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 infra-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 pre-integration, 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.