Second Order Differences of Cyclic Data and Applications in Variational Denoising

In many image and signal processing applications, as interferometric synthetic aperture radar (SAR), electroencephalogram (EEG) data analysis or color image restoration in HSV or LCh spaces the data has its range on the one-dimensional sphere $\mathbb S^1$. Although the minimization of total variation (TV) regularized functionals is among the most popular methods for edge-preserving image restoration such methods were only very recently applied to cyclic structures. However, as for Euclidean data, TV regularized variational methods suffer from the so called staircasing effect. This effect can be avoided by involving higher order derivatives into the functional. This is the first paper which uses higher order differences of cyclic data in regularization terms of energy functionals for image restoration. We introduce absolute higher order differences for $\mathbb S^1$-valued data in a sound way which is independent of the chosen representation system on the circle. Our absolute cyclic first order difference is just the geodesic distance between points. Similar to the geodesic distances the absolute cyclic second order differences have only values in [0,{\pi}]. We update the cyclic variational TV approach by our new cyclic second order differences. To minimize the corresponding functional we apply a cyclic proximal point method which was recently successfully proposed for Hadamard manifolds. Choosing appropriate cycles this algorithm can be implemented in an efficient way. The main steps require the evaluation of proximal mappings of our cyclic differences for which we provide analytical expressions. Under certain conditions we prove the convergence of our algorithm. Various numerical examples with artificial as well as real-world data demonstrate the advantageous performance of our algorithm.Comment: 32 pages, 16 figures, shortened version of submitted manuscrip

Constant temperature mashing at 72 °C for the production of beers with a reduced alcohol content in micro brewing systems

In this paper, we present a constant temperature mashing procedure where grist made of Pilsner malt is mashed-in directly in the temperature regime of alpha-amylase activity, thus omitting all conventional steps, followed by constant temperature mashing at 72 °C. The aim was to investigate an alternative mashing procedure for the production of alcohol-reduced beers. The mashing proceeds with a rapid buildup of sugars and is completed after 120 min at the latest, giving an iodine normal and clear wort. However, the distribution of the different sugars in the worts is strongly altered, in comparison to a more classical mashing procedure. The free amino nitrogen (FAN) concentration is sufficient for vivid fermentation with the bottom fermenting yeast Saccharomyces pastorianus TUM 34/70. The lag phase and initial fermentation performance of this yeast strain are comparable for conventionally and isothermally (72 °C) mashed wort. Under the given conditions the fermentation of the isothermally (72 °C) made wort is finished after 6 days whereas a conventional wort needs 4–5 days more to be completed. The alcohol concentration is remarkably reduced by isothermal mashing leading to roughly 3.4 vol.-% with an original gravity of 11°P whereas with a conventional mashing procedure 4.4 vol.-% are obtained for the same original gravity. In both cases the concentrations of the fermentation by-products are comparable. A preliminary comparison of tasteand foam stability did not show striking differences. Constant temperature mashing at 72 °C is a simple way to reduce the alcohol content of beer enriching it at the same time with non-fermentable sugars