Cone-beam image reconstruction using spherical harmonics

Image reconstruction. from cone-beam projections is required for both x-ray computed tomography (CT) and single photon emission computed tomography (S PECT), Grangeat's algorithm accurately performs cent-beam reconstruction pr ovided that Tuy's data sufficiency condition is satisfied and projections a re complete. The algorithm consists of three stages: (a) Forming weighted p lane integrals by calculating the line integrals on the cone-beam detector, and obtaining the first derivative of the plane integrals (3D Radon transf orm) by taking the derivative of the weighted plane integrals. (b) Rebinnin g the data and calculating the second derivative with respect to the normal to the plane. (c) Reconstructing the image using the 3D Radon backprojecti on. A new method for implementing the first stage of Grangeat's algorithm w as developed using spherical harmonics. The method assumes that the detecto r is large enough to image the whole object without truncation. Computer si mulations show that if the trajectory of the cone vertex satisfies Tuy's da ta sufficiency condition, the proposed algorithm provides an exact reconstr uction.