Augmented Slepians: Bandlimited Functions that Counterbalance Energy in Selected Intervals

Slepian functions provide a solution to the optimization problem of joint time-frequency localization. Here, this concept is extended by using a generalized optimization criterion that favors energy concentration in one interval while penalizing energy in another interval, leading to the "augmented" Slepian functions. Mathematical foundations together with examples are presented in order to illustrate the most interesting properties that these generalized Slepian functions show. Also the relevance of this novel energy-concentration criterion is discussed along with some of its applications

Optimal-Dimensionality Sampling on the Sphere: Improvements and Variations

For the accurate representation and reconstruction of band-limited signals on the sphere, an optimal-dimensionality sampling scheme has been recently proposed which requires the optimal number of samples equal to the number of degrees of freedom of the signal in the spectral (harmonic) domain. The computation of the spherical harmonic transform (SHT) associated with the optimal-dimensionality sampling requires the inversion of a series of linear systems in an iterative manner. The stability of the inversion depends on the placement of iso-latitude rings of samples along co-latitude. In this work, we have developed a method to place these iso-latitude rings of samples with the objective of improving the well-conditioning of the linear systems involved in the computation of the SHT. We also propose a multi-pass SHT algorithm to iteratively improve the accuracy of the SHT of band-limited signals. Furthermore, we review the changes in the computational complexity and improvement in accuracy of the SHT with the embedding of the proposed methods. Through numerical experiments, we illustrate that the proposed variations and improvements in the SHT algorithm corresponding to the optimal-dimensionality sampling scheme significantly enhance the accuracy of the SHT.Comment: 5 Pages, 4 figure

Fast directional spatially localized spherical harmonic transform

We propose a transform for signals defined on the sphere that reveals their localized directional content in the spatio-spectral domain when used in conjunction with an asymmetric window function. We call this transform the directional spatially localized spherical harmonic transform (directional SLSHT) which extends the SLSHT from the literature whose usefulness is limited to symmetric windows. We present an inversion relation to synthesize the original signal from its directional-SLSHT distribution for an arbitrary window function. As an example of an asymmetric window, the most concentrated band-limited eigenfunction in an elliptical region on the sphere is proposed for directional spatio-spectral analysis and its effectiveness is illustrated on the synthetic and Mars topographic data-sets. Finally, since such typical data-sets on the sphere are of considerable size and the directional SLSHT is intrinsically computationally demanding depending on the band-limits of the signal and window, a fast algorithm for the efficient computation of the transform is developed. The floating point precision numerical accuracy of the fast algorithm is demonstrated and a full numerical complexity analysis is presented.Comment: 12 pages, 5 figure

Fast directional correlation on the sphere with steerable filters

A fast algorithm is developed for the directional correlation of scalar band-limited signals and band-limited steerable filters on the sphere. The asymptotic complexity associated to it through simple quadrature is of order O(L^5), where 2L stands for the square-root of the number of sampling points on the sphere, also setting a band limit L for the signals and filters considered. The filter steerability allows to compute the directional correlation uniquely in terms of direct and inverse scalar spherical harmonics transforms, which drive the overall asymptotic complexity. The separation of variables technique for the scalar spherical harmonics transform produces an O(L^3) algorithm independently of the pixelization. On equi-angular pixelizations, a sampling theorem introduced by Driscoll and Healy implies the exactness of the algorithm. The equi-angular and HEALPix implementations are compared in terms of memory requirements, computation times, and numerical stability. The computation times for the scalar transform, and hence for the directional correlation, of maps of several megapixels on the sphere (L~10^3) are reduced from years to tens of seconds in both implementations on a single standard computer. These generic results for the scale-space signal processing on the sphere are specifically developed in the perspective of the wavelet analysis of the cosmic microwave background (CMB) temperature (T) and polarization (E and B) maps of the WMAP and Planck experiments. As an illustration, we consider the computation of the wavelet coefficients of a simulated temperature map of several megapixels with the second Gaussian derivative wavelet.Comment: Version accepted in APJ. 14 pages, 2 figures, Revtex4 (emulateapj). Changes include (a) a presentation of the algorithm as directly built on blocks of standard spherical harmonics transforms, (b) a comparison between the HEALPix and equi-angular implementation

Photosensor Characterization for the Cherenkov Telescope Array: Silicon Photomultiplier versus Multi-Anode Photomultiplier Tube

Photomultiplier tube technology has been the photodetector of choice for the technique of imaging atmospheric Cherenkov telescopes since its birth more than 50 years ago. Recently, new types of photosensors are being contemplated for the next generation Cherenkov Telescope Array. It is envisioned that the array will be partly composed of telescopes using a Schwarzschild-Couder two mirror design never built before which has significantly improved optics. The camera of this novel optical design has a small plate scale which enables the use of compact photosensors. We present an extensive and detailed study of the two most promising devices being considered for this telescope design: the silicon photomultiplier and the multi-anode photomultiplier tube. We evaluated their most critical performance characteristics for imaging gamma-ray showers, and we present our results in a cohesive manner to clearly evaluate the advantages and disadvantages that both types of device have to offer in the context of GeV-TeV gamma-ray astronomy.Comment: submitted to SPIE Optics+Photonics proceeding

Measurement of the earthshine polarization in the B, V, R, and I band as function of phase

The characterization of the polarimetric properties of the planet Earth is important for the interpretation of expected observations and the planning of future instruments. We present a multi-wavelengths and multi-phase set of benchmark values for the polarization signal of the integrated light from the planet Earth derived from new polarimetric observations of the earthshine back-scattered from the Moon's dark side. Using a new, specially designed wide field polarimeter we measured the fractional polarization of the earthshine in the B, V, R and I filters for Earth phase angles alpha between 30{\deg} and 110{\deg}. The phase dependence of the earthshine polarization is fitted by a function p x sin(alpha)^2. To determine the polarization of the planet Earth we correct our earthshine measurements by a polarization efficiency function for the lunar surface derived from measurements of lunar samples from the literature. The polarization of the earthshine decreases towards longer wavelengths and is about a factor 1.3 lower for the higher albedo highlands. For mare regions the measured maximum polarization is about 13 % at quadrature in the B band. The resulting fractional polarizations for Earth are 24.6 % for the B band, 19.1 % for the V band, 13.5 % for the R band, and 8.3 % for the I band. Together with literature values for the spectral reflectivity of Earth we obtain a contrast between the polarized flux of the Earth and the (total) flux of the Sun with an uncertainty of less than 20 % and we find that the best phase to detect an Earth twin is around an Earth phase alpha=65{\deg}. The polarimetric models of Earth-like planets from Stam (2008) are in qualitative agreement with our results but there are also significant differences which might guide more detailed computations.Comment: 14 pages, 14 figures, accepted for publication in Astronomy & Astrophysic