Spline driven: high accuracy projectors for 3D tomographic reconstruction from few projections
Résumé
Tomographic iterative reconstruction methods need a very thorough modeling of data. The core of this issue is the projectors's design, i.e. the numerical model of projection, is mostly influenced by the representation of the object of interest, decomposed on a basis of functions, and on the approximations made for the projection on the detector. Voxel driven and ray driven projection models, widely appreciated for their short execution time, are too coarse. Distance driven model has a better accuracy but also relies on strong approximations to project voxel basis functions. Cubic voxel basis functions are anisotropic, modeling accurately their projection is therefore computationally expensive. Smoother and more isotropic basis functions both better represent continuous functions and provide simpler projectors. This consideration has lead to the development of spherically symmetric volume elements, called blobs. Set apart their isotropy, blobs are often considered too computationally expensive in practice. We propose to use 3D B-splines, which are smooth piecewise polynomials, as basis functions. When the degree of these polynomials increases, their isotropy improves and projections can be computed regardless of their orientation. Thanks to their separability, very efficient algorithms can be used to decompose an image on B-spline basis functions. We approximate the projection of B-spline basis functions with a 2D separable model. The degree and the sampling of the B-splines can be chosen according to a tradeoff between approximation quality and computational complexity. We show on numerical experiments that with our accurate projector, the number of projections can be reduced while preserving a similar reconstruction quality. Used with cubic B-splines, our projector requires just twice as many operations as a model involving voxel basis functions. High accuracy projectors can enhance the resolution of existing systems, or can reduce the number of projections required to reach a given resolution, potentially reducing the dose absorbed by the patient.
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