Dynamic PET image reconstruction incorporating a median nonlocal means kernel method

Highlights

• This study proposed a median nonlocal means (MNLM)-based kernel method for dynamic PET image reconstruction.

• The kernel matrix is derived from median nonlocal means of pre-reconstructed composite images.

• The proposed approach shows the potential for low-count dynamic PET reconstruction.

Abstract

In dynamic positron emission tomography (PET) imaging, the reconstructed image of a single frame often exhibits high noise due to limited counting statistics of projection data. This study proposed a median nonlocal means (MNLM)-based kernel method for dynamic PET image reconstruction. The kernel matrix is derived from median nonlocal means of pre-reconstructed composite images. Then the PET image intensities in all voxels were modeled as a kernel matrix multiplied by coefficients and incorporated into the forward model of PET projection data. Then, the coefficients of each feature were estimated by the maximum likelihood method. Using simulated low-count dynamic data of Zubal head phantom, the quantitative performance of the proposed MNLM kernel method was investigated and compared with the maximum-likelihood method, conventional kernel method with and without median filter, and nonlocal means (NLM) kernel method. Simulation results showed that the MNLM kernel method achieved visual and quantitative accuracy improvements (in terms of the ensemble mean squared error, bias versus variance, and contrast versus noise performances). Especially for frame 2 with the lowest count level of a single frame, the MNLM kernel method achieves lower ensemble mean squared error (10.43%) than the NLM kernel method (13.68%), conventional kernel method with and without median filter (11.88% and 23.50%), and MLEM algorithm (24.77%). The study on real low-dose ¹⁸F-FDG rat data also showed that the MNLM kernel method outperformed other methods in visual and quantitative accuracy improvements (in terms of regional noise versus intensity mean performance).

Introduction

Positron emission tomography (PET), a powerful molecular imaging modality, can measure the spatial distribution of a radiotracer in vivo [[1], [2], [3]]. PET parametric images can directly quantify physiological or biochemical parameters of interest and can be reconstructed by direct or indirect methods. Direct methods [[4], [5], [6], [7], [8], [9]] incorporate the kinetic model [10,11] into the reconstruction and estimate parametric images directly from raw projection data. In comparison, indirect methods first reconstruct dynamic activity images from projection data of multiple time frames and then perform tracer kinetic modeling at the voxel or region of interest (ROI) level to obtain parametric images. For accurate tracer kinetic modeling, very short time frames are often required, and thus the counting statistics of each frame are limited [[12], [13], [14], [15], [16]].

In this paper, we focus on the reconstruction of dynamic PET images in indirect parametric imaging. For the reconstruction of the individual frame, the filtered back projection (FBP) method [[17], [18], [19]] often derives substantial noise in the reconstructed image. The maximum-likelihood expectation-maximization (MLEM) algorithm [20,21], combining the statistical models of measured data and the physical models of a detection system, can achieve better image quality than the FBP method. However, the MLEM estimate tends to converge to maximum likelihood (ML) with a noisy measurement, and thus the noise in reconstructed images tends to increase dramatically with the number of iterations [22]. Bayesian or maximum a posteriori (MAP) methods improve reconstructed image quality by incorporating image prior information [23], which can be obtained either from the PET image itself or from a co-registered high resolution anatomical image [[24], [25], [26], [27], [28], [29]]. The Bowsher method [30] selects neighboring pixels for each pixel according to a non-segmentation anatomical image and outperforms other anatomical priors in terms of quantitative performance and computational complexity [31]. Nonlocal regularization can calculate weights combining functional and anatomical information and thus may improve the estimate of image regions where mismatches between activity and anatomical images occur [[32], [33], [34]]. Other noteworthy Bayesian methods use joint entropy between PET and anatomical images as a regularization constraint [[35], [36], [37]]. Most of these methods employ explicit regularization to incorporate image prior information and require a convergent solution to achieve optimal performance.

Recently, Wang et al. proposed a kernel-based method [38] for PET imaging. The kernel-based image model is incorporated into the forward model of PET projection data, and thus the coefficients can be estimated by the maximum likelihood method. The kernel method has been successfully applied to dynamic PET imaging utilizing composite prior images [[38], [39], [40]], static PET image reconstruction using magnetic resonance (MR) anatomical information [41], PET imaging exploiting the hybrid PET and MR information [[42], [43], [44]], and direct PET parametric imaging that combines the spatial kernel and tracer kinetic models such as the spectral model [45] and Patlak graphical model [46]. The spatial kernel method tries to exploit the spatial correlation information of voxels to improve the image quality of individual frames [38], while other kernel methods [40,45,46] try to exploit both spatial and temporal correlation information of voxels to improve the kinetic parameter estimation of all dynamic frames. In our present study, we focus on the improvement of the spatial kernel from low-count dynamic data.

In a previous work [38], k-nearest-neighbor (kNN) was used in searching the k similar neighbors of each pixel for constructing the kernel matrix, which shares the same spirit of the nonlocal means (NLM) method with pixel intensity-based similarity measure other than patch. The nonlocal weights can also be estimated from the patch-based similarity measure of prior image. Notably, the quality of prior images can directly affect the construction of the kernel and consequently affect the image reconstruction quality of dynamic frames.

In the present study, median nonlocal means (MNLM) were introduced into the kernel matrix for PET image reconstruction. The median filter has been applied to clinical data for PET image reconstruction [[47], [48], [49], [50], [51]] and post-reconstruction filtering [52] because the median filter encourages locally monotonic smoothness while preserving abrupt changes such as edges. Especially in Ref. [52], the MNLM method could significantly improve the robustness of nonlocal similarity measurement affected by PET image noise and successfully applied to clinical data. Then the performance of the MNLM kernel method was compared with other methods in extensive simulation studies. The rest of this paper is organized as follows. We first introduce the kernel method framework for PET image reconstruction and describe how to construct an MNLM kernel in Section 2. We then present the computer simulation study in Section 3 and the real data study in Section 4 to validate the improvement of the MNLM kernel method over existing methods. Finally, discussions are in Section 5, and conclusions are in Section 6.