In this paper, an approach for fault detection and isolation filter design based on the reconstruction of the fault modes is presented. Input reconstruction and a very closely related concept, the system inversion is addressed to the fault detection and isolation problem. Moreover, the basic concept of that field, the fault detectability, output separability, and the fault detection and isolation, that is, fault diagnosis are also defined in terms of the system inversion. System invertibility can also be characterized as maximal rank condition which allows to prove the existence of FDI filters under minimal possible conditions.