto the sample and imaging the resulting coarser Moiré patterns containing the information about invisibly fine sample detail. With the help of computer reconstruction based on several such Moiré images, a high-resolution image of the sample can then be assembled.
Another way to obtain a high-resolution map of the sample is to utilize the blinking behavior inherent in most molecules, used to stain the sample. Recent methodological advances (Cox et al., Nature Methods 9, 195-200, 2012) enable us to create pointillist high-resolution maps of molecular locations in a living biological sample, even if in each of the required many individual images, these molecules are not individually discernible. Examples will be shown as a film of a cell at 30 millionths of millimeter resolution and 6 seconds between the individual movie frames.

Magnetic nanoparticles are used as contrast agents in magnetic resonance imaging (MRI) where they provide a negative contrast that can be used for the diagnosis of various diseases ranging from cardiovascular to oncological applications. Magnetic Particle Imaging (MPI) is an alternative technique that has the important advantage of providing positive contrast and in turn background free images enhancing the diagnostic value in medical applications. From a mathematical viewpoint the reconstruction of the particle distribution comes along with several computational challenges that have to be addressed in practice. In particular in real-time applications where the data acquisition rate is above 40 volumes/sec it is essential that the image reconstruction is not the bottleneck limiting the throughput of the entire signal chain. In this talk we will present matrix compression techniques that allow to significantly reduce the algorithmic complexity of image reconstruction and enab le processing the data in real-time. Beside acceleration techniques for image reconstruction the talk will introduce multi-spectral MPI reconstruction, which allows to simultaneously determine the particle concentration of different particle systems. The associated reconstruction problem is ill conditioned and requires appropriate regularization techniques.

The resolution of all microscopes is limited by diffraction. The observed signal is a convolution of the emitted signal with a low-pass kernel, the point-spread function (PSF) of the microscope. The frequency cut-off of the PSF is inversely proportional to the wavelength of light. Hence, the features of the object that are smaller than the wavelength of light are difficult to observe. In single-molecule microscopy the emitted signal is a collection of point sources, produced by blinking molecules. The goal is to recover the location of these sources with precision that is much higher than the wavelength of light. This leads to the problem of super-resolution of positive sources in the presence of noise. We show that the problem can be solved using convex optimization in a stable fashion. The stability of reconstruction depends on Rayleigh-regularity of the support of the signal, i.e., on how many point sources can occur within an interval of one wavelength. The stability estimate is complemented by a converse result: the performance of the convex algorithm is nearly optimal. I will also give a brief summary on the ongoing project, developed in collaboration with the group of Prof. W.E. Moerner, where we use the theoretical ideas to improve microscopes.