Optimal Regularized Reconstruction in Computerized Tomography

We consider regularized least squares solutions for the picture reconstruction problem in computerized tomography. The choice of the regularization parameter ρ is crucial, but the known methods (like cross-Validation) are expensive. This paper shows that the usual rotationally invariant geometry of tomographs greatly reduces their cost. In the case of the L₂-norm taken as the irregularity functional, this invariance was previously used to efficiently compute a solution f_ρ with an a priori fixed value of ρ. We present an algorithm which computes first an optimal value ρ ^ *, then fρ ^ * with only twice (four times in the Poisson noise situation) the number of operations and the same memory space as the computation of one solution with fixed ρ. In the case of any rotationally invariant norm and with a suitable discretization of the picture, we describe a similar algorithm which, after a first step which is roughly the standard back-projection of the data, has the same complexity as the previous one. Numerical applications show the performance of these adaptive reconstructions.