An Interpretation of the Linear Back-Projection Algorithm Used in Capacitance Tomography

J C Gamio and C Ortiz-Aleman

Instituto Mexicano del Petroleo, Eje Central L Cardenas 152, Mexico, DF, 07730, Mexico jgamio@imp.mx

ABSTRACT

The linear back-projection (LBP) algorithm was one of the first methods employed to perform image reconstruction in electrical capacitance tomography. Despite its relatively poor accuracy, it is a simple and fast procedure capable of real-time operation in many applications, and it has remained a popular choice. However, since it was first reported it has lacked a clear formal mathematical support. Its only justification has been that it was an adaptation of a method normally used in linear X-ray medical tomography, and the fact that it actually does produce useful (albeit only 'qualitative') images. In this paper, one way of interpreting LBP is presented. It is shown how LBP is actually based on the linearisation of a normalised form of the problem. More specifically, the normalised forward problem is approximated by means of a series of hyper-planes. The reconstruction matrix used in LBP is found to be a 'weighted' transpose of the linear operator (matrix) that defines the linearised-normalised forward problem. The rows of this latter matrix contain the information of the sensitivity maps used in LBP.

Keywords Capacitance tomography, Linear back-projection, Image reconstruction