Combines l2 data and gradient fitting in conjunction with l1 regularization

We are interested in minimizing functionals with l2 data and gradient fitting term and (absolute) l1 regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1d by reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution turns out to be a discrete polynomial spline whose knots coincide with the contact points. In 2d we modify Chambolle's algorithm to solve the minimization problem with absolute l1 norm and second order derivatives. This requires the application of fast cosine transforms. We demonstrate by numerical denoising examples that the l2 gradient fitting term can be used to avoid both edge blurring and staircasing effects

Splitting Methods in Image Processing

It is often necessary to restore digital images which are affected by noise (denoising), blur (deblurring), or missing data (inpainting). We focus here on variational methods, i.e., the restored image is the minimizer of an energy functional. The first part of this thesis deals with the algorithmic framework of how to compute such a minimizer. It turns out that operator splitting methods are very useful in image processing to derive fast algorithms. The idea is that, in general, the functional we want to minimize has an additive structure and we treat its summands separately in each iteration of the algorithm which yields subproblems that are easier to solve. In our applications, these are typically projections onto simple sets, fast shrinkage operations, and linear systems of equations with a nice structure. The two operator splitting methods we focus on here are the forward-backward splitting algorithm and the Douglas-Rachford splitting algorithm. We show based on older results that the recently proposed alternating split Bregman algorithm is equivalent to the Douglas-Rachford splitting method applied to the dual problem, or, equivalently, to the alternating direction method of multipliers. Moreover, it is illustrated how this algorithm allows us to decouple functionals which are sums of more than two terms. In the second part, we apply the above techniques to existing and new image restoration models. For the Rudin-Osher-Fatemi model, which is well suited to remove Gaussian noise, the following topics are considered: we avoid the staircasing effect by using an additional gradient fitting term or by combining first- and second-order derivatives via an infimal-convolution functional. For a special setting based on Parseval frames, a strong connection between the forward-backward splitting algorithm, the alternating split Bregman method and iterated frame shrinkage is shown. Furthermore, the good performance of the alternating split Bregman algorithm compared to the popular multistep methods is illustrated. A special emphasis lies here on the choice of the step-length parameter. Turning to a corresponding model for removing Poisson noise, we show the advantages of the alternating split Bregman algorithm in the decoupling of more complicated functionals. For the inpainting problem, we improve an existing wavelet-based method by incorporating anisotropic regularization techniques to better restore boundaries in an image. The resulting algorithm is characterized as a forward-backward splitting method. Finally, we consider the denoising of a more general form of images, namely, tensor-valued images where a matrix is assigned to each pixel. This type of data arises in many important applications such as diffusion-tensor MRI

Optimising Spatial and Tonal Data for PDE-based Inpainting

Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods

The final publication is available at link.springer.com. An Optimal Control Approach to Find Sparse Data for Laplace Interpolation

Abstract. Finding optimal data for inpainting is a key problem in the context of partial differential equation-based image compression. We present a new model for optimising the data used for the reconstruction by the underlying homogeneous diffusion process. Our approach is based on an optimal control framework with a strictly convex cost functional containing an L1 term to enforce sparsity of the data and non-convex constraints. We propose a numerical approach that solves a series of convex optimisation problems with linear constraints. Our numerical examples show that it outperforms existing methods with respect to quality and computation time

Na Duong (northern Vietnam) – an exceptional window into Eocene ecosystems from Southeast Asia

Today, the continental ecosystems of Southeast Asia represent a global biodiversity hotspot. From a deep-time perspective, however, very little is known about the formation of this hotspot. In particular, the evolutionary and biogeographic roots of numerous terrestrial biota are unknown, and ecosystemic information, especially for the Paleogene, is unavailable. The Na Duong Basin in northern Vietnam has yielded a remarkable diversity of Paleogene vertebrate, invertebrate, and plant fossils, and is thus one of the few localities in Southeast Asia allowing for multi-focused investigation of this period. We present stratigraphic, paleontologic, and paleoecologic results from the 220 m thick Na Duong section derived from magnetostratigraphy, biochronology, vertebrate, invertebrate, and plant taxonomy, and biomarker analysis. Only the upper 40 m of the section show the original magnetization, prohibiting any further magnetostratigraphic interpretation. The affinities of two newly described mammal species, Bakalovia orientalis nov. sp. (Anthracotheriidae) and Epiaceratherium naduongense nov. sp. (Rhinocerotidae), suggest an Eocene, late Bartonian to Priabonian age (39–35 Ma). High biodiversity is recorded for unionid mussels (five species), freshwater fishes (nine taxa, including Planktophaga minuta nov. gen. et sp.), turtles (five to six taxa), and crocodiles (three taxa), and long-term stability of Southeast Asian unionid and fish faunas is demonstrated. Fossil leaves, wood and resin document azonal and zonal vegetation; dipterocarp trees were identified from resin exudate spectroscopy. In-situ tree-stump horizons allow for calculation of tree density (600 specimens/ha) and maximum canopy height (35m); both values resemble those of recent Southeast Asian peat swamp forests. Environment changed abruptly from a swamp forest to a tropical to warm sub-tropical lake of fluctuating water depth. The strong biogeographic link between the Eocene mammal faunas from Na Duong and Europe highlights the importance of Southeast Asia as a source region for trans-continental mammal dispersal along the northern Tethys margin.Heutige kontinentale Ökosysteme Südost Asiens sind ein Hotspot der globalen Biodiversität, über dessen geohistorische Entstehung jedoch sehr wenig bekannt ist. So sind die evolutionären und biogeographischen Wurzeln der meisten terrestrischen Lebensformen unbekannt und Daten zu Ökosystemen, speziell aus dem Paläogen, fehlen. Das Becken von Na Duong im nördlichen Vietnam beherbergt bemerkenswert diverse Biota fossiler Pflanzen, Mollusken und Wirbeltiere und stellt für das Paläogen eine der wenigen Regionen Südost Asiens dar, welche einen multi-disziplinären Untersuchungsansatz erlauben. Auf der Grundlage von Paläomagnetik, Biochronologie, Wirbeltier-, Mollusken- und Pflanzen-Taxonomie, sowie der Analyse von Biomarkern präsentieren wir hier unsere stratigraphischen, paläontologischen und paläo-ökologischen Ergebnisse eines 220 m mächtigen Profils der Braunkohlengrube Na Duong. Nur die hangenden 40 m der Sektion zeigen die ursprüngliche Magnetisierung, wodurch eine magnetostratigraphische Interpretation verhindert wird. Die Verwandschaftsverhältnisse von zwei neu beschriebenen Säugetierarten, Bakalovia orientalis nov. sp. (Anthracotheriidae) und Epiaceratherium naduongense nov. sp. (Rhinocerotidae), verweisen auf ein eozänes Alter (oberes Bartonium bis Priabonium, 39-35 Ma). Starke biogeographische Bezüge beider Säugetiere aus Na Duong zu europäischen Arten unterstreichen die Bedeutung Südost Asiens als eine Quellregion paläogener transkontinentaler Säugetier Ausbreitungen entlang des nördlichen Randes der Tethys. Hohe Diversitäten werden für unionide Muscheln (fünf Arten), Süßwasserfische (9 Taxa, darunter Planktophaga minuta nov. gen. et sp.), Schildkröten (fünf bis sechs Taxa) und Krokodile (drei Taxa) belegt, wobei eine taxonomische Langzeit-Stabilität der Unioniden- und Fisch-Faunen Südost Asiens belegt wird. Fossile Blätter, Hölzer und Harze dokumentieren die azonale und zonale Vegetation, letztere repräsentiert durch spektroskopisch an Harzen belegte Zweiflügelfrucht-Gewächse (Dipterocarpaceae). In-situ Baumstubben-Horizonte erlauben die Berechnung der Baum-Dichte (600 per ha) und der maximalen Kronendach-Höhe (35 m), welches in guter Übereinstimmung zu heutigen südostasiatischen Sumpfwäldern steht. Die Umwelt veränderte sich abrupt von einem Sumpfwald zu einem tropisch bis warm-subtropischen See mit fluktuierender Wassertiefe

The state of the Martian climate

60°N was +2.0°C, relative to the 1981–2010 average value (Fig. 5.1). This marks a new high for the record. The average annual surface air temperature (SAT) anomaly for 2016 for land stations north of starting in 1900, and is a significant increase over the previous highest value of +1.2°C, which was observed in 2007, 2011, and 2015. Average global annual temperatures also showed record values in 2015 and 2016. Currently, the Arctic is warming at more than twice the rate of lower latitudes