Decoupling the equations of regularized tomography

∗ I thank Takeo Kanade for his insightful suggestions (especially on interpolation), and Branko Jaramaz for introducing me to the metal artifact reduction problem. The NSF Advanced Compu-tational Research program and the Heinz Endowment provided funding. Deferring discretization can occasionally change our perspective on imag-ing problems. To illustrate, we offer a reformulation of regularized com-puted tomography (CT) in which the large system of coupled equations for the unknown smoothed image is decoupled into many smaller and simpler equations, each for a separate projection. Regularized CT thus becomes a two-stage process of (nonhomogeneous) smoothing of the projections fol-lowed by filtered backprojection. As a by-product, the repeated forward and backprojections common in iterative image reconstruction are eliminated. Despite the computational simplification, we demonstrate that this method can be used to reduce metal artifacts in X-ray CT images. The decoupling of the equations results from postponing the discretization of image derivatives that realize the smoothness constraint, allowing for this constraint to be an-alytically “transferred ” from the image domain to the projection, or Radon, domain. Our analysis thus clarifies the role of image smoothness: it is an entirely intra-projection constraint.