  This paper considers the problem of adaptive estimation of a mean pattern in
a randomly shifted curve model. We show that this problem can be transformed
into a linear inverse problem, where the density of the random shifts plays the
role of a convolution operator. An adaptive estimator of the mean pattern,
based on wavelet thresholding is proposed. We study its consistency for the
quadratic risk as the number of observed curves tends to infinity, and this
estimator is shown to achieve a near-minimax rate of convergence over a large
class of Besov balls. This rate depends both on the smoothness of the common
shape of the curves and on the decay of the Fourier coefficients of the density
of the random shifts. Hence, this paper makes a connection between mean pattern
estimation and the statistical analysis of linear inverse problems, which is a
new point of view on curve registration and image warping problems. We also
provide a new method to estimate the unknown random shifts between curves. Some
numerical experiments are given to illustrate the performances of our approach
and to compare them with another algorithm existing in the literature.
