Image reconstruction using a generalized natural pixel basis

The solution q of the imaging equation Mq=FGq=p (F is the projector and G is a generalized backprojector) is determined using least squares, thus various basis functions can be used as an expansion for the reconstructed image. Here, a generalized natural pixel basis is chosen to allow flexibility in formulating the vector space for the solution q. The singular value decomposition (SVD) method is used to solve for q, and the final image is obtained by backprojecting q: ρ=Gq, and sampling ρ at a discrete array of points. Truncated parallel and non-truncated fan beam projection measurements were used to demonstrate that the solution q to Mq=FGq=p can be defined wherein, for example, if F is a fan beam projection operator, G can be a parallel backprojection operator defined base upon natural pixels. It is demonstrated that different backprojection geometries can give almost equivalent reconstructions of non-truncated projections. For truncated projections the estimation of q that covers the entire projection of the object is effective in reducing ring artifacts; however, using more projection bins is much more effective in preserving the resolution than is increasing the projection bin width. Also, a generalized natural pixel basis better models the geometric response of a collimator used in SPECT, therefore reconstructions of fan beam projections using generalized natural pixels are shown to have better resolution than those that use the filtered backprojection algorithm