We apply a new approach to quantifying galaxy morphology and identifying galaxy mergers to the rest-frame far-ultraviolet images of 82 z ~ 4 Lyman break galaxies (LBGs) and 55 1.2 < z < 1.8 emission-line galaxies in the GOODS and Ultra Deep Fields. We compare the distributions of the Gini coefficient (G), second-order moment of the brightest 20% of galaxy light (M20), and concentration (C) for high-redshift and low-redshift galaxies with average signal to noise per pixel > 2.5 and Petrosian radii > 0.3 arcsec. Ten of the 82 LBGs have M20 >= -1.1 and possess bright double or multiple nuclei, implying a major-merger fraction of star-forming galaxies ~ 10-25% at M_{FUV} < -20, depending on our incompleteness corrections. Galaxies with bulge-like morphologies (G >= 0.55, M20 < -1.6) make up ~ 30% of the z ~ 4 LBG sample, while the remaining ~ 50% have G and M20 values higher than expected for smooth bulges and disks and may be star-forming disks, minor-mergers or post-mergers. The star-forming z ~ 1.5 galaxy sample has a morphological distribution which is similar to the UDF z ~ 4 LBGs, with an identical fraction of major-merger candidates but fewer spheroids. The observed morphological distributions are roughly consistent with current hierarchical model predictions for the major-merger rates and minor-merger induced starbursts at z ~ 1.5 and ~4. We also examine the rest-frame FUV-NUV and FUV-B colors as a function of morphology and find no strong correlations at either epoch.
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