The unified model of active galactic nuclei (AGNs) has provided a successful explanation for the observed diversity of AGNs in the local universe. However, recent analysis of multi-wavelength spectral and image data suggests that the unified model is only a partial theory of AGNs, and may need to be augmented to remain consistent with all observations. Recent studies using high spatial resolution ground-and space-based observations of local AGNs show that Seyfert class and the "core" (r less than or similar to 1 kpc) host-galaxy morphology are correlated. Currently, this relationship has only been established qualitatively, by visual inspection of the core morphologies of low-redshift (z < 0.035) Seyfert host galaxies. We re-establish this empirical relationship in Hubble Space Telescope optical imaging by visual inspection of a catalog of 85 local (D < 63 Mpc) Seyfert galaxies. We also attempt to re-establish the core morphology-Seyfert class relationship using an automated, non-parametric technique that combines both existing classification parameter methods (the adapted CAS and G-M-20) and a new method which implements the Source Extractor software for feature detection in unsharp-mask images. This new method is designed explicitly to detect dust features in the images. We use our automated approach to classify the morphology of the AGN cores and determine that Sy2 galaxies visually appear, on average, to have more dust features than Sy1. With the exception of this "dustiness" however, we do not measure a strong correlation between the dust morphology and the Seyfert class of the host galaxy using quantitative techniques. We discuss the implications of these results in the context of the unified model.