Practical considerations for image-based PSF and blobs reconstruction in PET

Iterative reconstructions in positron emission tomography (PET) need a model relating the recorded data to the object/patient being imaged, called the system matrix (SM). The more realistic this model, the better the spatial resolution in the reconstructed images. However, a serious concern when using a SM that accurately models the resolution properties of the PET system is the undesirable edge artefact, visible through oscillations near sharp discontinuities in the reconstructed images. This artefact is a natural consequence of solving an ill-conditioned inverse problem, where the recorded data are band-limited. In this paper, we focus on practical aspects when considering image-based point-spread function (PSF) reconstructions. To remove the edge artefact, we propose to use a particular case of the method of sieves (Grenander 1981 Abstract Inference New York: Wiley), which simply consists in performing a standard PSF reconstruction, followed by a post-smoothing using the PSF as the convolution kernel. Using analytical simulations, we investigate the impact of different reconstruction and PSF modelling parameters on the edge artefact and its suppression, in the case of noise-free data and an exactly known PSF. Using Monte-Carlo simulations, we assess the proposed method of sieves with respect to the choice of the geometric projector and the PSF model used in the reconstruction. When the PSF model is accurately known, we show that the proposed method of sieves succeeds in completely suppressing the edge artefact, though after a number of iterations higher than typically used in practice. When applying the method to realistic data (i.e. unknown true SM and noisy data), we show that the choice of the geometric projector and the PSF model does not impact the results in terms of noise and contrast recovery, as long as the PSF has a width close to the true PSF one. Equivalent results were obtained using either blobs or voxels in the same conditions (i.e. the blob's density function being the same as the voxel-based PSF). From a practical point-of-view, the method can be used to perform fast reconstructions based on very simple models (compared to sinogram-based PSF modelling), producing artefact-free images with a better compromise between noise and spatial resolution than images reconstructed without or with under-estimated PSF. Besides, the method inherently limits the spatial resolution in the reconstructed images to the intrinsic one of the PET system.