Sankhya: The Indian Journal of Statistics

2001, Volume 63, Series B, Pt. 2, pp. 181--198

A WAVELET MIXTURE APPROACH TO THE ESTIMATION OF IMAGE DEFORMATION FUNCTIONS

By

T.R. DOWNIE, *University College London, London, U.K.*

and

B.W. SILVERMAN, *University of Bristol, Bristol, U.K.*

SUMMARY. Deriving a function that maps one image on to another similar image, or template, is a useful method in statistical image analysis. Such deformation functions can be modelled in terms of a suitable basis expansion. In a wavelet basis, it is reasonable to assume that most coefficients are zero, and this leads to a model where each wavelet coefficient has a mixture distribution with an atom of probability at zero. The model is used within a penalized least squares framework to fit a deformation to the image. The resulting deformation is explored by various visualization methods. The multiresolution property of the wavelet decomposition yields summary statistics of the deformation in terms of its energy at different scales and in different parts of the image. Inferences are then made on samples where each data point is itself an image. The methods are developed and illustrated in the context of a set of images of human femora, obtained from a palaeopathological study of osteoarthritis of the knee.

*AMS (1991) subject classification.* Primary 62H35; secondary 42C40.

*Key words and phrases. *Arthritis,
deformed template,
image analysis,
iterated conditional mode,
mixture distribution,
palaeopathology,
penalized least squares,
shape statistics,
wavelets.