Abstract
The stochastic structure of images, especially individual medical images as they are reconstructed nowadays from arrays of medical imaging sensors, is becoming steadily better understood. Less attention has been paid to the parallel notion of estimation error for the deformations that convey relations among these images, such as localized abnormality or growth prediction. The dominant current formalism for the biostatistics of deformations deals solely with the shape of a set of landmarks parameterizing the deformation, not otherwise with its behaviour inbetween the landmarks.
This paper attempts to fit a rigorous stochastic model for a deformation between landmarks and to assess the error of the fitted deformation. The relation between two images is modelled as a stochastic deformation, i.e. as an identity map plus a stochastic process whose value at every point is a vector-valued displacement.
There are two common strategies for fitting deformations given information at a set of landmarks. One involves minimizing a roughness penalty, e.g. for a thin-plate spline, and another involves prediction for a stochastic process, e.g. for a self-similar intrinsic random field. The stochastic approach allows parameter estimation and confidence limits for the predicted deformation. An application is presented from a study of breast images and how they deform as a function of the imaging procedure.
Similar content being viewed by others
References
F.L. Bookstein, “Principal warps: thin-plate splines and the decomposition of deformations,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 11, pp. 567–585, 1989.
F.L. Bookstein, Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge University Press, Cambridge, 1991.
J.-P. Chiles and P. Delfiner, Geostatistics: Modeling Spatial Uncertainty. Wiley, New York, 1999.
N. Cressie, Statistics for Spatial Data, Wiley, New York (2nd ed., 1993) 1991.
I.L. Dryden and K.V. Mardia, Statistical Shape Analysis. Wiley, Chichester, 1998.
C.A. Glasbey and K.V. Mardia, “A penalized likelihood approach to image warping (with discussion)”, J. Roy. Statistic. Soc., Vol. B 63, pp. 465–514, 2001.
J.V. Hajnal, D.L.G. Hill, and D.J. Hawkes (Eds), Medical Image Registration. CRC Press, London, 2001.
S.C. Joshi and M.M. Miller, “Landmark matching via large deformation diffeomorphisms”, IEEE Trans. Image Processing, Vol. 9, pp.1357–1370, 2000.
J.T. Kent, and K.V. Mardia, “The link between kriging and thin-plate splines”, In F.P. Kelly editor, Probability, Statistics and Optimization: a Tribute to Peter Whittle, Wiley, Chichester, pp. 325–339, 1994.
J.F. Krücker, G.L. LeCarpentier, J.B. Fowlkes, and P.L. Carson, “Rapid elastic image registration for 3-D ultrasound”, IEEE Trans. Medical Imaging, Vol. 21, pp. 1384–1394, 2002.
B.B. Mandelbrot, The Fractal Geometry of Nature. Freeman, San Francisco, 1982.
K.V. Mardia, J.T. Kent, and J.M. Bibby, Multivariate Analysis, Academic Press, London, 1979.
K.V. Mardia, and R.J. Marshall, “Maximum likelihood estimation of models for residual covariance in spatial regression”, Biometrika, Vol. 71, pp. 135–146, 1984.
K.V. Mardia, J.T. Kent, C.R. Goodall, and J.A. Little, “Kriging and splines with derivative informations”, Biometrika, Vol. 83, pp. 207–221, 1996.
S. Marsland, and C.J. Twining, “Constructing diffeomorphic representations for the groupwise analysis of non rigid registrations of medical images”, IEEE Trans. Medical Imaging, Vol. 23, pp. 1006–020, 2004.
G. Matheron, “The intrinsic random functions and their applications”, Advances in Applied Probability, Vol. 5, pp. 439–468, 1973.
C.R. Meyer, J.L. Boes, B. Kim, P. Bland, K.R. Zasandy, P.V. Ksion, K. Koral, K.A. Frey, and R.L. Wahl, “Demonstration of accuracy and clinical versality of mutual information for automatic multimodality image fusion using affine and thin-plate spline warped geometric deformations”, Medical Image Analysis, Vol. 1, pp. 195–206, 1997.
A.J. Singh, D. Goldgof, and D. Terzopoulos (Eds), Deformable Models in Medical Image Analysis. IEEE Computer Soc., Los Alamitos, California, 1998.
A.W. Toga, Brain Warping. Academic Press, San Diego, 1999.
G. Wahba, Spline Models for Observational Data, SIAM, Philadelphia, 1990.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mardia, K.V., Bookstein, F.L., Kent, J.T. et al. Intrinsic Random Fields and Image Deformations. J Math Imaging Vis 26, 59–71 (2006). https://doi.org/10.1007/s10851-006-7802-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-006-7802-5