Gabor Shearlets

In this paper, we introduce Gabor shearlets, a variant of shearlet systems, which are based on a different group representation than previous shearlet constructions: they combine elements from Gabor and wavelet frames in their construction. As a consequence, they can be implemented with standard filters from wavelet theory in combination with standard Gabor windows. Unlike the usual shearlets, the new construction can achieve a redundancy as close to one as desired. Our construction follows the general strategy for shearlets. First we define group-based Gabor shearlets and then modify them to a cone-adapted version. In combination with Meyer filters, the cone-adapted Gabor shearlets constitute a tight frame and provide low-redundancy sparse approximations of the common model class of anisotropic features which are cartoon-like functions.Comment: 24 pages, AMS LaTeX, 4 figure

The Brera Multi-scale Wavelet (BMW) ROSAT HRI source catalog. I: the algorithm

We present a new detection algorithm based on the wavelet transform for the analysis of high energy astronomical images. The wavelet transform, due to its multi-scale structure, is suited for the optimal detection of point-like as well as extended sources, regardless of any loss of resolution with the off-axis angle. Sources are detected as significant enhancements in the wavelet space, after the subtraction of the non-flat components of the background. Detection thresholds are computed through Monte Carlo simulations in order to establish the expected number of spurious sources per field. The source characterization is performed through a multi-source fitting in the wavelet space. The procedure is designed to correctly deal with very crowded fields, allowing for the simultaneous characterization of nearby sources. To obtain a fast and reliable estimate of the source parameters and related errors, we apply a novel decimation technique which, taking into account the correlation properties of the wavelet transform, extracts a subset of almost independent coefficients. We test the performance of this algorithm on synthetic fields, analyzing with particular care the characterization of sources in poor background situations, where the assumption of Gaussian statistics does not hold. For these cases, where standard wavelet algorithms generally provide underestimated errors, we infer errors through a procedure which relies on robust basic statistics. Our algorithm is well suited for the analysis of images taken with the new generation of X-ray instruments equipped with CCD technology which will produce images with very low background and/or high source density.Comment: 8 pages, 6 figures, ApJ in pres

SONAR Images Denoising

International audienc

Enhanced B-Wavelets via Mixed, Composite Packets

A modified B-wavelet construction with enhanced filter characteristics is considered. The design comprises a superposition of tessellated, integer dilated, ‘sister’ wavelet functions. We here propose a cascaded filter-bank realisation of this wavelet family together with some notable extensions. We prove that modifications of low-order members exist in the multiresolution subspace spanned by the half-translates of the original wavelets and hence that the resulting modified wavelet coefficients can be computed as convolutions of the undecimated original wavelet coefficients. Finite impulse response filters are thus designed and incorporated into a B-wavelet packet architecture such that the mainlobe-to-sidelobe ratio of the resulting wavelet filter characteristic is improved. This is achieved by designing the filters so that zeros are introduced near to the maxima of the harmonics. It is shown that the numbers of zeros can be balanced with the length of the corresponding filters by controlling the ‘modification order’. Several constructions are presented. We prove that two such constructions satisfy the perfect reconstruction property for all orders. The resulting modified wavelets preserve many of the properties of the original B-wavelets such as differentiability, number of vanishing moments, symmetry, anti-symmetry, finite support, and the existence of a closed form expression

Compact source detection in multi-channel microwave surveys: from SZ clusters to polarized sources

In this paper we describe the state-of-the art status of multi-frequency detection techniques for compact sources in microwave astronomy. From the simplest cases where the spectral behaviour is well-known (i.e. thermal SZ clusters) to the more complex cases where there is little a priori information (i.e. polarized radio sources) we will review the main advances and the most recent results in the detection problem.Comment: 13 pages, 4 figures. Accepted for publication in the Special Issue "Astrophysical Foregrounds in Microwave Surveys" of the journal Advances in Astronom