Evolutionary Optimization of Multiple Machine-Learned Objectives for PET Image Reconstruction

This article proposes optimizing machine-learned objective functions to estimate a set of images optimal for multiple tasks. Tomographic image reconstruction commonly relies on optimizing an objective function that dictates the goodness of an image estimate. We propose a novel use of deep learning in which we train models to perform blind estimation of the goodness of an image. The trained model that maps an image to a target figure of merit serves as the objective function. Using simulated PET images, we train convolutional neural networks (CNNs) to estimate for a single image the root mean square error (RMSE), signal-to-noise ratio, and full width half maximum (FWHM) of a point source. These figures of merit serve as surrogates for the common medical task performance in terms of quantitation, detection, and spatial resolution, respectively. These multiple objective functions are optimized using our recently proposed multiobjective evolutionary algorithm for PET image reconstruction. The evolutionary algorithm seeks to identify the Pareto-optimal front of image solutions, optimal for multiple objectives. Additionally, we propose a reference point method to focus the evolutionary optimization on regions of the multiobjective space to reduce computation. Likewise, we integrate optimality conditions to estimate how close a solution is to optimal without any prior knowledge about the Pareto-optimal front in an effort to guide termination conditions. The multiobjective approach is compared with conventional maximum-likelihood reconstruction with increasing explicit regularization and maximum a posteriori reconstruction with varying penalty strength. The approaches are tested with simulated PET data from a digital phantom and patient data. Results suggest that the proposed machine-learned objectives can estimate quantification, detection, and spatial resolution performance and that the evolutionary algorithm estimates a set of solutions with a good balance for convergence and diversity across different multiobjective combinations. In general, the evolutionary algorithm estimates images with superior performance compared to conventional techniques. Future work is needed to ensure and improve generalizability of the machine-learned objectives for different imaging conditions and to explore other target figures of merit for training machine-learned objectives. This proof-of-concept effort demonstrates a new use of deep learning for tomographic image reconstruction.