Null-space smoothing of tomographic images using TV norm minimization

Smoothing is desirable in tomographic imaging when it reduces the effects of noise in the data and is undesirable when it smooths a small feature such as a tumor or a lesion so much that they become undetectable. Linear algebra can be used to identify a significant problem associated with reconstruction from incomplete data set; namely, the rank of the resulting system matrix is less then full. To maximize its benefit and to minimize its harm, when smoothing is used in this case, it seems desirable to give more credence to the row-space component of the reconstruction than the null-space because the tomographic data contains only information about the row-space component of the object. The objective of the work presented here is to propose and demonstrate a method, which is called null-space smoothing, for achieving this. The Methodology used involved computer generated data. ART is used to reconstruct the row-space component of the Shepp and Logan phantom. By solving a convex optimization problem, an image in the null-space was added to the reconstruction so that the resulting image had a minimum TV norm; thus, leaving the row-space component unchanged. It is concluded that although null-space smoothing can produce smooth images with an unchanged row-space component, more work needs to be done in the future to demonstrate its usefulness with real data.