  Observational data from astronomical imaging surveys contain information
about a variety of source populations and environments, and its complexity will
increase substantially as telescopes become more sensitive. Even for existing
observations, measuring the correlations between point-like and diffuse
emission can be crucial to correctly inferring the properties of any individual
component. For this task information is typically lost, either because of
conservative data cuts, aggressive filtering or incomplete treatment of
contaminated data. We present the code PCAT-DE, an extension of probabilistic
cataloging designed to simultaneously model point-like and diffuse signals.
This work incorporates both explicit spatial templates and a set of
non-parametric Fourier component templates into a forward model of astronomical
images, reducing the number of processing steps applied to the observed data.
Using synthetic Herschel-SPIRE multiband observations, we demonstrate that
point source and diffuse emission can be reliably separated and measured. We
present two applications of this model. For the first, we perform point source
detection/photometry in the presence of galactic cirrus and demonstrate that
cosmic infrared background (CIB) galaxy counts can be recovered in cases of
significant contamination. In the second we show that the spatially extended
thermal Sunyaev-Zel'dovich (tSZ) effect signal can be reliably measured even
when it is subdominant to the point-like emission from individual galaxies.
