Resolution in Penalized-Likelihood Image Reconstruction
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Penalized-likelihood methods have been used widely in image reconstruction since they can model both the imaging system geometry and measurement noise very well. However, conventional penalized-likelihood image reconstructions are subject to anisotropic and shift-variant spatial resolution properties, which can complicate selection of the regularization parameter and make the analysis of the resulting images more difficult. The local impulse response is a resolution predictor, which may be used to quantify these shift-variant spatial resolution properties. We have derived a new formulation of the local impulse response for penalized-likelihood estimators, which is appropriate for a general class of imaging systems that acquire a finite number of measurements from a continuous object and reconstruct that object using a discrete model. We have developed fast techniques for evaluating both resolution and covariance predictors for emission tomography systems even when the geometric system model is inherently shift-variant. We have also developed practical methods based on these rapid predictions to provide for increased resolution control by designing an appropriate penalty function. The penalty function design allows for the specification of user-defined resolution properties like uniform resolution (i.e., both isotropic and shift-invariant). We show that these penalty design techniques can provide nearly uniform resolution even in intrinsically shift-variant imaging systems; whereas many traditional reconstruction techniques cannot fully compensate for the shift-variant effects. We discuss the relative resolution uniformity of different reconstruction methods and examine the relative noise performance of estimators for which the resolution properties are exactly matched. Among these matched estimators we find that the penalized-likelihood approach and the post-filtered maximum-likelihood approach often produce identical noise properties.