| Background:Positron emission tomography (PET) has been widely used in clinical diagnosis, scientific research and medicine experiments and become a major nuclear medicine imaging modality.PET is a kind of ECT (emission computed tomography). The basic principle of PET is y photon coincidences detection. By injecting radiopharmaceutical capable of producing positron (β+) into the patient body, the positron will be ejected from the radionuclide. After traveling a short distance, the positron annihilates with an electron and two y photons with 511keV energy will be generated. These two y photons will travel along a straight line in opposite direction and be detected by two detectors located at the two ends of the straight line. The scintillator in the detectors converts the y photons into light. The light is further converted into electronic signal and amplified by the photomultiplier tubes (PMTs) attached behind the scintillator. The straight line connecting the two detectors is named the line of response (LOR). All the detected coincidence events in all the LORs composite the projection of PET and are usually stored in sinogram. The purpose of PET image reconstruction is to estimate the distribution of annihilations in the human body from the sinogram and reflect the physiological or pathological status or the anatomy structures of the organs.However, in practice, due to the limited photon counts and acquisition time, electronic noise, and the defects of detectors, the projection data acquired by PET is usually very noisy and thus makes the PET image reconstruction an ill-posed problem. The analytical reconstruction method, such as the filtered backprojection (FBP), usually results in very noisy image. On the contrary, iterative reconstruction methods are capable of modeling the process of PET imaging and dealing with the corresponding factors accurately and usually resulting in better images. Due to this, iterative reconstruction methods are more popular in PET image reconstruction field.One of the mostly used iterative reconstruction methods is the maximum likelihood-expectation maximization (ML-EM) or its accelerated version ordered subsets (OS)-EM algorithm. ML-EM/OS-EM can model the physical factors and statistical properties of the counting process of PET system in a unified model, and reconstructs the images iteratively. Due to this, ML-EM/OS-EM has become the main reconstruction method in PET clinics. There are two steps in ML-EM/OS-EM:the E-step and the M-step. The E-step is the process of making expectation maximized and the M-step is the process computing the maximum likelihood.However, one of the main drawbacks of ML-EM/OS-EM method is the so-called "checkerboard effect" appears with the increasing of iteration numbers. To solve this problem, many methods have been proposed, such as to terminate the iterative process earlier, to filter the image during iteration process or after iteration, or to incorporate regularization into the reconstruction process by Bayesian prior framework and construct the maximum-a-posteriori (MAP) scheme. Green modified the EM algorithm and proposed the one-step-late (OSL)-EM method. OSL-EM method retains the E-step of the EM algorithm, but provides an approximate solution to the M-step. Lange further analyzed the convergence of OSL-EM under a specific set of sufficient conditions and investigated several Gibbs energy functions.Bayesian MAP method is able to introduce some priori knowledge of the image to be reconstructed into the reconstruction process and to regularize the ill-posed problem. The design of prior model is the key of Bayesian MAP method. A large amount of prior models have been proposed in literature, such as the Huber prior, the quadratic prior, the median root prior (MRP), the total variation (TV) prior, and the prior based on information theory, and so on. With the development of multimodality imaging devices, such as PET/CT and PET/MRI, utilizing the anatomical side information provided by CT or MRI images to guide the PET image reconstruction has been more appreciated and the advantages also have been proved.The traditional Gibbs priors are based on the interactions of local neighbors. Recently, non-local (NL) concept has attracted most attentions of the image processing researchers and interested the researchers in tomographic image reconstruction field too. The basic idea of NL method is to make use of the global similarity redundancy of nature image. In brief, NL method makes the following assumption that if two pixels are similar, then the two patches centered at these two pixels respectively, will be similar. Then the similarity comparison is transferred from the pixel-based method to the patch-based method. The most famous NL filter is the nonlocal-means (NLM) filtering.In the OSL-EM based Bayesian MAP framework, the gradient of the Gibbs energy function of current image estimate is needed. The prior is imposed on the image estimate through the computations of Gibbs energy function and its gradient from the k th image estimate. OSL-EM is the mostly used iterative method to solve the Bayesian MAP problem. Due to the "one-step-late" property, a full iteration of OSL-EM can be regarded as an error prediction/correction technique.The anisotropic diffusion (AD) filtering proposed by Perona and Malik is a partial differential (PDE)-based technique and is based on gradient of local neighbors. Barash et al found the relationship of AD and adaptive filtering and the bilateral filtering (BF). BF is the representation of a class of nonlinear filters. Zhou et al proposed inter-iteration-filtering (IIF)-MAP method based on the work of Barash et al. IIF-MAP method uses BF algorithm between two MAP iterations to smooth the image estimate and the result is encouraging. However, either the AD or the BF, only makes use of local neighbor interactions, e.g.,3×3 window size, and thus cannot utilize the global similarity information.The main work of this paper:1) We proposed a new Bayesian MAP prior model and evaluated it qualitatively and quantitatively through simulated experiments and real PET data experimentWe proposed a new NLM induced Bayesian prior and applied it to PET image reconstruction under the Bayesian MAP framework.NLM filter is a nonlinear adaptive filter. It can be regarded as an extension of AD or BF filter. NLM filter not only compares the value of single pixel point, but also searches the geometry similarity in a bigger neighborhood; thus, NLM is more robust than AD and BF. In this paper, based on the work of Barash et al and Zhou et al, we propose a new NLM filter induced prior for high quality PET image reconstruction under the Bayesian MAP framework. We named the method as NLMi-MAP.The present NLMi-MAP method contains the following steps. Firstly, some initializations are made and OSL-EM iteration starts. For the k th iteration image estimate λk, do NLM filtering and subtract the unfiltered λk from the NLM filtered result. The differential image is then feed back to the image reconstruction filter to generate a new image estimate (λk+1). Among these steps, NLM filtering can be thought of as a roughness prediction of the kth image estimate λk and is corresponding to the gradient computation of Gibbs energy function in the Bayesian MAP. It actually is an error prediction process. Feeding the error image back is the image correction process.To verify the present NLMi-MAP method, simulated experiments were carried out. A real PET data was also reconstructed.We simulated two numerical phantoms. One is the Shepp-Logan phantom and the other is the Hoffman phantom. The two phantoms are with different pixel values and different total counts. We generated the system matrix by analytical method. The ideal sinograms were generated by forward projecting the simulated phantom images. Poisson noise was then added and totally 31 independent noisy realizations were generated for each simulated phantom. We reconstructed all the simulated datasets to evaluate the present NLMi-MAP method.We first optimized the parameters of present method using the minimum square error (MSE) index. We investigated in detail the effects of regularization parameter β and the NLM filter parameters (i.e., the searching window ws, the patch wp, and the filtering parameter h). As no automatic parameter setting method is available currently, we found the optimal parameter values by trial and error using one of the noisy realizations. Then we reconstructed the rest 30 independent realizations of each phantom using the optimized parameter values. We computed the signal-to-noise ratio (SNR), the root mean square error (RMSE), the correlation coefficient (CORR), and the bias and standard deviation images. The horizontal and vertical profiles of each reconstruction results were plotted and analyzed. We also reconstructed a real PET dataset scanned with a hot-lesion phantom.The results indicate that the present NLMi-MAP method produces the maximum SNR, the minimum RMSE, and the maximum CORR among all the methods evaluated in this study. For the Shepp-Logan phantom, the SNR of the NLMi-MAP is 14.81, improved by 38.4% and 25.8% compared with those of FBP and ML-EM, respectively. Similarly, for the Hoffman phantom, the corresponding improvements are 37.2% and 17.4%, respectively. The RMSE of NLMi-MAP reached a stable minimum after about 50 iterations and the CORR reached a stable maximum after about 50 iterations too.The profile analysis indicates that the profiles of the present NLMi-MAP match those of the simulated phantoms better than other methods.It can be seen from the bias and standard deviation images that NLMi-MAP method retains the minimum bias and minimum standard deviation, and is good at reconstructing smoother image of uniform region.The reconstruction results of real PET data also indicate that the NLMi-MAP method is able to reconstruct smoother images of the uniform regions and in the meanwhile to preserve edges. The result of NLMi-MAP is superior to those of other methods by visual inspection. For the lack of reference image of the real PET data, we didn’t compute any quantitative index, such as the SNR, and didn’t do the corresponding comparison. Only visual inspection was done for real PET data in this study.2) We also extended the NLMi-MAP into anatomical side information guided version and tested it with simulated PET/MRI dataThe present NLMi-MAP method is also capable of incorporating anatomical side information. We extended the NLMi-MAP method into anatomical side information guided version. To accormadate the noisy data, we used median filtered NLM method and proposed the follwong variant methods of NLMi-MAP:the MNLMi-MAP and the corresponding anatomical side information guided method (AMNLMi-MAP).The MNLMi-MAP computes the NLM weights wm,j in the median filtered image and then does the NLMi-MAP reconstruction. The AMNLMi-MAP also computes the NLM weights wm,j in the median filtered image but modifies wm,j with anatomical side information provided by MRI image and then does the NLMi-MAP reconstruction.The results indicate that MNLMi-MAP and AMNLMi-MAP methods are capable of eliminating noise and preserving edges in the meanwhile. Among all the methods evaluated, whether hot lesion is embedded or not, the SNRs of MNLMi-MAP and AMNLMi-MAP are greater than those of other methods. The SNR was further improved when anatomical side information was used. The profile analysis indicates that the profiles of MNLMi-MAP and AMNLMi-MAP match that of the phantom well and the best match is produced by the AMNLMi-MAP method. This justifies the effect of anatomical side information in PET image reconstruction.Conclusion:In this study, we proposed a new NLM filtering induced prior and constructed the NLMi-MAP method for PET image reconstruction under Bayesian MAP framework. We tested the NLMi-MAP method with simulated and real PET data. Essentially, the NLMi prior approximates the gradient of the Gibbs energy function in the MAP framework. The theoretical basis of this approximation is the generalized nonlinear adaptive filtering scheme. We evaluated the present method qualitatively and quantitatively. The experimental results clearly indicated that the present method outperforms local neighbor-based MAP methods and FBP and ML-EM algorithms.We also extended the NLMi-MAP method to incorporate anatomical side information. We made use of median filtered NLM filtering to incorporate MRI anatomical side information and derived two variants of the NLMi-MAP method:the MNLMi-MAP and the AMNLMi-MAP. We evaluated the two methods using simulated PET/MRI data and the results clearly indicated the performance of the present NLMi-MAP method when MRI anatomical side information is used.The future work:In this study, we proposed a new NLM filtering induced prior model based on the nonlinear adaptive filtering theory and tested it with simulated and real PET datasets. The results clearly show the effectiveness of the proposed methods. However, we didn’t give any mathematical proof of the convergence of the proposed methods. Indeed, this is a difficult task at present. What’s more, the convergence of the Bayesian MAP is also related to the hyperparameter β; hence, the convergence proof is very difficult. Due to these reasons, we abandoned the convergence proof and only analyzed the performance in a limited iteration numbers. For the use of MRI anatomical side information, only one method was tested and evaluated. All the remaining problems will be our future research topics.
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