Simultaneous activity and attenuation estimation in TOF-PET with TV-constrained nonconvex optimization

TL;DR

An alternating direction method of multipliers framework is developed for nonsmooth biconvex optimization for inverse problems in imaging and this algorithm is extended by imposing total variation (TV) constraints on both the activity and attenuation map, resulting in the ADMM-TVSAA algorithm.

Abstract

An alternating direction method of multipliers (ADMM) framework is developed for nonsmooth biconvex optimization for inverse problems in imaging. In particular, the simultaneous estimation of activity and attenuation (SAA) problem in time-of-flight positron emission tomography (TOF-PET) has such a structure when maximum likelihood estimation (MLE) is employed. The ADMM framework is applied to MLE for SAA in TOF-PET, resulting in the ADMM-SAA algorithm. This algorithm is extended by imposing total variation (TV) constraints on both the activity and attenuation map, resulting in the ADMM-TVSAA algorithm. The performance of this algorithm is illustrated using the penalized maximum likelihood activity and attenuation estimation (P-MLAA) algorithm as a reference. Additional results on step-size tuning and on the use of unconstrained ADMM-SAA are presented in the previous arXiv submission: arXiv:2303.17042v1.

Figures (9)

• Figure 1: (left) Slice number 40 from the University of Washington Digital Reference Object: activity image in arbitrary units, and (right) attenuation map displayed in the gray scale window $[0.075,0.115]$ cm$^{-1}$. The dashed circle in the activity image indicates the activity distribution used for the investigation of SAA with interior data.
• Figure 2: Convergence of ADMM-TVSAA with noiseless TOF-PET data for the case of the full activity distribution (left) and the interior activity distribution (right). The data RMSE is normalized to the mean value of the TOF-PET data, and the activity/attenuation RMSEs are normalized to the mean values of their respective images.
• Figure 3: Reconstructed activity (left column) and attenuation (right column) images from noiseless data with ADMM-TVSAA at 50 (top row), 100 (middle row), and 5000 (bottom row) iterations. Because the result at 5000 iterations is visually indistinguishable from the test phantom the difference from the ground truth is displayed in the bottom row. The activity distribution is normalized to 1.0 for the maximum value.
• Figure 4: For the results with the interior activity distribution we only show iteration 5000 for the activty (left) and attenuation (right). The actual activity/attenuation images are shown in the top row, and the difference from ground truth is shown in the bottom row.
• Figure 5: Normalized standard deviation versus normalized bias of the activity (left) and attenuation (right) images as a function of iteration number computed empirically from 100 noise realizations for MLAA, MLAA with Huber penalties, and TV-constrained SAA. Normalization of bias and standard deviation is achieved by dividing by the mean value of the corresponding ground truth image. The labeled dots indicate the iteration numbers for the respective algorithm curves. For TV-constrained ADMM-SAA, curves are shown for activity and attenuation TV constraints set to $\gamma_\lambda = 1.0$ and $\gamma_\mu = 1.0$, respectively, where the constraint values are given as a fraction of the ground truth TV values. The Huber penalty parameters for P-MLAA are set so that the resulting activity and attenuation images have nearly the same TV values as the ground truth after 500 iterations.