A wavelet approach to curve alignment


J\'er\'emie Bigot

Laboratoire IMAG-LMC, Universit\'e Joseph Fourier 
BP 53, 38041 Grenoble Cedex 9, France



When studying some biological or physical process in different subjects, we
usually see that the observed paths have a common structural pattern. An
important matter consists in determining the typical shape of the observed
process or in testing whether there is any statistically significant difference
among two subsets of subjects. The final objective of this work is then to
generalize the concepts of analysis of variance for comparing multiple sets of
curves. \\


The presence of noise makes difficult the identification of the typical features
of a set of curves. Moreover, because of variations in dynamics and intensity
from one curve to another, the cross sectional average is usually not a good
estimator of the typical shape of a curve. Hence, to determine the typical
structure of a set of curves, it is better to find a common referential to
represent them. The approach proposed here, consists in finding, for each
observed curve, a transform in order to synchronize all the curves before
performing the average or any statistics.\\


The matching of two functions can be done by aligning individual locations of
corresponding structural points (landmarks) from one curve to another.  Wavelets
have successfully demonstrated their good localization properties of the
structure of a signal. In particular, it has been shown that one can characterize
the local structure of a signal by following the propagation across scales of the
zero-crossings and the modulus maxima of its continuous wavelet transform. A
nonparametric approach is proposed to estimate the zero-crossings and the wavelet
maxima of a noisy signal at various scales. In order to identify the landmarks of
the unknown signal, we introduce a new tool that computes the ``density'' of the
location of the zeros and the modulus maxima of a wavelet representation along
various scales. The modes of the resulting structural intensities are shown to be
located at the landmarks of the corresponding signal. Combined with bagging this
approach is also shown to be an effective technique for removing spurious
estimates. Finally a new method for aligning two functions based on the
structural intensities is proposed. The asymptotic convergence properties of the
resulting estimators are studied and illustrated by simulations. The usefulness
of the method is documented by applications to some curve registration problems
in the biomedical area. \\


{\bf Keywords :} Curve alignment, landmarks, continuous wavelet transform,
zero-crossings, wavelet maxima, feature extraction, non parametric estimation,
bagging.