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- Abstract
- I. Introduction
- II. FBP Formula for Triple-Source CBCT
- III. Simulation Results
- IV. Discussions and Conclusion
- References

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IEEE Trans Med Imaging. Author manuscript; available in PMC 2010 May 26.

Published in final edited form as:

PMCID: PMC2876985

NIHMSID: NIHMS193070

Jun Zhao, Department of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China;

Jun Zhao: nc.ude.utjs@oahznuj; Yannan Jin: moc.liamg@nijnannay; Yang Lu: nc.ude.utjs@gnayvl; Ge Wang: ude.tv@ggnaw

The publisher's final edited version of this article is available at IEEE Trans Med Imaging

See other articles in PMC that cite the published article.

Multisource cone-beam computed tomography (CT) is an attractive approach of choice for superior temporal resolution, which is critically important for cardiac imaging and contrast enhanced studies. In this paper, we present a filtered-backprojection (FBP) algorithm for triple-source helical cone-beam CT. The algorithm is both exact and efficient. It utilizes data from three inter-helix PI-arcs associated with the inter-helix PI-lines and the minimum detection windows defined for the triple-source configuration. The proof of the formula is based on the geometric relations specific to triple-source helical cone-beam scanning. Simulation results demonstrate the validity of the reconstruction algorithm. This algorithm is also extended to a multisource version for (2*N* + 1)-source helical cone-beam CT. With parallel computing, the proposed FBP algorithms can be significantly faster than our previously published multisource backprojection-filtration algorithms. Thus, the FBP algorithms are promising in applications of triple-source helical cone-beam CT.

The recently introduced Siemens’ dual-source computed tomography (CT) allows much-improved temporal resolution, and has received a major attention in the medical imaging field. A natural extension of the dual-source system is a triple-source cone-beam CT (CBCT) scanner for even better temporal resolution at an additional system cost. As a follow-up to our previously published work on backprojection-filtration (BPF) based triple-source CBCT, this paper presents an exact filtered-backprojection (FBP) algorithm for triple-source helical CBCT, which can be implemented using parallel-computing techniques much more efficiently than the BPF counterpart.

Many exact and efficient methods were developed for spiral/helical cone-beam CT over past years [1]–[15]. The focus loci range from specific to more general, and from single-source scanning to multisource configurations. In 2002, Katsevich introduced an exact filtered-backprojection (FBP) algorithm [1]. Then, Zou and Pan presented an exact backprojection-filtration (BPF) algorithm [2]. These algorithms utilize projection data from a PI-arc based on the so-called PI-line [16] and Tam–Danielsson window [16], [17]. The Katsevich FBP algorithm uses a 1-D Hilbert transform of derivatives of differential projection data within a window slightly larger than the Tam–Danielsson window. The merit of the Zou–Pan’s BPF formula is that it uses only the data in the Tam–Danielsson window and accommodates certain types of transverse data truncation.

To extend these results to the case of general scanning trajectories, Katsevich also presented a general scheme for constructing inversion algorithms [3]. Ye *et al.* [4], [5] proved the validity of the BPF formula in the case of a general scanning curve and derived a general Katsevich-type FBP formula [6]. Pack *et al.* [7], [8] also introduced the BPF and FBP formulas that can deal with discontinuous source curves. Other independent algorithms on general CBCT were also reported [46], [47]. Chen [9] gave an alternative derivation of the Katsvich cone-beam reconstruction formula based on the Tuy inversion formula [18]. Also, based on the Tuy formula, Zhao, Yu and Wang [10], [11] unified the above FBP and BPF formulas using appropriate operators.

Cardiac imaging and contrast enhanced studies are very important medical CT applications. Rapid data collection and exact reconstruction are highly desirable in the clinical settings. One approach to improve CT temporal resolution is to apply a half-scan technique [19]–[21]. However, further reduction of the source angular range is not generally possible since the data sufficiency condition [18] cannot be satisfied by a dataset less than a half-scan. Another approach is to use a multisource system [21]–[27], [12]–[15]. Theoretically, higher temporal resolution can be achieved as the number of sources becomes larger. The initial experience with the commercial dual-source CT system has been very encouraging in cardiac imaging and coronary angiography, which show that the dual-source CT generates high quality images with ultra high temporal resolution, making functional evaluation of the heart valves and myocardium clinically possible [23]–[26].

Unlike previous multisource systems that use either a circle scanning locus or an approximate reconstruction algorithm [21], [22], for the first time we recently proposed a generic design for exact triple-source helical CBCT [12]–[15]. Such an exact BPF reconstruction algorithm for triple-source helical CBCT [14], [15] was developed based on the Zhao window [12], [13] and the inter-helix PI-lines [12], [13]. The Zhao window is the minimum detection window for triple-source helical CBCT, which is a counterpart of the Tam–Danielsson window. Also, the role of the inter-helix PI-lines in triple-source helical CBCT is similar to that of the PI-lines in single source helical CBCT.

In this paper, we present an exact FBP algorithm for triple-source helical CBCT. The algorithm also uses data from the three inter-helix PI-arcs associated with the inter-helix PI-lines and corresponding Zhao windows. In comparison with our previous triple-source BPF algorithm, this triple-source FBP algorithm is advantageous in terms of image reconstruction speed when the parallel implementation is in place. Therefore, this algorithm is technically and clinically very attractive. The paper is organized as follows. In Section II, the FBP formula is derived by the roadmap in [10]–[13], [48], [49] and the geometric relations specific to triple-source helical CBCT. Section III describes the simulation results. In Section IV, relevant issues are discussed and conclusions are drawn.

Let a trajectory Γ be a piecewise differentiable curve in ^{3} described by **a**(*t*), *t* , and *f* an infinitely differentiable real integrable function with a compact support **Ω** ^{3}\Γ. ^{3}\Γ is the difference of sets ^{3} and Γ. The cone-beam transform *f* along Γ is defined by

$$\mathcal{D}f(\mathbf{a},\mathit{\theta})={\int}_{0}^{\infty}f(\mathbf{a}+s\mathit{\theta})ds,(\mathbf{a},\mathit{\theta})\in \mathrm{\Gamma}\times {\mathbb{S}}^{2}$$

(1)

where ** θ** is the direction of a ray emitted from the source

Let *f*(**x**) be an object density function to be reconstructed. Assume that this function is smooth and vanishes outside the object cylinder

$$\mathbf{\Omega}=\{\mathbf{x}=(x,y,z)\mid {x}^{2}+{y}^{2}\le {R}_{o}^{2},{z}_{min}\le z\le {z}_{max}\},0<{R}_{o}<R$$

(2)

where *R _{o}* is the radius of the object cylinder and

In the Cartesian coordinate system (*x*,*y*,*z*), the triple-helix trajectories can be expressed as

$$\{\begin{array}{l}{\mathbf{a}}_{1}(t)=(Rcos(t),Rsin(t),{\scriptstyle \frac{ht}{2\pi}})\hfill \\ {\mathbf{a}}_{2}(t)=(Rcos(t+{\scriptstyle \frac{2\pi}{3}}),Rsin(t+{\scriptstyle \frac{2\pi}{3}}),{\scriptstyle \frac{ht}{2\pi}})\hfill \\ {\mathbf{a}}_{3}(t)=(Rcos(t+{\scriptstyle \frac{4\pi}{3}}),Rsin(t+{\scriptstyle \frac{4\pi}{3}}),{\scriptstyle \frac{ht}{2\pi}})\hfill \end{array},R>0,h>0,t\in \mathbb{R}$$

(3)

where *R* stands for the distance from each X-ray source to the rotation axis (i.e., the *z*-axis), *h* the pitch of each helix, and *t* the rotation angle. Fig. 1 illustrates the triple-source helical CBCT configuration.

Illustration of a triple-source helical cone-beam scanning. The three X-ray sources are rotated around the *z*-axis along the helices **a**_{1}(*t*), **a**_{2}(*t*), and **a**_{3}(*t*), respectively. The helices **a**_{1}(*t*), **a**_{2}(*t*), and **a**_{3}(*t*) are on a cylinder of radius *R*. An object to **...**

Previously, we defined the inter-helix PI-lines and extended the traditional Tam–Danielsson window into the Zhao window in the case of triple helices [12], [13]. For convenience, here we restate the Zhao window as follows: for each source **a*** _{j}*(

Through any fixed **x** **Ω**, there exists one and only one inter-helix PI-line for any pair of the helices from the three helices defined by (3).

In the triple-helix case, there are three inter-helix PI-lines for a fixed **x** **Ω** and three corresponding inter-helix PI-arcs whose end points are the intersection points of the inter-helix PI-lines with the corresponding helices. The three inter-helix PI-arcs represent the source trajectory segments from which the sources illuminate the point **x**. Without loss of generality, we assume that the sources and the detectors are translated upwards and rotated counter-clockwise when being observed from the top. To describe the data completeness condition for a fixed **x** **Ω**, we use the Orlov method, i.e., consider direction curves on the unit sphere [29]. As shown in Fig. 2, the direction curves *C _{j}*,

inter-helix PI-arcs (coarse solid curve-segments)
$\stackrel{\u2322}{{\mathbf{a}}_{1}^{s}{\mathbf{a}}_{1}^{e}},\stackrel{\u2322}{{\mathbf{a}}_{2}^{s}{\mathbf{a}}_{2}^{e}}$ and
$\stackrel{\u2322}{{\mathbf{a}}_{3}^{s}{\mathbf{a}}_{3}^{e}}$ for **x**.

We introduce the Hilbert transform operator ^{βjk} for *j* {1, 2, 3}, and *k* {1, 2, 3}. It will become clear later that the index *k* determines the inter-helix PI line along which the filtration is done. We define *β** _{jk}* =

$${\mathit{\theta}}_{jk}(\gamma )=cos(\gamma ){\mathit{\alpha}}_{j}+sin(\gamma ){\mathit{\beta}}_{jk},\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}0\le \gamma <2\pi .$$

(4)

Then, we define the Hilbert transform ^{βjk}, depending on the mapping *α*_{j}*β** _{jk}*(

$$\begin{array}{c}{\mathcal{H}}^{{\mathit{\beta}}_{jk}}\zeta ({\mathbf{a}}_{j}(\text{t}),{\mathit{\alpha}}_{j})=-\frac{1}{\pi}{\int}_{0}^{2\pi}\frac{\zeta ({\mathbf{a}}_{j}(\text{t}),{\mathit{\theta}}_{jk}(\gamma ))}{sin\gamma}d\gamma ,\\ ({\mathbf{a}}_{j}(\text{t}),{\mathit{\alpha}}_{j})\in {\mathrm{\Gamma}}_{j}\times {\mathbb{S}}^{2}.\end{array}$$

(5)

Next, the backprojection operator is defined as

$$\begin{array}{c}{D}_{k}^{\#}\xi (\mathbf{x})=\sum _{j=1}^{3}{\eta}_{jk}\left({\int}_{{I}_{j}}\frac{\xi ({\mathbf{a}}_{j}(t),{\mathit{\alpha}}_{j}^{\mathbf{x}}(t))}{\left|\right|\mathbf{x}-{\mathbf{a}}_{j}(t)\left|\right|}dt\right),\\ \mathbf{x}\in \mathbf{\Omega}\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}({\mathbf{a}}_{j}(t),{\mathit{\alpha}}_{j}^{\mathbf{x}}(t))\in {\mathrm{\Gamma}}_{j}\times {\mathbb{S}}^{2}\end{array}$$

(6)

where *k* {1,2,3}

$${\mathit{\alpha}}_{j}^{\mathbf{x}}(t)=\frac{\mathbf{x}-{\mathbf{a}}_{j}(t)}{\left|\right|\mathbf{x}-{\mathbf{a}}_{j}(t)\left|\right|}\in {\mathbb{S}}^{2}$$

(7)

and

$${\eta}_{jk}={(-1)}^{\lfloor (j-k)mod3/2\rfloor}.$$

(8)

We list the following two properties of *δ*((**x** − **a*** _{j}*(

$$\frac{\delta ({\mathit{\alpha}}_{j}^{\mathbf{x}}(t)\xb7\mathit{\sigma})}{\left|\right|\mathbf{x}-{\mathbf{a}}_{j}(t)\left|\right|}=\delta ((\mathbf{x}-{\mathbf{a}}_{j}(t))\xb7\mathit{\sigma}),\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}j\in \{1,2,3\}$$

(9)

and

$${\int}_{{I}_{j}}G(t,\mathit{\sigma})\delta ((\mathbf{x}-{\mathbf{a}}_{j}(t))\xb7\mathit{\sigma})dt=\sum _{\{{\lambda}_{j}\in {I}_{j}\mid \mathbf{x}\xb7\mathit{\sigma}={\mathbf{a}}_{j}({\lambda}_{j})\xb7\mathit{\sigma}\}}\frac{G({\lambda}_{j},\mathit{\sigma})}{\mid {{\mathbf{a}}^{\prime}}_{j}({\lambda}_{j})\xb7\mathit{\sigma}\mid}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}j\in \{1,2,3\}.$$

(10)

Before stating the FBP reconstruction formula, we define

$$\mathbf{\Psi}=\left\{\mathit{\psi}|\mathit{\psi}=\frac{\mathbf{x}-{\mathbf{a}}_{k}^{s}}{\left|\right|\mathbf{x}-{\mathbf{a}}_{k}^{s}\left|\right|}\phantom{\rule{0.16667em}{0ex}}\text{or}\phantom{\rule{0.16667em}{0ex}}\mathit{\psi}=\frac{\mathbf{x}-{\mathbf{a}}_{k}^{e}}{\left|\right|\mathbf{x}-{\mathbf{a}}_{k}^{e}\left|\right|},k\in \{1,2,3\}\right\}.$$

(11)

[** FBP reconstruction formula**]

$$f(\mathbf{x})=\frac{1}{2\pi}{\mathcal{D}}_{k}^{\#}\circ {\mathcal{H}}^{{\mathit{\beta}}_{jk}}\circ \partial \circ \mathcal{D}f(\mathbf{x}),\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\text{k}\in \{1,2,3\}$$

(12)

*holds for all*
**x** **Ω**, *provided that the mapping *
${\mathit{\alpha}}_{j}^{\mathbf{x}}(t)\mapsto {\mathit{\beta}}_{jk}^{\mathbf{x}}({\mathit{\alpha}}_{j}^{\mathbf{x}}(t))={\mathit{\beta}}_{jk}({\mathit{\alpha}}_{j}^{\mathbf{x}}(t))\phantom{\rule{0.16667em}{0ex}}({\mathit{\alpha}}_{j}^{\mathbf{x}}(t)\in {\mathbb{S}}^{2}\backslash \mathbf{\Psi})$
*satisfies the normalization condition*

$$\sum _{j=1}^{3}\left({\eta}_{jk}\sum _{\{{\lambda}_{j}\in {\text{I}}_{\text{j}}:\mathbf{x}\xb7\mathit{\sigma}={\mathbf{a}}_{j}({\lambda}_{j})\xb7\mathit{\sigma}\}}sgn({\mathit{\beta}}_{jk}^{\mathbf{x}}({\lambda}_{j})\xb7\mathit{\sigma})sgn({{\mathbf{a}}^{\prime}}_{j}({\lambda}_{j})\xb7\mathit{\sigma})\right)=1,\text{a}.\text{e}.\text{in}\phantom{\rule{0.16667em}{0ex}}\mathit{\sigma}\in {\mathbb{S}}^{2}$$

(13)

The normalization condition (13) holds for

$$\begin{array}{l}{\mathit{\beta}}_{jk}^{\mathbf{x}}({\lambda}_{j})=\frac{{\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})\times [({\mathbf{a}}_{kmod3+1}^{e}-{\mathbf{a}}_{k}^{s})\times {\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})]}{\left|\right|{\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})\times [({\mathbf{a}}_{kmod3+1}^{e}-{\mathbf{a}}_{k}^{s})\times {\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})]\left|\right|}\\ \text{for}\phantom{\rule{0.16667em}{0ex}}{\lambda}_{j}\in {I}_{j}\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}\mathbf{x}\xb7\mathit{\sigma}={\mathbf{a}}_{j}({\lambda}_{j})\xb7\mathit{\sigma}.\end{array}$$

(14)

The index *k* determines the inter-helix PI line along which the filtration is done.

Note that in (14),
$\left|\right|{\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})\times [({\mathbf{a}}_{kmod3+1}^{e}-{\mathbf{a}}_{k}^{s})\times {\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})]\left|\right|\phantom{\rule{0.16667em}{0ex}}=0$ occurs for the source **a*** _{j}*(

In the following proof, we always assume *j* {1,2,3} and *k* {1,2,3}.

$${\mathcal{D}}_{k}^{\#}\circ {\mathcal{H}}^{{\mathit{\beta}}_{jk}}\circ \partial \circ \mathcal{D}f({\mathbf{a}}_{j}(t),{\mathit{\alpha}}_{j})=2\pi f(\mathbf{x}).$$

(15)

Extending the right side of (14), we have

$${\mathit{\beta}}_{jk}^{\mathbf{x}}({\lambda}_{j})=\frac{({\mathbf{a}}_{kmod3+1}^{e}-{\mathbf{a}}_{k}^{s})-[({\mathbf{a}}_{kmod3+1}^{e}-{\mathbf{a}}_{k}^{s})\xb7{\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})]\xb7{\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})}{\left|\right|{\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})\times [({\mathbf{a}}_{kmod3+1}^{e}-{\mathbf{a}}_{k}^{s})\times {\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})]\left|\right|}.$$

(16)

Note that

$$sgn\phantom{\rule{0.16667em}{0ex}}({\mathit{\beta}}_{jk}^{\mathbf{x}}({\lambda}_{j})\xb7\mathit{\sigma})=sgn\phantom{\rule{0.16667em}{0ex}}[({\mathbf{a}}_{kmod3+1}^{e}-{\mathbf{a}}_{k}^{s})\xb7\mathit{\sigma}].$$

(18)

Thus, the validity of (13) is equivalent to prove

$$[({\mathbf{a}}_{kmod3+1}^{e}-{\mathbf{a}}_{k}^{s})\xb7\mathit{\sigma}]\times \sum _{j=1}^{3}\left({\eta}_{jk}\sum _{\{{\lambda}_{j}\in {\text{I}}_{\text{j}}:\mathbf{x}\xb7\mathit{\sigma}={\mathbf{a}}_{j}({\lambda}_{j})\xb7\mathit{\sigma}\}}sgn({{\mathbf{a}}^{\prime}}_{j}({\lambda}_{j})\xb7\mathit{\sigma})\right)=1,\text{a}.\text{e}.\phantom{\rule{0.16667em}{0ex}}\text{in}\phantom{\rule{0.16667em}{0ex}}\mathit{\sigma}\in {\mathbb{S}}^{2}.$$

(19)

Let us prove (19) for *k* = 1 first. Following from the continuity of **a**′_{j}(*t*) and the change of variable *u* = (**x** − **a*** _{j}*(

Similarly, we can prove for (19) for *k* = 2 and *k* = 3. That completes the proof of Theorem 2. Although the above proof is based on Tuy’s formula, the same or similar results may be derived from other classic and recent findings on exact cone-beam reconstruction [3], [31]–[34], [46], [47].

Let us compare this FBP algorithm with our previous triple-source BPF algorithm [14], [15]. Both the algorithms are exact, based on the Zhao windows and the inter-helix PI-lines. It was reported that the FBP algorithm has some merits [8], [32], [33] in comparison with the BPF algorithm for single-source case. The triple-source FBP algorithm maintains these merits in comparison with its BPF counterpart. The most attractive feature of the triple-source FBP algorithm is its potential for practical applications when it is coupled with parallel computing techniques [35], [36]. With serial computing techniques, the triple-source FBP algorithm may be more computationally intensive than its BPF counterpart because the filtering direction on the detector plane depends not only on the inter-helix PI-line but also on the projection view while the filtering direction of the triple-source BPF algorithm just depends on the inter-helix PI-line. With pre-calculated interpolation weights and sufficient parallel processors, both the FBP algorithm and the BPF algorithm can be greatly accelerated for triple-source helical CBCT. As far as the computational efficiency is concerned, FBP-type algorithms have an advantage over BPF algorithms in terms of data flow. Qualitatively speaking, in the FBP framework each projection frame can be filtered, backprojected and discarded immediately, while in the BPF framework the inverse Hilbert filtering can only take place after the backprojection is done, and furthermore

$$\begin{array}{l}\sum _{j=1}^{3}\left({\eta}_{j1}\sum _{\{{\lambda}_{\text{j}}\in {\text{I}}_{\text{j}}\mid \mathbf{x}\xb7\mathit{\sigma}={\mathbf{a}}_{\text{j}}({\lambda}_{\text{j}})\xb7\mathit{\sigma}\}}sgn({{\mathbf{a}}^{\prime}}_{\text{j}}({\lambda}_{\text{j}})\xb7\mathit{\sigma})\right)\\ =\sum _{j=1}^{3}\left({\eta}_{j1}{\int}_{{\text{I}}_{\text{j}}}({{\mathbf{a}}^{\prime}}_{\text{j}}(t)\xb7\mathit{\sigma})\delta ((\mathbf{x}-{\mathbf{a}}_{\text{j}}(t))\xb7\mathit{\sigma})dt\right)=\frac{1}{2\pi}\sum _{j=1}^{3}\left({\eta}_{j1}{\int}_{{\text{I}}_{\text{j}}}({{\mathbf{a}}^{\prime}}_{\text{j}}(t)\xb7\mathit{\sigma}){\int}_{-\infty}^{\infty}{e}^{is(\mathbf{x}-{\mathbf{a}}_{\text{j}}(t))\xb7\mathit{\sigma}}\mathit{dsdt}\right)\\ =\frac{1}{2\pi}{\int}_{-\infty}^{\infty}\sum _{j=1}^{3}\left({\eta}_{j1}{\int}_{{\text{I}}_{\text{j}}}({{\mathbf{a}}^{\prime}}_{\text{j}}(t)\xb7\mathit{\sigma}){e}^{is(\mathbf{x}-{\mathbf{a}}_{\text{j}}(t))\xb7\mathit{\sigma}}\mathit{dtds}\right)\\ =-\frac{1}{2\pi}{\int}_{-\infty}^{\infty}\left({\int}_{(\mathbf{x}-{\mathbf{a}}_{1}^{s})\xb7\mathit{\sigma}}^{(\mathbf{x}-{\mathbf{a}}_{1}^{e})\xb7\mathit{\sigma}}{e}^{\mathit{isu}}du+{\int}_{(\mathbf{x}-{\mathbf{a}}_{2}^{s})\xb7\mathit{\sigma}}^{(\mathbf{x}-{\mathbf{a}}_{2}^{e})\xb7\mathit{\sigma}}{e}^{\mathit{isu}}du-{\int}_{(\mathbf{x}-{\mathbf{a}}_{3}^{s})\xb7\mathit{\sigma}}^{(\mathbf{x}-{\mathbf{a}}_{3}^{e})\xb7\mathit{\sigma}}{e}^{\mathit{isu}}du\right)ds\\ =\frac{i}{2\pi}{\int}_{-\infty}^{\infty}\frac{1}{s}[{e}^{is(\mathbf{x}-{\mathbf{a}}_{1}^{s})\xb7\mathit{\sigma}}-{e}^{is(\mathbf{x}-{\mathbf{a}}_{1}^{e})\xb7\mathit{\sigma}}+{e}^{is(\mathbf{x}-{\mathbf{a}}_{2}^{s})\xb7\mathit{\sigma}}-{e}^{is(\mathbf{x}-{\mathbf{a}}_{2}^{e})\xb7\mathit{\sigma}}-{e}^{is(\mathbf{x}-{\mathbf{a}}_{3}^{s})\xb7\mathit{\sigma}}+{e}^{is(\mathbf{x}-{\mathbf{a}}_{3}^{e})\xb7\mathit{\sigma}}]ds\\ =\frac{sgn((\mathbf{x}-{\mathbf{a}}_{1}^{s})\xb7\mathit{\sigma})-sgn((\mathbf{x}-{\mathbf{a}}_{1}^{e})\xb7\mathit{\sigma})+sgn((\mathbf{x}-{\mathbf{a}}_{2}^{s})\xb7\mathit{\sigma})-sgn((\mathbf{x}-{\mathbf{a}}_{2}^{e})\xb7\mathit{\sigma})-sgn((\mathbf{x}-{\mathbf{a}}_{3}^{s})\xb7\mathit{\sigma})+sgn((\mathbf{x}-{\mathbf{a}}_{3}^{e})\xb7\mathit{\sigma})}{2}\\ =sgn(({\mathbf{a}}_{2}^{e}-{\mathbf{a}}_{1}^{s})\xb7\mathit{\sigma})\end{array}$$

(20)

a complicated rebining procedure is needed to convert the reconstructed image from the inter-helix PI-line-associated system to the rectangular system. Clearly, the FBP scheme is two steps faster than the BPF counterpart, given sufficient computing resources. On the other hand, BPF algorithms have merits too. For example, BPF algorithms can support some transverse truncation in the projection data while FBP cannot. The recently developed interior reconstruction techniques have allowed exact reconstruction from purely local data assuming knowledge on a subregion in the volume of interest [30], [50], [54]. These promising techniques may be adapted for triple-source cardiac CT at much reduced radiation dose. Better scattering correction methods may also be used for the same purpose.

Our proposed triple-source FBP algorithm has theoretical and practical values, because it is an intermediate step towards a spatially invariant FBP method like the Katsevich helical cone-beam algorithm in the single source case [1], suits better for parallel computation, and also serves as the benchmark for development of approximate cone-beam algorithms in the triple-source case. Our ultimate goal is to develop exact and efficient FBP algorithms in the triple-source case. Because the FBP and BPF are quite different in the computational structure, in the identical imaging geometry separate papers are published on reconstruction methods in these formats; for example, BPF and FBP algorithms in the standard helical scanning case by Katsevich [1] and Pan’s group [2], [32], BPF and FBP algorithms in the general smooth curve scanning case by Wang’s group [5], [6]. We hope that this paper will be the first one on the FBP algorithm in the triple-source case, and be followed by more papers towards optimal imaging performance and highest computational efficiency. Actually, we recently developed a spatially invariant FBP method in the triple-source saddle scanning case [55].

Let us consider the reconstruction on the inter-helix PI-line
$\overline{{\mathbf{a}}_{k}^{s}{\mathbf{a}}_{kmod3+1}^{e}}$ for a fixed *k* {1, 2, 3}. The
${x}_{\pi \overline{k,kmod3+1}}$-axis is made along the inter-helix PI-line. Denote the object function at
${x}_{\pi \overline{k,kmod3+1}}$ on the inter-helix PI-line as
${f}_{\pi \overline{k,kmod3+1}}({x}_{\pi \overline{k,kmod3+1}},{t}_{k}^{s},{t}_{kmod3+1}^{e})$, where
${t}_{k}^{s}$ and
${t}_{kmod3+1}^{e}$ are the angular parameters of
${\mathbf{a}}_{k}^{s}$ and
${\mathbf{a}}_{kmod3+1}^{e}$, respectively.

The triple-source FBP algorithm can be implemented in the following steps.

- Step 1) Differentiate each projection with respect to variable
*t*, keeping the directions of involved X-rays the same for each differentiation operation. - Step 2) For each ${\mathit{\alpha}}_{j}^{\mathbf{x}}(t)\in {\mathbb{S}}^{2}\backslash \mathbf{\Psi}$, perform the Hilbert transform of the differentiated projection data along the direction of the cone-beam projection of the $\overrightarrow{{\mathbf{a}}_{k}^{s}{\mathbf{a}}_{kmod3+1}^{e}}$ on the corresponding detector plane.
- Step 3) Backproject the filtered data at ${x}_{\pi \overline{k,kmod3+1}}$ on the inter-helix PI-line to obtain ${f}_{\pi \overline{k,kmod3+1}}({x}_{\pi \overline{k,kmod3+1}},{t}_{k}^{s},{t}_{kmod3+1}^{e})$.
- Step 4) Transform the reconstructed image from the inter-helix PI-line based coordinate system into the Cartesian coordinate system. The linear interpolation technique is used. In fact, one does not necessarily need to backproject the filtered data onto the inter-helix PI lines. One can directly backproject onto an image matrix and needs no rebinning. This is one of the main advantages of FBP algorithms over BPF algorithms. Backprojecting the filtered data onto the inter-helix PI lines is one of possible ways, which is similar to backprojecting the filtered data onto the PI lines in [5].

In the simulation, three rectangular planar detectors were used. Fig. 3 shows a rectangular planar detector corresponding to the source **a**_{1}(*t*). The detector was so arranged that the line through the source and the detector center intersected the z-axis and detector perpendicularly. The parameters used in the simulation are listed in Table I. The filtered data are backprojected onto 512 inter-helix PI-lines for a fixed endpoint. There were 512 fixed endpoints for one reconstructed inter-helix PI-line surface, and 1536 reconstructed inter-helix PI-line surfaces in total. Both the 3-D Shepp–Logan phantom [37] and Defrise disk phantom [38] were used. These phantoms were respectively located at the center of the global coordinate system *x-y-z* within the same spherical support.

Detector plane and coordinate system for the triple-source FBP CBCT based on an inter-helix PI-line. *x* – *y* – *z* is the 3-D fixed-coordinate system,
${\mathbf{d}}_{1}^{1}-{\mathbf{d}}_{2}^{1}-{\mathbf{d}}_{3}^{1}$ the 3-D rotated coordinate system for the source **a**_{1}(*t*),
${x}_{}$ **...**

Fig. 4 presents the reconstructed images of the 3-D Shepp–Logan phantom in the inter-helix PI-line coordinates system. Fig. 5 gives the images reformatted into the Cartesian coordinate system. Fig. 6 shows the results for the Defrise phantom in the Cartesian coordinate system. All of the reconstructed images are in excellent agreement with the ideal phantom slices.

Representative reconstruction results with the Shepp–Logan phantom in the inter-helix PI-line based coordinate system. (a) and (b) Reconstructed slices
${f}_{\pi \overline{1,2}}({x}_{\pi \overline{1,2}},{t}_{1}^{s},{t}_{2}^{e})$ in inter-helix PI-line based coordinates, **...**

Representative reconstruction results with the Shepp–Logan phantom in the Cartesian coordinate system. (a)–(c) The reconstructed slices of the 3-D Shepp–Logan phantom at *x* = 0 cm, *y* = 0 cm, and *z* = −3.2 cm, respectively. **...**

Although the symmetrical arrangement of triple sources is preferred, it is not a necessary condition for deriving the triple-source FBP formula (12). For a more general configuration

$${\mathbf{a}}_{j}(t)=\left(Rcos(t+{\phi}_{j}),Rsin(t+{\phi}_{j}),\frac{ht}{2\pi}\right),R>0,h>0,t\in \mathbb{R}$$

(21)

where *j* {1, 2, 3} and 0 ≤ _{1} < _{2} < _{3}< 2π, the triple-source FBP formula (12) still holds exactly.

We can also extend the triple-source FBP formula (12) to perform exact reconstruction in the case of (2*N* + 1) sources, *N* . Let (2*N* + 1)-source helical loci be

$${\mathbf{a}}_{j}(t)=\left(Rcos(t+{\phi}_{j}),Rsin(t+{\phi}_{j}),\frac{ht}{2\pi}\right),R>0,h>0,t\in \mathbb{R}$$

(22)

where * _{j}* [0, 2π),

First, let us extend the concept of the Zhao windows into the case of (2*N* + 1)-source helical CB scanning. For the source **a*** _{j}*(

$$\begin{array}{c}{\mathcal{D}}_{k}^{\#}\xi (\mathbf{x})=\sum _{j=1}^{2N+1}{\eta}_{jk}\left({\int}_{{I}_{j}}\frac{\xi ({\mathbf{a}}_{j}(t),{\mathit{\alpha}}_{j}^{\mathbf{x}}(t))}{\left|\right|\mathbf{x}-{\mathbf{a}}_{j}(t)\left|\right|}dt\right)\\ \mathbf{x}\in \mathbf{\Omega}\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}({\mathbf{a}}_{j}(t),{\mathit{\alpha}}_{j}^{\mathbf{x}}(t))\in {\mathrm{\Gamma}}_{j}\times {\mathbb{S}}^{2}\end{array}$$

(23)

where *j* {1, 2,…, 2*N*+1}, and *k* {1, 2,…,2*N*+1}, *N* , and

$${\eta}_{jk}={(-1)}^{\lfloor (j-k)mod(2N+1)/(N+1)\rfloor}.$$

(24)

Finally, let

$$\mathbf{\Psi}=\left\{\mathit{\psi}\begin{array}{l}\mathit{\psi}={\scriptstyle \frac{\mathbf{x}-{\mathbf{a}}_{k}^{s}}{\left|\right|\mathbf{x}-{\mathbf{a}}_{k}^{s}\left|\right|}}\phantom{\rule{0.16667em}{0ex}}\text{or}\phantom{\rule{0.16667em}{0ex}}\mathit{\psi}={\scriptstyle \frac{\mathbf{x}-{\mathbf{a}}_{k}^{e}}{\left|\right|\mathbf{x}-{\mathbf{a}}_{k}^{e}\left|\right|}},\\ k\in \{1,2,\dots ,2N+1\},N\in \mathbb{N}\end{array}\right\}$$

(25)

then Theorem 3 can be extended to the following theorem for (2*N* + 1) sources.

[(2*N* + 1)**- source FBP reconstruction formula**]

$$\begin{array}{r}f(\mathbf{x})=\frac{1}{2\pi}{\mathcal{D}}_{k}^{\#}\circ {\mathcal{H}}^{{\mathit{\beta}}_{jk}}\circ \partial \circ \mathcal{D}f(\mathbf{x}),\\ k\in \{1,2,\dots ,2N+1\},N\in \mathbb{N}\end{array}$$

(26)

*holds for all*
**x** **Ω**, *provided that the mapping*
${\mathit{\alpha}}_{j}^{\mathbf{x}}(t)\mapsto {\mathit{\beta}}_{jk}^{\mathbf{x}}({\mathit{\alpha}}_{j}^{\mathbf{x}}(t))={\mathit{\beta}}_{jk}({\mathit{\alpha}}_{j}^{\mathbf{x}}(t))\phantom{\rule{0.16667em}{0ex}}({\mathit{\alpha}}_{j}^{\mathbf{x}}(t)\in {\mathbb{S}}^{2}\backslash \mathbf{\Psi})$
*satisfies the normalization condition: *

$$\sum _{j=1}^{3}\left({\eta}_{jk}\sum _{\{{\lambda}_{j}\in {\text{I}}_{\text{j}}:\mathbf{x}\mathit{\sigma}={\mathbf{a}}_{j}({\lambda}_{j})\xb7\mathit{\sigma}\}}sgn({\mathit{\beta}}_{jk}^{\mathbf{x}}({\lambda}_{j})\xb7\mathit{\sigma})sgn({{\mathbf{a}}^{\prime}}_{j}({\lambda}_{j})\xb7\mathit{\sigma})\right)=1,\text{a}.\text{e}.\phantom{\rule{0.16667em}{0ex}}\text{in}\phantom{\rule{0.16667em}{0ex}}\mathit{\sigma}\in {\mathbb{S}}^{2}.$$

(27)

The normalization condition (27) holds for

$$\begin{array}{l}{\mathit{\beta}}_{jk}^{\mathbf{x}}({\lambda}_{j})=\frac{{\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})\times \left[({\mathbf{a}}_{(k+N-1)mod(2N+1)+1}^{e}-{\mathbf{a}}_{k}^{s})\times {\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})\right]}{\Vert {\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})\times \left[({\mathbf{a}}_{(k+N-1)mod(2N+1)+1}^{e}-{\mathbf{a}}_{k}^{s})\times {\mathit{\alpha}}_{j}^{\mathbf{x}}({\lambda}_{j})\right]\Vert}\\ \text{for}\phantom{\rule{0.16667em}{0ex}}{\lambda}_{j}\in {I}_{j}\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}\mathbf{x}\xb7\mathit{\sigma}={\mathbf{a}}_{j}({\lambda}_{j})\xb7\mathit{\sigma}.\end{array}$$

(28)

for λ_{j}*I _{j}* and

Theorems 2 and 3 can also be extended to the cases of nonstandard helices in a similarly fashion.

Clearly, the more sources are used, the better the temporal resolution can be achieved, but the higher the system cost will be needed. How to place multiple sources and detectors under the physical constraints is a major issue in the system development, especially when more than two sources are used. Given the focus of this paper on the algorithm development, we would like only to point out that X-ray sources would be made more compact, more powerful and more cost-effective in the future to facilitate prototyping of multisource systems, and will not discuss the related technical issues further. While we have concentrated on the development of exact triple-source helical CBCT algorithms, we recognize that approximate CBCT algorithms should not be overlooked. As Tang *et al.* pointed out [44], [45], approximate CBCT algorithms have important merits in practical applications. In reference to the classic and recent findings on approximate CBCT [21]–[27], [39]–[45], the triple-source FBP algorithms presented in this paper may guide new development of approximate reconstruction algorithms for triple-source CBCT in both helical and circular scanning modes.

In conclusion, we have reported a triple-source FBP algorithm for helical CBCT. Our results allow exact and efficient image reconstruction with ultra-high temporal resolution. With adequate parallel computing hardware, our proposed triple-source FBP approach would run much faster than the BPF counterpart. Therefore, the triple-source FBP algorithm should be a good choice for practical helical CBCT. Our simulation results have verified our formulation, demonstrating excellent image quality. Finally, we have extended our work into the case of (2*N* + 1)-source helical CBCT to inspire or facilitate development of CT and micro-CT systems along this multi-source direction.

The authors would like to thank the anonymous reviewers for their comments and suggestions.

The definition of inter-helix PI-lines for the triple-source case is as follows:

Let *i*, *j* {1, 2,3}, *i* ≠ *j.*
**a*** _{j}(t)* is defined by (21). The inter-helix PI-line between helices

- The line intersects the two helices at
**a**and_{j}(t_{i})**a**respectively;_{j}(t_{j}), - The rotation angles
*t*and_{i}*t*corresponding to the two intersections satisfy the following condition_{j}

$$\{\begin{array}{ll}({t}_{j}+{\phi}_{j})-({t}_{i}+{\phi}_{i})\in (0,2\pi )\hfill & {\phi}_{j}-{\phi}_{i}>0\hfill \\ ({t}_{j}+{\phi}_{j})-({t}_{i}+{\phi}_{i})\in (-2\pi ,0)\hfill & {\phi}_{j}-{\phi}_{i}<0\hfill \end{array}.$$

(29)

For instance, if the three X-ray sources are symmetrically arranged, the three helices would be equally spaced. Thus the phases of the helices are

$${\phi}_{1}=0,{\phi}_{2}=\frac{2\pi}{3},{\phi}_{3}=\frac{4\pi}{3}.$$

(30)

Then, for the inter-helix PI-line between **a**_{1}(*t*) and **a**_{2}*(t),* the rotation angles at the intersection points should satisfy

$$0<\left({t}_{2}-{t}_{1}+\frac{2\pi}{3}\right)<2\pi .$$

(31)

The inter-helix PI-lines for (2*N* + 1)-source can be defined in the same way.

As shown in Fig. 7, in the triple-source case there are three inter-helix PI-lines
$\overline{{\mathbf{a}}_{1}^{s}{\mathbf{a}}_{2}^{e}},\overline{{\mathbf{a}}_{2}^{s}{\mathbf{a}}_{3}^{e}}$ and
$\overline{{\mathbf{a}}_{3}^{s}{\mathbf{a}}_{1}^{e}}$ for **x** **Ω**. The corresponding inter-helix PI-arcs are the coarse solid curve-segments
$\stackrel{\u2322}{{\mathbf{a}}_{1}^{s}{\mathbf{a}}_{1}^{e}},\stackrel{\u2322}{{\mathbf{a}}_{2}^{s}{\mathbf{a}}_{2}^{e}}$ and
$\stackrel{\u2322}{{\mathbf{a}}_{3}^{s}{\mathbf{a}}_{3}^{e}}$.

Define Zhao Window as the region of the scanning cylinder surface that is bounded by the helix turns {**a**_{j}_{mod3+1}(*t*) |(t−t_{0}+2π/3) [−2π ⎣*j*/3, 2π (1 − ⎣*j*/3)]} and {**a**_{(}_{j}_{+1) mod3+1}(*t*) |(t−t_{0} + 4π/3) [−2π ⎣*j*/2, 2π (1 − ⎣*j*/2)]}with **a*** _{j}*(

Fig. 8(a) shows the Zhao Window for the source position **a**_{1}(*t*_{0}) in the triple-source case. Fig. 9(a) shows the Zhao Window cone-beam projected onto the detector plane for triple-source. It is assumed that the detector plane is parallel to the axis of the spiral and is tangent to the spiral cylinder at the point opposite to the source.

Zhao Windows for the source position **a**_{1}(*t*_{0}) in the triple-source case (a) and quintuple-source case (b). The region bounded by the coarse solid curved lines is the Zhao Window.

Define Zhao Window as the region of the scanning cylinder surface that is bounded by the helix turns

$$\left\{{\mathbf{a}}_{(j+N-1)mod(2N+1)+1}(t)|\left(t-{t}_{0}+\frac{2N\pi}{(2N+1)}\right)\in \left[-2\pi \lfloor \frac{j}{[N+2]}\rfloor ,2\pi \left(1-\lfloor \frac{j}{[N+2]}\rfloor \right)\right]\right\}$$

and

$$\left\{{\mathbf{a}}_{(j+N)mod(2N+1)+1}(t)|\left(t-{t}_{0}+\frac{2(N+1)\pi}{(2N+1)}\right)\in \left[-2\pi \lfloor \frac{j}{[N+1]}\rfloor ,2\pi \left(1-\lfloor \frac{j}{[N+1]}\rfloor \right)\right]\right\}$$

with **a*** _{j}*(

Fig. 8(b) shows the Zhao Window for the source position **a**_{1}(*t*_{0}) in the quintuple-source case. Fig. 9(b) shows the Zhao Window cone-beam projected onto the detector plane for quintuple-source.

One concern with triple-source CT is that the improvement in temporal resolution may be offset by the increment in scattering artifacts. In fact, there were similar arguments against the dual-source system in comparison with the single-source counterpart prior to the introduction of the Siemens Definition dual-source CT scanner. Actually, the Siemens dual-source CT scanner has received a very positive market response, which is encouraging for our proposed triple-source extension. Philosophically, we do not believe that the source and detector technology will not be advanced to accommodate the triple source system sooner or later.

The Siemens dual-source CT has been claimed to have a dose benefit because the second detector is smaller, and the dose is reduced at the edge of the field-of-view. The dual-source CT scanner can deliver double the power but it needs only half the dose for cardiac applications—without any compromise in image quality [51]. It has been reported that cross scatter is smaller with the decreased collimation z-width or the decreased object size for dual-source CT [52]. These guidelines should be valuable for optimization of triple-source CT.

Furthermore, we underline that the height of the Zhao window for triple-source CT is of only one third of the Tam-Danielsson window (please refer to Fig. 8(a) and Fig. 9(a) in Appendix II). Theoretically, the total dose delivered to the object remains the same as in the single-source case. For each source-detector pair of the triple source system, we have a significantly smaller collimation in z-width, and the cross scattering effect should be proportionally smaller. In practice, to decrease cross scattering and improve temporal resolution, low or mediate pitch triple-source CT scanning may well be an optimal balance.

Another potential strategy is to utilize shutters so that at any time instant the object is only exposed to one or two sources, since the shutters can be rapidly opened or closed in an alternative fashion [53]. Other scattering correction methods may also be used for the same purpose, such as the recently developed interior reconstruction techniques which have allowed exact reconstruction from purely local data assuming knowledge on a subregion in the volume of interest [50]. These promising techniques may be adapted for triple source CT to improve temporal resolution significantly at an acceptable radiation dose level.

Jun Zhao, Department of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China.

Yannan Jin, Department of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China.

Yang Lu, Department of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China.

Ge Wang, Biomedical Imaging Division, Virginia Tech/Wake Forest University (VT-WFU) School of Biomedical Engineering and Science, Virginia Tech, Blacksburg, VA 24061 USA.

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