  We present an image-based method for comparing the structural properties of
galaxies produced in hydrodynamical simulations to real galaxies in the Sloan
Digital Sky Survey. The key feature of our work is the introduction of
extensive observational realism, such as object crowding, noise and viewing
angle, to the synthetic images of simulated galaxies, so that they can be
fairly compared to real galaxy catalogs. We apply our methodology to the
dust-free synthetic image catalog of galaxies from the Illustris simulation at
$z=0$, which are then fit with bulge+disc models to obtain morphological
parameters. In this first paper in a series, we detail our methods, quantify
observational biases, and present publicly available bulge+disc decomposition
catalogs. We find that our bulge+disc decompositions are largely robust to the
observational biases that affect decompositions of real galaxies. However, we
identify a significant population of galaxies (roughly 30\% of the full sample)
in Illustris that are prone to internal segmentation, leading to systematically
reduced flux estimates by up to a factor of 6, smaller half-light radii by up
to a factor of $\sim$ 2, and generally erroneous bulge-to-total fractions of
(B/T)=0.
