Regularization techniques for PSF-matching kernels - I. Choice of kernel basis

Abstract

We review current methods for building point spread function (PSF)‐matching kernels for the purposes of image subtraction or co‐addition. Such methods use a linear decomposition of the kernel on a series of basis functions. The correct choice of these basis functions is fundamental to the efficiency and effectiveness of the matching – the chosen bases should represent the underlying signal using a reasonably small number of shapes, and/or have a minimum number of user‐adjustable tuning parameters. We examine methods whose bases comprise multiple Gauss–Hermite polynomials, as well as a form‐free basis composed of delta‐functions. Kernels derived from delta‐functions are unsurprisingly shown to be more expressive; they are able to take more general shapes and perform better in situations where sum‐of‐Gaussian methods are known to fail. However, due to its many degrees of freedom (the maximum number allowed by the kernel size) this basis tends to overfit the problem and yields noisy kernels having large variance. We introduce a new technique to regularize these delta‐function kernel solutions, which bridges the gap between the generality of delta‐function kernels and the compactness of sum‐of‐Gaussian kernels. Through this regularization we are able to create general kernel solutions that represent the intrinsic shape of the PSF‐matching kernel with only one degree of freedom, the strength of the regularization λ. The role of λ is effectively to exchange variance in the resulting difference image with variance in the kernel itself. We examine considerations in choosing the value of λ, including statistical risk estimators and the ability of the solution to predict solutions for adjacent areas. Both of these suggest moderate strengths of λ between 0.1 and 1.0, although this optimization is likely data set dependent. This model allows for flexible representations of the convolution kernel that have significant predictive ability and will prove useful in implementing robust image subtraction pipelines that must address hundreds to thousands of images per night.

Spatially unresolved SED fitting can underestimate galaxy masses: a solution to the missing mass problem

“…The broadest PSF belonged to the F160W image, which had a PSF full-width-at-half-maximum (FWHM) of approximately 0.20 arcseconds. For each bandpass other than F160W, we convolved the images to match the broadest PSF by creating an 11 × 11 convolution kernel using the method described in Alard & Lupton (1998), employing a delta function at each pixel in the kernel as the set of basis functions (Becker et al 2012). The size of the kernel provided a good compromise, limiting the tendency of our choice of basis function to over-fit the noise in the wings of the PSF, but was still large enough to capture all the features of the transformation kernel.…”

The Lyman alpha reference sample

“…The final drizzled and registered images in each filter were then convolved with a delta-function-based kernel to match the PSFs (similar to the technique described in Becker et al (2012) using the filter with the widest PSF (F775W) as the reference. This is especially important when comparing the UV filters in ACS/SBC to the optical filters because the SBC PSF profile is significantly different from the UVIS PSF.…”

PyTorchDIA: A flexible, GPU-accelerated numerical approach to Difference Image Analysis

“…As noted by Becker et al (2012), while kernels modeled as a discrete array of pixels are very flexible, the consequent fidelity with the data can result in significant overfitting. To guard against excessively noisy kernels, we provide the option to regularise the loss with the addition of a penalty term.…”

“…This penalty term is derived from an approximation of the second derivative of the kernel surface (Becker et al 2012). Intuitively, it favours compact kernels, where adjacent kernel pixels should not vary too sharply.…”