2.1

Imaging model in PB-PCCT

In PB-PCCT, assuming the object’s attenuation of X-ray is weak, the 2D imaging model can be expressed as^{1, 8}

where vector g of size M denotes model data in PB-PCCT; vector f of size N denotes a 2D discrete image of the refractive index within the imaged object; is the discrete Laplacian operator on the image space; and system matrix of size M × N represents a discrete Radon transform from image space to data space. In this work, element h_ij of denotes the intersection between ray j and pixel i. For given corresponding to a LAR scanning configuration discussed below, we use Eq. (1) to generate model data g from numerical, realistic plant-seed phantom f on a grid of 190 x 200 square pixels of size 0.12 mm in Fig. 1a.⁹

2.2

Scan configuration

We consider a single-arc and a two-orthogonal-arc scanning configurations, ⁷ as shown in Figs. 1c and 1d, with a parallel-beam projection geometry, and the discrete Radon transform is used as the imaging model in PB-PCCT. We also assume that the former covers LAR of α_τ and that the latter includes two sub-arcs of LARs of α₁ and α₂, symmetric relative, respectively, to the y- and x-axis. In the work, without loss of generality, we consider the case with α_τ = α₁ + α₂ and α₁ = α₂. Furthermore, the detector array consists of 512 bins of size 0.046 mm. We generate noiseless data from the plant-seed phantom (i) with the single-arc configurations of α_τ = 60°, 90°, 120°, and 150°, with an angular interval of 0.5° per view, and (ii) with the two-orthogonal-arc configurations of 2α₁ = 2α₂ = 60°, 90°, 120°, and 150°. Based upon the noiseless data, we also generate noisy data which is equivalent to adding Gaussian noise to the intensity data,⁸ and the Gaussian noise has zero mean and standard deviation of 1.5% of the intensity for each detector bin.

2.3

Directional-TV reconstruction algorithm

Image reconstruction in PB-PCCT is equivalent to inverting the imaging model in Eq. (1). It is well-known that, while existing analytic algorithms can accurately reconstruct boundary images from data collected over FAR of 180°, they cannot yield accurate boundary images nor images from data collected over LARs that are considerably lower than 180°. Instead, we formulate the reconstruction problem from LAR data in PB-PCCT as an optimization problem and tailor the DTV algorithm to obtain images through solving the optimization problem.

We consider a convex optimization program containing a data-ℓ₂-norm fidelity and DTV constraints along the x- and y-directions, which is given by⁴

where vector g^[𝓜] of size M denotes discrete measured data; Φ(Δf − g^[𝓜]) denotes the ℓ₂-norm of the difference between Δf and measured data g^[𝓜]; Ψ(f) is the constraint term in the optimization program which includes three components: image DTV constraint along x-axis, image DTV constraint along y-axis, and image nonnegativity constraint; the DTV constraint applies an upper bound on an image DTV, which is defined as fi-norm of a vector obtained by calculating two-point differences of f along either x- or y-axis; and the image non-negativity constraint is to enforce non-negative values for each pixel in the image. In many of existing works,^{1, 10, 11} algorithms were developed largely for inverting in the imaging model, thus yielding the boundary image Δf, instead of image f itself. In this work, we develop and tailor the DTV algorithm to solve Eq. (2) directly for obtaining image f. Once image f is obtained, one can readily apply the Laplacian to it to obtain a boundary image, as shown in Fig. 1b. For references, we also reconstruct images by using the filtered-backprojection (FBP) algorithm.^{8, 12}

For evaluating image reconstructed, we first perform visual inspection of the images reconstructed. Additionally, we also perform a quantitative evaluation of image reconstructed from data over LAR by using the normalized root-mean-square-error (nRMSE) metric

where f^[recon] and f^[truth] denote the reconstructed and truth images. Metric nRMSE provides measures of quantitative accuracy between the reconstructed and truth images. We note that nRMSE(f^[recon]) approaches 0 as f^[recon] approaches f^[truth].

3.1

Reconstruction from noiseless data

We first conducted image reconstruction, as shown in Fig. 2, by using the DTV and FBP algorithms from noiseless data with the two-orthogonal-arc and single-arc configurations over total angular ranges α_τ = 60°, 90°, 120°, and 150°. In this case, measured data and model data are identical, i.e., g^[𝓜] = g = Δf. We first observe that FBP reconstructions from all LAR data contain significant artifacts such as leakage and distortion. It can also be observed that all DTV reconstructions of the two-orthogonal-arc configurations visually resemble the truth plant-seed phantom. However, DTV reconstructions of the single-arc configurations show visible artifacts for α_τ ≤ 90°. We also calculated nRMSEs of the DTV reconstructions and list the corresponding values in Table 1. It can be observed that the nRMSE increases as total-angular range α_τ decreases. In addition, the nRMSEs of DTV reconstructions for two-orthogonal-arc configurations are generally smaller than those of the single-arc configurations for a given α_τ.

Figure 2.

Images reconstructed by use of the FBP (rows 1&3) and DTV (rows 2&4) algorithm from noiseless data for single-arc configurations (rows 1&2) of α_τ = 150° (column 1), 120° (column 2), 90° (column 3), and 60° (column 4), respectively, and for two-orthogonal-arc configurations (rows 3&4) of α₁ = α₂ = 75° (column 1), 60° (column 2), 45° (column 3), and 30° (column 4), respectively. Display window: [0.0, 5×10^–7] AU.

Table 1.

nRMSEs of images reconstructed from noiseless data generated with single-arc and two-orthogonal-arc configurations of different ατ.

3.2

Reconstruction from noisy data

We then conducted image reconstruction with the DTV and FBP algorithms from noisy data g^[𝓜], as described in Sec. 2.2, by using the two-orthogonal-arc and the single-arc configurations covering total angular ranges α_τ ≤ 60°, 90°, 120°, and 150°, and show results in Fig. 3. Similarly to the noiseless-data study, there exist significant LAR artifacts in all FBP reconstructions. For the DTV reconstructions, in general, as the total angular coverage α_τ decreases, LAR artifacts increase in image reconstructions. We observe that DTV reconstructions with the two-orthogonal-arc configuration outperform those with the single-arc configuration in terms of artifacts reduction. This is understandable because the data model is less ill-conditioned for the former than for the latter, and thus the reconstruction of the former is less sensitive to data inconsistency, i.e., data noise, than that of the latter. In particular, DTV reconstructions with single-arc configurations considerably deteriorate for α_τ ≤ 90°, and some detailed structures cannot be identified. DTV reconstructions with two-orthogonal-arc configurations, however, can still reveal most of the structures for α_τ = 60°. We also conduct a quantitative analysis of the DTV reconstructions by computing their nRMSEs relative to the truth phantom, and show results in Table 2. We notice that, for DTV reconstructions from LAR data containing noise, the nRMSEs of reconstructions with the two-orthogonal-arc configuration are smaller than those of reconstructions with the single-arc configuration. Additionally, the nRMSE increases as the total angular range decreases.

Figure 3.

Images reconstructed by use of the FBP (rows 1&3) and DTV (rows 2&4) algorithm from noisy data for singlearc configurations (rows 1&2) of α_τ = 150° (column 1), 120° (column 2), 90° (column 3), and 60° (column 4), respectively, and for two-orthogonal-arc configurations (rows 3&4) of α₁ = α₂ = 75° (column 1), 60° (column 2), 45° (column 3), and 30° (column 4), respectively. Display window: [0.05, 0.9].

Table 2.

nRMSEs of the DTV reconstructions from noisy data generated with single-arc and two-orthogonal-arc configurations of different ατ.