Algorithm for hyperfast cone-beam spiral backprojection.

Cone-beam spiral backprojection is computationally highly demanding. At first sight, the backprojection requirements are similar to those of cone-beam backprojection from circular scans such as it is performed in the widely used Feldkamp algorithm. However, there is an additional complication: the illumination of each voxel, i.e. the range of angles the voxel is seen by the X-ray cone is a complex function of the voxel position. The weight function has no analytically closed form and must be numerically determined. Storage of the weights is prohibitive since the amount of memory required equals the number of voxels per spiral rotation times the number of projections a voxel receives contributions and therefore is in the order of 10(9) to 10(11) floating point values for typical spiral scans. We propose a new algorithm that combines the spiral symmetry with the ability of today's 64 bit CPUs to store large amounts of precomputed weights. Using the spiral symmetry in this way allows to exploit data-level parallelism and thereby to achieve a very high level of vectorization. An additional postprocessing step rotates these slices back to normal images. Our new backprojection algorithm achieves up to 24.6 Giga voxel updates per second (GUPS) on our systems that are equipped with two standard Intel X5570 quad core CPUs (Intel Xeon 5500 platform, 2.93 GHz, Intel Corporation). This equals the reconstruction of 410 images per second assuming each slice consists of 512 x 512 pixels, receiving contributions from 512 projections.

High performance cone-beam spiral backprojection with voxel-specific weighting.

Cone-beam spiral backprojection is computationally highly demanding. At first sight, the backprojection requirements are similar to those of cone-beam backprojection from circular scans such as it is performed in the widely used Feldkamp algorithm. However, there is an additional complication: the illumination of each voxel, i.e. the range of angles the voxel is seen by the x-ray cone, is a complex function of the voxel position. In general, one needs to multiply a voxel-specific weight w(x, y, z, alpha) prior to adding a projection from angle alpha to a voxel at position x, y, z. Often, the weight function has no analytically closed form and must be numerically determined. Storage of the weights is prohibitive since the amount of memory required equals the number of voxels per spiral rotation times the number of projections a voxel receives contributions and therefore is in the order of up to 10(12) floating point values for typical spiral scans. We propose a new algorithm that combines the spiral symmetry with the ability of today's 64 bit operating systems to store large amounts of precomputed weights, even above the 4 GB limit. Our trick is to backproject into slices that are rotated in the same manner as the spiral trajectory rotates. Using the spiral symmetry in this way allows one to exploit data-level paralellism and thereby to achieve a very high level of vectorization. An additional postprocessing step rotates these slices back to normal images. Our new backprojection algorithm achieves up to 17 giga voxel updates per second on our systems that are equipped with four standard Intel X7460 hexa core CPUs (Intel Xeon 7300 platform, 2.66 GHz, Intel Corporation). This equals the reconstruction of 344 images per second assuming that each slice consists of 512 x 512 pixels and receives contributions from 512 projections. Thereby, it is an order of magnitude faster than a highly optimized code that does not make use of the spiral symmetry. In its present version, the spiral backprojection algorithm is pixel-driven. A ray-driven version and a corresponding high performance forward projector can be easily designed. Thus, our findings can be used to speed up any type of image reconstruction algorithm (approximate or exact analytical algorithms and iterative algorithms) and therefore yield a versatile and valuable component of future image reconstruction pipelines.