MATHEMATICAL FOUNDATIONS OF ECG MAP RECONSTRUCTION

Abstract

The purpose of the paper is to provide a detailed account of mathematical methods for determining the system of orthogonal functions which allow a reconstruction of the ECG potential maps by means of a limited number of leads and for reconstructing the maps themselves. In the introduction the problem of the map reconstruction and its motivations are sketched. Section 2 illustrates the concept of canonical decomposition and presents a derivation of the Loeve Karhunen expansion starting from best mean square approximation. The truncation problem, the properties of the eigenvalues of the Fredholm equation and the meaning of the coefficients of the reconstruction are also examined. Section 3 deals with the use of spherical harmonics which is made throughout this paper. The advantages of using such functions are numerous: they also allow the construction of rotation invariant parameters and provide a powerful filtering technique of the data employed for building the eigenfunctions and moreover provide a simple way of solving the integral equations. Section 4 provides a detailed account of the application of the pseudoinverse method to the problem of reconstructing the complete map from a limited number of: leads and illustrates the leading criteria in choosing the lead locations, while Section 5 points out the fact that such a procedure for the reconstruction is equivalent to an orthogonal expansion by means of numerical integration formulas which are explained in Section 6. The equivalence of such approaches does not seem to have been pointed out in the literature. Section 7 is a brief sketch of the practical applications of the mathematics contained in this paper. A comprehensive report of such applications will be the object of a forthcoming paper.