Decoupling the equations of regularized tomography

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. 1 Background In the absence of noise, the basic problem of computerized tomography (CT) is to determine an unknown image f = f(x, y) from its (forward) projections, or Radon transform Rf , where (Rf)(t, θ) := ∫∫ f(x, y)δ(t−x cos θ−y sin θ)dxdy, where (x, y) are planar coordinates, t is the location along each projection, and θ ∈ [0, π) is the orientation of the projection. (In this paper we focus on the two dimensional problem with standard parallel-beam geometry, but the ideas readily extend to three dimensions and other scanning geometries.) Unfortunately, since