  Angular and spectral differential imaging is an observational technique of
choice to investigate the immediate vicinity of stars. The relative angular
motion and spectral scaling between on-axis and off-axis sources are exploited
by post-processing techniques to separate two components: the residual star
light and the light coming from surrounding objects such as circumstellar disks
or point-like objects. This paper introduces a new algorithm to jointly unmix
these two components and deconvolve disk images. The proposed algorithm is
based on a statistical modeling of the residual star light, accounting for its
spatial and spectral correlations. While critical, these correlations are not
modeled and compensated for by existing reconstruction algorithms. We leverage
dedicated shrinkage techniques to estimate the large number of parameters of
our model of correlations in a data-driven fashion. We show that the resulting
separable model of the spatial and spectral covariances captures very
accurately the star light, thus allowing its efficient suppression. We apply
our method on several datasets from the VLT/SPHERE instrument and compare the
performances against standard algorithms of the field (cADI, PCA). We show that
accounting for the multiple correlations of the data significantly enhances the
reconstruction quality, leading to better preservation of both the disk
morphology and photometry. Thanks to its unique joint spectral modeling of the
data, the proposed algorithm can reconstruct disks having a circular symmetry
(e.g., rings, spirals) at intensities one million times fainter than the star,
without resorting to additional reference datasets free from the off-axis
objects.
