Shapelets: I. A Method for Image Analysis

TL;DR

Shapelets introduce a linear, analytically tractable image representation for astronomy based on weighted Hermite polynomials around a Gaussian, connecting to the Quantum Harmonic Oscillator to enable simple Fourier behavior and operator-based transformations. The coefficients $f_{\mathbf n}$ yield direct, closed-form expressions for photometric and astrometric quantities and permit efficient convolution, noise handling, and coordinate transformations. The framework is demonstrated on HST data, achieving substantial compression (factors of $\sim$40–90) and enabling applications to archival cataloging, PSF correction, gravitational lensing, and deprojection, with a companion paper detailing shear recovery. Overall, shapelets offer a complementary, physics-inspired alternative to wavelets for scalable, high-precision analysis of large astronomical image datasets.

Abstract

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 localised 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 2-dimensional Quantum Harmonic Oscillator, and thus allow us to use the powerful formalism developed for this problem. Among their remarkable properties, they are invariant 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.

Figures (10)

• Figure 1: First few 1-dimensional basis functions $\phi_{n}(x)$.
• Figure 2: First few 2-dimensional Cartesian basis functions $\phi_{n_{1},n_{2}}$. The dark and light regions correspond to positive and negative values, respectively.
• Figure 3: Decomposition of a galaxy image found in the HDF. The original $60\times60$ pixel HST image (upper left-hand panel) can be compared with the reconstructed images with different maximum order $n=n_{1}+n_{2}$. The shapelet scale is chosen to be $\beta=4$ pixels. The lower right-hand panel ($n  20$) is virtually indistinguishable from the initial image.
• Figure 4: Shapelet coefficients for the image decomposition of the previous figure. Since the coefficient array is sparse, the images can be reconstructed from the few first largest coefficients.
• Figure 5: Reconstruction and compression of three HST galaxy images using shapelets. The left-hand column shows the orginal images extracted from the HDF and list $N_{\rm pix}$ their size in pixels. The right-hand column shows their reconstructed image from the $N_{\rm cof}$ largest coefficients (in absolute value) of their shapelet decomposition. Because the coefficient matrix is typically sparse, a large compression factor $N_{\rm pix}/N_{\rm cof}$ is achieved. The shapelet scale was chosen to be $\beta=4$ pixels in all 3 cases.