Potential equivalence of sinogram and image-domain penalized likelihood methods

In recent years, we and others have been exploring the use of penalized-likelihood sinogram-domain smoothing and restoration approaches for emission and transmission tomography. The strategy entails estimating the "ideal" line integrals needed for reconstruction of an activity or attenuation distribution from the set of noisy, potentially degraded tomographic measurements by maximizing a penalized-likelihood objective function. The objective function models the data statistics as well as any degradations that can be represented in the sinogram domain. The estimated line integrals can then be reconstructed by use of analytic reconstruction algorithms such as filtered backprojection (FBP). The motivation for this strategy was initially pragmatic: to provide a more computationally feasible alternative to fully iterative penalized-likelihood image reconstruction involving expensive backprojections and reprojections, while still obtaining some of the benefits of the statistical modeling employed in penalized-likelihood approaches. However, in this work we establish a potentially significant equivalence between the two approaches when the penalty functions are chosen appropriately in the two domains. We show that the two approaches can be made to produce very similar resolution properties and resolution-variance tradeoffs, but with a large advantage in computation time for the sinogram-domain approach.