  With the launch of the X-Ray Imaging and Spectroscopy Mission (XRISM) and the
advent of microcalorimeter detectors, X-ray astrophysics is entering in a new
era of spatially resolved high resolution spectroscopy. But while this new
generation of X-ray telescopes have much finer spectral resolutions than their
predecessors (e.g. XMM-Newton, Chandra), they also have coarser spatial
resolutions, leading to problematic cross-pixel contamination. This issue is
currently a critical limitation for the study of extended sources such as
galaxy clusters of supernova remnants. To increase the scientific output of
XRISM's hyperspectral data, we propose to fuse it with XMM-Newton data, and
seek to obtain a cube with the best spatial and spectral resolution of both
generations. This is the aim of hyperspectral fusion. In this article, we have
implemented an algorithm that jointly deconvolves the spatial response of XRISM
and the spectral response of XMM-Newton. To do so, we construct a forward model
adapted for instrumental systematic degradations and Poisson noise, then tackle
hyperspectral fusion as a regularized inverse problem. We test three methods of
regularization: low rank approximation with Sobolev regularization; low rank
approximation with 2D wavelet sparsity ; and 2D-1D wavelet sparsity. We test
our method on toy models constructed from hydrodynamic simulations of supernova
remnants. We find that our method reconstructs the ground truth well even when
the toy model is complex. For the regularization term, we find that while the
low rank approximation worked well as a spectral denoiser in models with less
spectral variability, it introduced a bias in models with more spectral
variability, in which case the 2D-1D wavelet sparsity regularization worked
best. After demonstrating a proof of concept in this article, we aim to apply
this method to real X-ray astrophysical data in the near future.
