Rendering tetrahedral meshes with higher-order attenuation functions for digital radiograph reconstruction

This paper presents a novel method for computing simulated x-ray images, or DRRs (digitally reconstructed radiographs), of tetrahedral meshes with higher-order attenuation functions. DRRs are commonly used in computer assisted surgery (CAS), with the attenuation function consisting of a voxelized CT study, which is viewed from different directions. Our application of DRRs is in intra-operative "2D-3D" registration, i.e., finding the pose of the CT dataset given a small number of patient radiographs. We register 2D patient images with a statistical tetrahedral model, which encodes the CT intensity numbers as Bernstein polynomials, and includes knowledge about typical shape variation modes. The unstructured grid is more suitable for applying deformations than a rectilinear grid, and the higher-order polynomials provide a better approximation of the actual density than constant or linear models. The intra-operative environment demands a fast method for creating the DRRs, which we present here. We demonstrate this application through the creation and use of a deformable atlas of human pelvis bones. Compared with other works on rendering unstructured grids, the main contributions of this work are: 1) Simple and perspective-correct interpolation of the thickness of a tetrahedral cell. 2) Simple and perspective-correct interpolation of front and back barycentric coordinates with respect to the cell. 3) Computing line integrals of higher-order functions. 4) Capability of applying shape deformations and variations in the attenuation function without significant performance loss. The method does not depend on for preintegration, and does not require depth-sorting of the visualized cells. We present imaging and timing results of implementing the algorithm, and discuss the impact of using higher-order functions on the quality of the result and the performance.