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Positron Emission Tomography » Model Based
Model Based Reconstruction of PET Images The image reconstruction methods that we have developed combine three components:
Images are reconstructed using these models through a MAP (maximum a posteriori) estimator that maximizes over the posterior probability for the image conditioned on the observed data. Our rationale for developing these methods is that the MAP approach has clear advantages over the two most widely used approaches to reconstruction of PET and SPECT images: filtered backprojection (FBP) and ordered subsets EM (OSEM) methods . The advantages of MAP with respect to FBP are based on our more accurate modeling of the physics and noise properties of coincidence detection. As detector size decreases, the number of events per detector drops and hence, so does the signal to noise ratio in the data; as this happens, it becomes increasingly important to consider the photon limited properties of the data when reconstructing the image. In addition, as detector size decreases, sinogram blurring due to photon-pair non-colinearity, positron range and inter-crystal scatter and penetration becomes more significant since the sinogram bins are more closely spaced. Accurate modeling of the physics of detection becomes increasingly important in cases where there are sufficient counts to achieve higher resolution. Thus we see advantages to accurate statistical and physical modeling at both ends of the spectrum: in high counts/voxel studies (such as small-animal microPET studies) higher resolution can be achieved through accurate physical modeling, while at low counts/voxel (.e.g. whole body studies) accurate statistical modeling allows reduced noise levels compared to FBP at matched resolutions. Differences between MAP and OSEM are more subtle. Both are based on the same basic Poisson model and both can use an accurate system model (although in practice, many OSEM implementations appear to use a simpler model, such as the linear-interpolated backprojection method commonly used in FBP). OSEM is a modification of the EM algorithm that computes a maximum likelihood (ML) estimate of the image. The ML estimator is generally unstable for PET systems and this instability is manifested by increasingly noisy reconstructions as the iterations proceed. Typically, in both EM and OSEM algorithms, the iterations are terminated before convergence to control noise propagation and some additional smoothing may be applied after the fact to further reduce noise. OSEM is not a true ML estimator since the data are divided into subsets over which the algorithm iterates. It can be shown that in general this algorithm will not converge and the solution is dependent on the order in which the data are visited. Thus the OSEM algorithm involves a number of heuristics which all affect the solution: number and order of subsets, number of iterations, and degree of post-reconstruction smoothing. In contrast, the MAP algorithm controls instability through the use of a regularizing or smoothing prior. The degree of smoothing is determined by a single smoothing parameter. Once the parameter is fixed, the solution of the problem is the image that maximizes the posterior probability and is (under certain conditions) unique.
Figure: This is an animation showing a maximum intensity projection of an F- bone scan of a mouse collected on the UCLA microPET scanner. Movie: F-Mousebone.mpg
Figure: Coronal sections through microPET FDG images of a rat brain reconstructed using MAP and FBP. Also show is the corresponding autoradiographic section. As shown in the figure below, the MAP images show improved contrast in white vs. grey matter compared to FBP and appear to be closer to that seen in the "gold standard" authoradiograph.
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