Regularization Graphs for Variational Imaging

Our perception of the world is, to a great extent, visual. Its manifestation, images, are an integral part of human culture. With today`s technology, images can digitally be acquired, processed and stored, enabling imaging as a scientific discipline. Imaging sciences increasingly affect our everyday lives, playing a role for digital photo and video technology, for instance capturing our holiday memories, and being a crucial part of modern diagnostic medical technologies such as computed tomography (CT) and magnetic resonance imaging (MRI). Mathematical imaging as scientific discipline comprises studying the translation of acquired data to a meaningful visual representation. This task is far from being trivial, in particular, in situations where there is no direct relation between visual representation and measured data. For tomography applications such as CT or MRI, where one aims at creating an image of the inside of a body, this is indeed the case. Additionally, difficulties arise from the measurements being corrupted by noise and potentially highly incomplete. The translation to a clean, accurate image, say, of the heart of a patient, requires solid mathematics and constitutes a challenging research problem. Variational methods contribute significantly to the progress towards the solution of such problems. They base on efficient mathematical models that can be transferred into computer programs performing the actual calculations. These models incorporate the relation between the image one aims to recover and the measurements, but also provide an abstract description of the qualitative properties the image satisfies. The latter, which is called regularization, is the key ingredient for solving the difficulties associated with perturbed and incomplete data. Its choice is challenging as it is the decisive factor for the performance of the method. Nonetheless, once a good regularization approach has been found, it can broadly be applied. In MRI, for instance, this allows to obtain reconstructions from only 10% of the data that is normally required and, consequently, for shorter scan time. Without appropriate regularization, such results would not be possible, making respective research an important topic within variational imaging. In the past years, great progress was achieved with seemingly very different regularization approaches. The effectiveness of those, however, is in fact driven by a similar underlying structure. This structure bears great potential in unifying and extending the state of the art. The project`s goal is to provide, in theory and application, such a unification and extension by virtue of regularization graphs. It is designed to be easily transferable into application, helping practitioners to push today`s limits for reconstruction in diverse fields of imaging sciences. Such advances could lay the foundations for concrete benefits, for instance, a scan-time reduction in MRI that enables new real- time applications.

The project was concerned with advancing regularization theory in imaging, a mathematical theory that is crucial for the solution of modern imaging reconstruction and processing problems. Its results cover the establishment of the novel concept of "regularization graphs", new efficient algorithms for the computational processing and reconstruction of digitally stored images as well as innovative applications in biomedical and nanoscale imaging. Images as the manifestation of visual perception are an integral part of human live. With today's technology, images can digitally be acquired, processed and stored, enabling imaging as a scientific discipline. Imaging sciences affect our everyday lives, playing a role in digital photo and video technology, and being a crucial part of modern diagnostic medical technologies such as computed tomography (CT) and magnetic resonance imaging (MRI). Mathematical imaging as scientific discipline comprises studying the translation of acquired data to a meaningful visual representation. This task can be difficult, in particular, in situations where there is no direct relation between visual representation and measured data. For tomography applications such as CT or MRI, where one aims at imaging the inside of a body, this is indeed the case. Additionally, difficulties arise from noisy or highly incomplete measurements. The translation to a clean, accurate image, say, of the heart of a patient, requires solid mathematics and constitutes a challenging research problem. So-called variational methods contribute significantly to the progress towards the solution of such problems. They base on efficient mathematical models that can be transferred into computer programmable algorithms. These models incorporate the relation between image and measurements, but also provide an abstract description of its qualitative properties. The latter, called regularization, is the key ingredient for solving the above-mentioned challenges. Its choice is subject to research as it is the decisive factor for the performance of the method. In MRI, for instance, appropriate regularization allows to obtain reconstructions with significantly reduced scan time. Such results would otherwise not be possible, making regularization theory an important topic within variational imaging. In the past years, great progress was achieved with seemingly very different regularization approaches. The effectiveness of those, however, is in fact driven by a similar underlying structure, which could be identified as the regularization graphs developed within the project. It showed, in particular, the potential of this structure in unifying and extending the state of the art. Its theoretic and algorithmic advances are easily transferable into application, helping practitioners to push today's limits for reconstruction in diverse fields of imaging sciences. Benefits for electron tomography and photoacoustic tomography were shown within the project, but also the foundations for other applications, for instance, a scan-time reduction in MRI that enables new real-time imaging, were laid.