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Modelling stochastic changes in curve shape, with an application to cancer diagnostics

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Asger Hobolth  and
Eva B. Vedel Jensen
Asger Hobolth*
Affiliation:
University of Aarhus
Eva B. Vedel Jensen*
Affiliation:
University of Aarhus
*
∗ Postal address: Laboratory for Computational Stochastics, Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark.
∗ Postal address: Laboratory for Computational Stochastics, Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark.
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Abstract

Often, the statistical analysis of the shape of a random planar curve is based on a model for a polygonal approximation to the curve. In the present paper, we instead describe the curve as a continuous stochastic deformation of a template curve. The advantage of this continuous approach is that the parameters in the model do not relate to a particular polygonal approximation. A somewhat similar approach has been used in Kent et. al. (1996), who describe the limiting behaviour of a model with a first-order Markov property as the landmarks on the curve become closely spaced; see also Grenander (1993). The model studied in the present paper is an extension of this model. Our model possesses a second-order Markov property. Its geometrical characteristics are studied in some detail and an explicit expression for the covariance function is derived. The model is applied to the boundaries of profiles of cell nuclei from a benign tumour and a malignant tumour. It turns out that the model with the second-order Markov property is the most appropriate, and that it is indeed possible to distinguish between the two samples.

Keywords

Cancer diagnosticscyclic stationaritydeformable templatesfeatureless objectsMarkov random fieldshapestochastic geometry

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 2 , June 2000 , pp. 344 - 362
Copyright
Copyright © Applied Probability Trust 2000 

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References

Anderson, T. W. (1958). The Statistical Analysis of Time Series. John Wiley, New York.Google Scholar
Cramer, H. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. John Wiley, New York.Google Scholar
Grenander, U. (1993). General Pattern Theory. Oxford University Press, Oxford.Google Scholar
Grenander, U. and Manbeck, K. M. (1993). A stochastic model for defect detection in potatoes. J. Comp. Graph. Statist. 2, 131–151.Google Scholar
Grenander, U. and Miller, M. I. (1994). Representations of knowledge in complex systems (with discussion). J. R. Statist. Soc. B 56, 549–603.Google Scholar
Hirsch, M. W. and Smale, S. (1974). Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, San Diego.Google Scholar
Hobolth, A. and Pedersen, J. (1999). Estimation in a class of circular Gaussian processes. Res. Rep. 8, Laboratory for Computational Stochastics, Institute of Mathematical Sciences, University of Aarhus. In preparation.Google Scholar
Hurn, M. A., Steinsland, I. and Rue, H. (1999). Parameter estimation for a deformable template model. Statist. Res. Rept. 99.02, University of Bath, UK.Google Scholar
Jensen, E. B. V. (1998). Local Stereology. World Scientific, Singapore.CrossRefGoogle Scholar
Jensen, E. B. V. and Sørensen, F. B. (1991). A note on stereological estimation of the volume-weighted second moment of particle volume. J. Microsc. 164, 21–27.CrossRefGoogle ScholarPubMed
Kass, M., Witkin, A. and Terzopoulos, D. (1988). Snakes: active contour models. Int. J. Computer Vision 1, 321–331.CrossRefGoogle Scholar
Kent, J. T., Mardia, K. V. and Walder, A. N. (1996). Conditional cyclic Markov random fields. Adv. Appl. Prob. 28, 1–12.CrossRefGoogle Scholar
Kent, J. T., Dryden, I. L. and Anderson, C. R. (1998). Using circulant symmetry to model featureless objects. Tech. Rep. STAT98/14, Department of Statistics, University of Leeds, UK.Google Scholar
Lauritzen, S. L. (1996). Graphical Models. Oxford University Press, Oxford.CrossRefGoogle Scholar
Rue, H. and Hurn, M. A. (1999). Bayesian object identification. Biometrika 86, 649–660. Statistics 6, Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway.Google Scholar
Rue, H. and Husby, O. K. (1998). Identification of partly destroyed objects using deformable templates. Statist. and Computing 8, 221–228.CrossRefGoogle Scholar
Rue, H. and Syversveen, A. R. (1998). Bayesian object recognition with Baddeley's delta loss. Adv. Appl. Prob. 30, 64–84.CrossRefGoogle Scholar
Sørensen, F. B. (1991). Stereological estimation of mean and variance of nuclear volume from vertical sections. J. Microsc. 162, 203–229.CrossRefGoogle ScholarPubMed