  This paper considers the problem of estimating a mean pattern in the setting
of Grenander's pattern theory. Shape variability in a data set of curves or
images is modeled by the random action of elements in a compact Lie group on an
infinite dimensional space. In the case of observations contaminated by an
additive Gaussian white noise, it is shown that estimating a reference template
in the setting of Grenander's pattern theory falls into the category of
deconvolution problems over Lie groups. To obtain this result, we build an
estimator of a mean pattern by using Fourier deconvolution and harmonic
analysis on compact Lie groups. In an asymptotic setting where the number of
observed curves or images tends to infinity, we derive upper and lower bounds
for the minimax quadratic risk over Sobolev balls. This rate depends on the
smoothness of the density of the random Lie group elements representing shape
variability in the data, which makes a connection between estimating a mean
pattern and standard deconvolution problems in nonparametric statistics.
