In this work, we provide a new mathematical formulation for the inverse problem in electrocardiography imaging. The novelty of this approach is that we take into account the missing measurements on the body surface. The electrocardiography imaging is formulated as a data completion problem for the Laplace equation. The Neumann boundary condition is given at the whole body surface. The difficulty comes from the fact that the Dirichlet boundary condition is only given on a part of the body surface. 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 use the method of factorization of elliptic boundary value problems combined with the finite difference 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.