Decoupling the equations of regularized tomography

Abstract

Deferring discretization can occasionally change our perspective on imaging problems. To illustrate, we offer a reformulation of regularized computed 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 followed 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 analytically "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.