Three-dimensional total variation norm for SPECT reconstruction

The total variation (TV) norm has been described in literature as a method for reducing noise in two-dimensional (2D) images. At the same time, the TV-norm is very good at recovering edges in images, without introducing ringing or edge artefacts. It has also been proposed as a 2D regularisation function in Bayesian reconstruction, implemented in an expectation maximisation (EM) algorithm, and called TV-EM. The TV-EM was developed for 2D SPECT imaging, and the algorithm is capable of smoothing noise while maintaining edges without introducing artefacts. We have extended the TV-norm to take into account the third spatial dimension, and developed an iterative EM algorithm based on the three-dimensional (3D) TV-norm, which we call TV3D-EM. This takes into account the correlation between transaxial sections in SPECT, due to system resolution. We have compared the 2D and 3D algorithms using reconstructed images from simulated projection data. Phantoms used were a homogeneous sphere, and a 3D head phantom based on the Shepp-Logan phantom. The TV3D-EM algorithm yielded somewhat lower noise levels than TV-EM. The noise in the TV3D-EM had similar correlation in transaxial and longitudinal sections, which was not the case for TV-EM, or any 2D reconstruction method. In particular, longitudinal sections from TV3D-EM were perceived as less noisy when compared to TV-EM. The use of 3D reconstruction should also be advantageous if compensation for distant dependent collimator blurring is incorporated in the iterative algorithm.