  (ABRIDGED) Motivated by the desire to reliably and automatically classify
structure of thousands of COSMOS galaxies, we present ZEST, the Zurich
Estimator of Structural Types. To classify galaxy structure, ZEST uses: (i)
Five non-parametric diagnostics: asymmetry, concentration, Gini coefficient,
2nd-order moment of the brightest 20% of galaxy pixels, and ellipticity; and
(ii) The exponent n of single--Sersic fits to the 2D surface brightness
distributions. To fully exploit the wealth of information while reducing the
redundancy present in these diagnostics, ZEST performs a principal component
(PC) Analysis. We use a sample of ~56,000 I<24 COSMOS galaxies to show that the
first three PCs fully describe the key aspects of the galaxy structure, i.e.,
to calibrate a three-dimensional classification grid of axis PC_1, PC_2, and
PC_3. We demonstrate the robustness of the ZEST grid on the z=0 sample of Frei
et al. (1996). The ZEST classification breaks most of the degeneracy between
different galaxy populations that affects morphological classifications based
on only some of the diagnostics included in ZEST. As a first application, we
present the evolution since z~1 of the Luminosity Functions of COSMOS galaxies
of early, disk and irregular galaxies and, for disk galaxies, of different
bulge-to-disk ratios. Overall, we find that the LF up to a redshift z=1 is
consistent with a pure-luminosity evolution (of about 0.95 magnitudes at z
\~0.7). We highlight however two trends, that are in general agreement with a
down-sizing scenario for galaxy formation: (1.) A deficit of a factor of about
two at z~0.7 of MB>-20.5 structurally--classified early--type galaxies; and
(2.) An excess of a factor of about three, at a similar redshift, of irregular
galaxies.
