Fourier, Gauss, Fraunhofer, Porod and the Shape from Moments Problem

We show how the Fourier transform of a shape in any number of dimensions can be simplified using Gauss's law and evaluated explicitly for polygons in two dimensions, polyhedra three dimensions, etc. We also show how this combination of Fourier and Gauss can be related to numerous classical problems in physics and mathematics. Examples include Fraunhofer diffraction patterns, Porods law, Hopfs Umlaufsatz, the isoperimetric inequality and Didos problem. We also use this approach to provide an alternative derivation of Davis's extension of the Motzkin-Schoenberg formula to polygons in the complex plane.Comment: 21 pages, no figure

M\"obius Invariants of Shapes and Images

Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to this problem is to identify the group of possible transformations of the object and to find invariants to the action of that group, meaning that the object has the same values of the invariants despite the action of the group. In this paper we study the invariants of planar shapes and images under the M\"obius group $\mathrm{PSL}(2,\mathbb{C})$, which arises in the conformal camera model of vision and may also correspond to neurological aspects of vision, such as grouping of lines and circles. We survey properties of invariants that are important in applications, and the known M\"obius invariants, and then develop an algorithm by which shapes can be recognised that is M\"obius- and reparametrization-invariant, numerically stable, and robust to noise. We demonstrate the efficacy of this new invariant approach on sets of curves, and then develop a M\"obius-invariant signature of grey-scale images

Perturbative reconstruction of a gravitational lens: when mass does not follow light

The structure and potential of a complex gravitational lens is reconstructed using the perturbative method presented in Alard 2007, MNRAS, 382L, 58; Alard 2008, MNRAS, 388, 375. This lens is composed of 6 galaxies belonging to a small group. The lens inversion is reduced to the problem of reconstructing non-degenerate quantities: the 2 fields of the perturbative theory of strong gravitational lenses. Since in the perturbative theory the circular source solution is analytical, the general properties of the perturbative solution can be inferred directly from the data. As a consequence, the reconstruction of the perturbative fields is not affected by degeneracy, and finding the best solution is only a matter of numerical refinement. The local shape of the potential and density of the lens are inferred from the perturbative solution, revealing the existence of an independent dark component that does not follow light. The most likely explanation is that the particular shape of the dark halo is due to the merging of cold dark matter halos. This is a new result illustrating the structure of dark halos at the scale of galaxies.Comment: Final version (Astronomy and Astrophysics in press

The Hurst Exponent of Fermi GRBs

Using a wavelet decomposition technique, we have extracted the Hurst exponent for a sample of 46 long and 22 short Gamma-ray bursts (GRBs) detected by the Gamma-ray Burst Monitor (GBM) aboard the Fermi satellite. This exponent is a scaling parameter that provides a measure of long-range behavior in a time series. The mean Hurst exponent for the short GRBs is significantly smaller than that for the long GRBs. The separation may serve as an unbiased criterion for distinguishing short and long GRBs.Comment: Accepted for publication in Monthly Notices of the Royal Astronomical Societ

A fast high-order solver for problems of scattering by heterogeneous bodies

A new high-order integral algorithm for the solution of scattering problems by heterogeneous bodies is presented. Here, a scatterer is described by a (continuously or discontinuously) varying refractive index n(x) within a two-dimensional (2D) bounded region; solutions of the associated Helmholtz equation under given incident fields are then obtained by high-order inversion of the Lippmann-Schwinger integral equation. The algorithm runs in O(Nlog(N)) operations where N is the number of discretization points. A wide variety of numerical examples provided include applications to highly singular geometries, high-contrast configurations, as well as acoustically/electrically large problems for which supercomputing resources have been used recently. Our method provides highly accurate solutions for such problems on small desktop computers in CPU times of the order of seconds

Efficient Spectral Domain MoM for the Design of Circularly Polarized Reflectarray Antennas Made of Split Rings

The method of moments (MoM) in the spectral domain is used for the analysis of the scattering of a plane wave by a multilayered periodic structure containing conducting concentric split rings in the unit cell. Basis functions accounting for edge singularities are used in the approximation of the current density on the split rings, which makes it possible a fast convergence of MoM with respect to the number of basis functions. Since the 2-D Fourier transforms of the basis functions cannot be obtained in closed-form, judicious tricks (controlled truncation of infinite summations, interpolations, etc.) are used for the efficient numerical determination of these Fourier transforms. The implemented spectral domain MoM software has been used in the design of a circularly polarized reflectarray antenna based on split rings under the local periodicity condition. The antenna has been analyzed with our spectral domain MoM software, with CST and with HFSS, and good agreement has been found among all sets of results. Our software has proven to be around 27 times faster than CST and HFSS

Adaptive transient solution of nonuniform multiconductor transmission lines using wavelets

Abstract—This paper presents a highly adaptive algorithm for the transient simulation of nonuniform interconnects loaded with arbitrary nonlinear and dynamic terminations. The discretization of the governing equations is obtained through a weak formula-tion using biorthogonal wavelet bases as trial and test functions. It is shown how the multiresolution properties of wavelets lead to very sparse approximations of the voltages and currents in typical transient analyzes. A simple yet effective time–space adaptive al-gorithm capable of selecting the minimal number of unknowns at each time iteration is described. Numerical results show the high degree of adaptivity of the proposed scheme. Index Terms—Electromagnetic (EM) transient analysis, multi-conductor transmission lines (TLs), wavelet transforms. I

Effect of hole-shape irregularities on photonic crystal waveguides

The effect of irregular hole shape on the spectrum and radiation losses of a photonic crystal waveguide is studied using Bloch-mode expansion. Deviations from a perfectly circular hole are characterized by a radius fluctuation amplitude and correlation angle. It is found that the parameter which determines the magnitude of the effect of disorder is the standard deviation of the hole areas. Hence, for a fixed amplitude of the radius fluctuation around the hole, those effects are strongly dependent on the correlation angle of the irregular shape. This result suggests routes to potentially improve the quality of photonic crystal structures.Comment: 3 pages, 3 figure