Optimized Bayes variational regularization prior for 3D PET images

Abstract

A new prior for variational Maximum a Posteriori regularization is proposed to be used in a 3D One-Step-Late (OSL) reconstruction algorithm accounting also for the Point Spread Function (PSF) of the PET system.

The new regularization prior strongly smoothes background regions, while preserving transitions. A detectability index is proposed to optimize the prior.

The new algorithm has been compared with different reconstruction algorithms such as 3D-OSEM + PSF, 3D-OSEM + PSF + post-filtering and 3D-OSL with a Gauss-Total Variation (GTV) prior.

The proposed regularization allows controlling noise, while maintaining good signal recovery; compared to the other algorithms it demonstrates a very good compromise between an improved quantitation and good image quality.

Introduction

Positron emission tomography (PET) and PET/computed tomography (CT) imaging systems have been greatly improved in the last few years both from the hardware and software point of view.

For example, all the physical effects limiting the spatial resolution of PET can be taken into account by a “global” Point Spread Function (PSF), as representative of the system response, which – if implemented in an iterative reconstruction algorithm – allows taking into account the system resolution loss and recovering it [1], [2], [3], [4].

Unfortunately, a common effect of iterative reconstruction techniques is the increase of noise as iterations proceed, due to the ill-posed nature of the reconstruction problem [5]. On the other hand, a high number of iterations are usually needed to recover a substantial percentage of the signal, especially when implementing PSF modelling [6]. A post-filtering applied to the image can limit noisy patterns, but it usually causes a loss of spatial resolution and a reduction in the quantification accuracy, which should obviously be avoided, particularly if a strategy of spatial resolution recovery is employed. Regularization techniques, instead, could help in controlling noise amplification during the reconstruction process (and, thus, they allow increasing the number of iterations), provided that the regularization method is able to preserve the spatial resolution as much as possible.

Many different regularization strategies have been proposed (see e.g. [7] for a review). The most widely used ones introduce a spatial interaction between voxels in the image – often by using Markov random field models – and thus associate, to the image, a global energy function whose value is based on local potentials evaluated on some subsets of the whole image. The effects of the regularization process depend on the characteristics of the local potential employed, which ideally should provide a smoothing behaviour inside each region and preserve the transitions between adjacent regions.

Among the different strategies to define the energy function, the variational approach [8], [9] allows taking into account not only the mean activity in the considered districts of the image, but also their rapidity of variation between adjacent regions by employing the gradients in the image.

Many different penalty functions have been proposed [10]. The most used Bayesian regularization strategy is based on the Gaussian prior, which smoothes all the gradient intensities with equal relative strength, resulting in a strong smoothing effect.

On the other hand, a commonly used prior for edge preservation is the Total Variation (TV) [11], which is able to maintain mainly unchanged the transitions between different regions in the image, but tends to produce staircasing effects in regions gradually changing in space.

A modification of the Gaussian prior is the Huber prior [12] or, equivalently in the variational framework, the Gauss-Total Variation (GTV) prior [10], which introduces a parameter to discriminate between different regularization behaviours to be applied respectively to background regions – Gaussian (GR) component – for noise suppression and signal regions–Total Variation component – for edge preservation.

Another promising strategy is represented by the generalized Gaussian or p-Gaussian prior (PR), as proposed in [13]: this regularization scheme has the capability of strongly smoothing background regions, while maintaining higher detail in signal regions.

In this work a new variational regularization prior has been proposed by modifying the p-Gaussian prior to reduce the resolution loss introduced by it. The prior has been implemented in a 3D Ordered Subsets Expectation Maximization (OSEM) reconstruction algorithm accounting for the PSF in the image space and validated, on phantom data and clinical images, by comparing it with OSEM + PSF images, post-filtered OSEM + PSF images and the images obtained with the Gauss-Total Variation prior. Moreover, since the presented regularization priors depend on different parameters, an optimization strategy (based on the qualitative and quantitative content of the images) has also been proposed and employed to compare the priors at their best performance.