*3.2. Image Reconstruction of the Chest Phantom*

#### 3.2.1. Verification Study with the Chest Phantom

A study is performed to first verify that (1) the DDD algorithm can invert the nonlinear model in Equation (3) and numerically accurately recover the basis sinogram *lkj* from noiseless low- and high-kVp data of the chest phantom acquired with the FAR scan of 360◦; and (2) the DTV algorithm developed and tailored can reconstruct numerically accurate basis images and VMIs from noiseless basis sinogram. In Figure 3a, we display the VMI at 100 keV [29], along with a zoomed-in view, and show in Figure 3b their differences from the truth counterparts in Figure 2c (top row). The result confirms that the two-step method can yield accurate reconstructions from noiseless FAR data. In an attempt to demonstrate possible LAR artifacts associated with the phantom, we apply both the DTV and FBP algorithms to reconstructing images from noiseless basis sinogram acquired with a SA scan of LAR *ατ* = 30◦, and display them in Figure 3c,d, respectively. It can be observed that the two-step method can significantly reduce the LAR artifacts observed in the FBP image.

**Figure 3.** Row 1: (**a**) VMI of the chest phantom at 100 keV obtained with the two-step method from FAR data, (**b**) difference between the VMI in (**a**) and its truth in Figure 2c, and VMIs at 100 keV obtained with (**c**) the two-step method and (**d**) the FBP algorithm from noiseless data acquired over a SA of LAR 30◦; Row 2: zoomed-in views of their corresponding images in row 1. The zoomed-in area is enclosed by the rectangular box depicted in the VMI in (**a**). Display windows [0, 0.22] cm−<sup>1</sup> for columns (**a**,**c**,**d**), and [−10−4, 10−4] cm−<sup>1</sup> for column (**b**).

#### 3.2.2. Image Reconstruction from Noiseless Data Acquired with SA and TOA Scans of LARs

We subsequently apply the two-step method verified to reconstructing basis images of water and iodine from noiseless data of the chest phantom collected in SA or TOA scans of *ατ* = 20◦, 30◦, 45◦, 60◦, 90◦, 120◦, 150◦, and 180◦. In Figure 4, we display VMIs at 100 keV, along with their zoomed-in views within the ROI, reconstructed for the SA (rows 1&2 and 5&6) and TOA (rows 3&4 and 7&8) scans. It can be observed that the two-step method yields visually comparable images for these scans of LARs, revealing quantitatively possible performance upper bounds of the two-step method in accurate image reconstruction, i.e., numerically accurately inverting Equation (3), for SA and TOA scans of LARs studied in the work.

From VMIs in Figure 4, we compute nRMSEs and PCCs, which are displayed in row 1 of Figure 5. Using the method described in Section 2.5, we also estimate iodine concentrations in ROIs 1–4 indicated in the top row of Figure 2c, and plot them as functions of LARs in row 1 of Figure 6. These results reveal that, from the chest phantom noiseless data collected over the range of LARs as low as 20◦, the two-step method can yield VMIs visually and quantitatively close to the reference VMIs from FAR data of 360◦ in terms of PCC and estimated iodine concentrations. Regarding metric nRMSE, it increases as LAR decreases, mainly due to the increasing null spaces present in the system matrices of the LAR scans, while TOA scans can lower nRMSE by an order of magnitude especially for small LARs as compared to SAs of the same LAR.

**Figure 4.** VMIs (rows 1, 3, 5, and 7), along with their respective zoomed-in views (rows 2, 4, 6, and 8), of the chest phantom at 100 keV obtained from noiseless data over SAs (rows 1&2 and 5&6) and TOAs (rows 3&4 and 7&8) of LAR 20◦, 30◦, 45◦, 60◦, 90◦, 120◦, 150◦, and 180◦, respectively, by use of the two-step method. Display window: [0, 0.22] cm−1.

**Figure 5.** Metrics nRMSE and PCC, computed over VMIs of the chest phantom from noiseless data in Figure 4 (row 1) and those from noisy data in Figure 7 (row 2) as functions of LARs *ατ* for SA (blue, dashed) and TOA (red, solid) scans. The horizontal lines (black, dotted) indicate the reference values from FAR data of 360◦.

**Figure 6.** Iodine concentrations, along with their respective error bars, in ROIs 1–4 (from left to right) within the chest phantom, as functions of LARs *ατ* for SA (blue, dashed) and TOA (red, solid) scans, estimated from basis images reconstructed from noiseless (row 1) and noisy (row 2) data by use of the two-step method.

3.2.3. Image Reconstruction from Noisy Data Acquired with SA and TOA Scans of LARs

We repeat the study by applying the DTV algorithm to noisy data of the chest phantom for SA and TOA scans considered in the noiseless study above. In Figure 7, we display VMIs at 100 keV, along with their zoomed-in views. For the noise levels considered, it can be observed that (1) LAR artifacts can be amplified by noise, (2) LAR artifacts are reduced substantially in VMIs for SA and TOA scans of LARs *ατ* ≥ 120◦, and (3) TOA scans can more effectively suppress LAR artifacts than SA scans for the chest phantom and noise level studied in the work. Such observations may provide insights into the design of practical procedures for image reconstruction from LAR data that contain additional inconsistencies. We note that no processing is applied to the data or images reconstructed in our study.

**Figure 7.** VMIs (rows 1, 3, 5, and 7), along with their respective zoomed-in views (rows 2, 4, 6, and 8), of the chest phantom at 100 keV obtained from noisy data over SAs (rows 1&2 and 5&6) and TOAs (rows 3&4 and 7&8) of LAR 20◦, 30◦, 45◦, 60◦, 90◦, 120◦, 150◦, and 180◦, respectively, by use of the two-step method. Display window: [0, 0.22] cm−1.

Similar to the noiseless-data study, we compute nRMSEs and PCCs from VMIs in Figure 7, and plot them as functions of LARs in row 2 of Figure 5. We also estimate iodine concentrations and plot them as functions of LAR in row 2 of Figure 6. In the noisy-data study, error bars, i.e., standard deviations, are calculated over the chest-phantom ROIs indicated in Figure 2c, and they are plotted in row 2 of Figure 6. The horizontal lines (black, dotted) indicate the reference values from FAR data of 360◦. Quantitative results of PCC appear consistent with the visual inspection, suggesting that VMI images for *ατ* ≥ 120◦ in SA and *ατ* ≥ 90◦ in TOA scans visually resemble the reference VMI obtained from noisy FAR data, and the degree of resemblance drops understandably as LAR decreases. The estimation accuracy of iodine concentration for *ατ* ≥ 90◦ remains comparable to those obtained from the reference images reconstructed from noisy FAR data.

#### *3.3. Image Reconstruction of the Suitcase Phantom*

Next, we repeat the studies in Section 3.2 with the suitcase phantom. We show in Figure 8a the VMI and its zoomed-in view reconstructed from FAR data and in Figure 8b their differences from the truth counterparts in Figure 2c (bottom row). The result again confirms the reconstruction accuracy of the two-step method using the suitcase phantom, which is of different complexity and structure to the chest phantom. To reveal the LAR artifacts associated with the suitcase phantom, we apply the DTV and FBP algorithms to reconstruct images from noiseless basis sinogram over an SA of *ατ* = <sup>30</sup>◦ and display them in Figure 8c,d, respectively. It can be observed that the LAR artifacts in the FBP image are almost eliminated in the image reconstructed by use of the two-step method.

**Figure 8.** Row 1: (**a**) VMI of the suitcase phantom at 40 keV obtained with the two-step method from FAR data, (**b**) difference between the VMI in (**a**) and its truth in Figure 2c, VMIs at 40 keV obtained with (**c**) the two-step method and (**d**) the FBP algorithm from data acquired over a SA of LAR 30◦; and row 2: zoomed-in views of their corresponding images in row 1. The zoomed-in area is enclosed by the rectangular box depicted in the VMI in (**a**). Display windows [0.1, 0.65] cm−<sup>1</sup> for columns (**a**,**c**,**d**), and [−10−4, 10−4] cm−<sup>1</sup> for column (**b**).

3.3.1. Image Reconstruction from Noiseless Data Acquired with SA and TOA Scans of LARs

Next, we apply the algorithm verified to reconstructing basis images of PE and KN from noiseless data of the suitcase phantom collected in SA or TOA scans of *ατ* = 14◦, 20◦, 30◦, 60◦, 90◦, 120◦, 150◦, and 180◦. The lowest LAR studied for the suitcase phantom, 14◦, is smaller than that for the chest phantom, 20◦. In Figure 9, we display the VMIs at 40 keV, along with their zoomed-in views, reconstructed from data collected with SA (rows 1&2 and 5&6) and TOA (rows 3&4 and 7&8) scans. It can be observed that the two-step method yields almost visually identical images for these LARs, revealing possible performance upper bounds of the method in numerically accurately inverting Equation (3) for scans with SA and TOA of LARs.

From VMIs in Figure 9, we compute nRMSEs and PCCs, and display them in row 1 of Figure 10. Using the method described in Section 2.5, we also estimate effective atomic numbers in ROIs 3–6 indicated in bottom row of Figure 2c, and plot them as functions of LARs in row 1 of Figure 11. These results reveal that, from the suitcase phantom noiseless data collected over the range of LARs as low as 14◦, the two-step method can yield VMIs visually and quantitatively close to the reference VMIs from FAR data of 360◦ in terms of PCC and estimated effective atomic numbers. With regard to metric nRMSE, it increases as LAR decreases, largely due to the increasing null spaces in the system matrices of the LAR scans, while TOA scans can lower nRMSE by an order of magnitude especially for small LARs as compared to SAs of the same LAR.

3.3.2. Image Reconstruction from Noisy Data Acquired with SA and TOA Scans of LARs

We apply the two-step method to reconstructing images from noisy data of the suitcase phantom collected over the same LARs in SA and TOA. In Figure 12, we display the VMIs at 40 keV, along with their zoomed-in views. For the suitcase phantom, LAR artifacts are substantially reduced in VMIs from data collected over *ατ* ≥ 90◦ in SA and *ατ* ≥ 60◦ in TOA scans. Similar to the chest phantom results, TOA configurations can more effectively suppress LAR artifacts than SA ones, especially recovering the distorted edges around the circular and elliptical disks, for the suitcase phantom under noise level studied in the work.

**Figure 9.** VMIs (rows 1, 3, 5, and 7), along with their respective zoomed-in views (rows 2, 4, 6, and 8), of the suitcase phantom at 40 keV obtained from noiseless data acquired over SAs (rows 1&2 and 5&6) and TOAs (rows 3&4 and 7&8) of LAR 14◦, 20◦, 30◦, 60◦, 90◦, 120◦, 150◦, and 180◦, respectively, by use of the two-step method. Display window: [0.1, 0.65] cm−1.

**Figure 10.** Metrics nRMSE and PCC, computed over VMIs of the suitcase phantom from noiseless data in Figure 9 (row 1) and those from noisy data in Figure 12 (row 2) as functions of LARs *ατ* for SA (blue, dashed) and TOA (red, solid) scans. The horizontal lines (black, dotted) indicate the reference values from FAR data of 360◦.

**Figure 11.** Effective atomic numbers of (**a**) water, (**b**) ANFO, (**c**) Teflon, and (**d**) PVC, along their respective error bars, within the suitcase phantom estimated as functions of LAR *ατ* for SA (blue, dashed) and TOA (red, solid) scans, computed from basis images reconstructed from noiseless (row 1) and noisy (row 2) data by use of the two-step method.

We compute nRMSEs and PCCs from VMIs in Figure 12, and plot them as functions of LARs in row 2 of Figure 10. We also estimate effective atomic numbers and plot them as functions of LAR in row 2 of Figure 11. In the noisy-data study, error bars, i.e., standard deviations, are calculated over the suitcase-phantom ROIs indicated in Figure 2c, and they are plotted in row 2 of Figure 11. The quantitative results suggest that VMI images visually resemble the reference VMI from FAR data for noisy LAR data collected over *ατ* ≥ 60◦ in SA and *ατ* ≥ 14◦ in TOA scans, and the resemblance decreases understandably as LAR decreases and that the estimation accuracy from noisy LAR data collected over *ατ* ≥ 60◦ is comparable to those from the FAR data.

**Figure 12.** VMIs (rows 1, 3, 5, and 7), along with their respective zoomed-in views (rows 2, 4, 6, and 8), of the suitcase phantom at 40 keV obtained from noisy data acquired over SAs (rows 1&2 and 5&6) and TOAs (rows 3&4 and 7&8) of LAR 14◦, 20◦, 30◦, 60◦, 90◦, 120◦, 150◦, and 180◦, respectively, by use of the two-step method. Display window: [0.1, 0.65] cm−1.

#### **4. Discussion**

In this work, we have investigated and developed a two-step method for image reconstruction from low- and high-kVp data collected with SA and TOA scans of LARs in DECT. The method combines the DDD and DTV algorithms to effectively compensate for both BH and LAR artifacts, yielding accurate VMIs and physical-quantity estimation. For the study conditions such as phantoms and noise levels considered, visual inspection of VMIs at energies of interest indicates that the method can yield from noiseless LAR data VMIs that are visually comparable to the reference VMI from FAR data, and from noisy LAR data VMIs with reduced BH and LAR artifacts; and quantitative observations can be made that the accurate estimation of physical quantities such as iodine concentrations and effective atomic numbers can be obtained for noiseless data of LAR as low as 20◦ and for noisy data of LAR as low as 60◦. For the SA and TOA scans of the same total angular range studied, the latter appear to yield more accurate images and estimations of physical quantities than the former, due to the improved conditioning of the system matrix.

We used two distinct phantoms, i.e., chest and suitcase phantoms, of varying complexity levels and structures of different application interest. The chest phantom contains lung tissue, airways, and blood vessels within the pulmonary anatomy, while the suitcase phantom contains various materials of interest in baggage screening. Results of the numerical study indicate that the effectiveness of the two-step method, like any other algorithm, is understandably dependent on the anatomic complexity of an object imaged with varying contrast and spatial resolution. Results from the suitcase phantom are less impacted, in terms of image artifacts and quantitative accuracy of the estimated physical quantities, by the decreasing LAR than the chest phantom, possibly due to its structure and the noise levels in the data. We have studied additional phantoms of different anatomies, and corroborative observations can be made.

In the work, we have investigated the DTV algorithm for numerically accurately solving the optimization program in Equation (4) with DTV constraints. Additionally, we have conducted noisy data studies to provide some preliminary insights into the stability of the two-step method in the presence of data inconsistencies. While a fixed total number of quanta is used for Poisson noise simulation, the visualization of VMIs and estimation accuracy of physical quantities obtained can be dependent on the noise levels and characteristics of different applications. In addition, other sources of inconsistency, such as metal, scatter, imperfect spectra, low- and high-kVp X-ray mismatch, and decomposition error, may also impact the reconstruction quality and estimation accuracy. Blooming artifacts usually stem from highly attenuating materials present in the patient, such as metal implants and calcification plaques. While it is important to investigate the effectiveness of the two-step method in studies containing these physical effects, such an investigation nevertheless is beyond the scope of this work, and the proposed method may be used as the basis for future investigative efforts that focus on correcting other physical factors in DECT with LAR data.

The studies and results in this work may provide insights into the possible development for practical approaches to reducing radiation dose and scanning time and to avoiding collision between the moving gantry of the scanner and the imaged object in clinical and industrial applications. One limitation of the proposed two-step method is the requirement of completely overlapping arcs of low- and high-kVp scans, imposed by the data-domain decomposition step. This can be avoided by performing the image-domain decomposition in a two-step method [30]; however, linear data models are usually assumed and the nonlinear BH effect is not explicitly corrected for, which may impact the quantitative accuracy of the reconstruction. On the other hand, one-step methods [25] may accommodate LAR scanning configurations with partially or non-overlapping arcs of low- and high-kVp scans, while using the non-linear data model and correcting for the BH effect. Therefore, future investigations will include studies on one-step methods for DECT reconstruction with LAR data. It is worthy of a separate, comprehensive investigation, since existing studies on one-step methods focus largely on full- or short-angular-range scans and leverage image constraints, such as TV, not specifically designed for LAR data [15,25].

#### **5. Conclusions**

In this work, we investigated and developed a two-step method to reconstruct images accurately from low- and high-kVp LAR data by correcting for both BH and LAR effects in DECT. Numerical studies conducted reveal that the two-step method can yield VMIs with reduced BH and LAR artifacts, and estimation of physical quantities with improved accuracy, and that for SA and TOA scans with identical total LARs, the latter generally yields more accurate image reconstruction and physical-quantity estimation than the former. Results and knowledge acquired in the work on accurate image reconstruction in LAR DECT may give rise to further understanding and insights into the practical design of LAR scan configurations and reconstruction procedures for DECT applications. Future works will investigate the impact of additional inconsistencies and the one-step method for accommodating non-overlapping scans in DECT with LAR data.

**Author Contributions:** Conceptualization, B.C. and X.P.; methodology, B.C. and Z.Z.; software, B.C. and Z.Z.; validation, B.C., Z.Z. and D.X.; formal analysis, B.C.; investigation, B.C.; resources, X.P.; data curation, B.C. and T.G.-S.; writing—original draft preparation, B.C.; writing—review and editing, B.C., Z.Z., D.X., E.Y.S., T.G.-S. and X.P.; visualization, B.C.; supervision, X.P.; project administration, B.C.; funding acquisition, T.G.-S., E.Y.S. and X.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded in part by NIH R01 Grant Nos. EB026282 and EB023968, and R21 Grant No. CA263660-01A1. The contents of this article are solely the responsibility of the authors and do not necessarily represent the official views of the National Institutes of Health.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **Appendix A. Pseudo-Code of the DTV Algorithm**

#### **Algorithm A1** Pseudo-code of the DTV algorithm for solving Equation (4)

1: INPUT: **L***k*, *tkx*, *tky*, A, *ρ* 2: *<sup>L</sup>* ← ||K||2, *<sup>τ</sup>* <sup>←</sup> *<sup>ρ</sup>*/*L*, *<sup>σ</sup>* <sup>←</sup> 1/(*ρL*), *<sup>ν</sup>*<sup>1</sup> ← ||A||2/||D*x*||2, *<sup>ν</sup>*<sup>2</sup> ← ||A||2/||D*y*||2, *<sup>μ</sup>* ← ||A||2/||I||<sup>2</sup> 3: *n* ← 0 4: INITIALIZE: **<sup>b</sup>**(0), **<sup>w</sup>**(0), **<sup>p</sup>**(0), **<sup>q</sup>**(0), and **<sup>t</sup>** (0) to zero 5: **<sup>b</sup>**¯ (0) <sup>←</sup> **<sup>b</sup>**(0) 6: **repeat** 7: **<sup>w</sup>**(*n*+1) = (**w**(*n*) <sup>+</sup> *<sup>σ</sup>*(A**b**¯ (*n*) <sup>−</sup> **<sup>L</sup>**))/(<sup>1</sup> <sup>+</sup> *<sup>σ</sup>*) 8: **<sup>p</sup>**(*n*) <sup>=</sup> **<sup>p</sup>**(*n*) <sup>+</sup> *σν*1D*x***b**¯ (*n*) **<sup>q</sup>**(*n*) <sup>=</sup> **<sup>q</sup>**(*n*) <sup>+</sup> *σν*2D*y***b**¯ (*n*) 9: **<sup>p</sup>**(*n*+1) <sup>=</sup> **<sup>p</sup>**(*n*) <sup>−</sup> *<sup>σ</sup>* **<sup>p</sup>**(*n*) <sup>|</sup>**p**(*n*)<sup>|</sup> -1ball*<sup>ν</sup>*1*tkx* (|**p**(*n*)<sup>|</sup> *<sup>σ</sup>* ) **<sup>q</sup>**(*n*+1) <sup>=</sup> **<sup>q</sup>**(*n*) <sup>−</sup> *<sup>σ</sup>* **<sup>q</sup>**(*n*) <sup>|</sup>**q**(*n*)<sup>|</sup> -1ball*<sup>ν</sup>*2*tky* (|**q**(*n*)<sup>|</sup> *<sup>σ</sup>* ) 10: **t** (*n*+1) = neg(**<sup>t</sup>** (*n*) + *σμ***b**¯ (*n*)) 11: **<sup>b</sup>**(*n*+1) <sup>=</sup> **<sup>b</sup>**(*n*) <sup>−</sup> *<sup>τ</sup>*(A**w**(*n*+1) <sup>+</sup> *<sup>ν</sup>*1D *<sup>x</sup>* **<sup>p</sup>**(*n*+1) <sup>+</sup> *<sup>ν</sup>*2D *<sup>y</sup>* **<sup>q</sup>**(*n*+1) <sup>+</sup> *<sup>μ</sup>***<sup>t</sup>** (*n*+1)) 12: **<sup>b</sup>**¯ (*n*+1) <sup>=</sup> <sup>2</sup>**b**(*n*+1) <sup>−</sup> **<sup>b</sup>**(*n*) 13: *<sup>n</sup>* <sup>←</sup> *<sup>n</sup>* <sup>+</sup> <sup>1</sup> 14: **until** the convergence conditions are satisfied 15: OUTPUT: **<sup>b</sup>**(*n*) as the estimate of **<sup>b</sup>***<sup>k</sup>*

In the pseudo-code of the derived algorithm instance, the definitions of the auxiliary variables, including matrices <sup>K</sup> and <sup>I</sup> and vectors **<sup>w</sup>**(*n*), **<sup>p</sup>**(*n*), **<sup>q</sup>**(*n*), **<sup>p</sup>**(*n*), **<sup>q</sup>**(*n*), and **<sup>t</sup>** (*n*), and operators, including || · ||2, neg(·), -1ball*β*(·), and <sup>|</sup>**q**(*n*)|, are intentionally kept consistent with those used in Ref. [6]. In each reconstruction from a set of basis data in an SA and TOA scan, the DTV algorithm reconstructs the basis images through solving Equation (4) until the convergence conditions described in Ref. [5] are satisfied numerically.

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