Characterization of the Inner Knot of the Crab: The Site of the Gamma-ray Flares?

One of the most intriguing results from the gamma-ray instruments in orbit has been the detection of powerful flares from the Crab Nebula. These flares challenge our understanding of pulsar wind nebulae and models for particle acceleration. We report on the portion of a multiwavelength campaign using Keck, HST, and Chandra concentrating on a small emitting region, the Crab's inner knot, located a fraction of an arcsecond from the pulsar. We find that the knot's radial size, tangential size, peak flux, and the ratio of the flux to that of the pulsar are correlated with the projected distance of the knot from the pulsar. A new approach, using singular value decomposition for analyzing time series of images, was introduced yielding results consistent with the more traditional methods while some uncertainties were substantially reduced. We exploit the characterization of the knot to discuss constraints on standard shock-model parameters that may be inferred from our observations assuming the inner knot lies near to the shocked surface. These include inferences as to wind magnetization, shock shape parameters such as incident angle and poloidal radius of curvature, as well as the IR/optical emitting particle enthalpy fraction. We find that while the standard shock model gives good agreement with observation in many respects, there remain two puzzles: (a) The observed angular size of the knot relative to the pulsar--knot separation is much smaller than expected; (b) The variable, yet high degree of polarization reported is difficult to reconcile with a highly relativistic downstream flow.Comment: 46 pages, 14 figures, submitted to the Astrophysical Journa

Direct and Inverse Computational Methods for Electromagnetic Scattering in Biological Diagnostics

Scattering theory has had a major roll in twentieth century mathematical physics. Mathematical modeling and algorithms of direct,- and inverse electromagnetic scattering formulation due to biological tissues are investigated. The algorithms are used for a model based illustration technique within the microwave range. A number of methods is given to solve the inverse electromagnetic scattering problem in which the nonlinear and ill-posed nature of the problem are acknowledged.Comment: 61 pages, 5 figure

Improvements In computed tomography perfusion output using complex singular value decomposition and the maximum slope algorithm

OBJECTIVE: Determine if complex singular value decomposition (cSVD) used as preprocessing in the maximum slope algorithm reduces image noise of resultant physiologic parametric images. Noise will be decreased in the parametric maps of cerebral blood flow (CBF), cerebral blood volume (CBV) as compared to the same algorithm and data set with no cSVD applied. MATERIALS AND METHODS: A set of 10 patients (n=15) underwent a total combined 15 CT perfusion studies upon presenting with stroke symptoms. It was determined these patients suffered from occlusions resulting in a prolonged arrival time of blood to the brain. DICOM data files of these patients scans were selected based on this increased arrival delay. We compared the output of estimation calculations for cerebral blood flow (CBF), and cerebral blood volume (CBV), using preprocessing cSVD against the same scan data with no preprocessing cSVD. Image noise was assessed through the calculation of the standard deviation within specific regions of interest copied to specific areas of grey and white matter as well as CSF space. A decrease in the standard deviation values will indicate improvement in the noise level of the resultant images.. Results for the mean value within the regions of interest are expected to be similar between the groups calculated using cSVD and those calculated under the standard method. This will indicate the presence of minimal bias. RESULTS: Between groups of the standard processing method and the cSVD method standard deviation (SD) reductions were seen in both CBF and CBV values across all three ROIs. In grey matter measures of CBV, SD was reduced an average of 0.0034 mL/100g while measures of CBF saw SD reduced by an average of 0.073 mL/100g/min. In samples of white matter, standard deviations of CBV values were reduced on average by 0.0041mL/100g while CBF SD's were reduced by 0.073 mL/100g/min. CSF ROIs in CBV calculations saw SD reductions averaging 0.0047 mL/100g and reductions of 0.074 mL/100g/min in measures of CBF. Bias within CBV calculations was at most minimal as determined by no significant changes in mean calculated values. Calculations of CBF saw large downward bias in the mean values. CONCLUSIONS: The application of the cSVD method to preprocessing of CT perfusion imaging studies produces an effective method of noise reduction. In calculations of CBV, cSVD noise reduction results in overall improvement. In calculations of CBF, cSVD, while effective in noise reduction, caused mean values to be statistically lower than the standard method. It should be noted that there is currently no evaluation of which values can be considered more accurate physiologically. Simulations of the effect of noise on CBF showed a positive correlation suggesting that the CBF algorithm itself is sensitive to the level of noise

Measuring dark matter ellipticity of Abell 901/902 using Particle Based Lensing

We present a non-parametric measure of the ellipticity and the alignment of the dark matter halos in Abell 901/902 supercluster. This super-cluster is a system of four separate peaks in a $0.5^{\circ}\times0.5^{\circ}$ field of view. We map the mass distribution of each individual peak using an improved version of Particle Based Lensing (PBL) and measure the ellipticity of the dark matter halos associated with two of the peaks directly from the mass map and by fitting them to a singular isothermal ellipse. The parametric and non-parametric measurements are consistent for A901b while the position angle for the Southwest Group is different for the two techniques. We account for this discrepancy to substructure present in the Southwest Peak. We estimate an axis ratio of $0.37\pm 0.1$ for A901b and $0.54^{+0.08}_{-0.09}$ for the Southwest Group.Comment: submitted to APJ, 25 pages, 13 figure

Probing the distribution of dark matter in the Abell 901/902 supercluster with weak lensing

We present a weak shear analysis of the Abell 901/902 supercluster, composed of three rich clusters at z=0.16. Using a deep R-band image from the 0.5 x 0.5 degree MPG/ESO Wide Field Imager together with supplementary B-band observations, we build up a comprehensive picture of the light and mass distributions in this region. We find that, on average, the light from the early-type galaxies traces the dark matter fairly well, although one cluster is a notable exception to this rule. The clusters themselves exhibit a range of mass-to-light (M/L) ratios, X-ray properties, and galaxy populations. We attempt to model the relation between the total mass and the light from the early-type galaxies with a simple scale-independent linear biasing model. We find M/L_B=130h for the early type galaxies with zero stochasticity, which, if taken at face value, would imply Omega_m < 0.1. However, this linear relation breaks down on small scales and on scales equivalent to the average cluster separation (approximately 1 Mpc), demonstrating that a single M/L ratio is not adequate to fully describe the mass-light relation in the supercluster. Rather, the scatter in M/L ratios observed for the clusters supports a model incorporating non-linear biasing or stochastic processes. Finally, there is a clear detection of filamentary structure connecting two of the clusters, seen in both the galaxy and dark matter distributions, and we discuss the effects of cluster-cluster and cluster-filament interactions as a means to reconcile the disparate descriptions of the supercluster.Comment: 23 pages, 19 figures. ApJ, accepte

First order algorithms in variational image processing

Variational methods in imaging are nowadays developing towards a quite universal and flexible tool, allowing for highly successful approaches on tasks like denoising, deblurring, inpainting, segmentation, super-resolution, disparity, and optical flow estimation. The overall structure of such approaches is of the form ${\cal D}(Ku) + \alpha {\cal R} (u) \rightarrow \min_u$ ; where the functional ${\cal D}$ is a data fidelity term also depending on some input data $f$ and measuring the deviation of $Ku$ from such and ${\cal R}$ is a regularization functional. Moreover $K$ is a (often linear) forward operator modeling the dependence of data on an underlying image, and $\alpha$ is a positive regularization parameter. While ${\cal D}$ is often smooth and (strictly) convex, the current practice almost exclusively uses nonsmooth regularization functionals. The majority of successful techniques is using nonsmooth and convex functionals like the total variation and generalizations thereof or $\ell_1$-norms of coefficients arising from scalar products with some frame system. The efficient solution of such variational problems in imaging demands for appropriate algorithms. Taking into account the specific structure as a sum of two very different terms to be minimized, splitting algorithms are a quite canonical choice. Consequently this field has revived the interest in techniques like operator splittings or augmented Lagrangians. Here we shall provide an overview of methods currently developed and recent results as well as some computational studies providing a comparison of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure

Wavelets and their use

This review paper is intended to give a useful guide for those who want to apply discrete wavelets in their practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous proofs of mathematical statements are omitted, and the reader is just referred to corresponding literature. The multiresolution analysis and fast wavelet transform became a standard procedure for dealing with discrete wavelets. The proper choice of a wavelet and use of nonstandard matrix multiplication are often crucial for achievement of a goal. Analysis of various functions with the help of wavelets allows to reveal fractal structures, singularities etc. Wavelet transform of operator expressions helps solve some equations. In practical applications one deals often with the discretized functions, and the problem of stability of wavelet transform and corresponding numerical algorithms becomes important. After discussing all these topics we turn to practical applications of the wavelet machinery. They are so numerous that we have to limit ourselves by some examples only. The authors would be grateful for any comments which improve this review paper and move us closer to the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh