[Paper Review] A hybrid Fourier--Real Gaussian Mixture method for fast galaxy--PSF convolution

Dustin Lang|arXiv (Cornell University)|Dec 31, 2020

Galaxies: Formation, Evolution, Phenomena8 references4 citations

TL;DR

This paper presents a hybrid Fourier–real Gaussian mixture method for fast, accurate convolution of galaxy profiles with pixelized point-spread functions (PSFs). By approximating common galaxy profiles (e.g., de Vaucouleurs, Sérsic) as mixtures of Gaussians, the method leverages the Fourier transform’s efficiency for high-frequency components and real-space convolution for low-frequency components, avoiding aliasing and super-resolution costs while maintaining exactness within the mixture approximation.

ABSTRACT

I present a method for the fast convolution of a model galaxy profile by a point-spread function (PSF) model represented as a pixel grid. The method relies upon three observations: First, most simple radial galaxy profiles of common interest (deVaucouleurs, exponential, Sersic) can be approximated as mixtures of Gaussians. Second, the Fourier transform of a Gaussian is a Gaussian, thus the Fourier transform of a mixture-of-Gausssian approximation of a galaxy can be directly evaluated as a mixture of Gaussians in Fourier space. Third, if a mixture component would result in Fourier-space aliasing, that component can be evaluated in real space. For mixture components to be evaluated in Fourier space, we use the FFT for the PSF model, direct evaluation of the Fourier transform for the galaxy, and the inverse-FFT to return the result to pixel space. For mixture components to be evaluated in real space---which only happens when the mixture components is much larger than the PSF---we use a simple Gaussian approximation of the PSF, perform the convolution analytically, and evaluate in real pixel space. The method is fast and exact (to the extent that the mixture-of-Gaussians approximation of the galaxy profile is exact) as long as the pixelized PSF model is well sampled. This Fourier method can be seen as a way of applying a perfect low-pass filter to the (typically strongly undersampled) galaxy profile before convolution by the PSF, at exactly the Nyquist frequency of the PSF pixel model grid. In this way, it avoids the computational expense of a traditional super-resolution approach. This method allows the efficient use of pixelized PSF models (ie, a PSF represented as a grid of pixel values) in galaxy forward model-fitting approaches such as the Tractor.

Motivation & Objective

• To address the computational bottleneck of convolving undersampled galaxy profiles with pixelized PSFs in astronomical image simulations.
• To eliminate the need for super-resolution rendering or expensive look-up tables in Fourier-space galaxy profile convolution.
• To enable exact, fast convolution of galaxy profiles with arbitrary PSF models represented as pixel grids.
• To support scalable forward modeling in large surveys by minimizing computational cost while preserving accuracy.

Proposed method

• Approximate common galaxy surface brightness profiles (e.g., de Vaucouleurs, Sérsic, exponential) as finite mixtures of Gaussians.
• Apply the Fourier transform to the mixture: since the Fourier transform of a Gaussian is a Gaussian, the galaxy’s Fourier transform becomes a mixture of Gaussians in Fourier space.
• For mixture components that would cause aliasing when transformed via FFT, evaluate their convolution directly in real space using analytical Gaussian convolution.
• Use the FFT to compute the PSF’s Fourier transform, multiply in Fourier space with the galaxy’s Fourier transform, and apply the inverse FFT to return to pixel space.
• For components with large spatial extent (low frequency), use direct real-space convolution with a Gaussian-approximated PSF to avoid high-frequency aliasing.
• Combine results from Fourier and real-space components to produce a final, aliased-free, and exact convolution result within the mixture approximation.

Experimental results

Research questions

• RQ1 Can a hybrid approach combining Fourier and real-space convolution achieve both speed and accuracy in galaxy-PSF convolution?
• RQ2 To what extent can standard galaxy profiles be accurately approximated as mixtures of Gaussians for computational purposes?
• RQ3 How can aliasing from undersampled galaxy profiles be avoided without resorting to super-resolution rendering?
• RQ4 Can the method be applied efficiently to large-scale survey data with diverse PSF models?
• RQ5 Does the method eliminate the need for pre-computed look-up tables for Sérsic profiles in Fourier-space convolution?

Key findings

• The method achieves exact convolution (within the mixture-of-Gaussians approximation) and avoids aliasing by applying a perfect low-pass filter at the Nyquist frequency of the PSF grid.
• The approach eliminates the need for super-resolution rendering, significantly reducing computational cost compared to traditional methods.
• The method enables efficient forward modeling of general Sérsic profiles at scale, with interpolation across Sérsic indices avoiding the need for per-index look-up tables.
• The technique has been successfully deployed in the DESI Legacy Imaging Surveys, processing over two billion sources across 20,000 square degrees of imaging data.
• The method supports fast evaluation of the exponential and Sérsic profiles in Fourier space using analytical expressions, with performance accelerated by fast exp approximations.
• The implementation in the Tractor code demonstrates that the hybrid approach maintains high accuracy while reducing computational overhead in large-scale photometric surveys.