Shapelets: I. a method for image analysis

By Alexandre Refregier. 2001

293 citations recorded by the source corpus.

We present a new method for the analysis of images, a fundamental task in observational astronomy. It is based on the linear decomposition of each object in the image into a series of localized basis functions of different shapes, which we call ‘shapelets’. A particularly useful set of complete and orthonormal shapelets is that consisting of weighted Hermite polynomials, which correspond to perturbations around a circular Gaussian. They are also the eigenstates of the two-dimensional quantum harmonic oscillator, and thus allow us to use the powerful formalism developed for this problem. One of their special properties is their invariance under Fourier transforms (up to a rescaling), leading to an analytic form for convolutions. The generator of linear transformations such as translations, rotations, shears and dilatations can be written as simple combinations of raising and lowering operators. We derive analytic expressions for practical quantities, such as the centroid (astrometry), flux (photometry) and radius of the object, in terms of its shapelet coefficients. We also construct polar basis functions which are eigenstates of the angular momentum operator, and thus have simple properties under rotations. As an example, we apply the method to Hubble Space Telescope images, and show that the small number of shapelet coefficients required to represent galaxy images lead to compression factors of about 40 to 90. We discuss applications of shapelets for the archival of large photometric surveys, for weak and strong gravitational lensing and for image deprojection.

Subfields: Geochemistry and Petrology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c52ce80aea87be477654ce477c747ab64ca0ce6d

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Citations in the corpus

Shear calibration biases in weak-lensing surveys (2003, methodology): “We note that the polynomials are related to Refregier (2003)’s polar Hermite polynomials according to pq(r cos φ, r sin φ) = H p,q (r ) eimφ .”
THE NEXT GENERATION VIRGO CLUSTER SURVEY (NGVS). I. INTRODUCTION TO THE SURVEY (2012, methodology): “The second is based on Shapelet (Refregier 2003) decomposition of stars, again selected based on SExtractor parameters, followed by a fit of the Shapelet coefficients over the entire mosaic to enable the reconstruction of model PSFs at any given position.”
Weighing simulated galaxy clusters using lensing and X-ray (2009, methodology): “In short, the code uses a set of real galaxies decomposed into shapelets (Refregier 2003) to model the source morphologies on a synthetic sky.”
Origins of weak lensing systematics, and requirements on future instrumentation (or knowledge of instrumentation) (2012, background): “Derivations using weighted moments are complicated and involve mixing of higher order moments (KSB; RRG; Kaiser 2000; Réfrégier 2003; Melchior et al. 2011).”
CFHTLenS: Improving the quality of photometric redshifts with precision photometry (2011, methodology): “Both steps take advantages of some of the mathematical properties of the shapelet formalism (Refregier 2003).”
The Intrinsic Alignment of Galaxies and its Impact on Weak Gravitational Lensing in an Era of Precision Cosmology (2014, background): “…(1991); Kaiser (1992); Blandford and Narayan (1992); Schneider et al. (1992); Mellier (1999); Bartelmann and Schneider (2001); Wittman (2002); Refregier (2003b); van Waerbeke and Mellier (2003); Schneider et al. (2006b); Hoekstra and Jain (2008); Munshi et al. (2008); Heavens (2009);…”
Lensing and x-ray mass estimates of clusters (simulations) (2012, methodology): “In short, the code uses a set of real galaxies decomposed into shapelets (Refregier 2003) to model the source morphologies on a synthetic sky.”
Combining Strong and Weak Gravitational Lensing in Abell 1689 (2006, methodology): “These perturbations are modeled as shapelets (Refregier 2003), adding flexibility in the modeling and thus accommodating a broad range of density profiles from cuspy, using a NFWprofile, to various modifications of it.”
Observational probes of cosmic acceleration (2012, methodology): “The “shapelet” basis (Refregier, 2003; Refregier and Bacon, 2003), derived from energy eigenstates of a 2D quantum harmonic oscillator, is useful in both types of methods.”
Gravitational lensing analysis of the Kilo-Degree Survey (2015, methodology): “To describe the PSF, we model the stars with shapelet expansions following Refregier (2003), and use these to construct and apply a suitable spatially varying convolution kernel.”
Galaxy–galaxy weak lensing in the Sloan Digital Sky Survey: intrinsic alignments and shear calibration errors (2004, methodology): “1 I N T RO D U C T I O N Weak gravitational lensing has emerged as a powerful tool for directly measuring the matter distribution in the Universe (e.g. Bartelmann & Schneider 2001; Refregier 2003a).”
Systematic errors in weak lensing: application to SDSS galaxy-galaxy weak lensing (2005, methodology): “The re-Gaussianization scheme is exact to first order in PSF nonGaussianity, however higher order approaches have been proposed based on expansions of the galaxy and PSF in orthogonal functions (Bernstein & Jarvis 2002; Refregier 2003a).”
The Shear Testing Programme – I. Weak lensing analysis of simulated ground-based observations (2005, methodology): “…been developed [Rhodes et al. 2000 (RRG); Kaiser 2000 (K2K); Smith 2000 (ELLIPTO); Bridle et al. 2002 (Im2shape); Bernstein & Jarvis 2002 (BJ02); Refregier 2003 (shapelets); Massey & Refregier 2005 (polar shapelets)] and implemented in cosmic shear analyses [see for example Wittman et al. 2001…”
H0LiCOW – XIII. A 2.4 per cent measurement of H0 from lensed quasars: 5.3σ tension between early- and late-Universe probes (2019, background): “The primary difference between the codes is that GLEE reconstructs the source on a pixelized grid with regularization, whereas LENSTRONOMY describes the source as a parametrized profile with additional shapelet functions (Refregier 2003; Birrer et al. 2015).”
Deep GALEX Imaging of the COSMOS HST Field: A First Look at the Morphology of z ~ 0.7 Star-forming Galaxies (2007, methodology): “Following along this path, people have devised and applied several ingenious algorithms for morphological classification, the most popular being the Sersic index (Sérsic 1968), bulgeto-disk decomposition (Simard 1998; Peng et al. 2002), and shapelet decomposition (Refregier 2003).”
H0LiCOW – IX. Cosmographic analysis of the doubly imaged quasar SDSS 1206+4332 and a new measurement of the Hubble constant (2018, methodology): “(ii) Option DOUBLE_SERSIC+2nmax: Additionally to DOU- BLE_SERSIC we add shapelet functions (Refregier 2003; Birrer et al. 2015) with maximal polynomial order nmax = 2 centered at the quasar and with free scale parameter, β. (iii) Option DOUBLE_SERSIC+5nmax: Addition of maximal polynomial order nmax…”