PET Reconstruction With Non-Negativity Constraint in Projection Space: Optimization Through Hypo-Convergence

Abstract

Standard positron emission tomography (PET) reconstruction techniques are based on maximum-likelihood (ML) optimization methods, such as the maximum-likelihood expectation-maximization (MLEM) algorithm and its variations. Most of these methodologies rely on a positivity constraint on the activity distribution image. Although this constraint is meaningful from a physical point of view, it can be a source of bias for low-count/high-background PET, which can compromise accurate quantification. Existing methods that allow for negative values in the estimated image usually utilize a modified loglikelihood, and therefore break the data statistics. In this work we propose to incorporate the positivity constraint on the projections only, by approximating the (penalized) log-likelihood function by an adequate sequence of objective functions that are easily maximized without constraint. This sequence is constructed such that there is hypo-convergence (a type of convergence that allows the convergence of the maximizers under some conditions) to the original log-likelihood, hence allowing us to achieve maximization with positivity constraint on the projections using simple settings. A complete proof of convergence under weak assumptions is given. We provide results of experiments on simulated data where we compare our methodology with the alternative direction method of multipliers (ADMM) method, showing that our algorithm converges to a maximizer which stays in the desired feasibility set, with faster convergence than ADMM. We also show that this approach reduces the bias, as compared with MLEM images, in necrotic tumors-which are characterized by cold regions surrounded by hot structures-while reconstructing similar activity values in hot regions.

Multi-Channel Convolutional Analysis Operator Learning for Dual-Energy CT Reconstruction

“…q) is repeated until either a convergence criterion is met. A more detailed explanation can be found in Bousse, Courdurier, Émond, Thielemans, Hutton, Irarrazaval & Visvikis (2020). We utilized the implementation proposed in Zhu, Byrd, Lu & Nocedal (1997).…”

Multi-channel convolutional analysis operator learning for dual-energy CT reconstruction

“…q) is repeated until a convergence criterion is met. A more detailed explanation can be found in Bousse et al (2020). We utilized the implementation proposed in Zhu et al (1997).…”

Local-Mean Preserving Post-Processing Step for Non-Negativity Enforcement in PET Imaging: Application to 90Y-PET

“…Previous works have been focused on finding new algorithms that require non-negativity in the data space while being unconstrained in the image space [15], [16]. Such methods partially solve the physical inconsistencies, but negative voxels may still be present in the image.…”