A new representation and projection model for tomography, based on separable B-splines

Data modelization in tomography is a key point for iterative reconstruction. The design of the projector, i.e. the numerical model of projection, is mostly influenced by the representation of the object of interest, decomposed on a discrete basis of functions. Standard projector models are voxel or ray driven; more advanced models such as distance driven, use simple staircase voxels, giving rise to modelization errors due to their anisotropic behaviour. Moreover approximations made at the projection step amplify these errors. Though a more accurate projection could reduce approximation errors, characteristic functions of staircase voxels constitute a too coarse basis for representing a continuous function. As a result, pure modelization errors still hold. Spherically symmetric volume elements (blobs) have already been studied to eradicate such errors, but at the cost of increased complexity, because they require some tuning parameters for adapting them to this use. We propose to use 3D B-splines, which are piecewise polynomials, as basis functions. When the degree of these polynomials is sufficiently high, they are very close from being with a spherical symmetry, i.e. blobs, avoiding projection inconsistencies, while keeping local influence and separability property. B-splines are considered, in sampling theory, as the almost optimal functions for the discretization of a continuous signal, not necessarily band- limited, potentially allowing to reduce the angular sampling of the data without any loss of quality. We show that the projection of B-splines can be approximated rather accurately by a separable function, independent from the angle of projection, easier to integrate on detector pixels. The higher the degree of the used B-splines, the better the quality of the approximation, but also the larger the number of required operations. Thanks to these approximations, a convenient trade- off between the need of accuracy and a fast calculation can be obtained. This h- s resulted in the implementation of a more accurate numerical projector, which can deal with a reduced angular sampling without loss of performance. The additional computation cost is also efficiently reduced. We have studied the quality of enhancement involved by this projector on 2D iterative reconstructions of a Shepp-Logan phantom, from a small number of fan beam projections. Reconstructions have been performed by optimization methods, minimizing the squared data residuals with a regularization term, using an efficient Quasi- Newton optimization algorithm.