2.1

LAR scanning configuration with an offset-detector

In the work, the X-ray transform⁵ in a discrete form is used as the imaging model, and image reconstruction is equivalent to inverting the imaging model. Clearly, the ill-conditionedness of the imaging model depends upon the LAR extent and distribution over which knowledge of the discrete X-ray transform (DXT), i.e., the imaging model, is available. Without loss of generality, we consider a circular fan-beam configuration with an offset-detector, as shown in Fig. 1a. From the head image in Fig. 1b, truncated data generated with the configuration over an FAR of 360° are shown in Fig. 1c. On the other hand, an LAR scanning configuration with an offset-detector is shown in row 1 of Fig. 2a, which consists of two orthogonal arc components, separated by 90°, indicated by arcs α₁ and α₂, where α₁ and α₂ indicate their angular ranges. We refer to this configuration as a two-orthogonal-arc (TOA) LAR configuration, and thus as a single-arc (SA) LAR configuration if α₁ = 0 or α₂ = 0 as displayed in row 2 of Fig. 2a. We investigated image reconstruction from data, as shown in Figs. 2b-2d, collected over the TOA and SA scanning configurations, of the same total angular range, with an offset-detector.

Figure 1.

(a) A circular fan-beam scanning configuration with an offset-detector in which O and L denote the rotation center and the length of the detector-offset. (b) The numerical head phantom, and (c) data collected over an FAR of 360°. It can be observed that the data are truncated in the right-hand side of the data space in (c). Display windows are [0.15, 0.3] cm⁻¹ and [0, 3], respectively.

Figure 2.

(a) Two-orthogonal-arc (row 1) and single-arc (row 2) scanning configurations with an offset-detector. Data generated with the two-orthogonal-arc scanning configurations (row 1) over LARs (α₁, α₂) = (75°, 75°) (b), (67.5°, 67.5°) (c), and (60°, 60°) (d), and with the single-arc scanning configurations (row 2) over LARs α_τ = 150° (b), 135° (c), and 120° (d). Display window is [0, 3].

In the circular fan-beam configuration, the distances from the X-ray source to the detector and to the rotation center are 60 and 40 cm, respectively. The detector consists of 754 elements with a size of 0.2 mm. A detector-offset L = 30 mm is used for yielding an extended FOV. The head phantom shown in Fig. 1b consists of 200×200 pixels of size 0.8×0.8 mm².

Using the head phantom, we first generate data, as shown in Fig. 1c over an FAR of 360° with an angular interval of 0.5° per view. For each view, the generated data are truncated on one side of the detector as a result of the detector-offset. We consider TOA scanning configurations with LARs (α₁, α₂) = (75°, 75°), (67.5°, 67.5°), and (60°, 60°), and use them to generate data from the head phantom, as shown in row 1 of Figs. 2b-2d. For comparison, we have also investigated image reconstruction from data of the head phantom generated over SAs with LARs α_τ = α₁ + α₂ = 150°, 135°, 120°, as shown in row 2 of Figs. 2b-2d.

In addition, using noiseless LAR data generated above, we created noisy LAR data by adding the Poisson noise, corresponding to 10⁷ noise equivalent quanta (NEQ) per detector bin in air scans, and we subsequently performed image reconstructions from the noisy LAR data.

2.2

Image reconstruction

For CT scans over TOA or SA LAR with an offset-detector, the image reconstruction can be formulated as the solution to a constrained optimization program in which a weighted data-ℓ₂ fidelity is minimized under image-DTV constraints along the x- and y-directions. The optimization program is given by

where vector g^[ℳ] of size M denotes discrete measured data acquired with two-orthogonal-arc or single-arc LAR scanning configuration; vector f of size N is a 2D discrete image; f_i is the entry i of f; the system matrix of size M × N representing the X-ray transform, with element h_j,i representing the contribution of pixel i to ray j; W is an M × M diagonal matrix in which each diagonal element represents a positive weighting factor for a corresponding X-ray measurement; ||·||_p indicates the ℓ_p-norms of the input vector for p = 1 and 2; matrices 𝒟_x and 𝒟_y of size N × N denote two-point differences along the x- and y-axes, respectively; vectors 𝒟_xf and 𝒟_yf are of size N; and parameters t₁ and t₂ depict the upper bounds on the DTV constraints along x- and y-directions.

Weighting matrix W is introduced for controlling the numerical effect of the discontinuity of truncated data on image reconstruction. Different designs of weighting matrix W may result in different reconstruction programs, leading to different reconstructions when data are inconsistent with the imaging model.

Based upon the general primal-dual optimization framework,^{6, 7} a DTV algorithm was developed previously for solving the constrained optimization program minimizing data-ℓ₂ fidelity without weighting matrix W under DTV constraints.^{3, 4, 8, 9} In this work, we tailored the DTV algorithm to solve Eq. (1) including a weighting factor W for image reconstruction from truncated data acquired with the offset-detector.

3.1

Reconstruction from noiseless data

Using the tailored DTV algorithm, we first reconstruct images of the head phantom from noiseless data generated over a number of TOAs with LARs (α₁, α₂) = (75°, 75°), (67.5°, 67.5°), and (60°, 60°), (i.e., data in row 1 of Figs. 2b-2d), and display the reconstructed images in row 1 of Fig. 3. In an attempt to compare the differences between TOA and SA LAR reconstructions from the offset-detector data (i.e., truncated data), we also perform reconstructions from noiseless data collected over several SAs of LARs α_τ = 150°, 135°, 120° (i.e., data in row 2 of Fig. 2b-2d), and show the reconstructed images in row 2 of Fig. 3. The FDK reconstructions from the truncated TOA LAR data are shown in row 3 of Fig. 3 for demonstrating the artifacts as the combined result of LAR and data truncation issues.

Figure 3.

Row 1: images reconstructed by use of the DTV algorithm from noiseless data of TOA scanning configurations over LARs (α₁, α₂) = (75°, 75°) (a), (67.5°, 67.5°) (b), and (60°, 60°) (c); row 2: images reconstructed by use of the DTV algorithm from noiseless data of SA scan configurations over LARs α_τ = 150° (a), 135° (b), and 120° (c); and row 3: images reconstructed by use of the FDK algorithm from noiseless data of TOA scanning configurations used for obtaining images in row 1. Display window is [0.0, 0.6] cm⁻¹.

It can be observed that FDK reconstructions contain significant LAR artifacts that overwhelm soft-tissue contrast of interest, and tooth structures in the reconstructed images are difficult to be discerned. DTV reconstructions from both TOA and SA data appear visually comparable to the phantom image and the LAR artifacts observed in the corresponding FDK reconstructions are significantly reduced. Although the DTV reconstructions for SA scanning configurations show clearly the tooth structures without much of the artifacts in the FDK images, there remain residual artifacts around the spine region. As the LAR α_τ is reduced from 150° to 120°, the artifacts become more significant, caused by the combination of data-truncation and LAR. Conversely, the DTV reconstructions for the TOA scanning configurations remain visually identical to the phantom image and largely free of the artifacts.

3.2

Reconstruction from noisy data

In order to investigate the robustness of the DTV reconstruction from truncated TOA and SA LAR data, we repeat the study in Sec. 3.1 with noisy data and show the results in Fig. 4. Again, for noisy LAR data considered, the DTV reconstructions for both the TOA and SA scans yield images with significantly reduced LAR artifacts as compared to the FDK reconstructions. Also, the TOA configurations appear to yield images with minimum artifacts as compared to that obtained for the SA configurations, consistent with the observation made above.

Figure 4.

Row 1: images reconstructed by use of the DTV algorithm from noisy data of two-orthogonal-arc scanning configurations over LARs (α₁, α₂) = (75°, 75°) (a), (67.5°, 67.5°) (b), and (60°, 60°) (c); row 2: images reconstructed by use of the DTV algorithm from noisy data of single-arc scanning configurations over LARs α_τ=150° (a), 135° (b), and 120° (c); and row 3: images reconstructed by use of the FDK algorithm from noisy data of two-orthogonal-arc scanning configurations used for obtaining images in row 1. Display window is [0.0, 0.6] cm⁻¹.