Towards a complete description of three-dimensional filtered backprojection

Filtered backprojection (FBP) is an algorithm to perform image reconstruction from x-ray (line integral) projection data. In the 3D problem, the filter function is not unique due to redundancy of the projection data. Valid filters are those satisfying a general filter equation. FBP is a linear, shift invariant (LSI) algorithm irrespective of filter applied. Conversely, any reconstruction algorithm that is LSI is equivalent to FBP with a particular filter. An LSI algorithm is presented, and the equivalent FBP formulation derived. Noise properties of LSI algorithms are completely determined by the filter in its equivalent FBP representation. The complete classification of FBP filters is still an open problem. Null filters are introduced as an approach to solving the general filter equation. Null filters are solutions to the homogeneous form of the general equation, and reconstructions using null filters result in images consisting entirely of noise. New theoretical results obtained using the null filters approach are presented. (author)