Cone-beam filtered backprojection image reconstruction using a factorized weighting function

We present several new families of mathematically exact cone-beam image reconstruction algorithms for a general source trajectory that fulfills Tuy's data sufficiency condition. The basic structure of the new algorithms is to reconstruct images via filtered backprojection (FBP) with a 1-D shift-invariant filter. Specifically, the general weighting function w(x,k̂;t) for redundant data was decomposed into three components w₁(x,k̂), w₂(x,t), and sgn[k̂·y′(t)], viz. w(x,k̂;t) =[w₁(x,k̂)w₂(x,t)sgn(k̂-y(t))]. Based upon the normalization condition of the weighting function, the first component w₁ (x, k̂) may be calculated using the second component w₂(x, t) Thus, the design of the weighting function was reduced to the selection of the second component w₂(x, t). Using this scheme, it has been demonstrated that, for a given scanning configuration, one may develop infinitely many different, exact cone-beam FBP image reconstruction algorithms. To demonstrate how this general procedure may be used to develop FBP image reconstruction algorithms, a two-concentric-circle scanning configuration is discussed in detail. Numerical simulations have been conducted to validate several of the derived image reconstruction algorithms. Several possible scan strategies are presented, and the possibility of performing multiple reconstructions with different scan configurations to reduce image noise is described. Noise properties also have been numerically studied for the implemented image reconstruction algorithms, then compared with two other shift-invariant FBP reconstruction algorithms.