Introduction:

Shapelets are a complete, orthonormal set of 2D basis functions constructed from Laguerre or Hermite polynomials weighted by a Gaussian. A linear combination of these functions can be used to model any image, in a similar way to Fourier or wavelet synthesis. The shapelet decomposition is particularly efficient for images localised in space, and provide a high level of compression for individual galaxies in astronomical data. The basis has many elegant mathematical properties that make it convenient for image analysis and processing. The formalism was first introduced to astronomy by Alexandre Refregier & David Bacon, and a related method has also been independently suggested by Gary Bernstein & Mike Jarvis.

This shapelets web site is maintained by Richard Massey.

IDL shapelets software:

The IDL shapelets software requires IDL v5.4 or above. A stand-alone set of core routines is available, including everything you need to decompose images into shapelets and to manipulate them. Routines to generate multicolour simulated astronomical images are also now documented and publicly available. We would be interested to collaborate with anyone requiring more advanced applications.

Published shapelets papers:

• Polar Shapelets
  Massey R. & Refregier A. 2005, MNRAS 363, 197
• Shapelets: I. A Method for Image Analysis
  Refregier A. 2003, MNRAS 338 35

• Laguerre 2D-functions and their application in quantum optics
  Wünsche A. 1998, J. Phys. A: Math. Gen., 31, 8267
• General Hermite and Laguerre two-dimensional polynomials
  Wünsche A. 2000, J. Phys. A: Math. Gen., 33, 1603
• Image Simulation with Shapelets
  Massey R., Refregier A., Conselice C. & Bacon D. 2004, MNRAS 348 214
• Modal decomposition of astronomical images with application to shapelets
  Berry R., Hobson M. & Withington S. 2004, MNRAS , 354, 199
• Optical coherence tomography of skin for measurement of epidermal thickness by shapelet-based image analysis
  Weissman J., Hancewicz T. & Kaplan P., 2004, Optics Express 12, 23, 5760
• Morphological Classification of Galaxies by Shapelet Decomposition in the SDSS
  Kelly B. & McKay T. 2004, AJ, 127, 625
• Morphological Classification of Galaxies by Shapelet Decomposition in the SDSS II: Multiwavelength Classification
  Kelly B. & McKay T. 2005, AJ, 129, 1287
• Shapelets correlated with surface normals produce surfaces
  Kovesi P., 2005, Computer Vision ICCV 2, 994
• Representation of Sun Spots with Shapelets
  Young C., Gallagher P., Ireland J. & McAteer R., 2005, American Geophysical Union SMSP, 11, 7
• Properties and possibilities of quantum shapelets
  Coffey M., 2006, JPhysA, 39, 877
• General integrated field spectrograph simulation
  Tilquina A., Bonissent A., Aumeunier M., Prieto E., Ealet A., Gerdes D. & Macaire C., 2006 New Astronomy Reviews 50, 4, 275
• Reliable Shapelet Image Analysis
  Melchior P., Meneghetti M. & Bartelmann M., 2006, A&A 463, 1215
• GaaP: PSF- and aperture-matched photometry using shapelets
  Kuijken K., 2006, A&A submitted
• Simultaneously band and space limited functions in two dimensions, and receptive fields of visual neurons
  Victor J. & Knight B. 2003, Springer Applied Mathematical Sciences Series, p. 375-420. Eds: E.Kaplan, J. Marsden, and K. R. Sreenivasan. Springer
• Responses of V1 Neurons to Two-Dimensional Hermite Functions
  Victor J., Mechler F., Repucci M., Purpura K. & Sharpee T. 2005, J. Neurophysiol 95, 379
• Contextual modulation of V1 receptive fields depends on their spatial symmetry
  Sharpee T. & Victor J. 2008, J. Comp. Neuroscience 26, 203
• Laminar and Orientation-Dependent Characteristics of Spatial Nonlinearities: Implications for the Computational Architecture of Visual Cortex
  Victor J., Mechler F., Ohiorhenuan I., Schmid A. & Purpura K. 2009, J. Neurophysiol 102, 3414

• Shapelets: II. A Method for Weak Lensing Measurements
  Refregier A. & Bacon D. 2003, MNRAS 338 48
• Shapes and Shears, Stars and Smears: Optimal Measurements for Weak Lensing
  Bernstein G. & Jarvis M. 2002, AJ, 123, 2
• Shape Reconstruction and Weak Lensing Measurement with Interferometers: A Shapelet Approach
  Chang T.-C. & Refregier A. 2002, ApJ, 570, 447
• The Galaxy Octopole Moment as a Probe of Weak Lensing Shear Fields
  Goldberg D. & Natarajan P. 2002, ApJ, 564, 65
• Optimal Galaxy Shape Measurements for Weak Lensing Applications using the HST ACS
  Park Y., Casertano S. & Ferguson H. 2003, ApJ, 600, L159
• Weak Lensing from Space II: Dark Matter Mapping
  Massey R., Rhodes J., Refregier A. et al. 2004, AJ, 127, 3089
• Weak Lensing by Large-Scale Structure with the FIRST Radio Survey
  Chang T.-C., Refregier A. & Helfand D., 2004, ApJ, 617, 794
• Extraction of relative magnification matrices from VLBI observations of gravitational lens systems
  Bähren L., Schneider P. & King L., 2002, evlb conf. 201
• Observations of cluster substructure using weakly lensed sextupole moments
  Irwin J. & Shmakova M. 2003
• Galaxy-Galaxy Flexion: Weak Lensing to Second Order
  Goldberg D. & Bacon D. 2004, ApJ, 619, 741
• Shapelets "multiple multipole" shear measurement methods
  Massey R., Refregier A. & Bacon D., 2005, IAU symp. 225, 31
• Mass modeling of Abell 1689 Advanced Camera for Surveys observations with a perturbed Navarro-Frenk-White model
  Zekser K., White R., Broadhurst T., Benítez N., Ford H., Illingworth G., Blakeslee J., Postman M., Jee M. & Coe D., 2006, ApJ, 640, 639
• Shears from shapelets
  Kuijken K., 2006, A&A 456, 827
• Weak gravitational shear and flexion with polar shapelets
  Massey R., Rowe B., Refregier A., Bacon D. & Berge J., 2007, MNRAS 380, 229
• Measuring Flexion
  Goldberg D. & Leonard A., 2007, ApJ 660, 1003
• The Shear TEsting Programme 2: Factors affecting high precision weak lensing analyses
  Massey R. et al. 2007, MNRAS 376, 13

Online resources:

• IDL shapelets software
• Animations showing the process of shapelet decomposition in HDF galaxies
• Flowchart describing the process of simulating images with shapelets < GIF | PS | PDF >
• Flowchart without background image (smaller files) < PS | PDF >
• Flexion measurement (second order gravitational lensing)
• Examples of deep simulated images from space < GIF | FITS >
• Simulated ground-based images for the Shear TEsting Program < STEP1 | STEP2 | spaceSTEP >
• SNAP weak lensing predictions.

Talks:

• Shapelets: A New Method to Measure Galaxy Shapes Refregier, Chang & Bacon 2002. Proceedings of the Workshop "The Shapes of Galaxies and their Halos", Yale, May 2001
• Shapelets "Multiple Multipole" Measurement Method Massey, Refregier, & Bacon 2004. Proceedings of the Workshop "Impact of gravitaional lensing on cosmology", Lausanne, July 2004
• Simulated Images with Shapelets 6.4Mb Powerpoint file from IoA seminar, October 2002
• Shapelets for Shear Surveys 8.3Mb Powerpoint file from "Structure Evolution and Cosmology", ESO Santiago, October 2002

People:

• David Bacon
• Joel Bergé
• Gary Bernstein
• Dave Goldberg
• Mike Jarvis
• Konrad Kuijken
• Phil Marshall
• Richard Massey
• Alexandre Refregier
• Barnaby Rowe
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