We present two new non-parametric methods for quantifying galaxy morphology: the relative distribution of the galaxy pixel flux values (the Gini coefficient or G) and the second-order moment of the brightest 20% of the galaxys flux (M20). We test the robustness of G and M20 to decreasing signal-to-noise and spatial resolution, and find that both measures are reliable to within 10% at average signal-to-noise per pixel greater than 3 and resolutions better than 1000 pc and 500 pc, respectively. We have measured G and M20, as well as concentration (C), asymmetry (A), and clumpiness (S) in the rest-frame near-ultraviolet/optical wavelengths for 150 bright local normal Hubble type galaxies (E-Sd) galaxies and 104 0.05 < z < 0.25 ultra-luminous infrared galaxies (ULIRGs).We find that most local galaxies follow a tight sequence in G-M20-C, where early-types have high G and C and low M20 and late-type spirals have lower G and C and higher M20. The majority of ULIRGs lie above the normal galaxy G-M20 sequence, due to their high G and M20 values. Their high Gini coefficients arise from very bright nuclei, while the high second-order moments are produced by multiple nuclei and bright tidal tails. All of these features are signatures of recent and on-going mergers and interactions. We also find that in combination with A and S, G is more effective than C at distinguishing ULIRGs from the normal Hubble-types. Finally, we measure the morphologies of 45 1.7 < z < 3.8 galaxies from HST NICMOS observations of the Hubble Deep Field North. We find that many of the z $sim$ 2 galaxies possess G and A higher than expected from degraded images of local elliptical and spiral galaxies, and have morphologies more like low-redshift single nucleus ULIRGs.
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