  We describe application of the `shapelet' linear decomposition of galaxy
images to morphological classification using images of $\sim$ 3000 galaxies
from the Sloan Digital Sky Survey. After decomposing the galaxies we perform a
principal component analysis to reduce the number of dimensions of the shapelet
space to nine. We find that each of these nine principal components contains
unique morphological information, and give a description of each principal
component's contribution to a galaxy's morphology. We find that galaxies of
differing Hubble type separate cleanly in the shapelet space. We apply a
Gaussian mixture model to the 9-dimensional space spanned by the principal
components and use the results as a basis for classification. Using the mixture
model, we separate galaxies into seven classes and give a description of each
class's physical and morphological properties. We find that several of the
mixture model classes correlate well with the traditional Hubble types both in
their morphology and their physical parameters (e.g., color, velocity
dispersions, etc.). In addition, we find an additional class of late-type
morphology but with high velocity dispersions and very blue color; most of
these galaxies exhibit post-starburst activity. This method provides an
objective and quantitative alternative to traditional and subjective visual
classification.
