The influence of vibrational relaxation on the operation of the 119 μm CH3OH laser

Measurements of the pressure dependence of the relaxation rate of vibrationally excited molecules in the 119 m CH₃OH laser have been correlated with the pressure and pump power dependence of the output power in the framework of a rate-equation model for which numerical and approximate analytical solutions have been obtained. Both diffusion and collisional de-excitation are important, and substantial deviations from thermal equilibrium populations are deduced, consistent with an operating pressure range which increases with pump power.     

A unified characterization of reproducing systems generated by a finite family

This article presents a general result from the study of shift-invariant spaces that characterizes tight frame and dual frame generators for shift-invariant subspaces of L²(ℝⁿ). A number of applications of this general result are then obtained, among which are the characterization of tight frames and dual frames for Gabor and wavelet systems.     

Quantitative analysis of thin samples by differential absorption imaging using a laser-plasma soft X-ray source

We consider the application of the differential absorption imaging technique in the soft X-ray region. The surface-density maps of test samples containing bromine, resulting from the application of the differential absorption imaging technique, are presented. Images of the samples were obtained in an X-ray monochromatic projection imaging scheme based on a spherically bent crystal using line emission close to the $L_2$ edge of bromine from a microplasma produced by focusing a nanosecond laser on the surface of an aluminum target.     

Detection of Singularities by Discrete Multiscale Directional Representations

One of the most remarkable properties of the continuous curvelet and shearlet transforms is their sensitivity to the directional regularity of functions and distributions. As a consequence of this property, these transforms can be used to characterize the geometry of edge singularities of functions and distributions by their asymptotic decay at fine scales. This ability is a major extension of the conventional continuous wavelet transform which can only describe pointwise regularity properties. However, while in the case of wavelets it is relatively easy to relate the asymptotic properties of the continuous transform to properties of discrete wavelet coefficients, this problem is surprisingly challenging in the case of discrete curvelets and shearlets where one wants to handle also the geometry of the singularity. No result for the discrete case was known so far. In this paper, we derive non-asymptotic estimates showing that discrete shearlet coefficients can detect, in a precise sense, the location and orientation of curvilinear edges. We discuss connections and implications of this result to sparse approximations and other applications.     

Shearlet Smoothness Spaces

The shearlet representation has gained increasingly more prominence in recent years as a flexible and efficient mathematical framework for the analysis of anisotropic phenomena. This is achieved by combining traditional multiscale analysis with a superior ability to handle directional information. In this paper, we introduce a class of shearlet smoothness spaces which is derived from the theory of decomposition spaces recently developed by L. Borup and M. Nielsen. The introduction of these spaces is motivated by recent results in image processing showing the advantage of using smoothness spaces associated with directional multiscale representations for the design and performance analysis of improved image restoration algorithms. In particular, we examine the relationship of the shearlet smoothness spaces with respect to Besov spaces, curvelet-type decomposition spaces and shearlet coorbit spaces. With respect to the theory of shearlet coorbit space, the construction of shearlet smoothness spaces presented in this paper does not require the use of a group structure.     

Characterization of Piecewise-Smooth Surfaces Using the 3D Continuous Shearlet Transform

One of the most striking features of the Continuous Shearlet Transform is its ability to precisely characterize the set of singularities of multivariable functions through its decay at fine scales. In dimension n=2, it was previously shown that the continuous shearlet transform provides a precise geometrical characterization for the boundary curves of very general planar regions, and this property sets the groundwork for several successful image processing applications. The generalization of this result to dimension n=3 is highly nontrivial, and so far it was known only for the special case of 3D bounded regions where the boundary set is a smooth 2-dimensional manifold with everywhere positive Gaussian curvature. In this paper, we extend this result to the general case of 3D bounded regions with piecewise-smooth boundaries, and show that also in this general situation the continuous shearlet transform precisely characterizes the geometry of the boundary set.     

Representation of Fourier Integral Operators Using Shearlets

Traditional methods of time-frequency and multiscale analysis, such as wavelets and Gabor frames, have been successfully employed for representing most classes of pseudodifferential operators. However, these methods are not equally effective in dealing with Fourier Integral Operators in general. In this article, we show that the shearlets, recently introduced by the authors and their collaborators, provide very efficient representations for a large class of Fourier Integral Operators. The shearlets are an affine-like system of well-localized waveforms at various scales, locations and orientations, which are particularly efficient in representing anisotropic functions. Using this approach, we prove that the matrix representation of a Fourier Integral Operator with respect to a Parseval frame of shearlets is sparse and well-organized. This fact recovers a similar result recently obtained by Candès and Demanet using curvelets, which illustrates the benefits of directional multiscale representations (such as curvelets and shearlets) in the study of those functions and operators where traditional multiscale methods are unable to provide the appropriate geometric analysis in the phase space. The second author was supported in part by a National Science Foundation grant DMS 0604561.     

Geometric Separation in $$\mathbb {R}^3$$ R 3

Journal of Fourier Analysis and Applications - The geometric separation problem, initially posed by Donoho and Kutyniok (Commun Pure Appl Math 66:1–47, 2013), aims to separate a distribution...     

Time-Frequency Analysis of Pseudodifferential Operators

 In this paper we apply a time-frequency approach to the study of pseudodifferential operators. Both the Weyl and the Kohn–Nirenberg correspondences are considered. In order to quantify the time-frequency content of a function or distribution, we use certain function spaces called modulation spaces. We deduce a time-frequency characterization of the twisted product of two symbols σ and τ, and we show that modulation spaces provide the natural setting to exactly control the time-frequency content of from the time-frequency content of σ and τ. As a consequence, we discuss some boundedness and spectral properties of the corresponding operator with symbol . (Received 27 December 1999; in final form 9 November 2000)     

Magnetically guided fast electrons in cylindrically compressed matter

Fast electrons produced by a 10 ps, 160 J laser pulse through laser-compressed plastic cylinders are studied experimentally and numerically in the context of fast ignition. K(α)-emission images reveal a collimated or scattered electron beam depending on the initial density and the compression timing. A numerical transport model shows that implosion-driven electrical resistivity gradients induce strong magnetic fields able to guide the electrons. The good agreement with measured beam sizes provides the first experimental evidence for fast-electron magnetic collimation in laser-compressed matter.