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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
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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