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Hybrid source translation scanning mode for interior tomography

Song Ni, HaiJun Yu, Jie Chen, ChuanJiang Liu, and FengLin Liu

Open Access Open Access

Abstract

Interior tomography is a promising technique that can be used to image large objects with high acquisition efficiency. However, it suffers from truncation artifacts and attenuation value bias due to the contribution from the parts of the object outside the ROI, which compromises its ability of quantitative evaluation in material or biological studies. In this paper, we present a hybrid source translation scanning mode for interior tomography, called hySTCT—where the projections inside the ROI and outside the ROI are finely sampled and coarsely sampled respectively to mitigate truncation artifacts and value bias within the ROI. Inspired by our previous work—virtual projection-based filtered backprojection (V-FBP) algorithm, we develop two reconstruction methods—interpolation V-FBP (iV-FBP) and two-step V-FBP (tV-FBP)—based on the linearity property of the inverse Radon transform for hySTCT reconstruction. The experiments demonstrate that the proposed strategy can effectively suppress truncated artifacts and improve the reconstruction accuracy within the ROI.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Micro-computed tomography (micro-CT), which noninvasively captures the three-dimensional internal microstructure of objects at a high spatial resolution, has been widely used in fields of materials science [1], microelectronics [2], geology [3], non-destructive testing [4] and geoscience [5]. However, achieving high spatial resolution in objects with a transversal diameter larger than the field of view (FOV) is a challenge. Interior tomography is an effective alternative that only illuminates the region-of-interest (ROI) within the object at high resolution. Besides, interior tomography can mitigate undesirable interference outside the ROI. For instance, in 3D representations of lithium-ion battery electrodes (which can assist in understanding and ultimately improving battery performance), interior tomography is a promising technique to alleviate the effect of high-attenuating elements on the image quality of the low-contrast carbon-binder domain [6,7].

However, interior tomography faces some intractable problems, such as non-unique and unstable solutions and truncation artifacts, which significantly compromise its application value. Over the decades, the techniques developed for solving these problems can fall into five categories: projection extrapolations, analytical algorithms, iterative algorithms, deep learning, and extra scan techniques. Projection extrapolation methods mitigate truncation artifacts by extrapolating the truncated data via a smooth function, such as symmetric mirroring [8], cosine function fitting [9], water cylinder extrapolation [10], or learned function [11,12], which are qualitatively appealing but quantitatively imprecise due to the lack of consistency in sinogram extrapolation. Analytical algorithms such as differentiate back-projection (DBP) enable an exact and stable reconstruction if a sub-region inside the ROI is known [13–15]; however, such a known sub-region is not always readily unavailable and even impossible to find in some cases. Iterative algorithms like total variation minimization permit a unique and stable result when the object is essentially piecewise constant or polynomial [16–19] which is also a non-universal in practical applications. Recently, the newborn deep learning techniques provide a new way to solve the truncation problem [20–23] but can be challenging to deploy in practice due to factors including lack of generalization, interpretability, and massive training data. The extra scan technique is a promising way to provide artifact-free ROI images with acceptable accuracy, whose mathematical principle is that the external projections outside the ROI still produce a non-zero 2D local image inside the ROI.

To subtract contributions of external regions from truncated projections, a low-resolution scan of the entire object by changing the magnification or by using a big low-resolution scouting detector is a widely used technique [24–28]. Changing the magnification requires to solve the problem of projection registration due to inconsistent magnifications [24–26], which can increase the computational cost and damage the high-frequency information. Using a scouting detector can ease this problem but increase the system’s cost and complexity [27]. Dual-FOV optical coupling detection system can simultaneously acquire the low-resolution projections of the entire object and the high-resolution but truncated projections of the ROI by dividing the lights into two sub-optical paths [28]; however, it suffers from the problem of low detection efficiency and high noise level. In our previous work, to avoid changing the magnification and increasing the system complexity, we employ the multi-source-translation CT (mSTCT) scanning mode to collect the low-resolution projections of the entire object, and the rotation scanning mode (the source-detector is fixed, while the object is rotated) to acquire the high-resolution projections of the ROI [29]. However, merging projections from two different scanning modes still produces errors.

In this work, we propose a hybrid source translation scanning mode for interior tomography, in which a dense source translation sampling is used to acquire high-resolution projections within the ROI, and a coarse source translation sampling is used to acquire low-resolution projections outside the ROI, called hySTCT. The hySTCT can avoid the process of merging projections since both projections are acquired using the same scanning mode with the same magnification. This work is inspired by our previous work—mSTCT scanning mode [30]. Each source sampling position only illuminates part of the object, resulting in projection truncation. Thus, we regroup the sampled attenuated rays divergent from each source position to those divergent from each detector unit, forming non-truncated virtual projections and further deriving a virtual projection-based FBP (V-FBP) algorithm [31]. However, mSTCT faces the problem of low acquisition efficiency. hySTCT that only focuses on interior ROI imaging can effectively improve efficiency.

The rest of the paper is organized as follows. Section 2 briefly introduces the theory of the interior tomography. Section 3 presents the proposed hySTCT scanning mode and its reconstruction algorithm. Section 4 investigates the numerical and physical experiment results. Section 5 discusses the hySTCT. Section 6 concludes the paper.

2. Theory

Considering a 2D function $f({x,y} ),\sqrt {{x^2} + {y^2}} \le R$, its Radon transform can be written as

$$\begin{array}{{cc}} {\mathrm{{\cal R}}f({r,\varphi } )= \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {f({x,y} )\delta ({x\cos \varphi + y\sin \varphi - r} )\textrm{d}x\textrm{d}y,} } }&{r \in [{ - R,R} ],\varphi \in [{0,\pi } )} \end{array}. $$

The corresponding interior Radon transform is ${{{\cal R}}_{in}}f({r,\varphi } ),|r |\le {R_0}$, with ${R_0}({{R_0} < R} )$ being the ROI radius. Natterer [32] proved that, for any reconstructed image from the interior dataset, it can be viewed as an exact reconstruction from a complete dataset $\mathrm{{\cal R}}f({r,\varphi } ),|r |\le R$ and an exterior dataset ${\mathrm{{\cal R}}_{ex}}f({r,\varphi } ),{R_0} < |r |\le R$, that is

$${\mathrm{{\cal R}}^{ - 1}}({{\mathrm{{\cal R}}_{in}}f} )= {\mathrm{{\cal R}}^{ - 1}}({\mathrm{{\cal R}}f} )- {\mathrm{{\cal R}}^{ - 1}}({{\mathrm{{\cal R}}_{ex}}f} ), $$
where, ${\mathrm{{\cal R}}^{ - 1}}$ is the inverse Radon transform. Although ${\mathrm{{\cal R}}_{ex}}f({r,\varphi } )= 0,|r |\le {R_0}$, it still produces a non-zero 2D local image ${({{\mathrm{{\cal R}}^{ - 1}}({{\mathrm{{\cal R}}_{ex}}f} )} )_{in}}$ inside the ROI, which is the reason for the truncation artifacts and non-uniqueness in interior tomography, as illustrated in Fig. 1.

 figure: Fig. 1.

Fig. 1. Diagram of the interior tomography. The first row shows densely sampled projections and sparsely sampled projections outside ROI. The second, third, and fourth rows describes Eq. (2), (3), and (4), respectively.

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Generally, if we know a portion of the exterior dataset ${\mathrm{\tilde{{\cal R}}}_{ex}}f({r,\varphi } ),{R_0} < |r |\le R$, we can then obtain an estimation of the global image via

$$\hat{f} = {\mathrm{{\cal R}}^{ - 1}}({{\mathrm{{\cal R}}_{in}}f} )+ {\mathrm{{\cal R}}^{ - 1}}({{{\mathrm{\tilde{{\cal R}}}}_{ex}}f} ). $$

Although the partial exterior dataset cannot produce high-quality images outside the ROI, it enables improving image quality inside the ROI, including truncation artifacts reduction and reconstruction accuracy improvements. The improved ROI can be extracted from Eq. (3) by

$${\hat{f}_{in}} = {({{\mathrm{{\cal R}}^{ - 1}}({{\mathrm{{\cal R}}_{in}}f} )} )_{in}} + {({{\mathrm{{\cal R}}^{ - 1}}({{{\mathrm{\tilde{{\cal R}}}}_{ex}}f} )} )_{in}}. $$

Figure 1 shows the significantly improved ROI reconstruction quality when the partial exterior dataset is introduced. Therefore, several techniques were developed to collect a portion of the exterior dataset, including low-resolution scans over the entire sample [26] and a few full and low-dose scans over the whole sample [33]. These strategies produce a multiple-resolution global image, as shown in the third row of Fig. 1, with high resolution inside the ROI and low resolution outside the ROI. In this paper, we proposed to use a mSTCT scanning mode to obtain the low-resolution exterior projections to improve the image quality of the ROI.

3. Method

3.1 mSTCT and hySTCT imaging

Figure 2 reviews the imaging model of mSTCT, which consists of multiple STCTs with different translation angles (Fig. 2(a)). In each STCT, the object is located close to the X-ray source, while the X-ray source equidistantly translates along a straight line parallel to the fixed flat-panel detector (FPD) to collect the projection data over the whole object (Fig. 2(b)). We can express the discrete source location on the trajectory in each STCT as

$$\begin{array}{{cc}} {{{\vec{S}}_\theta } = ({n\Delta \lambda , - l} )\left( {\begin{array}{{cc}} {\cos \theta }&{ - \sin \theta }\\ {\sin \theta }&{\cos \theta } \end{array}} \right),}&{n\Delta \lambda \in [{ - s,s} ]} \end{array}, $$
where $\theta $ is the translation angle, $\Delta \lambda $ is the source sampling interval, $n \in \left\{ { - {\textstyle{N \over 2}}, - {\textstyle{N \over 2}} + 1, \cdots ,{\textstyle{N \over 2}}} \right\}$, and $N = {{2s} / {\Delta \lambda }}$. Here, s is the half source translation distance to cover the whole object of radius R, with
$$R = \frac{{sh - dl}}{{\sqrt {{{({l + h} )}^2} + {{({s + d} )}^2}} }}, $$
where l is the source-to-object distance; h is the detector-to-object distance; d is the half-detector length. When $\Delta \lambda < {{\Delta u} / {2k}}$ ($\Delta u$ is the size of detector unit), mSTCT can avoid loss of resolution, resulting in a huge number of projections per STCT if a large object is scanned. Therefore, we propose to mainly focus on ROI imaging and develop a hySTCT scanning mode.

 figure: Fig. 2.

Fig. 2. The schematic illustration of mSTCT and hySTCT. (a) 3D geometry of mSTCT, (b) 2D geometry of mSTCT. (c) 2D geometry of hySTCT.

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In hySTCT, the projections within the ROI are finely sampled, while the projections outside the ROI are coarsely sampled to mitigate the contributions of external regions from the ROI. Specifically, the X-ray source is sampled in a non-equidistant manner, which can be mathematically expressed as

$$\begin{array}{{cc}} {{{\vec{S}^{\prime}}_\theta } = ({n\kappa \Delta \lambda , - l} )\left( {\begin{array}{{cc}} {\cos \theta }&{ - \sin \theta }\\ {\sin \theta }&{\cos \theta } \end{array}} \right),}&{\left\{ {\begin{array}{{cc}} {\kappa = 1,}&{n\kappa \Delta \lambda \in [{ - {s_0},{s_0}} ]}\\ {\kappa \ge 1,}&{n\kappa \Delta \lambda \in [{ - s, - {s_0}} ]\cup [{{s_0},s} ]} \end{array}} \right.} \end{array}$$
where $\kappa \in {\mathbb N}$ is an undersampling factor, $n \in \left\{ { - {\textstyle{{{N_0} + {N_1}} \over 2}}, - {\textstyle{{{N_0} + {N_1}} \over 2}} + 1, \cdots ,{\textstyle{{{N_0} + {N_1}} \over 2}}} \right\},\ {N_1} = {2({s - {s_0}} )} /$$ {\kappa \Delta \lambda }$, and ${N_0} = {{2{s_0}} / {\Delta \lambda }}$. Here, ${s_0}$ is the half source translation distance to cover the ROI of radius R0, with
$${R_0} = \frac{{{s_0}h - dl}}{{\sqrt {{{({l + h} )}^2} + {{({{s_0} + d} )}^2}} }}. $$

This acquisition mode results in two projection datasets: a finely sampled interior projection dataset with source interval $\Delta \lambda $,

$$\begin{array}{{cc}} {p_\theta ^{\Delta \lambda }({\lambda ,u} ),}&{|\lambda |\le {s_0}} \end{array}, $$
and a coarsely sampled exterior projection dataset with source interval $\kappa \Delta \lambda$,
$$\begin{array}{{cc}} {p_\theta ^{\kappa \Delta \lambda }({\lambda ,u} ),}&{{s_0} < |\lambda |\le s} \end{array}, $$
where $\lambda$ is the index of the X-ray source, u is the index of the detector unit.

Here, the difference between mSTCT and hySTCT:

  • (1) When $\kappa = 1$ in $[{ - s, - {s_0}} )\cup ({{s_0},s} ]$, hySTCT becomes mSTCT;
  • (2) When $\kappa = \infty $ in $[{ - s, - {s_0}} )\cup ({{s_0},s} ]$, hySTCT only collects projections inside the ROI;

3.2 Analytic reconstruction algorithms

In mSTCT, we denote a real projection view as a 1D function ${p_\theta }({\lambda , \cdot } )$, which is a set of X-rays divergent from the source position (Fig. 3(a)). Since all projection views in mSTCT are truncated, we developed a virtual projection-based FBP (V-FBP) algorithm for reconstruction [31]. The virtual projection view is a set of X-rays divergent from the detector element (Fig. 3(b)), denoted as a 1D function ${\tilde{p}_\theta }({u, \cdot } )$. The relationship between rays in the real projection view and the virtual projection view is that

$${p_\theta }({\lambda ,u} )\textrm{ = }{\tilde{p}_\theta }({u,\lambda } )$$

 figure: Fig. 3.

Fig. 3. Illustrations of the truncated real projection view (a) and the non-truncated virtual projection view (b).

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Then, V-FBP can be written as

$${f_\theta }({x,y} )= \frac{{{{({l + h} )}^2}}}{{{{({x\sin \theta - y\cos \theta + h} )}^2}}}\int\limits_{ - d}^{ + d} {\int\limits_{ - s}^{ + s} {\frac{{{w_\theta }({\lambda ,u} )}}{{\sqrt {{{({l + h} )}^2} + {{({\lambda - u} )}^2}} }}{p_\theta }({\lambda ,u} )q({\lambda^{\prime} - \lambda } )\textrm{d}\lambda \textrm{d}u} }, $$
with $\lambda ^{\prime} = {\textstyle{{u({x\sin \theta - y\cos \theta - l} )+ ({l + h} )({x\cos \theta + y\sin \theta } )} \over {x\sin \theta - y\cos \theta + h}}}$. Here, ${w_\theta }$ is a weighting function for redundant data, q is the kernel of the ramp-filter. The final reconstruction is the superposition of results from all STCTs. The V-FBP reconstruction algorithm can be implemented as follows. We discretely represent ${p_\theta }({\lambda ,u} )$ as
$$\begin{array}{{cc}} {{p_\theta }({n\Delta \lambda ,m\Delta u} ),}&{n ={-} {\textstyle{N \over 2}}, - {\textstyle{N \over 2}} + 1, \cdots ,{\textstyle{N \over 2}}} \end{array};{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} m ={-} {\textstyle{M \over 2}}, - {\textstyle{M \over 2}} + 1, \cdots ,{\textstyle{M \over 2}}. $$

The projection after geometry weighting and redundancy weighting is

$${p_{\theta w}}({n\Delta \lambda ,m\Delta u} )= \frac{{l + h}}{{\sqrt {{{({l + h} )}^2} + {{({n\Delta \lambda - m\Delta u} )}^2}} }} \cdot {w_\theta }({n\Delta \lambda ,m\Delta u} )\cdot {p_\theta }({n\Delta \lambda ,m\Delta u} ). $$

Its 1D Fourier transform concerning the variable $\lambda $ is

$$\begin{array}{{cc}} {{P_{\theta w}}({k\Delta \xi ,m\Delta u} )= \Delta \lambda \sum\limits_{n ={-} {\textstyle{N \over 2}}}^{{\textstyle{N \over 2}}} {{p_{\theta w}}({n\Delta \lambda ,m\Delta u} ){\textrm{e}^{ - 2\pi \textrm{i}({n\Delta \lambda k\Delta \xi } )}}} ,}&{k ={-} {\textstyle{N \over 2}}, - {\textstyle{N \over 2}} + 1, \cdots ,{\textstyle{N \over 2}}} \end{array}, $$
with $\Delta \xi = {\textstyle{1 \over {N\Delta \lambda }}}$. Then, the projection after high-pass filtering is
$${p^{\prime}_{\theta w}}({n\Delta \lambda ,m\Delta u} )\approx \int\limits_{ - {\xi _m }}^{{\xi _m }} {{P_{\theta w}}({\xi ,u} )|\xi |{\textrm{e}^{2\pi \textrm{i}\lambda \xi }}\textrm{d}\xi } = \Delta \xi \sum\limits_{k ={-} {\textstyle{N \over 2}}}^{{\textstyle{N \over 2}}} {{P_{\theta w}}({j\Delta \xi ,m\Delta u} )|{j\Delta \xi } |{\textrm{e}^{2\pi \textrm{i}({n\Delta \lambda j\Delta \xi } )}}}. $$
with ${\xi _m} = {\textstyle{1 \over {2\Delta \lambda }}}$. The reconstruction image from each STCT is
$${f_\theta }({x,y} )\approx \frac{{\Delta u({l + h} )}}{{{{({x\sin \theta - y\cos \theta + h} )}^2}}}\sum\limits_{m ={-} {\textstyle{M \over 2}}}^{{\textstyle{M \over 2}}} {{{p^{\prime}}_{\theta w}}({\lambda^{\prime}({x,y} ),m\Delta u} )}. $$

The final reconstruction image for mSTCT is obtained by

$$f({x,y} )\approx \sum\limits_{i = 1}^T {{f_{{\theta _i}}}({x,y} )} , $$
where T is the number of STCTs.

We denote the V-FBP reconstruction algorithm as a reconstruction operator $V\textrm{ - }FBP({\cdot} )$. Obviously, $V\textrm{ - }FBP({\cdot} )$ is an inverse Radon transform. Based on Eq. (3), the projections from hySTCT can be reconstructed via

$$\hat{f} = V\textrm{ - }FBP\left( {\begin{array}{{cc}} {p_\theta^{\Delta \lambda }({\lambda ,u} ),}&{|\lambda |\le {s_0}} \end{array}} \right) + V\textrm{ - }FBP\left( {\begin{array}{{cc}} {p_\theta^{\kappa \Delta \lambda }({\lambda ,u} ),}&{{s_0} < |\lambda |\le s} \end{array}} \right), $$
which is termed as two-step V-FBP (tV-FBP). According to the linearity property of $V\textrm{ - }FBP$ operator, two terms can integrate into one term as
$$\hat{f} = V\textrm{ - }FBP\left( {\begin{array}{{cc}} {p_\theta^{\Delta \lambda }({\lambda ,u} )+ p_\theta^{\kappa \Delta \lambda }({\lambda ,u} ),}&{|\lambda |\le s} \end{array}} \right). $$

However, Eq. (20) cannot be directly implemented due to the inconsistent sampling interval. Here, we can use an interpolation operator $I({\cdot} )$ to interpolate the coarsely sampled projections, that is

$$\bar{p}_\theta ^{\Delta \lambda }({\lambda ,u} )= I({p_\theta^{\kappa \Delta \lambda }({\lambda ,u} )} ). $$

Then, Eq. (20) becomes

$${\hat{f}_{in}} = V\textrm{ - }FBP\left( {\begin{array}{{cc}} {p_\theta^{\Delta \lambda }({\lambda ,u} )+ \bar{p}_\theta^{\Delta \lambda }({\lambda ,u} ),}&{|\lambda |\le s} \end{array}} \right). $$
which is termed as interpolation V-FBP (iV-FBP). For visualizing the proposed algorithm Fig. 4 shows its specific reconstruction process.

4. Experiments

4.1 Numerical experiments

To assess the performance of the proposed hySTCT and its reconstruction algorithms tV-FBP and iV-FBP on the interior tomography, we conduct the numerical experiment on a Shepp-Logan phantom with a size of 8.57 mm × 8.57 mm. The simulated geometry parameters: source-to-object distance 15 mm, object-to-detector distance 190 mm, detector size 1024 × 1024 pixels, the size of a single detector pixel 0.127 mm ×0.127 mm, the interval translation angle 36.5 degrees, the number of STCTs 5. Here, we set the source translation distance 2s to 30 mm to cover the whole phantom, and set $2{s_0}$ to 15 mm to obtain an ROI of radius 2.1 mm. Inside the ROI, the source sampling interval is set as 15 µm, resulting in 1000 interior projections per STCT.

Firstly, to investigate the effects of exterior projections on the ROI reconstruction accuracy, we set the undersampling factor to ∞, 20, 10, 5, 2.5, and 1, resulting in the number of exterior projections being 0, 50, 100, 200, 400, and 1000. When the undersampling factor is ∞, hySTCT does not collect information outside the ROI; when the undersampling factor is 1, hySTCT becomes mSTCT and collects projections with the same sampling interval over the entire object. All data are reconstructed by tV-FBP, iV-FBP, respectively.

Secondly, to highlight the efficacy of hySTCT, we compared it with our previous work—mSTCT-PC—under the same geometry parameters. In mSTCT-PC, 1500 projections were uniformly collected over 360 degrees using RCT scanning mode (object rotates to acquire projections), to acquire information inside the ROI with a fixed radius of 4.5 mm; 21 × 5 projections were collected using mSTCT scanning mode to acquire information over the whole object. In hySTCT, to obtain a ROI with a radius of 4.5 mm, $2{s_0}$ should be 20.5 mm; therefore, 1366 interior projections were acquired at 15µm source sampling interval. Besides, the number of the exterior projections was set as 0 and 50, respectively.

We employ three evaluation metrics to quantitatively evaluate the image quality of the ROI, including the root-mean-square-error (RMSE), peak signal-to-noise ratio (PSNR), and structural similarity index (SSIM). In general, a better result means a lower RMSE and a higher SSIM and PSNR.

4.2 Physical experiments

To demonstrate the feasibility of hySTCT, we conduct the physical experiments on a prototype micro-CT system, which consists of a micro-focus X-ray source (L10321, Hamamatsu, Japan), a flat-panel detector (PaxScan1313DX, Varian, USA), a rotation platform (RGV100BL, Newport, USA) and a translation platform (M-ILS250PP, Newport, USA) as shown in Fig. 5. In our experiments, the X-ray source works at 60 kV, 70µA. The geometry parameters are consistent with those in the numerical experiment. We scanned a specimen of bamboo toothpicks, with radius being 5 mm. We set the source translation distance 2s to 20 mm to cover the entire specimen, and set $2{s_0}$ to 12 mm to obtain an ROI of radius 1.2 mm. Inside the ROI, the source sampling interval is 10 µm, leading to 1200 interior projections per STCT. Outside the ROI, the source sampling interval is ∞, 160, 80, 40, 20, and 10 µm, resulting in the number of exterior projections being 0, 50, 100, 200, 400, and 800.

 figure: Fig. 4.

Fig. 4. The diagram of V-FBP operator, tV-FBP, and iV-FBP reconstruction algorithms.

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 figure: Fig. 5.

Fig. 5. Practical experiment platform.

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To verify the performance of hySTCT, we conduct numerical and physical experiments. The program is written in MATLAB and tested on a computer with an Intel Core i7-4790 CPU @ 3.60 GHz. The experiments mainly consist of image quality evolution among different sparse sampling views and comparison among methods.

5. Results

5.1 Results of numerical experiments

5.1.1 Different numbers of exterior projections

Figure 6 displays the global reconstruction images of tV-FBP and iV-FBP concerning different numbers of exterior projections in hySTCT scanning mode. For both reconstruction methods, severe artifacts are present in the results without exterior projections (Fig. 6(f) and (l)). Drawing on the exterior projections, the artifacts in the results are significantly mitigated. As the number of exterior projections increases, the artifacts gradually decrease. Figure 6 also shows that, when the number of exterior projections is small, tV-FBP performs better on artifacts suppression than iV-FBP (Fig. 6(e) and (k)), due to the inaccurately interpolated projections outside the ROI in iV-FBP. As the number of exterior projections increases, the difference between tV-FBP and iV-FBP becomes smaller and smaller. For computational consumption, iV-FBP costs about 13s, while tV-FBP costs about 24s, because it performs V-FBP reconstruction two times.

 figure: Fig. 6.

Fig. 6. Global reconstructions by the tV-FBP and iV-FBP methods with different numbers of outside projections: (a)-(f) are reconstructions of tV-FBP with 1000, 400, 200, 100, 50, and 0 outside projections; (g)-(l) are the counterparts of iV-FBP.

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Figure 7 compares the ROI image quality in terms of RMSE and the difference image between the phantom and the result. It shows that, without the information outside the ROI, considerable value bias is present inside the ROI due to the truncated projection data. With the information outside the ROI, the value bias is significantly reduced, which can be demonstrated by both RMSE values and difference images. In tV-FBP reconstructed ROI, the RMSE value considerably decreases as the number of exterior projections increases. However, in iV-FBP reconstructed ROI, the RMSE value at N1 = 50 is comparable to that at N1 = 1000; the RMSE value at N1 = 1000 equals that at N1 = 200. This indicates that, compared with mSTCT, hySTCT in tandem with iV-FBP can reduce hundreds of projections per STCT with the same image quality in the ROI, thus improving the acquisition efficiency.

 figure: Fig. 7.

Fig. 7. Interior tomography by the tV-FBP and iV-FBP methods with different numbers of outside projections: (a)-(f) and (m)-(r) are ROI images with 1000, 400, 200, 100, 50, and 0 outside projections; (g)-(l) and (s)-(x) are the corresponding difference images.

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Table 1 lists the quantitative evaluation metrics calculated on the ROI. For both tV-FBP and iV-FBP, as the number of exterior projections increases from 0 to 1000, the RMSE value decreases from 0.0452 to 0.061; the PSNR increases from 19.8378 to 41.8115; the SSIM increases from 0.5751 to 0.9940, indicating the significant improvement of image quality inside the ROI. Besides, iV-FBP works better than tV-FBP especially when the number of exterior projections is 50.

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Table 1. Quantitative evolution metrics of ROI image

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5.1.2 Methods in comparison

Figure 8 compares hySTCT with RCT and mSTCT-PC for ROI imaging. As for RCT, we can observe two differences: first, at a fixed magnification, the size of the ROI is fixed in RCT while that can be adjusted by changing the translation distance of the source in hySTCT; second, the effects caused by the truncated projections are different in two different scanning modes—they are in the form of cupping artifacts at the ROI edges and value biases over the ROI in RCT, in contrast to value biases over the ROI and slight artifacts at the ROI edges in hySTCT. As for mSTCT-PC, the RMSE value and difference image show that hySTCT achieves better accuracy and smaller value bias than mSTCT-PC. This is because mSTCT-PC inevitably introduces some errors when merging the projection data from two different scanning modes (RCT and mSTCT). Besides, these errors are greater than the errors caused by the interpolation in iV-FBP. Figure 8 also indicates that the reconstruction accuracy decreases with the degrees of data truncation increasing in hySTCT.

 figure: Fig. 8.

Fig. 8. Interior tomography via different methods. The first row shows ROI images, while the second row is the corresponding difference images. The former four columns display ROI images with a FOV being 4.5 mm in radius, while the last two columns show ROI images with a smaller FOV being 2.1 mm in radius.

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5.2 Results of practical experiments

5.2.1 Different numbers of exterior projections

Figure 9 displays the global reconstruction images on realistic data. The results echo the observation that the information outside the ROI enables a better global reconstruction image, and tV-FBP achieves better performance in recovering the global reconstruction image when the number of projections outside the ROI is small.

 figure: Fig. 9.

Fig. 9. Global reconstructions by the tV-FBP and iV-FBP methods with different numbers of outside projections: (a)-(f) are tV-FBP reconstructions with 800, 400, 200, 100, 50, and 0 outside projections; (g)-(l) are the counterparts of iV-FBP.

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In practical experiments, the image obtained by mSTCT serves as the reference for calculating quantitative evaluation metrics and difference images. Figure 10 shows the ROI reconstruction. For both tV-FBP and iV-FBP, the projection data outside the ROI can notably alleviate the artifacts at the ROI edges and the value bias inside the ROI. Compared with tV-FBP, iV-FBP can better suppress the streak artifacts caused by truncated data at the ROI edges but will produce slight artifacts inside the ROI due to interpolation.

 figure: Fig. 10.

Fig. 10. ROI images reconstructed by the tV-FBP and iV-FBP methods with different numbers of outside projections: (a)-(f) and (m)-(r) are ROI images with 800, 400, 200, 100, 50, and 0 outside projections; (g)-(l) and (s)-(x) are the corresponding difference images.

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Table 2 summarizes quantitative comparison results for ROI images over various numbers of outside projections, which quantitatively reinforces our observation. Compared to results with and without exterior projections, ROI image quality has greatly improved for both two methods. When the exterior projections increase from 0 to 400, RMSE decreases from 0.2230 to 0.0097 for tV-FBP, and from 0.2230 to 0.0398 for iV-FBP. When the exterior projections further increase from 50 to 400, iV-FBP method performs more robust than tV-FBP with smaller fluctuation of values, but tV-FBP can further improve image accuracy with gradually increased outside projections.

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Table 2. Quantitative evolution metrics of ROI image

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5.2.2 Methods in comparison

Figure 11 shows the ROI reconstructions of different methods. Visually, the proposed method with exterior data outperforms other methods in ROI image accuracy without obvious bias in difference image. The ROI image of RCT scan suffers from severe cupping artifacts because of truncated projection. The mSTCT-PC method improves the ROI image by completing ROI truncated projection. Without exterior data, the proposed method also suffers from some artifacts at the edges but is much slighter compared to the other two methods. As the proposed method can flexibly modify FOV only by changing source sampling distance, we also display ROI images reconstructed by the proposed method with a smaller FOV being 1.2 mm in radius. It also demonstrates that the exterior data sparsely sampled by hySTCT can greatly alleviate truncated artifacts and value bias of ROI images.

 figure: Fig. 11.

Fig. 11. ROI images reconstructed via different scanning modes. The first row shows ROI images, while the second row displays the corresponding difference images. The former four columns display ROI images with a FOV being 3.0 mm in radius, while the last two columns show ROI images with a smaller FOV being 1.2 mm in radius.

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6. Discussion

In our previous work, we proposed a mSTCT that enables to enlarge the FOV by controlling the X-ray source translation, and developed a V-FBP reconstruction algorithm to solve the problem of truncated data in each projection view. However, as the FOV enlarges, the number of projections increases, resulting in low acquisition efficiency. In some applications, we are only interested in a certain region of the object. Therefore, in this paper, we proposed a hySTCT scanning mode for interior ROI imaging by shortening the X-ray source translation distance, finally reducing the number of projections. However, the truncation effects are inevitable in ROI imaging. Since the truncation effects are mainly contributed by the exterior dataset, we proposed to use hySTCT to sparsely collect projection data outside the ROI, so as to reduce the influence of the exterior dataset on the image quality of the ROI without sacrificing acquisition efficiency. Since the exterior sparse dataset and interior dense dataset are obtained under the same hySTCT scanning mode with the same geometry parameters, complex data fusion is unnecessary. We can directly employ the linearity of the inverse Radon transform to obtain the reconstruction image. Due to the different sampling rates between the exterior dataset and interior dataset, the reconstructed global image has multi-resolution—high resolution inside the ROI, and low resolution outside the ROI. Although the quality outside the ROI is low, the quality inside the ROI can be significantly improved due to the introduced exterior dataset.

Compared with RCT to acquire the interior dataset, hySTCT collecting the interior dataset via mSTCT has the following advantages: firstly, hySTCT permits to change in the size of the ROI by controlling the source translation; secondly, the effects in hySTCT reconstruction images caused by the truncated data are smaller, mainly presenting as the streak artifacts at the edge of the ROI and the value bias inside the ROI. Compared with mSTCT-PC to acquire the interior dataset via RCT and the exterior dataset via mSTCT, hySTCT acquiring the interior dataset and the exterior dataset via the same mode with the same parameters can avoid loss of accuracy and resolution caused by data fusion.

Building on the linearity of the V-FBP operator, we developed two algorithms—tV-FBP, iV-FBP—for hySTCT reconstruction. The experiments show that, for both tV-FBP and iV-FBP, the exterior projections outside the ROI can notably alleviate the artifacts at the ROI edges and the value bias inside the ROI. Compared with tV-FBP, iV-FBP can better suppress the artifacts caused by truncated data, since the interpolation is used to estimate the missing exterior projections. Therefore, iV-FBP is more robust to the number of collected exterior projections. However, more accurate estimation is available with the help of deep learning techniques, which may further decrease the number of exterior projections needed for high ROI reconstruction accuracy.

7. Conclusion

In this paper, we proposed a hySTCT scanning mode for interior tomography and developed two corresponding reconstruction algorithms—tV-FBP, iV-FBP. The experiments demonstrated that this mode enables reconstructing ROI with variable size, fewer truncation artifacts, and acceptable accuracy. Besides, the results showed that iV-FBP is more robust with variable complemented outside projections than tV-FBP. This approach may be beneficial for micro-scale interior imaging tasks, such as battery imaging, in which the inner active particles, binder, and pore phases directly affect the battery performance. The proposed method may assist in structure analysis and performance improvement.

Funding

National Key Research and Development Program of China (2022YFF0706400); National Natural Science Foundation of China (62171067); Natural Science Foundation of Chongqing (CSTB2022NSCQ-MSX1311).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (11)
Fig. 1.
Fig. 1. Diagram of the interior tomography. The first row shows densely sampled projections and sparsely sampled projections outside ROI. The second, third, and fourth rows describes Eq. (2), (3), and (4), respectively.

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Fig. 2.
Fig. 2. The schematic illustration of mSTCT and hySTCT. (a) 3D geometry of mSTCT, (b) 2D geometry of mSTCT. (c) 2D geometry of hySTCT.

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Fig. 3.
Fig. 3. Illustrations of the truncated real projection view (a) and the non-truncated virtual projection view (b).

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Fig. 4.
Fig. 4. The diagram of V-FBP operator, tV-FBP, and iV-FBP reconstruction algorithms.

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Fig. 5.
Fig. 5. Practical experiment platform.

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Fig. 6.
Fig. 6. Global reconstructions by the tV-FBP and iV-FBP methods with different numbers of outside projections: (a)-(f) are reconstructions of tV-FBP with 1000, 400, 200, 100, 50, and 0 outside projections; (g)-(l) are the counterparts of iV-FBP.

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Fig. 7.
Fig. 7. Interior tomography by the tV-FBP and iV-FBP methods with different numbers of outside projections: (a)-(f) and (m)-(r) are ROI images with 1000, 400, 200, 100, 50, and 0 outside projections; (g)-(l) and (s)-(x) are the corresponding difference images.

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Fig. 8.
Fig. 8. Interior tomography via different methods. The first row shows ROI images, while the second row is the corresponding difference images. The former four columns display ROI images with a FOV being 4.5 mm in radius, while the last two columns show ROI images with a smaller FOV being 2.1 mm in radius.

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Fig. 9.
Fig. 9. Global reconstructions by the tV-FBP and iV-FBP methods with different numbers of outside projections: (a)-(f) are tV-FBP reconstructions with 800, 400, 200, 100, 50, and 0 outside projections; (g)-(l) are the counterparts of iV-FBP.

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Fig. 10.
Fig. 10. ROI images reconstructed by the tV-FBP and iV-FBP methods with different numbers of outside projections: (a)-(f) and (m)-(r) are ROI images with 800, 400, 200, 100, 50, and 0 outside projections; (g)-(l) and (s)-(x) are the corresponding difference images.

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Fig. 11.
Fig. 11. ROI images reconstructed via different scanning modes. The first row shows ROI images, while the second row displays the corresponding difference images. The former four columns display ROI images with a FOV being 3.0 mm in radius, while the last two columns show ROI images with a smaller FOV being 1.2 mm in radius.

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Tables (2)

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Table 1. Quantitative evolution metrics of ROI image

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Table 2. Quantitative evolution metrics of ROI image

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Equations (22)

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R f ( r , φ ) = ∫ − ∞ + ∞ ∫ − ∞ + ∞ f ( x , y ) δ ( x cos ⁡ φ + y sin ⁡ φ − r ) d x d y , r ∈ [ − R , R ] , φ ∈ [ 0 , π ) .
R − 1 ( R i n f ) = R − 1 ( R f ) − R − 1 ( R e x f ) ,
f ^ = R − 1 ( R i n f ) + R − 1 ( R ~ e x f ) .
f ^ i n = ( R − 1 ( R i n f ) ) i n + ( R − 1 ( R ~ e x f ) ) i n .
S → θ = ( n Δ λ , − l ) ( cos ⁡ θ − sin ⁡ θ sin ⁡ θ cos ⁡ θ ) , n Δ λ ∈ [ − s , s ] ,
R = s h − d l ( l + h ) 2 + ( s + d ) 2 ,
S → ′ θ = ( n κ Δ λ , − l ) ( cos ⁡ θ − sin ⁡ θ sin ⁡ θ cos ⁡ θ ) , { κ = 1 , n κ Δ λ ∈ [ − s 0 , s 0 ] κ ≥ 1 , n κ Δ λ ∈ [ − s , − s 0 ] ∪ [ s 0 , s ]
R 0 = s 0 h − d l ( l + h ) 2 + ( s 0 + d ) 2 .
p θ Δ λ ( λ , u ) , | λ | ≤ s 0 ,
p θ κ Δ λ ( λ , u ) , s 0 < | λ | ≤ s ,
p θ ( λ , u )  =  p ~ θ ( u , λ )
f θ ( x , y ) = ( l + h ) 2 ( x sin ⁡ θ − y cos ⁡ θ + h ) 2 ∫ − d + d ∫ − s + s w θ ( λ , u ) ( l + h ) 2 + ( λ − u ) 2 p θ ( λ , u ) q ( λ ′ − λ ) d λ d u ,
p θ ( n Δ λ , m Δ u ) , n = − N 2 , − N 2 + 1 , ⋯ , N 2 ; m = − M 2 , − M 2 + 1 , ⋯ , M 2 .
p θ w ( n Δ λ , m Δ u ) = l + h ( l + h ) 2 + ( n Δ λ − m Δ u ) 2 ⋅ w θ ( n Δ λ , m Δ u ) ⋅ p θ ( n Δ λ , m Δ u ) .
P θ w ( k Δ ξ , m Δ u ) = Δ λ ∑ n = − N 2 N 2 p θ w ( n Δ λ , m Δ u ) e − 2 π i ( n Δ λ k Δ ξ ) , k = − N 2 , − N 2 + 1 , ⋯ , N 2 ,
p θ w ′ ( n Δ λ , m Δ u ) ≈ ∫ − ξ m ξ m P θ w ( ξ , u ) | ξ | e 2 π i λ ξ d ξ = Δ ξ ∑ k = − N 2 N 2 P θ w ( j Δ ξ , m Δ u ) | j Δ ξ | e 2 π i ( n Δ λ j Δ ξ ) .
f θ ( x , y ) ≈ Δ u ( l + h ) ( x sin ⁡ θ − y cos ⁡ θ + h ) 2 ∑ m = − M 2 M 2 p ′ θ w ( λ ′ ( x , y ) , m Δ u ) .
f ( x , y ) ≈ ∑ i = 1 T f θ i ( x , y ) ,
f ^ = V  -  F B P ( p θ Δ λ ( λ , u ) , | λ | ≤ s 0 ) + V  -  F B P ( p θ κ Δ λ ( λ , u ) , s 0 < | λ | ≤ s ) ,
f ^ = V  -  F B P ( p θ Δ λ ( λ , u ) + p θ κ Δ λ ( λ , u ) , | λ | ≤ s ) .
p ¯ θ Δ λ ( λ , u ) = I ( p θ κ Δ λ ( λ , u ) ) .
f ^ i n = V  -  F B P ( p θ Δ λ ( λ , u ) + p ¯ θ Δ λ ( λ , u ) , | λ | ≤ s ) .
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