  Neutral hydrogen (HI) emission exhibits complex morphology that encodes rich
information about the physics of the interstellar medium (ISM). We apply the
scattering transform (ST) to characterize HI emission structure via a set of
compact and interpretable coefficients, and find a connection between HI
emission morphology and HI cold neutral medium (CNM) phase content. Where HI
absorption measurements are unavailable, the HI phase structure is typically
estimated from the emission via spectral line decomposition. Here we present
the first probe of CNM content using measures solely derived from HI emission
spatial information. We apply the scattering transform to GALFA-HI data at high
Galactic latitudes (|b|>30 deg), and compare the resulting coefficients to CNM
fraction measurements derived from archival HI emission and absorption spectra.
We quantify the correlation between the ST coefficients and measured CNM
fraction (fCNM), and find that HI emission morphology encodes substantial
fCNM-correlating information, and that ST-based metrics for small-scale
linearity are particularly predictive of fCNM. This is further corroborated by
the enhancement of $I_{857}/N_{HI}$ ratio with larger ST measures of
small-scale linearity. These results are consistent with the picture that
regions with higher CNM content are more populated with small-scale filamentary
HI structures. Our work illustrates a physical connection between HI morphology
and phase content, and suggests that future phase decomposition methods can be
improved by making use of both HI spectral and spatial information.
