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.

1. INTRODUCTION

Positron emission tomography (PET) has become an important modality in medical and molecular imaging. However, PET images often suffer from low spatial resolution in most applications. The spatial resolution can be limited by several factors, such as finite crystal width d, scanner diameter D due to noncollinearity, effective source size s (including positron range), and block effect b. An empiric formula proposed by Moses and Derenzo states that the imaging spatial resolution can be given by

$${\displaystyle \mathrm{SR}\approx 1.25\sqrt{{\left(d/2\right)}^2+{\left(0.0022D\right)}^2+s^2+b^2}.}$$ (1)

The constant factor 1.25 is due to the image reconstruction algorithm.¹ The crystal width is still a main resolution limiting factor in most PET applications.² Although reducing the crystal width is a simple way to improve the spatial resolution,^3,4 it can be cost prohibitive with current manufacturing processes, the small crystals can be underutilized or wasted for general-purpose imaging applications, and intercrystal penetration may become a dominating factor. On the other hand, most PET systems are designed for general-purpose applications, and they cannot provide the required resolution for some specific region-of-interest (ROI) imaging.

Super-sampling acquisition can effectively improve the spatial resolution of the reconstructions from an existing PET system with fixed crystal width.^5,6 It is known that data acquisition in general PET systems with discrete crystals is undersampled related to the frequency content of the object;⁷ it is possible to mitigate the spatial-resolution loss due to the block effect by increasing the spatial sampling without decreasing the crystal width.⁶ Super resolution has been applied in PET with mechanical motion of the scanner or the object, e.g., scanner wobbling,^8,9 dichotomic motion,¹⁰ and object shifting or rotating.^5,6,11 The rationale behind super resolution is that the multiple data acquisitions of the object with proper translations or rotations can provide complementary information and increased spatial sampling, which can be fused by super-resolution algorithms.

In PET reconstruction, there is a trend toward incorporating as many physical effects as possible (e.g., resolution modeling, attenuation and scatter correction, motion correction) in the forward projection and following the Poisson statistics model for the measured data in the reconstruction algorithm. This can extract the most information from the measured data and improve the image quality in the reconstruction images. However, most super-resolution techniques in emission tomography are based on the iterative backprojection (IBP) method,¹² which separates the super-resolution step from the tomographic reconstruction step.^5,11 Sinogram-restoration approaches can also be applied to oversampled low-resolution (LR) sinograms to enhance spatial resolution with scanner-wobbling techniques.^9,13,14 The super-resolution image is reconstructed from either the separately constructed high-resolution (HR) sinograms or the multiple reconstructed low-resolution images, instead of directly from the PET measured data. The separated reconstructions may degrade the reconstructed super-resolution image, and the super-sampling techniques can be underutilized due to the nonlinearity of iterative algorithms and the photon-limited nature of PET imaging. A PET supersets data framework was introduced to unify the treatments of both unintentional motion and intentional motion in image reconstruction.¹⁵ The measured data sets with different motion need to be mapped to the supersets data (binned or list format) by applying the motion correction in projection space, which can introduce the interpolation error prior to image reconstruction.

To fully take the advantage of the super-sampling techniques, we introduce a general super-sampling data acquisition model and develop a unified framework for efficient image reconstructions from super-sampled data sets. The proposed super-sampling methods use the original measured data instead of the preprocessed supersets data, which can introduce additional interpolation error.¹⁵ The object motion, as well as the down- and upsampling, is incorporated in the forward and backward projections at each iteration. For the first time in emission tomography, we give a unified super-sampling reconstruction framework to efficiently reconstruct super-resolution images without compromising the image quality, and this is also the main contribution of this work. Specifically, we extend the classical MLEM and Landweber algorithms to reconstruct super-resolution images directly from the super-sampled data sets. We also derive a very efficient backprojection-filtration-like (BPF-like) method for super-sampling reconstruction. In addition, we explore other variant methods: the separate super-sampling resolution-modeling reconstruction and the reconstruction without downsampling. All proposed super-resolution methods can reconstruct super-resolution images simultaneously from the multiple data acquisitions (binned or list format data). The super-sampling techniques with the proposed super-resolution reconstruction algorithms can find applications in preclinical and clinical PET ROI imaging.

2. SUPER-SAMPLING RECONSTRUCTIONS

2.A. Super-sampling data acquisition

By applying object motion in a PET system, multiple data sets can be acquired with different subpixel shifts, from which a HR image can be reconstructed. The LR measured data sets (list-mode data or sinograms) include blurring due to finite crystal width, positron range, noncollinearity, etc. We use a lexicographical stacked column vector f ∈ ℝ^n_h to denote the object digitized into n_h voxels. The expectation of the mth acquisition data g_m is related to the unknown object f through

$${\displaystyle {\overline{\mathit{g}}}_m={\mathcal{H}}_m\mathit{f}+{\mathit{r}}_m=\mathcal{P}\mathcal{D}{\mathcal{T}}_m\mathit{f}+{\mathit{r}}_m,m=0,1,\dots ,M-1,}$$ (2)

where ℋ_m = 𝒫𝒟𝒯_m, 𝒫 ∈ ℝ^n_d×n_ℓ is the tomographic forward projector or system matrix, 𝒟 ∈ ℝ^n_ℓ×n_h is the downsampling operator, 𝒯_m ∈ ℝ^n_h×n_h is the shifting/wobbling operator at the mth acquisition and, M is the number of super-sampling acquisitions. The shifting operator 𝒯_m is also unitary, i.e., ${\mathcal{T}}_m^{-1}={\mathcal{T}}_m^T$. We use lexicographical vectors g_m and r_m ∈ ℝ^n_d to denote, respectively, the mth data set and the background events such as randoms and scatters. We use n_d, n_ℓ, and n_h to denote, respectively, the numbers of LORs, the number of pixels in high- and low-resolution images. Figure 1 shows a possible sampling scheme for super-sampling data acquisitions. Note here that 𝒫 and 𝒟 are independent of motion. 𝒟 becomes an identity matrix when the downsampling factor is 1, and a larger tomographic matrix needs to be created at the expense of more computation. We use Eq. (2) to model the physical processes for data acquisition. The objective is to derive efficient and useful methods based on the model to reconstruct high-resolution images from the super-sampled data sets.

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.

The multiple linear models in Eq. (2) can be rearranged into a large linear model

$${\displaystyle \left[\begin{array}{c}\hfill {\overline{\mathit{g}}}_0\hfill \\ \hfill {\overline{\mathit{g}}}_1\hfill \\ \hfill ⋮\hfill \\ \hfill {\overline{\mathit{g}}}_{M-1}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \mathcal{P}\mathcal{D}{\mathcal{T}}_0\hfill \\ \hfill \mathcal{P}\mathcal{D}{\mathcal{T}}_1\hfill \\ \hfill ⋮\hfill \\ \hfill \mathcal{P}\mathcal{D}{\mathcal{T}}_{M-1}\hfill \end{array}\right]\mathit{f}+\left[\begin{array}{c}\hfill {\mathit{r}}_0\hfill \\ \hfill {\mathit{r}}_1\hfill \\ \hfill ⋮\hfill \\ \hfill {\mathit{r}}_{M-1}\hfill \end{array}\right]}$$ (3)

or equivalently in matrix form

$${\displaystyle \overline{\mathit{g}}=\mathcal{A}\mathit{f}+\mathit{r}.}$$ (4)

The tomographic forward projector 𝒫 is often the geometrical projector, i.e., the line/strip (or tube in 3D) integral model 𝒢. The tomographic projector can also incorporate resolution modeling in image space and sinogram space with 𝒫 = 𝒢ℛ and 𝒫 = ℬ𝒢, respectively.^16,17 ℛ and ℬ denote the image- and sinogram-space blurring operators, respectively.

2.B. MLEM super-resolution reconstruction

The measurements g_m in PET can be well modeled as a conditionally independent Poisson vector with mean ${\overline{\mathit{g}}}_m$. After omitting the constants that are independent of f, the log-likelihood function of the measurements g_m can be given by

$${\displaystyle ℒ\left({\mathit{g}}_0,{\mathit{g}}_1\dots ,{\mathit{g}}_{M-1}|\mathit{f}\right)=\sum _{m=0}^{M-1}\sum _{i=0}^{n_d-1}\left({\left[{\mathit{g}}_m\right]}_{i}log{\left[{\mathcal{H}}_m\mathit{f}+{\mathit{r}}_m\right]}_i-{\left[{\mathcal{H}}_m\mathit{f}+{\mathit{r}}_m\right]}_i\right)=\sum _{t=0}^{M{n}_d-1}\left({\left[\mathit{g}\right]}_{t}log{\left[\mathcal{A}\mathit{f}+\mathit{r}\right]}_t-{\left[\mathcal{A}\mathit{f}+\mathit{r}\right]}_t\right),}$$ (5)

where the second equality follows from the fact ${\left[\mathit{g}\right]}_t={\left[{\mathit{g}}_m\right]}_i$ with index t = m × n_d + i. The expectation–maximization (EM) algorithm¹⁸ can be used to reconstruct the super-resolution image from the multiple data sets. After applying the EM algorithm to the large linear model, the super-resolution MLEM can be given by

$${\displaystyle {\hat{\mathit{f}}}^{(k+1)}=\frac{{\hat{\mathit{f}}}^{\left(k\right)}}{{\mathcal{A}}^T\mathbf{1}}{\mathcal{A}}^T\left[\frac{\mathit{g}}{\mathcal{A}{\hat{\mathit{f}}}^{\left(k\right)}+\mathit{r}}\right]=\frac{{\hat{\mathit{f}}}^{\left(k\right)}}{\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\mathbf{1}}\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\left[\frac{{\mathit{g}}_m}{{\mathcal{H}}_m{\mathit{f}}^{\left(k\right)}+{\mathit{r}}_m}\right],}$$ (6)

where 1 ∈ ℝ^n_d is a vector with all elements equal to one, and Hadamard component notation is used for convenience.¹⁹ Equation (6) gives the extension of MLEM that is applicable for super-sampling reconstructions. At each iteration, there are M independent forward and backward projections that can be easily implemented using parallel computing to improve the speed. The downsampling in the forward projection reduces the size of the image, from which the projection is generated using a small-size tomographic matrix. In the backward projection, the adjoint of downsampling, i.e., the upsampling, increases the size of tomographic back-projected images and produces high-resolution images. The details of downsampling and its adjoint are given in Appendix A.

2.C. Landweber super-resolution reconstruction

When the measurements g_m can be approximately modeled as independent and identically distributed Gaussian vector with covariance matrix σ²ℐ, one can obtain super-resolution images by minimizing the following data-agreement functional²⁰

$${\displaystyle Q_{\mathrm{data}}\left({\mathit{g}}_0,{\mathit{g}}_1\dots ,{\mathit{g}}_{M-1}|\mathit{f}\right)=\frac{1}{2{\sigma }^2}\sum _{m=0}^{M-1}{∥{\mathit{g}}_m-{\mathcal{H}}_m{\mathit{f}}^{\left(k\right)}-{\mathit{r}}_m∥}^2.}$$ (7)

So, we can iteratively reconstruct super-resolution images using the Landweber algorithm²¹

$${\displaystyle {\hat{\mathit{f}}}^{(k+1)}={\hat{\mathit{f}}}^{\left(k\right)}+\lambda {\mathcal{A}}^T\left[\mathit{g}-\mathcal{A}{\hat{\mathit{f}}}^{\left(k\right)}-\mathit{r}\right]={\hat{\mathit{f}}}^{\left(k\right)}+\lambda \sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\left[{\mathit{g}}_m-{\mathcal{H}}_m{\mathit{f}}^{\left(k\right)}-{\mathit{r}}_m\right],}$$ (8)

where λ is a positive relaxation factor. To guarantee convergence, λ should be chosen in the range of 0  <  λ  <  2/σ_max, where σ_max is the largest eigenvalue of the Gram matrix 𝒜^T𝒜. From Eq. (2), one can easily obtain

$${\displaystyle {\mathcal{A}}^T\mathcal{A}=\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}{\mathcal{H}}_m=\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}{\mathcal{D}}^T{\mathcal{P}}^T\mathcal{P}\mathcal{D}{\mathcal{T}}_m.}$$ (9)

From Appendix C, σ_max can be approximated by

$${\displaystyle {\sigma }_{max}\approx \underset{j}{max}\left({\mathbf{1}}^T{\mathcal{A}}^T\mathcal{A}{\mathit{e}}_j\right)=\underset{j}{max}\left({\mathbf{1}}^T{\mathcal{P}}^T\mathcal{P}{\mathit{e}}_j\right),}$$ (10)

where e_j is the jth unit vector. If the relaxation factor λ is too large, the Landweber algorithm will diverge. If λ is too small, it may take a long time for the algorithm to converge. Instead of testing different values for λ, we can use Eq. (10) to quickly determine the relaxation factor by selecting λ = 2η/σ_max, where 0 < η ≤ 1. We choose η = 0.5 in the simulated reconstructions to ensure the convergence of the Landweber algorithm.

2.D. BPF-like super-resolution reconstruction

We can rewrite Eq. (8) as follows:

$${\displaystyle {\hat{\mathit{f}}}^{(k+1)}=\lambda \sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\left({\mathit{g}}_m-{\mathit{r}}_m\right)+\left[\mathit{I}-\lambda \sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\hspace{0.17em}{\mathcal{H}}_m\right]{\hat{\mathit{f}}}^{\left(k\right)}.}$$ (11)

If we choose f⁽⁰⁾ as an initial image, it is easy to show by induction that

$${\displaystyle {\hat{\mathit{f}}}^{\left(k\right)}={\left[\mathit{I}-\lambda \sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\hspace{0.17em}{\mathcal{H}}_m\right]}^k{\hat{\mathit{f}}}^{\left(0\right)}+\lambda \sum _{\ell =0}^{k-1}{\left[\mathit{I}-\lambda \sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\hspace{0.17em}{\mathcal{H}}_m\right]}^{\ell }\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\left({\mathit{g}}_m-{\mathit{r}}_m\right).}$$ (12)

After assuming a zero initial image, we obtain

$${\displaystyle {\hat{\mathit{f}}}^{\left(k\right)}=\frac{\mathit{I}-{\left(\mathit{I}-\lambda \sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}{\mathcal{H}}_m\right)}^k}{\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}{\mathcal{H}}_m}\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\left({\mathit{g}}_m-{\mathit{r}}_m\right).}$$ (13)

Here, we used the well-known relation

$${\displaystyle \mathit{I}+\mathit{X}+{\mathit{X}}^2+\cdots +{\mathit{X}}^{k-1}=\frac{\mathit{I}-{\mathit{X}}^k}{\mathit{I}-\mathit{X}}}$$ (14)

with symmetric matrix $\mathit{X}=\mathit{I}-\lambda \sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\hspace{0.17em}{\mathcal{H}}_m$. We assume I − X is invertible, i.e., the null space of I − X contains only the zero vector. We use fractional matrix form 1/(I − X) to represent ${\left(\mathit{I}-\mathit{X}\right)}^{-1}$ without confusion because matrix I − X^k commutes with I − X and thus commutes with ${\left(\mathit{I}-\mathit{X}\right)}^{-1}$. Note that when k → ∞, Eq. (13) simply becomes the least-squares solution. When k = 1, Eq. (13) is just the backprojection without filtering. For a finite k, the fraction term in Eq. (13) is a product of an inverse filter and a low-pass filter in the frequency domain. The inverse filter is just the ramp filter used for an FBP or BPF reconstruction; the low-pass filter is the Landweber filter.²² It is worth noting that the generalized inverse can be used when I − X is not invertible. In such cases, the solution can be unbounded and the regularizer of priori knowledge can be applied to obtain meaningful approximations of the true solution.

Direct calculation of the first term in Eq. (13) can be computationally intensive due to the large size of the system matrix and the presence of the matrix inverse. It is well known that 𝒫^T𝒫 can be well approximated as shift invariant for objects inside the scanner FOV or by a BCCB (block circulant with circulant blocks) matrix.^20,23 The BCCB matrix can be diagonalized by a 2D FFT matrix (Kronecker product of the 1D FFT matrix) with eigenvalues given by the Fourier transform of one column of the matrix with an appropriate phase shift. With the shift-invariant approximation, Eq. (13) becomes an efficient algorithm

$${\displaystyle {\hat{\mathit{f}}}^{\left(k\right)}={\mathit{\ell }}^{\left(k\right)}*\left[\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}{\mathcal{D}}^T{\mathcal{P}}^T\left({\mathit{g}}_m-{\mathit{r}}_m\right)\right],}$$ (15)

where ${\mathit{\ell }}^{\left(k\right)}$ is a variant Landweber filter. From Appendix B, the frequency response $L^{\left(k\right)}\left(\mathit{\rho }\right)$ with image-space resolution modeling is given by

$${\displaystyle L^{\left(k\right)}\left(\mathit{\rho }\right)=\frac{\mathcal{F}\left[\frac{\mathit{I}-{\left(\mathit{I}-\lambda \sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\hspace{0.17em}{\mathcal{H}}_m\right)}^k}{\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}\hspace{0.17em}{\mathcal{H}}_m}\right]{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\frac{\sqrt{M}\mathit{\rho }}{{\left|R\left(\sqrt{M}\mathit{\rho }\right)\right|}^{2}D\left(\mathit{\rho }\right)}\left[\mathbf{1}-{\left(\mathbf{1}-\lambda \frac{{\left|R\left(\sqrt{M}\mathit{\rho }\right)\right|}^{2}D\left(\mathit{\rho }\right)}{\sqrt{M}\mathit{\rho }}\right)}^k\right],}$$ (16)

where ℱ is the 2D Fourier transform matrix, e_j is the jth unit vector, $D\left(\mathit{\rho }\right)$ is given by Eq. (B15) in Appendix B, and $R\left(\mathit{\rho }\right)$ is the frequency response of the image-space blurring. With the shift-invariant approximation, Eq. (15) gives an efficient super-resolution reconstruction with its polyphase block diagram shown in Fig. 2(b).

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.

The super-sampling Landweber filter Eq. (16) has a tuning parameter k that is equivalent to the number of iterations in the Landweber algorithm. It is a product of the ramp filter and the Landweber filter; it is similar to, but different from, the Metz filter which was used by King et al., to boost the midfrequency content while depressing high-frequency noise in nuclear medicine imaging.²⁴ When M = 1, Eq. (16) becomes a Landweber-windowed ramp filter, and this filter was also used in an FBP reconstruction with noise characteristics similar to that of an iterative Landweber algorithm by Zeng.²⁵

3. VARIANT RECONSTRUCTIONS

3.A. Separate super-sampling resolution modeling

Resolution modeling in tomographic reconstruction by incorporating the physics and resolution-blurring effects of the PET acquisition can improve the spatial resolution of the reconstructed image.¹⁷ However, resolution modeling makes the tomographic reconstruction more ill-posed and leads to notable edge/ringing artifacts. These issues can be mitigated by super-sampling acquisitions. Using image-space resolution modeling 𝒫 = 𝒢ℛ, we can rewrite Eq. (2) as

$${\displaystyle {\overline{\mathit{g}}}_m=\mathcal{G}\mathcal{R}\mathcal{D}{\mathcal{T}}_m\mathit{f}+{\mathit{r}}_m,m=0,1,\dots ,M-1.}$$ (17)

Instead of reconstructing f from g_m directly, we can divide Eq. (17) into two problems: tomographic reconstruction and super-sampling resolution modeling,

$${\displaystyle {\overline{\mathit{g}}}_m=\mathcal{G}{\mathit{z}}_m+{\mathit{r}}_m,}$$ (18)

$${\displaystyle {\mathit{z}}_m=\mathcal{R}\mathcal{D}{\mathcal{T}}_m\mathit{f},m=0,1,\dots ,M-1.}$$ (19)

Since the tomographic reconstruction problem is well-posed or only mildly ill-posed, we can use an existing reconstruction package, e.g., MLEM or OS-EM , to reconstruct multiple low-resolution images z_m with very fast convergence speed. Using MLEM, the low-resolution images can be reconstructed as

$${\displaystyle {\hat{\mathit{z}}}_m^{(k+1)}=\frac{{\hat{\mathit{z}}}_m^{\left(k\right)}}{{\mathcal{G}}^T\mathbf{1}}{\mathcal{G}}^T\left[\frac{{\mathit{g}}_m}{\mathcal{G}{\mathit{z}}_m^{\left(k\right)}+{\mathit{r}}_m}\right],m=0,1,\dots ,M-1.}$$ (20)

The reconstruction with resolution modeling is a highly ill-posed problem because high-frequency information is highly attenuated in the blurring process;^17,26 however, we have the multiple low-resolution images with different shifts to provide complementary information to combat the ill-posed nature of the reconstruction with resolution modeling and reduce the aliasing errors compared to only a single acquisition. Though sinogram-space resolution modeling can be used, here we adopt image-space resolution modeling because it is more straightforward to incorporate into the super-sampling reconstruction.

After modifying Eq. (6), we can reconstruct a high-resolution image from z_m using the MLEM algorithm as

$${\displaystyle {\hat{\mathit{f}}}^{(k+1)}={\hat{\mathit{f}}}^{\left(k\right)}\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}{\mathcal{D}}^T{\mathcal{R}}^T\left(\frac{{\mathit{z}}_m}{\mathcal{R}\mathcal{D}{\mathcal{T}}_m{\mathit{f}}^{\left(k\right)}}\right).}$$ (21)

The blurring matrix can always be normalized such that each column’s sum is one, ℛ^T1 = 1 (i.e., the volume under each PSF in the positive half-space is one). One can then verify that the sensitivity image satisfies $\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}{\mathcal{D}}^T{\mathcal{R}}^T\mathbf{1}=\mathbf{1}$ and is omitted in Eq. (21). For spatial-invariant blurring ℛf = κ*f, Eq. (21) can be simplified as

$${\displaystyle {\hat{\mathit{f}}}^{(k+1)}={\hat{\mathit{f}}}^{\left(k\right)}\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}\left[\overline{\mathit{\kappa }}*\left(\frac{{\mathit{z}}_m}{\mathit{\kappa }*\left({\mathcal{T}}_m{\mathit{f}}^{\left(k\right)}\downarrow M\right)}\right)\uparrow M\right],}$$ (22)

where $\overline{\mathit{\kappa }}\left(x,y\right)=\mathit{\kappa }\left(-x,-y\right)$ and its frequency response is the complex conjugate of that of κ and is given by Eq. (B6).²² Here we used ↓M and ↑M, as alternatives to 𝒟 and 𝒟^T, to denote the down- and upsampling by a factor of M, respectively. Similar to Eq. (8), the high-resolution image can be reconstructed from z_m using the Landweber algorithm as

$${\displaystyle {\hat{\mathit{f}}}^{(k+1)}={\hat{\mathit{f}}}^{\left(k\right)}+\lambda \sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}{\mathcal{D}}^T{\mathcal{R}}^T\left[{\mathit{z}}_m-\mathcal{R}\mathcal{D}{\mathcal{T}}_m{\mathit{f}}^{\left(k\right)}\right].}$$ (23)

After examining the above Landweber method, one can find it is similar to, but different from, the IBP method.¹² With spatial-invariant blurring, Eq. (23) can be simplified as

$${\displaystyle {\hat{\mathit{f}}}^{(k+1)}=\left({\mathit{e}}_0-\lambda \mathit{\varphi }\right)*\hspace{0.17em}{\hat{\mathit{f}}}^{\left(k\right)}+\lambda \sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}\left[\overline{\mathit{h}}*z_m\uparrow M\right].}$$ (24)

Similar to Eq. (B14), the frequency response $\Phi \left(\mathit{\rho }\right)$ of ϕ can be expressed by the frequency responses $R\left(\mathit{\rho }\right)$ and $D\left(\mathit{\rho }\right)$ as

$${\displaystyle \Phi \left(\mathit{\rho }\right)=\frac{\mathcal{F}\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}{\mathcal{D}}^T{\mathcal{R}}^T\mathcal{R}\mathcal{D}{\mathcal{T}}_m{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}={\left|R\left(\sqrt{M}\mathit{\rho }\right)\right|}^{2}D\left(\mathit{\rho }\right).}$$ (25)

After modifying Eq. (15), we can reconstruct high-resolution ${\hat{\mathit{f}}}^{\left(k\right)}$ from low-resolution images z_m using the BPF-like method

$${\displaystyle {\hat{\mathit{f}}}^{\left(k\right)}={\mathit{\ell }}^{\left(k\right)}*\left[\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}\left(\overline{\mathit{h}}*{\mathit{z}}_m\uparrow M\right)\right],}$$ (26)

where the frequency response $L^{\left(k\right)}\left(\mathit{\rho }\right)$ of ${\mathit{\ell }}^{\left(k\right)}$ is

$${\displaystyle L^{\left(k\right)}\left(\mathit{\rho }\right)=\frac{\mathbf{1}-{\left(\mathbf{1}-{\left|R\left(\sqrt{M}\mathit{\rho }\right)\right|}^{2}D\left(\mathit{\rho }\right)\right)}^k}{{\left|R\left(\sqrt{M}\mathit{\rho }\right)\right|}^{2}D\left(\mathit{\rho }\right)}.}$$ (27)

3.B. Reconstructions without downsampling

The super-sampling data acquisition model in Eq. (2) can lead to efficient reconstruction algorithms with reduced size of the tomographic matrix 𝒫. In some imaging scenarios, one may want to further improve the image quality of the reconstructed image at a high computational cost. Here, we provide a trade-off between computing efficiency and image quality by choosing a different factor in the downsampling matrix 𝒟. One can choose a downsampling factor that is smaller than the number of shifting positions M and compute a larger high-fidelity tomographic matrix 𝒫 to improve the image quality. In the extreme case, one can choose unity as the factor, i.e., no downsampling with 𝒟 = ℐ. The MLEM, Landweber, and BPF-like methods in Eqs. (6), (8), and (15) can be used to reconstruct images with improved image quality using downsampling matrix 𝒟 with a small factor and a fine tomographic matrix 𝒫. The largest eigenvalue σ_max and the variant Landweber filter, respectively, in Eqs. (10) and (16) need corresponding changes according to the different factors in the downsampling matrix 𝒟. In the case 𝒟 = ℐ, σ_max can be approximated by

$${\displaystyle {\sigma }_{max}\approx \underset{j}{max}\left({\mathbf{1}}^T{\mathcal{A}}^T\mathcal{A}{\mathit{e}}_j\right)=M\underset{j}{max}\left({\mathbf{1}}^T{\mathcal{P}}^T\mathcal{P}{\mathit{e}}_j\right),}$$ (28)

and the frequency response of the variant Landweber filter becomes

$${\displaystyle L^{\left(k\right)}\left(\mathit{\rho }\right)=\mathit{\rho }/M\left[\mathbf{1}-{\left(\mathbf{1}-\lambda M\mathit{\rho }\right)}^k\right].}$$ (29)

4. SIMULATIONS AND RESULTS

To assess the performance of the super-sampling methods in PET, we performed simulated reconstructions with a generic 2D PET system. The sinogram has 139 angles uniformly spaced over 180° and 65 radial bins per angle with bin size of 2 mm covering an FOV of 146 mm based on a small animal PET scanner. In this study, we simulated a simple tomographic system with system matrix generated analytically using a strip-integral model with 2 mm LOR width and 2 mm center-to-center spacing (accounting for finite crystal with finite detector resolution). Scatter, randoms, and attenuation were not included in the simulation. We included accurate object shifting, down/up sampling, and resolution modeling based on a Gaussian kernel. We generated the downsampling matrix 𝒟 with different factors by averaging small image blocks of different sizes and the blurring matrix ℛ with a 2D Gaussian kernel of 1 mm FWHM to simulate image-space blurring effects, such as noncollinearity and positron range.

The image FOV is 128 × 128 mm², and the pixel size can be 1.0 and 0.5 mm depending on the size of the reconstructed images. We computed the variant Landweber filter for an impulse at the center of the FOV and compared it with the analytic approximation Eq. (16). Figure 3 shows the comparisons of the components of the computed Landweber filter and its analytic approximation. The downsampling factor was 2 × 2, the Landweber parameter k was 1024, and the image size was 128 × 128. We selected relaxation factor λ = 1/σ_max = 4.3 × 10⁻⁵ by calculating the largest eigenvalue σ_max using Eq. (10). The comparisons showed good agreement between the computed filter and the analytic formulas. The slight deviation between Figs. 3(d) and 3(h) is due to the finite angular sampling and LOR width.

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.

We performed computer simulations to evaluate the super-resolution methods. A resolution phantom with uniform background of 110 mm was used in the simulation. The phantom consists of a large number of rods with diameters of 1.2, 1.6, 2.4, and 3.2 mm. The rods are evenly spaced with center-to-center distance twice their diameters. The contrast of hot rods to background is 4:1. We digitized the phantom into 1024 × 1024 with pixel size of 0.125 mm for simulating the data acquisitions. The phantom was moved to different positions (1² and 2²) over a 1 × 1 mm² region for super-sampling acquisitions. For example, 2² indicates 4 data acquisitions with positions shifted (0, 0), (−0.5, 0), (0, − 0.5), and (−0.5, − 0.5) mm. The digitized phantom was shifted into different positions and blurred with a 2D Gaussian kernel of 1 mm FWHM before the forward projection. Then the shifted and blurred phantom was projected into sinograms; Poisson noise was added to the sinograms with expected total counts of 100 M over all positions.

For image reconstruction, we computed two versions of the geometric projection matrix using the strip-integral model with corresponding pixel sizes of 1.0 and 0.5 mm, which we call coarse matrix and fine matrix, respectively. We also generated two corresponding versions of image-space blurring matrix. We applied the MLEM, Landweber, and BPF-like algorithms to reconstruct the images from the super-sampled sinograms. The initial image is a uniform image for the MLEM reconstruction and is a zero image for the Landweber and BPF-like reconstructions. The images were reconstructed using both the coarse and the fine matrices for efficient reconstruction and improved image quality, respectively. The downsampling matrices with different factors were used for the efficient reconstructions using the coarse system matrix. For these coarse reconstructions, the images were reconstructed with different sizes from different super-sampling data sets, e.g., the 128 × 128 images with pixel size of 1 mm and the 256 × 256 images with pixel size of 0.5 mm were reconstructed from 1 data set and 4 data sets, respectively. For the reconstructions using the fine system matrix, all images were 256 × 256 with pixel size of 0.5 mm. We ran up to 4096 iterations for all reconstructions to ensure that convergence can be achieved for all reconstructions.

Figure 4 shows the comparisons of the reconstruction types using the coarse and fine system matrices with 1 data set and 4 data sets for visual inspection. In Fig. 4(a), the images sizes are 128 × 128 and 256 × 256 for the left and right images, respectively; the coarse system matrix and downsampling matrix were used in the reconstructions for efficiency. In Fig. 4(b), the images were reconstructed in 256 × 256; the fine system matrix without downsampling were used to improve image quality at the cost of more computation. From Fig. 4, we see that the images reconstructed from different methods have different noise textures. We selected the uniform background within the 110 mm disk after excluding the hot-rod regions as the background ROI. The background variability, as an index of background noise, was calculated from the standard deviation of the pixel values in the background ROI. The images in Fig. 4 have approximately matched background variability of 0.2 by selecting different numbers of iterations (iteration-number equivalent parameter k for the BPF-like reconstructions). We see from Fig. 4 that the MLEM reconstructs are better than the Landweber and the BPF-like reconstructions in terms of resolving the small hot rods with matched background variability. The BPF-like method produces the least appealing images with some background artifacts; however, it is the most efficient method that can reduce the reconstruction time by a few orders of magnitudes depending on the total number of iterations. For all three methods, the reconstructions from the 4 super-sampled data sets yield better images in terms of resolving the small hot rods than that from the 1 data set at the same count level.

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.

For a more quantitative comparison, we used contrast recovery coefficient (CRC) versus background variability as image-quality metric to compare the three types of reconstructions with different super-sampled data sets. The CRC is defined as

$${\displaystyle \mathrm{CRC}=\frac{C_{\mathrm{hot}}/C_{\mathrm{bkgd}}-1}{a_{\mathrm{hot}}/a_{\mathrm{bkgd}}-1},}$$ (30)

where C_hot and C_bkgd are the average activity measured in the reconstructed images in the signal and background ROIs, and a_hot/a_bkgd is the ratio of the true activities in the signal and background ROIs.²⁷ We calculated the CRCs for the four types of hot rods with different diameters. The signal ROIs were the hot-rod regions and fractional weightings were assigned to pixels that are partially inside the signal ROIs. The calculated CRCs versus background variability for the three types of reconstructions using the coarse and the fine system matrix are shown in Figs. 5 and 6, respectively. There are 32 markers in each plot and each marker represents 128 iterations. The CRCs shown in Figs. 5 and 6 were calculated from the two types of hot rods with diameters of 1.6 and 2.4 mm, respectively. Without surprise, we can see from Figs. 5 and 6 that the MLEM reconstructions with both the coarse and fine system matrices are better than the Landweber reconstructions, and the Landweber reconstructions are better than the BPF-like reconstructions by comparing the CRC with fixed background variability level. The reconstructions with 4 super-sampled data sets always have higher CRCs than that with 1 data set at a fixed background variability and the same count level. After comparing Figs. 5(a) and 6(a), we see that the reconstructions of the 1.6 mm hot rods using the fine system matrix have higher CRCs and better noise performance than the reconstructions using the coarse system matrix, especially, the MLEM and Landweber reconstructions with 4 data sets.

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.

To investigate the effect of different count levels for the super-sampling reconstructions, we performed reconstructions using the MLEM algorithm with different expected total counts of 4, 40, and 100M. For the super-sampling case, the total counts are evenly distributed among the 4 data sets, e.g., each data set has approximately 10M counts when the number of expected total counts is 40M. To fully demonstrate the performance improvement of the super-sampling reconstruction, we used the fine system matrix, i.e., the same matrix for the reconstructions shown in Fig. 4(b). The comparison of reconstructed images from 1 and 4 data set(s) with different total counts are shown in Fig. 7 for visual inspection. The second row is the same as the top row in Fig. 4(b). The background variabilities for different count levels are proportional to the noise level based on the Poisson statistics; i.e., the background variabilities are 0.63, 0.2, and 0.13 for the expected total counts 4, 40, and 100M, respectively. The images from 1 and 4 data set(s) have approximately matched background for each count level by selecting different iteration numbers.

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.

We show in Fig. 8 the calculated CRCs versus background variability for the reconstructions with different count levels for a quantitative comparison. All curves for the reconstructions with different counts are well separated between the 4 data sets and 1 data set, which means that the super-sampling reconstructions always outperform the reconstructions with 1 data set for all count levels. However, the CRC difference at fixed background variability becomes small for the low count level of 4M. One interpretation of Fig. 8 is that the CRC versus background variability curve for the low count case is approximately a stretched version of that for the high count case. There is larger CRC difference at a higher background variability after more iterations; however, the reconstructed image can be noisy and may not be very useful for visual diagnostics because the resolution improvement is buried in the background noise. To control background noise for the low count reconstruction, we normally stop at an earlier iteration, and there can be limited resolution improvement of the super-sampling reconstruction with low count. So the resolution improvement of super-sampling reconstruction can be affected by the count levels, and in general, a larger resolution improvement can be obtained with more counts and more iterations.

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.

5. DISCUSSION

We used Eq. (2) to model the physical process of the super-sampling data acquisition. The tomographic projector 𝒫 and shifting operator 𝒯_m are essential building blocks. 𝒟 can have different downsampling factors in simulations and reconstructions; it downsamples the shifted object 𝒯_mf to match the input size of the tomographic projector 𝒫. When the object is continuous, 𝒟 can be considered as a continue-to-discrete operator, which performs the Nyquist–Shannon sampling and generates a digitized approximation of the continue object. With image-space resolution modeling, Eq. (2) naturally becomes Eq. (17). Using the Nobel identity in Eq. (B10), we have ℛ𝒟 = 𝒟ℛ^h, where ℛ^h is the corresponding resolution-modeling matrix with fine grid. Using the fine matrix ℛ^h, we have a modified super-sampling model,

$${\displaystyle {\overline{\mathit{g}}}_m={\mathcal{H}}_m^h\mathit{f}+{\mathit{r}}_m=\mathcal{G}\mathcal{D}{\mathcal{R}}^h{\mathcal{T}}_m\mathit{f}+{\mathit{r}}_m,m=0,1,\dots ,M-1.}$$ (31)

Most derived reconstruction formulas can be applied to the modified model by just replacing ℋ_m with ${\mathcal{H}}_m^h$ or replacing ℛ𝒟 with 𝒟ℛ^h. For example, Eq. (23) becomes the IBP method¹² with the modified model.

Sinogram-space super-sampling can also be applied to improve the spatial resolution. It is possible to manipulate the LOR detection process and acquire different data sets with different sinogram manipulations. In general, we have the following sinogram-space super-sampling model,

$${\displaystyle {\overline{\mathit{g}}}_m=\mathcal{D}{\mathcal{S}}_m\mathcal{P}\mathit{f}+{\mathit{r}}_m,m=0,1,\dots ,M-1,}$$ (32)

where 𝒮_m denotes the sinogram-manipulation operator. A perfect example of the sinogram-space super-sampling is the collimated-PET method.^28,29 A PET collimator is designed to partially mask detector crystals to detect LORs within fractional crystals. Multiple collimator-encoded sinograms are measured with different collimation configurations. The PET collimation can be modeled by 𝒟𝒮_m. The collimated-PET is a sinogram super-sampling method, and it is an effective technique to improve spatial resolution with not only increased linear and angular samplings but also the reduced effective crystal width. With sinogram-space resolution modeling 𝒫 = ℬ𝒢, we can divide Eq. (32) into two problems

$${\displaystyle {\overline{\mathit{g}}}_m=\mathcal{D}{\mathcal{S}}_m\mathcal{B}\mathit{y}+{\mathit{r}}_m,m=0,1,\dots ,M-1,}$$ (33)

$${\displaystyle \mathit{y}=\mathcal{G}\mathit{f}.}$$ (34)

So, we can solve the high-resolution sinogram y from Eq. (33) first, and then perform the reconstruction from the high-resolution sinogram y to reconstruct image f using conventional MLEM or OS-EM. We have developed an LOR-interleaving reconstruction to efficiently solve Eq. (33) for collimated PET.²⁹

Resolution modeling in tomographic reconstruction can also boost spatial resolution of the reconstruction images. A highly ill-posed inverse problem needs be solved to boost the spatial resolution, which can produce Gibbs artifacts near the edge of the object.^26,30 Super-sampling increases the spatial sampling naturally, so it can increase spatial resolution by reducing the resolution-degradation factors caused by undersampling, e.g., the block effect in PET imaging with discrete crystals. Resolution modeling can be incorporated into super-sampling reconstructions to further improve the spatial resolution. Super-sampled data sets from different positions provide complementary information of the object, which can mitigate the ill-posedness and make the reconstruction more stable compared with a single data set. The spatial-invariant resolution modeling was implemented in the simulated super-sampling reconstructions, and the edge artifacts cannot be fully eliminated with super-sampling. We can still observe some edge artifacts in the MLEM reconstructions from both 1 data set and 4 data sets using the coarse system matrix in Fig. 4(a).

We used the super-sampling scheme of uniformly shifting the object in small steps over a unit region (1 × 1 mm²) in the simulated reconstructions to demonstrate super-sampling can improve spatial resolution using the reconstruction algorithms. We can obviously see from Fig. 4 the improved spatial resolution in the reconstructions from 4 data sets compared to the reconstructions from 1 data set. We also performed reconstructions with 16 data sets, and the image quality can be further improved compared to the reconstructions from 4 data sets. However, the improvement is smaller than the improvement from 4 data sets to 1 data set. This implies that the most benefits can be obtained with only a few data sets from a limited amounts of motion positions. It should be stressed that the super-sampling scheme (shifting step and grid) is not optimized; super-sampling optimization is our further work and is expected to further improve the spatial resolution. In deriving the BPF-like algorithm, we assumed that the projector–backprojector 𝒫^T𝒫 is shift invariant; however, it is not exactly shift invariant, especially at locations close to the FOV boundary. Our simulations showed that the BPF-like super-resolution reconstruction has pronounced background artifacts, and the noise textures are different from that of the MLEM and Landweber reconstructions. Smoothing can be used to reduce the background artifacts, but it can degrade the spatial resolution. We derive the BPF-like algorithm as an alternative to the iterative algorithms since it can significantly reduce the reconstruction time.

The proposed algorithms can find different application based on the balance between improving spatial resolution and keeping the reconstruction time low. We implemented the super-resolution methods with spatially invariant resolution modeling for a generic 2D PET. It is straightforward to implement the MLEM and Landweber super-resolution algorithms for 3D PET reconstructions. The BPF-like is developed for 2D PET reconstruction; and it can be combined seamlessly with the Fourier rebinning method³¹ to provide 3D PET reconstructions. The super-sampled data sets can be efficiently stored in list-mode format,³² and the list-mode version of the MLEM and Landweber super-resolution algorithms can be implemented to reconstruct the multiple data sets in list-mode format.³³ The computational cost for shifting, down/up sampling, and shift-invariant blurring in the MLEM super-resolution reconstruction is negligible compared with that of tomographic forward and backward projection. So the speed of the MLEM super-resolution reconstruction is comparable to that of the conventional MLEM reconstruction. The MLEM super-resolution reconstruction can be accelerated by applying the ordered subsets and rescaled block-iterative strategies, similarly as proposed for MLEM reconstructions.^34,35 Furthermore, the relatively independent data structures from multiple data sets allow efficient parallel implementation that can take advantage of the massively parallel CPU or GPU processors.

The super-sampling techniques can be applied to clinical PET scanners to improve the spatial resolution, e.g., stepping of a deliberately misaligned patient.³⁶ A typical PET scanner acquires data of a patient from several overlapped bed positions, with the bed aligned with the scanners axial direction. By introducing a vertical stepping and a near-continuous axial translation, multiple data sets, each with a unique position relative to the scanners LORs, are acquired. High-resolution images can be reconstructed from these super-sampled data sets using the proposed super-resolution algorithms. In principle, the nonrigid-body motion of the object can also be implemented in this reconstruction framework for modeling realistic unintentional motion during acquisition for super-resolution reconstruction with motion correction. The spatially variant resolution modeling, physics effect corrections (attenuation, scatter and random corrections, crystal normalization) are readily incorporated in the MLEM and Landweber super-resolution algorithms to further improve the image quality of the reconstructed images from a 3D PET with super-sampling acquisition, which is also our future work. The super-sampling reconstructions with attenuation correction require the attenuation–correction factors for each position, which can be computed by either calculating the forward projection of the shifted attenuation image or transforming the precalculated attenuation–correction factors for a reference position. Similarly, only one single-scatter simulation is required since the scatter distribution is unchanged relative to the patient body and then its transformation can be determined. Finally, random events can be estimated from the delayed-channel acquisition. Attenuation factors, scatter, and random events can be naturally incorporated in the forward projection model and corrected in the reconstruction algorithms.

Super-sampling reconstruction can improve the spatial resolution compared to regular reconstruction at the same total counts, and the improvement is higher at larger total counts. The super-sampling techniques with the proposed super-resolution reconstruction algorithms can be very useful for imaging applications that demand a higher spatial resolution than the scanner otherwise offers. The super-sampling techniques can find applications in the ROI imaging by adding a super-resolution mode in a future PET scanner with builtin hardware to support super-sampling acquisition and the implementation of the proposed algorithms to reconstruct images with improved spatial resolution.³⁷ To support the super-sampling acquisition, a smart patient bed allowing accurate translations in three perpendicular axes and a synchronization mechanism between data acquisition and bed motion are required. One super-sampling scheme is the stepping of a deliberately misaligned bed. The synchronization mechanism, e.g., sending digital pulses to PET scanner and recording them as control-events in list-mode data, can separate different super-sampled data sets from different positions. The position for each data set, controlled by the smart bed, is accurately known (positions may be predetermined or optimized). The proposed super-resolution algorithm can be employed to reconstruct a higher resolution image than offered by any single data set.

6. CONCLUSION

In this paper, we developed a unified reconstruction framework that can reconstruct super-resolution images from the super-sampled data sets with known acquisition motion. We extended MLEM and Landweber algorithms to reconstruct high-resolution images from multiple super-sampled low-resolution data sets. We also derived a BPF-like method for super-sampling reconstruction. Furthermore, we explored variant reconstructions: the separate super-sampling resolution-modeling reconstruction and the reconstruction without downsampling to improve image quality at the cost of more computation. We evaluated the three types of algorithms with different super-sampled data sets based on simulated reconstructions of a resolution phantom with different count levels. We observed that all three algorithms can significantly and consistently achieve increased CRCs and reduce background artifacts with super-sampled data sets compared with a single data set at the same count levels. The MLEM method achieves better image quality than the Landweber method, which in turn achieves image quality better than the BPF-like method. The reconstructions from super-sampled data sets using a fine system matrix yield improved image quality compared with the reconstructions using a coarse system matrix. Super-sampling reconstructions with different count levels showed that the more spatial-resolution improvement can obtained with higher count at a larger iteration number. In summary, 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 ROI PET imaging applications.

APPENDIX A: DOWNSAMPLING OPERATOR AND ITS ADJOINT

Downsampling and upsampling are extensively used in super-sampling reconstruction. Conventionally downsampling keeps one sample out of a block and discards the remaining samples. This method can be less robust in super-sampling reconstruction since it discards the other samples.³⁸ Here we use downsampling by averaging, e.g., downsample by a factor of 2 in 1D has matrix form,

$${\displaystyle \left[\begin{array}{cccccc}\hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \frac{1}{2}\hfill & \hfill \frac{1}{2}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \frac{1}{2}\hfill & \hfill \frac{1}{2}\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill \\ \hfill \hfill \end{array}\right]\left[\begin{array}{c}\hfill ⋮\hfill \\ \hfill x\left[0\right]\hfill \\ \hfill x\left[1\right]\hfill \\ \hfill x\left[2\right]\hfill \\ \hfill x\left[3\right]\hfill \\ \hfill ⋮\hfill \end{array}\right]=\left[\begin{array}{c}\hfill ⋮\hfill \\ \hfill \frac{x\left[0\right]+x\left[1\right]}{2}\hfill \\ \hfill \frac{x\left[2\right]+x\left[3\right]}{2}\hfill \\ \hfill ⋮\hfill \end{array}\right].}$$ (A1)

In the spatial domain, we can write the downsample $y\left[n\right]$ from $x\left[n\right]$ as

$${\displaystyle y\left[n\right]=\mathcal{D}x\left[n\right]=\frac{1}{M}\sum _{m=0}^{M-1}x\left[nM+m\right].}$$ (A2)

In the frequency domain, the Fourier transform $Y\left(\omega \right)$ of the downsample $y\left[n\right]$ can be given by

$${\displaystyle Y\left(\omega \right)=\sum _{n=-\infty }^{+\infty }\left(\frac{1}{M}\sum _{\ell =0}^{M-1}x\left[nM+\ell \right]\right)e^{-j\omega n}=\frac{1}{M}\sum _{\ell =0}^{M-1}\sum _{k=-\infty }^{+\infty }x\left[k+\ell \right]\left(\frac{1}{M}\sum _{m=0}^{M-1}{e}^{j2\pi mk/M}\right)e^{-j\omega (k/M)}=\frac{1}{M^2}\sum _{\ell =0}^{M-1}\sum _{m=0}^{M-1}\sum _{k=-\infty }^{+\infty }x\left[k+\ell \right]e^{-j\left(\omega -2\pi m/M\right)k}=\frac{1}{M^2}\sum _{m=0}^{M-1}\left(\sum _{\ell =0}^{M-1}{e}^{j\left(\omega -2\pi m/M\right)\ell }\right)X\left(\frac{\omega -2\pi m}{M}\right)=\frac{1}{M^2}\sum _{m=0}^{M-1}\frac{1-e^{j\omega }}{1-e^{j\left(\omega -2\pi m\right)/M}}X\left(\frac{\omega -2\pi m}{M}\right).}$$ (A3)

From Eq. (A3), we see that this downsample is equivalent to filtering the input signal with a rectangle filter of length M then downsampling by removing the other samples.³⁸ And the frequency response of the rectangular average filter is $\left(1-e^{j\omega M}\right)/\left(M-M{e}^{j\omega }\right)$.

Interestingly, upsampling is the adjoint of downsampling, which is similar to the backward projection is the adjoint of the forward projection in iterative tomographic reconstruction. Conventionally, upsampling with zero insertion is used, we use upsampling with replication, e.g., upsample by a factor of 2 in 1D has matrix form,

$${\displaystyle \frac{1}{2}\left[\begin{array}{cccc}\hfill \ddots \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill \ddots \hfill & \hfill 1\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill 1\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill 1\hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill 1\hfill & \hfill \ddots \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \ddots \hfill \\ \hfill \hfill \end{array}\right]\left[\begin{array}{c}\hfill ⋮\hfill \\ \hfill y\left[0\right]\hfill \\ \hfill y\left[1\right]\hfill \\ \hfill ⋮\hfill \end{array}\right]=\left[\begin{array}{c}\hfill ⋮\hfill \\ \hfill \frac{1}{2}y\left[0\right]\hfill \\ \hfill \frac{1}{2}y\left[0\right]\hfill \\ \hfill \frac{1}{2}y\left[1\right]\hfill \\ \hfill \frac{1}{2}y\left[1\right]\hfill \\ \hfill ⋮\hfill \end{array}\right].}$$ (A4)

The upsampling matrix is just the transpose of the downsampling matrix in Eq. (A1). We can write the upsample $x\left[n\right]$ from $y\left[n\right]$ as

$${\displaystyle x\left[n\right]={\mathcal{D}}^{T}y\left[n\right]=\frac{1}{M}y\left[⌊\frac{n}{M}⌋\right],}$$ (A5)

where the floor function $⌊\cdot ⌋$ gives the largest integer less than or equal to its argument. In frequency domain, the Fourier transform $X\left(\omega \right)$ of the upsample $x\left[n\right]$ can be given by

$${\displaystyle X\left(\omega \right)=\frac{1}{M}\sum _{n=-\infty }^{\infty }y\left[⌊\frac{n}{M}⌋\right]e^{-j\omega n}=\frac{1}{M}\sum _{k=-\infty }^{+\infty }\sum _{m=0}^{M-1}y\left[k\right]e^{-j\omega \left(kM+m\right)}=\frac{1}{M}\left(\sum _{m=0}^{M-1}{e}^{-j\omega m}\right)\left(\sum _{k=-\infty }^{+\infty }y\left[k\right]e^{-j\omega kM}\right)=\frac{1}{M}\frac{1-e^{-jM\omega }}{1-e^{-j\omega }}Y\left(M\omega \right).}$$ (A6)

One can easily verify that downsampling after upsampling gives a scalar matrix, i.e.,

$${\displaystyle \mathcal{D}{\mathcal{D}}^T=\left(\downarrow M\right)\left(\uparrow M\right)=\frac{1}{M}\mathcal{I},}$$ (A7)

which in turn implies a trigonometric identity,

$${\displaystyle \frac{1}{M^2}\sum _{m=0}^{M-1}\frac{1-cos\omega }{1-cos\left(\right(\omega -2\pi m)/M)}=1.}$$ (A8)

In general, upsampling after downsampling causes information loss or aliasing with overlapped spectra in the Fourier domain. However, it has the property of a mathematical projector

$${\displaystyle {\left({\mathcal{D}}^T\mathcal{D}\right)}^2={\mathcal{D}}^T\left(\mathcal{D}{\mathcal{D}}^T\right)\mathcal{D}=\frac{1}{M}{\mathcal{D}}^T\mathcal{D},}$$ (A9)

and it only causes a scalar effect for a upsampled signal. The down and upsampling for multidimensions can be performed separately for each dimension, which implies that the Kronecker product can be used to construct multidimensional down- and upsampling operators from the 1D operator.³⁹

APPENDIX B: FREQUENCY RESPONSE OF $\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}{\mathcal{H}}_m$

Let e_j represent the jth unit vector, and we first define the local frequency response of operator 𝒪 as

$${\displaystyle {\hat{\mathit{h}}}_j=\frac{\mathcal{F}\mathcal{O}{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j},}$$ (B1)

where ℱ denotes the 2D Fourier transform. For a shift-invariant operator, the frequency is independent of the spatial index j and $\hat{\mathit{h}}=\mathcal{F}\mathcal{O}{\mathit{e}}_j/\mathcal{F}{\mathit{e}}_j=\mathcal{F}\mathcal{O}{\mathit{e}}_0$, where e₀ is an impulse at the center of the FOV whose frequency response is one.

The tomographic projector 𝒫 is just the geometric projector 𝒢 without resolution modeling, and 𝒫 = 𝒢ℛ with image-space resolution modeling. The geometric projector 𝒢 is a digitized version of Radon transform operator. It is well known²⁰ that 𝒢^T𝒢 can be well approximated by a shift-invariant system with point spread function 1/r, i.e.,

$${\displaystyle {\mathcal{G}}^T\mathcal{G}\mathit{f}=\frac{1}{\mathit{r}}*\hspace{0.17em}\mathit{f},}$$ (B2)

where * denotes 2D convolution and vector r is a vectorization of 2D samples of r by stacking the columns similar to f. If we let u and v denote the 2D spatial frequency coordinates and define $\rho =\sqrt{u^2+v^2}$. Since the 2D Fourier transform or Hankel transform of 1/r is 1/ρ, so we have

$${\displaystyle {\mathcal{G}}^T\mathcal{G}={\mathcal{F}}^{-1}\mathrm{diag}\left[\frac{1}{\mathit{\rho }}\right]\mathcal{F}.}$$ (B3)

From Eqs. (B2) and (B3), the frequency response of 𝒢^T𝒢 can be given by

$${\displaystyle \frac{\mathcal{F}{\mathcal{G}}^T\mathcal{G}{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\frac{1}{\mathit{\rho }}.}$$ (B4)

With image-space resolution modeling 𝒫 = 𝒢ℛ, we have 𝒫^T𝒫 = ℛ^T𝒢^T𝒢ℛ. When ℛ is spatial invariant, it can be diagonalized by a 2D FFT matrix as 𝒢^T𝒢. So we have the frequency response of 𝒫^T𝒫,

$${\displaystyle \frac{\mathcal{F}{\mathcal{P}}^T\mathcal{P}{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\frac{\mathcal{F}{\mathcal{G}}^T\mathcal{G}{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}\frac{\mathcal{F}{\mathcal{R}}^T{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}\frac{\mathcal{F}\mathcal{R}{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}.}$$ (B5)

We denote $R\left(\mathit{\rho }\right)\equiv \mathcal{F}\mathcal{R}{\mathit{e}}_j/\mathcal{F}{\mathit{e}}_j$. Since ℛ is spatial invariant and real, we have

$${\displaystyle \frac{\mathcal{F}{\mathcal{R}}^T{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\frac{\mathcal{F}{\mathcal{R}}^{*}{\mathcal{F}}^{-1}\mathcal{F}{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\frac{{\left({\mathcal{F}}^{-*}\mathcal{R}{\mathcal{F}}^{*}\right)}^{*}\mathcal{F}{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\frac{{\left(\mathcal{F}\mathcal{R}{\mathcal{F}}^{-1}\right)}^{*}\mathcal{F}{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\frac{\mathrm{diag}\left[\overline{R\left(\mathit{\rho }\right)}\right]\mathcal{F}{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\overline{R\left(\mathit{\rho }\right)}.}$$ (B6)

Here, $\overline{R\left(\mathit{\rho }\right)}$ is the complex conjugate of $R\left(\mathit{\rho }\right)$. Putting Eqs. (B4) and (B6) into Eq. (B5), we have

$${\displaystyle \Gamma \left(\mathit{\rho }\right)\equiv \frac{\mathcal{F}{\mathcal{P}}^T\mathcal{P}{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\frac{{\left|R\left(\mathit{\rho }\right)\right|}^2}{\mathit{\rho }}.}$$ (B7)

The object projections are acquired with different shifting positions described by 𝒯_m. For a rectangular grid shifting ${\mathcal{T}}_{m}f\left[x,y\right]=f\left[x-m_1,y-m_2\right]$, m₁ = 0, 1, …, M₁ − 1, m₂ = 0, 1, …, M₂ − 1 in 2D. The frequency response of $\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}{\mathcal{D}}^T\mathcal{D}{\mathcal{T}}_m$ can be given by

$${\displaystyle \frac{\mathcal{F}\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}{\mathcal{D}}^T\mathcal{D}{\mathcal{T}}_m{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\frac{1}{M_1^2{M}_2^2}\frac{{sin}^2\hspace{0.17em}{M}_1\pi \mathit{u}}{{sin}^2\pi \mathit{u}}\frac{{sin}^2\hspace{0.17em}{M}_2\pi \mathit{v}}{{sin}^2\pi \mathit{v}}.}$$ (B8)

Since each dimension can be performed separately, we only prove the 1D case. For an arbitrary input 1D signal q with frequency response $Q\left(\omega \right)$, we have $\left[\mathcal{F}{\mathcal{T}}_m\mathit{q}\right]\left(\omega \right)=e^{-j\omega m}Q\left(\omega \right)$. After using Eqs. (A3) and (A6), we obtain

$${\displaystyle \left[\mathcal{F}\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}{\mathcal{D}}^T\mathcal{D}{\mathcal{T}}_m\mathit{q}\right]\left(\omega \right)=\frac{1}{M^3}\frac{1-e^{-jM\omega }}{1-e^{-j\omega }}\times \sum _{\ell =0}^{M-1}\frac{1-e^{jM\omega }}{1-e^{j\left(\omega -(2\pi \ell /M)\right)}}\left(\sum _{m=0}^{M-1}{e}^{-j(2\pi m\ell /M)}\right)Q\left(\omega -\frac{2\pi \ell }{M}\right)=\frac{1}{M^2}\frac{1-e^{-jM\omega }}{1-e^{-j\omega }}\frac{1-e^{jM\omega }}{1-e^{j\omega }}Q\left(\omega \right)=\frac{1}{M^2}\frac{{sin}^{2}M\pi u}{{sin}^2\pi u}Q\left(\omega \right),}$$ (B9)

where ω = 2πu. After performing the same operations for the other dimension, we prove Eq. (B8). After applying the Nobel identity³⁸

$${\displaystyle \left(\uparrow M\right)H\left(\omega \right)=H\left(M\omega \right)\left(\uparrow M\right)}$$ (B10)

and Eq. (B7) to Eq. (9), we have

$${\displaystyle \sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}{\mathcal{D}}^T{\mathcal{P}}^T\mathcal{P}\mathcal{D}{\mathcal{T}}_m=\mathcal{K}\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}{\mathcal{D}}^T\mathcal{D}{\mathcal{T}}_m,}$$ (B11)

where

$${\displaystyle \frac{\mathcal{F}\mathcal{K}{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\Gamma \left(\sqrt{M_1^2{\mathit{u}}^2+M_2^2{\mathit{v}}^2}\right).}$$ (B12)

After putting Eqs. (B8) and (B12) into Eq. (B11), we obtain the frequency response of $\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}{\mathcal{H}}_m$ as

$${\displaystyle \frac{\mathcal{F}\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}{\mathcal{H}}_m{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\frac{\mathcal{F}\mathcal{K}{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}\frac{\mathcal{F}\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}{\mathcal{D}}^T\mathcal{D}{\mathcal{T}}_m{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\frac{{\left|R\left(\sqrt{M_1^2{\mathit{u}}^2+M_2^2{\mathit{v}}^2}\right)\right|}^2}{\sqrt{M_1^2{\mathit{u}}^2+M_2^2{\mathit{v}}^2}}\odot \frac{1}{M_1^2{M}_2^2}\frac{{sin}^2\hspace{0.17em}{M}_1\pi \mathit{u}}{{sin}^2\pi \mathit{u}}\frac{{sin}^2\hspace{0.17em}{M}_2\pi \mathit{v}}{{sin}^2\pi \mathit{v}}.}$$ (B13)

When $M_1^2=M_2^2=M$, Eq. (B13) becomes

$${\displaystyle \frac{\mathcal{F}\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{H}}}{\mathcal{H}}_m{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}=\frac{{\left|R\left(\sqrt{M}\mathit{\rho }\right)\right|}^{2}D\left(\mathit{\rho }\right)}{\sqrt{M}\mathit{\rho }},}$$ (B14)

where

$${\displaystyle D\left(\mathit{\rho }\right)=\frac{1}{M^2}\frac{{sin}^2\sqrt{M}\pi \mathit{u}}{{sin}^2\pi \mathit{u}}\frac{{sin}^2\sqrt{M}\pi \mathit{v}}{{sin}^2\pi \mathit{v}}.}$$ (B15)

APPENDIX C: CALCULATION OF SPECTRAL RADIUS σ_max

From Eq. (B2), we know that a BCCB matrix can be diagonalized by a 2D FFT matrix with eigenvalues given by the frequency response defined in Eq. (B1). The Gram matrix 𝒜^T𝒜 can be well approximated by a BCCB matrix and all elements are non-negative. So its largest eigenvalue σ_max can be approximately given by

$${\displaystyle {\sigma }_{max}\approx \underset{j}{max}\left(max\left|\frac{\mathcal{F}{\mathcal{A}}^T\mathcal{A}{\mathit{e}}_j}{\mathcal{F}{\mathit{e}}_j}\right|\right)=\underset{j}{max}\left(max\left|\mathcal{F}{\mathcal{A}}^T\mathcal{A}{\mathit{e}}_j\right|\right)=\underset{j}{max}\left({\mathbf{1}}^T{\mathcal{A}}^T\mathcal{A}{\mathit{e}}_j\right).}$$ (C1)

In the above equation, we used $\left|\mathcal{F}{\mathit{e}}_j\right|=1$, i.e., the absolute value of the phase shift is one. We also used that the largest absolute entry of a Fourier transform of a non-negative vector is its DC component, i.e., $\left|\mathcal{F}\mathit{h}\right|\le {\mathbf{1}}^T\mathit{h}$ for a non-negative vector h. Applying Eq. (9), we have

$${\displaystyle {\mathbf{1}}^T{\mathcal{A}}^T\mathcal{A}{\mathit{e}}_j={\mathbf{1}}^T\left[\sum _{m=0}^{M-1}\underset{m}{\overset{T}{\mathcal{T}}}{\mathcal{D}}^T{\mathcal{P}}^T\mathcal{P}\mathcal{D}{\mathcal{T}}_m\right]{\mathit{e}}_j=\sum _{m=0}^{M-1}{\left(\mathcal{D}{\mathcal{T}}_m\mathbf{1}\right)}^T{\mathcal{P}}^T\mathcal{P}\left(\mathcal{D}{\mathcal{T}}_m{\mathit{e}}_j\right)=\sum _{m=0}^{M-1}{\mathbf{1}}^T{\mathcal{P}}^T\mathcal{P}\frac{1}{M}{\mathit{e}}_{j_d}={\mathbf{1}}^T{\mathcal{P}}^T\mathcal{P}{\mathit{e}}_{j_d},}$$ (C2)

where index j_d is the downsampled version of index j, and $j_d=⌊(j+m)/M⌋=⌊j/M⌋$ in the 1D case. After combining Eqs. (C1) and (C2), we prove Eq. (10).