  The shapelets method for image analysis is based upon the decomposition of
localised objects into a series of orthogonal components with convenient
mathematical properties. We extend the "Cartesian shapelet" formalism from
earlier work, and construct "polar shapelet" basis functions that separate an
image into components with explicit rotational symmetries. These frequently
provide a more compact parameterisation, and can be interpreted in an intuitive
way. Image manipulation in shapelet space is simplified by the concise
expressions for linear coordinate transformations; and shape measures
(including object photometry, astrometry and galaxy morphology estimators) take
a naturally elegant form. Particular attention is paid to the analysis of
astronomical survey images, and we test shapelet techniques upon real data from
the Hubble Space Telescope. We present a practical method to automatically
optimise the quality of an arbitrary shapelet decomposition in the presence of
observational noise, pixellisation and a Point-Spread Function. A central
component of this procedure is the adaptive choice of the shapelet expansion's
scale size and truncation order. A complete software package to perform
shapelet image analysis is made available on the world-wide web at
http://www.astro.caltech.edu/~rjm/shapelets/ .
