Image reconstructions from super-sampled data sets with resolution modeling in PET imaging

Abstract

Purpose: Spatial resolution in positron emission tomography (PET) is still a limiting factor in many imaging applications. To improve the spatial resolution for an existing scanner with fixed crystal sizes, mechanical movements such as scanner wobbling and object shifting have been considered for PET systems. Multiple acquisitions from different positions can provide complementary information and increased spatial sampling. The objective of this paper is to explore an efficient and useful reconstruction framework to reconstruct super-resolution images from super-sampled low-resolution data sets.

Methods: The authors introduce a super-sampling data acquisition model based on the physical processes with tomographic, downsampling, and shifting matrices as its building blocks. Based on the model, we extend the MLEM and Landweber algorithms to reconstruct images from super-sampled data sets. The authors also derive a backprojection-filtration-like (BPF-like) method for the super-sampling reconstruction. Furthermore, they explore variant methods for super-sampling reconstructions: the separate super-sampling resolution-modeling reconstruction and the reconstruction without downsampling to further improve image quality at the cost of more computation. The authors use simulated reconstruction of a resolution phantom to evaluate the three types of algorithms with different super-samplings at different count levels.

Results: Contrast recovery coefficient (CRC) versus background variability, as an image-quality metric, is calculated at each iteration for all reconstructions. The authors observe that all three algorithms can significantly and consistently achieve increased CRCs at fixed background variability and reduce background artifacts with super-sampled data sets at the same count levels. For the same super-sampled data sets, the MLEM method achieves better image quality than the Landweber method, which in turn achieves better image quality than the BPF-like method. The authors also demonstrate that the reconstructions from super-sampled data sets using a fine system matrix yield improved image quality compared to the reconstructions using a coarse system matrix. Super-sampling reconstructions with different count levels showed that the more spatial-resolution improvement can be obtained with higher count at a larger iteration number.

Conclusions: The authors developed a super-sampling reconstruction framework that can reconstruct super-resolution images using the super-sampling data sets simultaneously with known acquisition motion. The super-sampling PET acquisition using the proposed algorithms provides an effective and economic way to improve image quality for PET imaging, which has an important implication in preclinical and clinical region-of-interest PET imaging applications.

FIG. 1.

A simple illustration to explain the concept of super-sampling data acquisition. The four sets of symbols represent four sets of sample points of an object with different shifts. By fusing the information contained within all four data sets, we obtain an image with improved spatial resolution.

FIG. 2.

The polyphase block diagrams for the forward model of the super-sampling data acquisitions and the corresponding BPF-like reconstruction. The background events r_m are ignored in the block diagrams for conciseness. (a) Forward model of super-sampling acquisition. (b) The BPF-like super-sampling reconstruction.

FIG. 3.

An example calculation and analytic approximation of the frequency response of the variant Landweber filter. The top row shows the Fourier transform of the responses of the central impulse e₀, and the bottom row shows the corresponding analytic approximations. The downsampling factor in this example M = 2 × 2, and the image size is 256 × 256 pixels. The variant Landweber filter in (d) ${\mathit{\ell }}^{\left(k\right)}={\left(\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\hspace{0.17em}{\mathcal{H}}_m\right)}^{-1}[\mathit{I}-{(\mathit{I}-\lambda \sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\hspace{0.17em}{\mathcal{H}}_m)}^k]{\mathit{e}}_0$, and the parameter k  =  1024.

FIG. 4.

Comparison of the reconstructed images using the three methods (top: MLEM, middle: Landweber, and bottom: BPF-like). In both (a) and (b), the left and right images were reconstructed from 1 data set and 4 super-sampled data sets, respectively. The images in (a) and (b) are reconstructed using the coarse and fine system matrices for efficient reconstruction and improved image quality, respectively. The images have approximately matched background variability of 0.2 by selecting different iteration number. (a) Reconstructions using the coarse system matrix, the sizes of the images are 128  ×  128 (left) and 256  ×  256 (right). (b) Reconstructions using the fine system matrix, the size of image is 256  ×  256.

FIG. 5.

Calculated CRCs versus background variability for the three types of reconstructions of the two different sizes of hot rods using the coarse system matrix, i.e., the system matrix used the reconstructions shown in Fig. 4(a). Each marker represents steps of 128 iterations. (a) 1.6 mm hot rods. (b) 2.4 mm hot rods.

FIG. 6.

Calculated CRCs versus background variability for the three types of reconstructions of the two different sizes of hot rods using the fine system matrix, i.e., the system matrix used reconstruction shown in Fig. 4(b). Each marker represents steps of 128 iterations. (a) 1.6 mm hot rods. (b) 2.4 mm hot rods.

FIG. 7.

Comparison of reconstructed images from 1 data set (left) and 4 data sets (right) with different total counts using the MLEM algorithm with the fine system matrix. The expected total counts in the reconstructions from top to bottom are 4, 40, and 100M, respectively. The background variabilities are proportional to the noise level based on the Poisson statistics. The images in each row have approximately matched background variability by selecting different iteration number.

FIG. 8.

Calculated CRCs versus background variability for the reconstructions of the two sizes of hot rods with different count levels. The reconstructions with background variability of 0.2 are shown in Fig. 7. Each marker represents steps of 128 iterations. (a) 1.6 mm hot rods. (b) 2.4 mm hot rods.