This talk introduces the concept and applications of consistency conditions in inverse problems with redundant data. We consider inverse problems modelled by some linear operator or matrix A, such as the Radon transform in tomography. The inverse problem consists in solving an equation Ax=y for x given measured data y. In many applications this equation admits a solution only if the data satisfy a set of equations denoted C(y)=0, referred to as the consistency conditions. If the data are noise free and the operator A accurately models the imaging system, the data are by definition consistent because the “exact” object x satisfies Ax=y, and in that case C(y)=0. In practice however the consistency conditions are not satisfied; they can then be used to estimate some vector of parameters p of the imaging system (typically calibration parameters) by solving C(p, y)=0 for p, where C(y, p) denotes the consistency condition corresponding to the parameter p. After a general introduction to the concept, we will review a variety of examples pertaining to 2D and 3D tomography. Applications will be briefly described.
Prof. Michel Defrise received the Ph.D. degree in theoretical physics from the University of Brussels in 1981. He was a visiting professor in the Department of Radiology of the University of Geneva in 1992 and 1993. He is currently a research professor in the Department of Nuclear Medicine at the VUB University Hospital in Brussels. He has participated actively in the advancement of 3-D PET methodology. His current research interests include 3-D image reconstruction in nuclear medicine (PET and SPECT) and in CT.