The purpose of this thesis is to investigate the mathematical properties of some models which are currently used in image processing. Generalizing an approach by S.J. Osher, L. Rudin and E. Fatemi, we decompose an image f ∈ L 2 (R 2) as a sum u + v where u belongs to some functional Banach space E while v belongs to L 2 (R 2). The Banach space E is aimed at modeling the objects contained in the given image and the optimal decomposition minimizes the energy J(u) = u E + λ f − u 2 2. The main difficulty is to choose an adapted Banach space E. The common choices are E =Ḃ 1,1 1 (R 2) which leads to the well-known Donoho's wavelet thresholding or E = BV (R 2) the space of functions of bounded variations. The latter choice is the Osher Rudin Fatemi algorithm. These two choices are suffering from severe drawbacks. In the first case, sharp edges are erased. The second choice does not lead to a wavelet thresholding. That is why we propose E =Ḃ 1,∞ 1 (R 2) which yields sharp edges and is given by wavelet thresholding. This is the the two first parts of the thesis. In the third part, we investigate the mathematical properties of the Osher-Vese newest algorithm which keeps track of the textured components.

BackgroundObstructive sleep apnea (OSA) is a risk factor for perioperative complications but data on anesthesia regimen are scarce.MethodsIn patients with established or strongly suspected OSA, we assessed in a prospective, randomized design the effects on nocturnal apnea-hypopnea-index (AHI) and oxygen saturation (SpO2) of propofol/remifentanil or sevoflurane/remifentanil based anesthesia. Patients were selected by a history for OSA and / or a positive STOP – questionnaire and received general anesthesia using remifentanil (12 μg/kg/h) combined either with propofol (4-6 mg/kg/h, n = 27) or sevoflurane (approx. 2.2 vol% endtidal, n = 27). AHI and SpO2 were measured during the nights before and after anesthesia.ResultsThere were no differences in AHI between anesthetic regimens nor between the pre- and postoperative nights (propofol: 8.6 h− 1 (median, CI: 3.6–21.9) vs. 7.9 h− 1 (1.8–28.8); p = 0.97; sevoflurane: 3.8 h− 1 (1.8–7.3) vs. 2.9 h− 1 (1.2–9.5); p = 0.85). Postoperative minimum SpO2 (propofol: 80.7% ± 4.6, sevoflurane: 81.6 ± 4.6) did not differ from their respective preoperative baselines (propofol: 79.6% ± 6.5; p = 0.26, sevoflurane: 80.8% ± 5.2; p = 0.39). Even in patients with a preanesthetic AHI > 15, nocturnal AHI remained unchanged postoperatively.ConclusionThus, in a cohort of patients with suspected or confirmed OSA undergoing surgery of moderate duration and severity neither the volatile agent sevoflurane nor the intravenous anesthetic propofol altered nocturnal AHI or oxygen saturation, when combined with the short acting opioid remifentanil.Trial registrationGerman Clinical Trials Register, DRKS00005824 retrospectively registered on 03/12/2014.

In this work, we compare two models that were proposed to improve the well-known model of Rudin, Osher, and Fatemi [

Obstructive sleep apnea (OSA) is a risk factor for perioperative complications, but many OSA patients present undiagnosed. While polysomnography (PSG) is the "gold standard" for diagnosis, its application is technology-intense, time-consuming, expensive, and requires specialists, often delaying surgery. Thus, miniaturized devices were developed for OSA screening aimed at ruling out major OSA while measuring a lesser number of biological signals. We evaluated the accuracy of a photoplethysmography (PPG)-based device for OSA detection. 48 patients with established or strongly suspected (STOP-Questionnaire) OSA scheduled for surgery underwent in their preoperative nights parallel recordings by PPG and a classic polygraphy (PG) devices (SomnoLab2®). We compared the diagnostic accuracy of the PPG in diagnosing mild [Apnea-/Hypopnea-Index (AHI) 5-14 events/h] and moderate-to-severe OSA (AHI > 15). PPG and PG-derived AHI correlated significantly (r = 0.85, p < 0.0001) and high area under curve (AUC) in receiver operator characteristics (ROC) values were seen for both AHI thresholds (0.93 and 0.95, respectively). For an AHI > 5, sensitivity was 100%, specificity 44%, positive predictive value (PPV) 62%, negative predictive value (NPV) 100%, likelihood ratio (LHR) 1.79, and Cohen κ was 0.43. For an AHI > 15, sensitivity was 92%, specificity 77%, PPV 60%, NPV 96%, LHR 4.04, and Cohen κ was 0.59. In a typical perioperative cohort of confirmed and suspected OSA patients, PPG reliably detected OSA patients while showing some false-positive results. Such devices are helpful for preoperative OSA screening.