A hybrid Fourier--Real Gaussian Mixture method for fast galaxy--PSF convolution

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.