The Hilbert transform on the two-sphere: A spectral characterization

The analytic signal is an important representation in one dimensional signal processing. Its generalization to two dimensions is the monogenic signal. The properties of the analytic and the monogenic signal in the Fourier domain are well known. A generalization to the sphere is given by the Hilbert transform on the sphere known from Clifford analysis. Nonetheless no spectral characterization exists and therefore prohibits an interpretation. We derive the spherical harmonic coefficients of the Hilbert transform on the sphere and give a series expansion. It will turn out that it acts as a differential operator on the spherical harmonic basis functions of the Laplace equation solution, analogously to the Riesz transform in two dimensions. This allows an interpretation of the Hilbert transform suitable for signal processing of signals naturally arising on the two-sphere. We show that the scale space naturally arising is a Poisson scale space in the unit ball. In addition the obtained interpretation of the Hilbert transform is used for orientation analysis of plane waves. This representation is justified as a novel signal model on the sphere which can be used to construct intensity and rotation invariant feature detectors in a scale-space concept

On the Analysis and Decomposition of Intrinsically One-Dimensional Signals and their Superpositions

Computer and machine vision tasks can roughly be divided into a hierarchy of processing steps applied to input signals captured by a measuring device. In the case of image signals, the first stage in this hierarchy is also referred to as low-level vision or low-level image processing. The field of low-level image processing includes the mathematical description of signals in terms of certain local signal models. The choice of the signal model is often task dependent. A common task is the extraction of features from the signal. Since signals are subject to transformations, for example camera movements in the case of image signals, the features are supposed to fulfill the properties of invariance or equivariance with respect to these transformations. The chosen signal model should reflect these properties in terms of its parameters. This thesis contributes to the field of low-level vision. Local signal structures are represented by (sinusoidal) intrinsically one-dimensional signals and their superpositions. Each intrinsically one-dimensional signal consists of certain parameters such as orientation, amplitude, frequency and phase. If the affine group acts on these signals, the transformations induce a corresponding action in the parameter space of the signal model. Hence, it is reasonable, to estimate the model parameters in order to describe the invariant and equivariant features. The first and main contribution studies superpositions of intrinsically one-dimensional signals in the plane. The parameters of the signal are supposed to be extracted from the responses of linear shift invariant operators: the generalized Hilbert transform (Riesz transform) and its higher-order versions and the partial derivative operators. While well known signal representations, such as the monogenic signal, allow to obtain the local features amplitude, phase and orientation for a single intrinsically one-dimensional signal, there exists no general method to decompose superpositions of such signals into their corresponding features. A novel method for the decomposition of an arbitrary number of sinusoidal intrinsically one-dimensional signals in the plane is proposed. The responses of the higher-order generalized Hilbert transforms in the plane are interpreted as symmetric tensors, which allow to restate the decomposition problem as a symmetric tensor decomposition. Algorithms, examples and applications for the novel decomposition are provided. The second contribution studies curved intrinsically one-dimensional signals in the plane. This signal model introduces a new parameter, the curvature, and allows the representation of curved signal structures. Using the inverse stereographic projection to the sphere, these curved signals are locally identified with intrinsically one-dimensional signals in the three-dimensional Euclidean space and analyzed in terms of the generalized Hilbert transform and partial derivatives therein. The third contribution studies the generalized Hilbert transform in a non-Euclidean space, the two-sphere. The mathematical framework of Clifford analysis proposes a further generalization of the generalized Hilbert transform to the two-sphere in terms of the corresponding Cauchy kernel. Nonetheless, this transform lacks an intuitive interpretation in the frequency domain. A decomposition of the Cauchy kernel in terms of its spherical harmonics is provided. Its coefficients not only provide insights to the generalized Hilbert transform on the sphere, but also allow for fast implementations in terms of analogues of the convolution theorem on the sphere

4-Year Results from the RAPID-PsA Phase 3 Randomised Placebo-Controlled Trial of Certolizumab Pegol in Psoriatic Arthritis

Objective: To report the efficacy, patient-reported, radiographic and safety outcomes of 4 years\u27 certolizumab pegol (CZP) treatment in patients with psoriatic arthritis (PsA). Methods: RAPID-PsA (NCT01087788) was double-blind and placebo-controlled to Week 24, dose-blind to Week 48 and open-label (OL) to Week 216. Patients were randomised 1:1:1 to either placebo or CZP 200 mg every 2 weeks (Q2W) or 400 mg every 4 weeks (Q4W) (following 400 mg at Weeks 0/2/4). Patients randomised to CZP continued their assigned dose in the OL period. Patients randomised to placebo were re-randomised to CZP 200 mg Q2W or 400 mg Q4W (post-loading dose) at Week 16 (early escape) or after the double-blind phase. We present observed and imputed data; missing values were imputed using non-responder imputation (NRI) for categorical and last observation carried forward (LOCF) for continuous measures. Results: 409 patients were randomised; 20% (54/273) of Week 0 patients randomised to CZP had prior anti-tumour necrosis factor (TNF) exposure; 67% (183/273) completed 216 weeks. By Week 48, 60.4% of patients achieved Disease Activity Index for Psoriatic Arthritis low disease activity or remission, which was maintained; 66.3% achieved these outcomes at Week 216 (NRI). At Weeks 48 and 216, 39.2% of patients achieved minimal disease activity (NRI). 75% reduction in Psoriasis Area and Severity Index responses were 65% and 52% at Weeks 48 and 216 (NRI). Total resolution rates for enthesitis, dactylitis and nail psoriasis, at 4 years, were 71%, 81% and 65%, respectively (LOCF). Structural damage progression was low over 4 years\u27 treatment. No new safety signals were identified after Week 96. Conclusions: CZP efficacy in treating PsA was maintained over 4 years, in patients both with and without prior anti-TNF exposure, with no new safety signals identified

Flux ramp modulation based MHz frequency-division dc-SQUID multiplexer

We present a MHz frequency-division dc-SQUID multiplexer that is based on flux ramp modulation and a series array of $N$ identical current-sensing dc-SQUIDs with tightly coupled input coil. By running a periodic, sawtooth-shaped current signal through an additional modulation coil being tightly, but non-uniformly coupled to the individual SQUIDs, the voltage drop across the array changes according to the superposition of the flux-to-voltage characteristics of the individual SQUIDs within each cycle of the modulation signal. In this mode of operation, an input signal injected in the input coil of one of the SQUIDs and being quasi-static within a time frame adds a constant flux offset and leads to a phase shift of the associated SQUID characteristics. The latter is inherently proportional to the input signal and can be inferred by channelizing and down-converting the sampled array output voltage. Using a prototype multiplexer as well as a self-developed high-speed readout electronics for real-time phase determination, we demonstrate the simultaneous readout of four signal sources with MHz bandwidth per channel.Comment: The article has been submitted to Applied Physics Letter

Bioimpedance spectroscopy for assessment of volume status in patients before and after general anaesthesia

BackgroundTechnically assisted assessment of volume status before surgery may be useful to direct intraoperative fluid administration. We therefore tested a recently developed whole-body bioimpedance spectroscopy device to determine pre- to postoperative fluid distribution.MethodsUsing a three-compartment physiologic tissue model, the body composition monitor (BCM, Fresenius Medical Care, Germany) measures total body fluid volume, extracellular volume, intracellular volume and fluid overload as surplus or deficit of 'normal' extracellular volume. BCM-measurements were performed before and after standardized general anaesthesia for gynaecological procedures (laparotomies, laparoscopies and vaginal surgeries). BCM results were blinded to the attending anaesthesiologist and data analysed using the 2-sided, paired Student's t-test and multiple linear regression.ResultsIn 71 females aged 45 ± 15 years with body weight 67 ± 13 kg and Duration of anesthesia 154 ± 69 minutes [corrected] duration of anaesthesia 154 ± 68 min, pre- to postoperative fluid overload increased from -0.7 ± 1.1 L to 0.1 ± 1.0 L, corresponding to -5.1 ± 7.5% and 0.8 ± 6.7% of normal extracellular volume, respectively (both pConclusionsRoutine intraoperative fluid administration results in a significant, and clinically meaningful increase in the extracellular compartment. BCM-measurements yielded plausible results and may become useful to guide intraoperative fluid therapy in future studies

Exploitation of the Timing Capabilities of Metallic Magnetic Calorimeters for a Coincidence Measurement Scheme

In this report, we compare two filter algorithms for extracting timing information using novel metallic magnetic calorimeter detectors, applied to the precision X-ray spectroscopy of highly charged ions in a storage ring. Accurate timing information is crucial when exploiting coincidence conditions for background suppression to obtain clean spectra. For X-rays emitted by charge-changing interactions between ions and a target, this is a well-established technique when relying on conventional semiconductor detectors that offer a good temporal resolution. However, until recently, such a coincidence scheme had never been realized with metallic magnetic calorimeters, which typically feature much longer signal rise times. In this report, we present optimized timing filter algorithms for this type of detector. Their application to experimental data recently obtained at the electron cooler of CRYRING@ESR at GSI, Darmstadt is discussed