Using the Notion of Copula in Tomography

Abstract

In 1917 Johann Radon introduced the Radon transform which is used in 1963 by A. M. Cormack for application in the context of tomographic image reconstruction. He proposed to reconstruct the spatial variation of the material density of the body from X-Ray images (radiographies) for different directions. Independently G. N. Hounsfield derived an algorithm and built the first medical CT scanner. Basically the idea of the X-ray CT is to get an image of the interior structure of an object by X-raying the object from many different directions. The mathematical problem is then estimating a multivariate function from its line integrals. Four year before Cormack's idea, Abe Sklar introduced a theory in the context of Statistics called copula. Shortly copulas are functions that link multivariate distributions to theirs univariate marginal functions. It appeared that copulas captivated all dependence structure concerning the marginal functions and offer a wide range of parametric family model which could be used as a model for the joint distribution function. This statistical problem is the same as in Tomography, because a marginal density is obtained from a line integral of its joint distribution. In the particular case of only given horizontal and vertical projections corresponding to a given two marginal functions, we link the theory of copula to tomography via the Radon transform and Sklar's theorem. The result we propose seems to be new as mathematical approach to solve this tomographic inverse problem.