Characterization and computation of canonical tight windows for Gabor frames

Let $(g_{nm})_{n,m\in Z}$ be a Gabor frame for $L_2(R)$ for given window $g$. We show that the window $h^0=S^{-1/2} g$ that generates the canonically associated tight Gabor frame minimizes $\|g-h\|$ among all windows $h$ generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in each domain the canonical $h^0$ is expressed using functional calculus for Gabor frame operators. Furthermore, we derive a Wiener-Levy type theorem for rationally oversampled Gabor frames. Finally, a Newton-type method for a fast numerical calculation of \ho is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples

Scaling regimes and critical dimensions in the Kardar-Parisi-Zhang problem

We study the scaling regimes for the Kardar-Parisi-Zhang equation with noise correlator R(q) ~ (1 + w q^{-2 \rho}) in Fourier space, as a function of \rho and the spatial dimension d. By means of a stochastic Cole-Hopf transformation, the critical and correction-to-scaling exponents at the roughening transition are determined to all orders in a (d - d_c) expansion. We also argue that there is a intriguing possibility that the rough phases above and below the lower critical dimension d_c = 2 (1 + \rho) are genuinely different which could lead to a re-interpretation of results in the literature.Comment: Latex, 7 pages, eps files for two figures as well as Europhys. Lett. style files included; slightly expanded reincarnatio

On Lerch's transcendent and the Gaussian random walk

Let $X_1,X_2,...$ be independent variables, each having a normal distribution with negative mean $-\beta<0$ and variance 1. We consider the partial sums $S_n=X_1+...+X_n$, with $S_0=0$, and refer to the process $\{S_n:n\geq0\}$ as the Gaussian random walk. We present explicit expressions for the mean and variance of the maximum $M=\max\{S_n:n\geq0\}.$ These expressions are in terms of Taylor series about $\beta=0$ with coefficients that involve the Riemann zeta function. Our results extend Kingman's first-order approximation [Proc. Symp. on Congestion Theory (1965) 137--169] of the mean for $\beta\downarrow0$. We build upon the work of Chang and Peres [Ann. Probab. 25 (1997) 787--802], and use Bateman's formulas on Lerch's transcendent and Euler--Maclaurin summation as key ingredients.Comment: Published at http://dx.doi.org/10.1214/105051606000000781 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Short- and long-term experience in pulmonary vein segmental ostial ablation for paroxysmal atrial fibrillation*

Introduction: Segmental ostial pulmonary vein isolation (PVI) is considered a potentially curative therapeutic approach in the treatment of paroxysmal atrial fibrillation (PAF). There is only limited data available on the long-term effect of this procedure. Methods: Patients (Pts) underwent a regular clinical follow up visit at 3, 6 and 24 months after PVI. Clinical success was classified as complete (i.e. no arrhythmia recurrences, no antiarrhythmic drug), partial (i.e. no/only few recurrences, on drug) or as a failure (no benefit). The clinical responder rate (CRR) was determined by combining complete and partial success. Results: 117 patients (96 male, 21 female), aged 51±11 years (range 25 to 73) underwent a total of 166 procedures (1.4/patient) in 2-4 pulmonary veins (PV). 115 patients (98%) had AF, 2 patients presented with regular PV atrial tachycardia. ,109/115 patients. exhibited PAF as the primary arrhythmia (versus persistent AF). A total of 113 patients with PVI in the years 2001 to 2003 were evaluated for their CRR after 6 (3) months. A single intervention was carried out in 63 patients (55.8%), two interventions were performed in 45 patients (39.8%) and three interventions in 5 patients (4.4%). The clinical response demonstrated a complete success of 52% (59 patients), a partial success of 26% (29 patients) and a failure rate of 22% (25 patients), leading to a CRR of 78% (88 patients). Ostial PVI in all 4 PVs exhibited a tendency towards higher curative success rates (54% versus 44% in patients with 3 PVs ablated for the 6 month follow up). Long-term clinical outcome was evaluated in 39 patients with an ablation attempt at 3 PVs only (excluding the right inferior PV in our early experience) and a mean clinical follow up of 21±6 months. At this point in time the success rate was 41% (complete, 16 patients) and 21% (partial, 8 patients), respectively, adding up to a CRR of 62% (24 patients). In total, 20 patients (17.1%) had either a single or 2 (3 patients, 2.6%) complications independent of the number of procedures performed with PV stenosis as the leading cause (7.7%). Conclusion: The CRR of patients with medical refractory PAF in our patient cohort is 78% at the 6 month follow up. PV stenosis is the main cause for procedure-related complications. Ablation of all 4 PV exhibits a tendency towards higher complete success rates despite equal CRR. Calculation of the clinical response after a mid- to long-term follow of 21±6 months in those patients with an ostial PVI in only 3 pulmonary veins (sparing the right inferior PV) shows a further reduction to 62%, exclusively caused by a drop in patients with a former partial success. To evaluate the long-term clinical benefit of segmental ostial PVI in comparison with other ablation techniques, more extended follow up periods are mandatory, including a larger study cohort and a detailed description of procedural parameters

Locally Adaptive Frames in the Roto-Translation Group and their Applications in Medical Imaging

Locally adaptive differential frames (gauge frames) are a well-known effective tool in image analysis, used in differential invariants and PDE-flows. However, at complex structures such as crossings or junctions, these frames are not well-defined. Therefore, we generalize the notion of gauge frames on images to gauge frames on data representations $U:\mathbb{R}^{d} \rtimes S^{d-1} \to \mathbb{R}$ defined on the extended space of positions and orientations, which we relate to data on the roto-translation group $SE(d)$, $d=2,3$. This allows to define multiple frames per position, one per orientation. We compute these frames via exponential curve fits in the extended data representations in $SE(d)$. These curve fits minimize first or second order variational problems which are solved by spectral decomposition of, respectively, a structure tensor or Hessian of data on $SE(d)$. We include these gauge frames in differential invariants and crossing preserving PDE-flows acting on extended data representation $U$ and we show their advantage compared to the standard left-invariant frame on $SE(d)$. Applications include crossing-preserving filtering and improved segmentations of the vascular tree in retinal images, and new 3D extensions of coherence-enhancing diffusion via invertible orientation scores