Hyperfast general-purpose 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, α) prior to adding a projection from angle α to a voxel at position x, y, z. (The only exception to this rule is when only data from within the Tam window shall be backprojected which then implies a low dose usage due to unused projection data.) 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 10⁹ to 10¹¹ 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 and 64 bit operating systems to store large amounts of precomputed weights, even above the 4 GB limit. Our new backprojection algorithm achieves up to 17 giga voxel updates per second (GUPS) on a standard four-socked hexa core CPU (Intel Xeon 7300 platform, 2.66 GHz, Intel Corporation). This equals the reconstruction of 344 images per second assuming each slice consists of 512×512 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. Our backprojection algorithm can be used to speed up any type of image reconstruction algorithm (approximate or exact analytical algorithms and iterative algorithms) and th- - erefore is a versatile and valuable component of future image reconstruction pipelines.