Analysis of the ECGI inverse problem solution with respect to the measurement boundary size and the distribution of noise

In this work, we analyze the influence of adding a body surface missing data on the solution of the electrocardiographic imaging inverse problem. The difficulty comes from the fact that the measured Cauchy data is provided only on a part of the body surface and thus a missing data boundary is adjacent to a measured boundary. In order to construct the electrical potential on the heart surface, we use an optimal control approach where the unknown potential at the external boundary is also part of the control variables. We theoretically compare this case to the case where the Dirichlet boundary condition is given on the full accessible surface. We then compare both cases and based on the distribution of noise in the measurements, we conclude whether or not it is worth to use all the data. We use the method of factorization of elliptic boundary value problems combined with the finite element method. We illustrate the theoretical results by some numerical simulations in a cylindrical domain. We numerically study the effect of the size of the missing data zone on the accuracy of the inverse solution.