Home | About | Journals | Submit | Contact Us | Français |

**|**HHS Author Manuscripts**|**PMC2885857

Formats

Article sections

- Abstract
- I. Introduction
- II. Background Materials
- III. General Approach
- IV. First Fast FBP Algorithm
- V. Second Fast FBP Algorithm
- VI. Analysis on the Quasi-Exactness
- VII. Numerical Results
- VIII. Discussions and Conclusion
- References

Authors

Related links

IEEE Trans Med Imaging. Author manuscript; available in PMC 2010 June 15.

Published in final edited form as:

Published online 2009 November 17. doi: 10.1109/TMI.2009.2035617

PMCID: PMC2885857

NIHMSID: NIHMS199632

Yang Lu,^{*} Alexander Katsevich, Jun Zhao, Member IEEE, Hengyong Yu, Senior Member, IEEE, and Ge Wang, Fellow, IEEE

Yang Lu, Department of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (Email: nc.ude.utjs@gnayvl)

Asterisk indicates corresponding author

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.

Cardiac computed tomography (CT) has been improved over past years, but it still needs improvement for higher temporal resolution in the cases of high or irregular cardiac rates. Given successful applications of dual-source cardiac CT scanners, triple-source cone-beam CT seems a promising mode for cardiac CT. In this paper, we propose two filtered-backprojection algorithms for triple-source helical cone-beam CT. The first algorithm utilizes two families of filtering lines. These lines are parallel to the tangent of the scanning trajectory and the so-called *L* lines. The second algorithm utilizes two families of filtering lines tangent to the boundaries of the Zhao window and *L* lines, respectively, but it eliminates the filtering paths along the tangent of the scanning trajectory, thus reducing the required detector size greatly. The first algorithm is theoretically exact for *r <* 0.265 *R* and quasi-exact for 0.265 *R* ≤ *r <* 0.495 *R*, and the second algorithm is quasi-exact for *r* < 0.495 *R*, where *r* and *R* denote the object radius and the trajectory radius, respectively. Both algorithms are computationally efficient. Numerical results are presented to verify and showcase the proposed algorithms.

Thanks to the modern detection technique, projection data can now be collected on a 2-D array of detectors [1], which is much faster than with the conventional single- or multi-row detectors. The more detector rows we use, the larger the longitudinal coverage we have, and the faster the data acquisition is for volumetric scanning. However, the detector array cannot be too large in a cost and dose efficient way, and the mechanical rotation of an X-ray source has already approached its limit. One alternative solution is to use multi-source computed tomography (CT) architecture, which led to the emergence of Siemens' dual-source CT [2]. Given successful applications of dual-source cardiac CT scanners, triple-source cone-beam CT (CBCT) seems a promising mode for cardiac CT [3].

For the dual-source scanner, the minimum rotation interval is 90° plus the fan angle *α* compared to 180° + *α* in a third-generation geometry. Since the fan angle *α* is relatively small, there is > 40% reduction in acquisition time. As a result, the dual-source scanner reduces the data acquisition time to 0.35 s, yielding 83 ms temporal resolution with cardiac gating.

There are major problems with current cardiac CT including dual-source CT [2]. Actually, even in the favorable cases, retrospectively reconstructed cardiac images still suffer from substantial motion blurring for patients who have high and irregular heartbeats because in practice each projection sector covers a projection angular range of a substantial length. Within such an angular range, the heart moves appreciably, especially when it is not in a relative stationary phase. As a benchmark, we routinely achieve ~0.3 mm spatial resolution in spiral CT of the temporal bone where the motion magnitude is much less than that of the heart [4], [5]. On the other hand, the spatial resolution with cardiac CT is at best in the millimeter domain.

As a natural extension to dual-source CT, triple-source CT promises to bring temporal resolution down to 0.2 s and hopefully to be as fast as electron-beam CT. Interestingly, for helical cone-beam reconstruction [6], [7], the trinity (triple-source architecture) is better than the duality (dual-source architecture) because the triple-source helical scan allows a perfect mosaic of longitudinally truncated cone-beam data to satisfy the Orlov condition and yields better noise performance than the dual-source counterpart [8].

Since the 1-D Hilbert transform can be obtained on a generalized PI-line/chord by backprojecting the weighted differential projection data, which was first proposed by Gel'fand and Graev [9] and generalized in the 2D SPECT case [10] and rediscovered in the 2D and 3D CT cases [11], we previously formulated a backprojection-filtration (BPF) approach for triple-source helical CBCT [12] and a corresponding “slow” filtered-backprojection (FBP) algorithm [13]. In this paper, we present two “fast” exact/quasi-exact FBP algorithms in this important case. The algorithms are not specifically designed for cardiac gating which is widely used in cardiac CT. However, the triple-source scanner achieves a data acquisition speed three times faster than the conventional CT, thus the temporal resolution is significantly increased and may be applied without gating in multiple dynamic cases such as perfusion studies in which gating is generally not possible.

This paper is organized as follows. In Section II, we summarize the general reconstruction formulae and the notations of the inter-PI lines and inter-PI arcs. In Section III, we analyze the intersections of the plane through a reconstruction point and the inter-PI arcs associated with the point. In Section IV, we derive the filtering directions and backprojection coefficients for our first algorithm. In Section V, we present the second algorithm after defining a different weight function. In Section VI, we analyze the percentage of the incorrectly weighted Radon planes. In Section VII, we describe the numerical results to validate our algorithms. In Section VIII, we discuss the relevant issues and conclude the paper.

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

$$\begin{array}{l}\Omega =\{x=({x}_{1},{x}_{2},{x}_{3})|{x}_{1}^{2}+{x}_{2}^{2}\le {r}^{2},{x}_{3\text{min}}\le {x}_{3}\le {x}_{3\text{max}},\}0<r<R\end{array}$$

(1)

where *r* is the radius of the object cylinder and *R* the radius of the scanning cylinder on which a scanning trajectory resides. In the Cartesian coordinate system (*x*_{1}, *x*_{2}, *x*_{3}), the triple-helix trajectories can be expressed as

$$\{\begin{array}{l}{y}_{1}(s)=(R\text{cos}s,R\text{sin}s,\frac{h}{2\pi}s)\hfill \\ {y}_{2}(s)=(R\text{cos}(s+\frac{2}{3}\pi ),R\text{sin}(s+\frac{2}{3}\pi ),\frac{h}{2\pi}s)\hfill \\ {y}_{3}(s)=(R\text{cos}(s+\frac{4}{3}\pi ),R\text{sin}(s+\frac{4}{3}\pi ),\frac{h}{2\pi}s).\hfill \end{array}$$

(2)

where *h* > 0 is the pitch of each helix, and *s* is the rotation angle. Fig. 1 illustrates the triple-source helical CBCT geometry.

Geometry of triple-source helical CBCT. Three X-ray sources are rotated around the *x*_{3}-axis along the helices *y*_{1}(*s*), *y*_{2}(*s*), and *y*_{3}(*s*), respectively. The helices *y*_{1}(*s*), *y*_{2}(*s*), and *y*_{3}(*s*) are on a cylinder of radius *R*. An object to be reconstructed is confined **...**

Previously, we defined inter-helix PI-lines (for simplicity, we call them inter-PI lines thereafter) and extended the traditional Tam–Danielsson window to the Zhao window in the case of triple helices [14]. Specifically, for each source position *y _{j}*(

Recall that an inter-PI line for *y _{j}*(

*Theorem 1:* Through any fixed *x* Ω, there exists one and only one inter-PI line for any pair of the three helices defined by (2).

In the triple-helix case, there are three inter-PI lines for a fixed *x* Ω and corresponding inter-helix PI-arcs whose end points are along the corresponding helices and share the intersection points of the inter-PI lines. The three inter-PI arcs represent the source trajectory arcs along which the sources illuminate the point *x* (Fig. 3).

In 2003, Katsevich proposed a general scheme for constructing inversion algorithms for CBCT [16]. It can be stated as follows:

$$\begin{array}{ll}f(x)=\hfill & -\frac{1}{4{\pi}^{2}}{\int}_{I}\sum _{m}\frac{{c}_{m}(s,x)}{|x-y(s)|}\hfill \\ & \times \underset{0}{\overset{2\pi}{\int}}\frac{\partial}{\partial q}{D}_{f}(y(q),\text{cos}\gamma \beta (s,x)\hfill \\ & +\text{sin}\gamma {\alpha}^{\perp}(s,x,{\theta}_{m})){|}_{q=s}\frac{d\gamma}{\text{sin}\gamma}ds\hfill \end{array}$$

(3)

$$\beta (s,x):=\frac{x-y(s)}{|x-y(s)|}$$

(4)

$${\alpha}^{\perp}(s,x,\theta ):=\beta (s,x)\times \alpha (s,x,\theta )$$

(5)

$${c}_{m}(s,x):=\underset{\epsilon \to {0}^{+}}{\text{lim}}(\varphi (s,x,{\theta}_{m}+\epsilon )-\varphi (s,x,{\theta}_{m}-\epsilon ))$$

(6)

$$\varphi (s,x,\theta ):=\text{sgn}(\alpha \cdot \dot{y}(s))n(s,x,\alpha )$$

(7)

$$I:=\underset{l=1}{\overset{{L}_{c}}{\cup}}\left[{a}_{l},{b}_{l}\right]\to {\mathbb{R}}^{3},I\ni s\to y(s)\in {\mathbb{R}}^{3}$$

$$|\dot{y}(s)|\phantom{\rule{0.2em}{0ex}}\ne 0\phantom{\rule{0.2em}{0ex}}\text{on}I$$

where *D _{f}*(

The aforementioned general inversion formula can be applied to any trajectory that satisfies Tuy's condition, but only when the weight function *n*(*s*, *x, α*) is well designed can the inversion formula have a shift-invariant filtering structure. To derive fast exact FBP algorithms for triple-source helical CBCT, our general approach involves the following key concepts of and analyses on the inflection line, *A−*, T−, *L*-, and *B _{s}*-curves.

On the detector plane, the boundaries of the Zhao window are expressed as

$$u(s)=\frac{D\text{sin}s}{1-\text{cos}s},\phantom{\rule{0.2em}{0ex}}\upsilon (s)=\frac{Dh}{2\pi R}\frac{(s+\Delta s)}{(1-\text{cos}s)}.$$

(8)

where *D* is the distance between the detector and the source, *s* is the angular parameter relative to the corresponding source position, Δ*s = −*2/3*π* and Δ*s =* −4/*3π* are for the top and bottom boundaries respectively. Then

$$\dot{u}(s)=\frac{D}{\text{cos}s-1}$$

(9)

$$\ddot{u}(s)=\frac{D\text{sin}s}{{(\text{cos}s-1)}^{2}}$$

(10)

$$\dot{v}(s)=\frac{Dh}{2\pi R}\frac{[1-\text{cos}s-(s+\Delta s)\text{sin}s]}{{(1-\text{cos}s)}^{2}}$$

(11)

$$\begin{array}{l}\ddot{v}(s)=\frac{Dh}{2\pi R{(1-\text{cos}s)}^{3}}[(\text{cos}s-1)(s+\Delta s)\text{cos}s+2\text{sin}s(\text{cos}s+s\text{sin}s+\Delta s\text{sin}s-1)]\end{array}$$

(12)

$$\frac{{d}^{2}v}{d{u}^{2}}=\frac{\ddot{v}(s)\dot{u}(s)-\ddot{u}(s)\dot{v}(s)}{{[\dot{u}(s)]}^{3}}=\frac{h}{2\pi D}(s+\Delta s-\text{sin}s).$$

(13)

The inflection point exists when *d*^{2}*v*/*du*^{2}
*=* 0. Thus, we obtain *s _{u} =* 2.6053 and

$$\begin{array}{ll}\frac{dv}{du}& =\frac{\dot{v}(s)}{\dot{u}(s)}\\ & =h\frac{[1-\text{cos}s-(s+\Delta s)\text{sin}s]}{2\pi R(\text{cos}s-1)}\\ & =\frac{h}{2\pi R}\text{cos}s\end{array}$$

(14)

Because cos *s _{u}* = cos

To construct an appropriate weight function, at first we must know how Radon planes intersect with the trajectories. Recall that the number of intersection points only changes when a Radon plane is tangent to the trajectory or contains one PI line/inter-PI line. Hence, if we find all such Radon planes, we can determine the distribution of the intersection points. Since each plane is uniquely determined by its normal vector, in the following sections we use unit vectors instead of the Radon planes. Let us define several key notions introduced in [17]. An *A*-curve consists of all unit vectors orthogonal to an inter-PI line. A *T*-curve consists of all unit vectors

$$\alpha (s)=\pm \frac{(x-y)(s))\times \dot{y}(s)}{\left|(x-y(s))\times \dot{y}(s)\right|}$$

(15)

where *s* belongs to an inter-PI arc. Actually, the *A*-curve represents all Radon planes containing one inter-PI line, and the *T*-curve represents all Radon planes tangent to the trajectory. Since there are three inter-PI lines and three inter-PI arcs for a fixed *x*, there are accordingly three *A*-curves and three *T*-curves. Similar to [17], let us use spherical coordinates (*θ*_{1}, *θ*_{2}) to describe these curves on the unit sphere

$$\begin{array}{l}\alpha =(\text{cos}{\theta}_{1}\text{sin}{\theta}_{2},\text{sin}{\theta}_{1}\text{sin}{\theta}_{2},\text{cos}{\theta}_{2}),-\pi \le {\theta}_{1}\le \pi ,0\le {\theta}_{2}\le \pi .\end{array}$$

(16)

With the identification (*θ*_{1}, *θ*_{2}) ((*θ*_{1}+*π*) _{mod 2}* _{π}*,

As an example, the *A*-curves and *T*-curves of point *x*
** =** (0.1, 0, 0) are illustrated in Fig. 5, where

Visualization of the domains delimited by the *A*-curves and *T*-curves on the surface of the unit sphere. (a) *A*-curves and *T*-curves in spherical coordinates for *x* = (0.1, 0, 0); (b), (c), and (d) the zoom-in versions of the areas bounded by the bottom left, **...**

The *A*-curves and *T*-curves divide the surface of the unit sphere into several connected domains, in each of which all the planes through *x* have the same number of intersection points (IPs) with the inter-PI arcs of *x*. Given an object point and one source trajectory, the number of IPs changes only when a Radon plane is tangent to the trajectory or contains an endpoint of the trajectory. The *A*-curve represents all planes containing the endpoints of the trajectory, and the *T*-curve represents all planes tangent to the trajectory. If we pick any Radon plane and rotate it around one direction, the normal vector of this plane forms a curve on the unit sphere. Clearly, only when this curve intersects with the *A*-curve or *T*-curve does the number of IPs change. Thus, the *A* -curve and *T*-curve define the boundaries of different domains in which the number of IPs is constant. The distribution of IPs over the inter-PI arcs is listed in Table I.

Distribution of IPs on Each of the Domains Delimited by the *A*-Curves and *T*-Curves on the Surface of the Unit Sphere. Symbol “-” Indicates no IP on the Corresponding Inter-PI Arc

To determine the distribution of IPs, we first pick a vector *α*(*θ*_{1}, *θ*_{2}) in each domain, and then generate the plane through *x* and perpendicular to *α*(*θ*_{1}, *θ*_{2}), and compute numerically the number of IPs.

By construction, a *T*-curve always starts from an *A*-curve and ends on another *A*-curve. It can be seen from Fig. 5 that a *T*-curve is possibly not smooth at some point *a _{c}*, but the limits of unit tangent vectors at
${a}_{c}^{-}$ and
${a}_{c}^{+}$ are equal. Such a point

The diagram we plot in spherical coordinates deforms smoothly as a function of *x*. The new diagram is equivalent to the old one if the distribution of IPs remains the same. An essential change could happen when three boundaries intersect each other at one point, which is called “critical event.” A “critical event” may happen in the following seven cases.

- Three
*A*-curves intersect at one point. - Three
*T*-curves intersect at one point. - Two
*A*-curves and one*T*-curve intersect at one point. - One
*A*-curve and two*T*-curves intersect at one point. - A
*T*-curve becomes tangent to an*A*-curve at a point of nonsmoothness (i.e., cusp). - When the order of tangency (i.e., the zero derivatives of this order) at the beginning of the
*T*-curve is increased, the*T*-curve reemerges on the other side of the*A*-curve. *T*-curve develops a smooth dent and becomes tangent to an*A*-curve.

From Lemmas 2–3, we know Case 1, 2, 4, 6, 7 do not occur for *r* < 0.495*R* and Case 5 does not occur for *r* < 0.265*R*. Case 3 is possible. If Case 3 takes place, the tangency of *T*-curve and *A*-curve will move across another *A*-curve, then one domain disappears. For example, when *x* = (0.1, 0, 0) gradually changes to *x* = (0, −0.15, 0), in Fig. 5(d) the tangency of *T*_{3} and *A*_{2} will move across *A*_{3}, and domain *D*_{10} disappears [see Fig. 7(a)]. Case 5 is possible for *r* ≥ 0.265*R*. If it takes place, a *T*-curve will only intersect *A*-curves at the endpoints. That is, the cusp of that *T*-curve and one domain disappear [see Fig. 7(b)].

The *L*-curve is used to split the domain *D*_{4} into subdomains, making the weight function *n* continuous across all the *A*-curves. This is the key requirement, which will allow us to develop efficient reconstruction algorithms (see Section IV below). Thus, the *L*-curve should not go across an *A*-curve. Let us fix *x* and run *s* over the three inter-PI arcs, *x^* forms a trajectory on the detector plane. Because at the endpoint of the inter-PI arc the line connecting *y _{i}*(

For *r* ≥ 0.265*R*, one or more cusps will disappear if “critical event Case 5” occurs, then the *L*-curve may start from the intersection of *T*-curve and *A*-curve, and end at one *A*-curve. Also, the *L*-curve may start from one cusp and end at one *A*-curve or start from the intersection of *T*-curve and *A*-curve, and end at one *A*-curve. For example, see Fig. 9, *L*_{1} starts from the intersection of *T*_{3} and *A*_{2}, and ends at the intersection of *T*_{2} and *A*_{2}; *L*_{2} starts from one point on *A*_{2}, and ends at the cusp of *T*_{1}; *L*_{3} starts from the cusp of *T*_{1}, and ends at one point on *A*_{2}.

L-curves for *x* = (0.2, −0.3, 0). (a) is the full diagram shows different regions spit by *A*–, *T*- and *L*-curves; (b), (c), (d) are the zoom-in versions of the regions bounded by the up, bottom right, and bottom left circles in (a).

Whatever the endpoints of *L*-curves are, the *L*-curves intersects at one point, i.e., *θ*_{2} = *π* or 0 in the spherical coordinates (which corresponds to the plane containing *x* and parallel to the *x*_{1} − *x*_{2} plane). Then *D*_{4} is split into several sub-domains. If the endpoints of *L*-curves are cusps, by Lemma 5, each sub-domain contains only one *A*-curve. If not, small “line segments” on *A*-curves may appear and the sub-domains may contain more than one *A*-curve [see Fig. 9(b)].

A *B _{s}*-curve consists of all unit vectors perpendicular to

(a) *B*_{s}-curve is tangent to a *T*-curve in *D*_{11} for the source on *y*_{1}(*s*). (b) *B*_{s}-curve is tangent to a *T* -curve in *D*_{5} for the source on *y*_{1}(*s*). (c) *B*_{s}-curve passes across the second *T*-curve when the source on *y*_{1}(*s*).

The jumps across an *A*-curve can only be of two types: from a 1-IP domain to a 3-IP domain and from a 3-IP domain to a 5-IP domain. Note that the *B _{s}*-curve is tangent from the inside to

Now, we are ready to design our first fast FBP algorithm for triple-source helical CBCT. We will start with specifying the weight function *n*(*s*, *x*, *α*). Then, we will determine the filtering directions by finding the discontinuities of (*s*, *x*, *θ*) := sgn(*α* · (*s*))*n*(*s*, *x*, *α*). Then, the backprojection coefficients can be calculated according to (6). Once the filtering lines and the backprojection coefficients are determined, we can use (3) to reconstruct the object.

To have an efficient FBP structure, the weight function *n*(*s*, *x*, *α*) should be continuous across all *A*-curves. Thus, the weight function can be defined as shown in Table II. The values in Table II are the weights assigned to IPs. For example, in the *D*_{1} domain the Radon plane has only one IP on the inter-PI segment
$\stackrel{\u2040}{{s}_{1}^{s}{s}_{1}^{e}}$. Accordingly, we assign weight 1 to this IP and use “-” to indicate that there is no IP on the inter-PI segments
$\stackrel{\u2040}{{s}_{1}^{s}{s}_{1}^{e}}$,
$\stackrel{\u2040}{{s}_{2}^{s}{s}_{2}^{e}\phantom{\rule{0.2em}{0ex}}}$, and
$\stackrel{\u2040}{{s}_{3}^{s}{s}_{3}^{e}}$. In the *D*_{11} domain the Radon plane has three IPs on
$\stackrel{\u2040}{{s}_{1}^{s}{s}_{1}^{e}}$, one IP on
$\stackrel{\u2040}{{s}_{2}^{s}{s}_{2}^{e}\phantom{\rule{0.2em}{0ex}}}$, and one IP on
$\stackrel{\u2040}{{s}_{3}^{s}{s}_{3}^{e}}$. Thus, we assign weight 1 to one IP on
$\stackrel{\u2040}{{s}_{1}^{s}{s}_{1}^{e}}$ and weight 0 to all other IPs.

To find the backprojection coefficients, let us pick a representative point in each area, determine the discontinuities of (*s*, *x*, *θ*) := sgn(*α* · (*s*))*n*(*s*, *x*, *α*), and extend the results by continuity to the entire area. A discontinuity of sgn occurs only when a *B _{s}*-curve is tangent to a

By construction, the weight function *n* is continuous across all inter-PI lines. A discontinuity of *n* occurs only when a *B _{s}*-curve intersects a

Fig. 13(a) and (b) summarizes the filtering lines and the backprojection coefficients discussed above. In these figures,
${L}_{0}^{\text{'}}$ is the line parallel to *L*_{0} and *L _{l}* denotes the

(a) Filtering lines in the case of *x^* *G*_{1} *G*_{3} for the first fast FBP algorithm. (b) Filtering lines in the case of *x^* *G*_{2} for the first fast FBP algorithm.

Again, the design of our second fast FBP algorithm starts with specifying new weights (Table III). By construction, *n*(*s*, *x*, *α*) is continuous across all inter-PI lines. More importantly, a swap of two IPs takes place when a *B _{s}*-curve becomes tangent to a

A discontinuity of *n* can occur only when a *B _{s}*-curve intersects a

Filtering lines for our two fast FBP algorithms when *x^* is above *L*_{it} · *L*_{t}_{2} and *L*_{t}_{1} are for the first and second algorithms, respectively.

The reconstruction formula for the second algorithm is very similar to that for the first algorithm. The only difference lies in the selection of the filtering lines. For clarity, our second fast FBP algorithm is illustrated in Fig. 16. Because the filtering paths along the tangent of the scanning trajectory are eliminated, the required detector area is reduced by at least 30% (Fig. 14).

By Lemma 3, there are two types of “line segments” according to different critical events. First, let us consider the “line segment” related to a critical event in Case 3. Recall that before entering *D*_{4} the *B _{s}*-curve will be tangent to a

Since the algorithms are not always exact, we are interested in estimating what percentage of the Radon planes is incorrectly calculated. If the Radon planes with approximate weighting only have a small percentage, the algorithms can be considered quasi-exact, and we can still reconstruct with high image quality.

First, let us consider the incorrectly weighted planes caused by the critical event in Case 3. It appears in the area *r* < 0.495*R*. Let us fix *x* for *r* < 0.495*R*, denote the intersection of the Radon plane and the detector plane as *L*(*θ*), *θ* [0, 2*π*], run *s* over the three inter-PI arcs, and see what happens with *x^* and *L*(*θ*). Based on the discussion in Section III-E, for *x^* in *G*_{2}, if the critical event occurs, the *B _{s}*-curve will first intersect a

Then, let us consider the incorrectly weighted planes caused by the critical event in Case 5. Recall that the *L*-curves are used to split the domain *D*_{4} into sub-domains, making the weight function *n* continuous across all the *A*-curves, and the cusps are the starting and ending points of the *L*-curves. If the endpoints of the *L*-curves are not the cusps, there will be small fractions (or “line segments”) on the *A*-curves, making the weight function *n* discontinuous across them and ruining the FBP structure of our algorithms. It possibly occurs for *r* ≥ 0.265*R*. As discussed above, the *B _{s}* -curve will first enter a 1-IP domain from a 3-IP domain across the “line segment,” and then pass through an

To verify and showcase our proposed fast FBP algorithms, numerical tests were performed using the Clock phantom. This phantom consists of ellipses, as parameterized in Tables IV. In the simulations, the origin of the reconstruction coordinate system was set to the center of each phantom. The spherical phantom support was of 375 mm for the experiment. Three sources were arranged uniformly along a circle with their corresponding detectors on the opposite side. The source-detector distance was 1000 mm. Projections were generated from 1000 view angles while the sources and the detectors were constantly moved along three helixes in one turn. The helix was of 750 mm in radius and 100 mm in pitch. The detector plane consisted of 1300 × 200 detection elements of 1.0 × 1.0 mm^{2}.

The implementation of our algorithms consists of the following steps.

- Step 1) Differentiate each projection with respect to variable
*s*. - Step 2) For each
*y*(_{i}*s*),*i*{1,2,3} perform the Hilbert transform of derivative data along the given filtering directions on the corresponding detector plane. - Step 3) Backproject the filtered data on the inter-PI segments to reconstruct the object point.

A more detailed description of the proposed algorithm is similar to [20] except for the following differences: 1) in triple-helices geometry the filtered data are backprojected on inter-PI segments; 2) there are two families of filtering lines for each algorithm so that each point on the detector plane will be filtered twice. Since our algorithms allow shift-invariant filtration, all results are in Cartesian coordinates directly, and there is no coordinate transform like what we have used in the slow-FBP algorithm [13] or BPF algorithm [12].

The algorithms were coded in MATLAB and executed on a regular PC (Intel Core2 Duo CPU 3.06 GHz, 4 GB RAM). Reconstructed images are shown in Fig. 17. Our numerical results show that in the case of *r* = 0.495*R* both two algorithms produced high quality images.

Although our previously published BPF algorithm for triple-source helical CBCT can indeed produce excellent image quality, FBP algorithms (either “slow” or “fast”) are computationally desirable for several reasons, such as being amendable for parallel processing as discussed in [13]. In particular, while the computational structures of our BPF algorithm and FBP algorithms are quite similar, the FBP algorithms avoid densely sampled intermediate reconstruction in the PI-line-based coordinate system, and more importantly they can reconstruct a region of interest (ROI) or volume of interest (VOI) much more efficiently than the BPF counterpart. Note that ROI/VOI reconstruction is very common in medical imaging. A related technology called “interior tomography” is being actively developed to target this type of problems [21], [22]. Then, an interesting possibility would be to develop tripe-source interior CBCT.

The proposed two fast exact/quasi-exact FBP algorithms for triple-source helical CBCT have their advantages and disadvantages. From the perspective of exact reconstruction, the first algorithm is more desirable than the second algorithm because it is not affected by critical events in Case 3. However, in terms of efficient data acquisition, it requires a larger detector area than the second algorithm. In the medical CT field, the rectangular detector shape is most popular, and the helical pitch may be varied case by case. Therefore, it is practically possible to have projection data for reconstruction using either or both of our two fast FBP algorithms. A detailed comparison of these two algorithms should be done in terms of image quality and radiation dosage. Other designs for fast FBP in the triple-source helical CBCT are also possible.

One concern with triple-source CT is the increment in scattering artifacts, which was addressed in our earlier paper for the BPF formulation [13]. Although the height of the Zhao window for triple-source helical CT is one third of the Tam-Danielsson window, our proposed fast FBP formulation requires additional data beyond the Zhao window. In this case, the total dose delivered to the patient can be controlled to be comparable to that in the single-source case, and the total amount of scattering would be similar as well. Another way is to utilize source multiplexing so that not all the sources are on simultaneously. Other scattering correction means, such as interior tomography [21], [22], may be applied as well.

In conclusion, based on our previous triple-source helical CBCT work and guided by the Katsevich general reconstruction scheme, we have proposed two fast FBP algorithms for triple-source helical CBCT. The proposed algorithms utilize the inter-PI line and inter-PI arcs, and have a shift-invariant filtering structure. Unlike our slow-FBP algorithm performing filtration spatial-variantly line by line, the proposed fast-FBP algorithms filter projection data spatial-invariantly view by view, representing a significant computational benefit. Since triple-source helical CBCT may triple temporal resolution, it seems a promising mode for cardiac CT and other applications, and our proposed algorithms may find applications in this context.

This work was supported in part by the National Science Foundation of China (30570511 and 30770589), the National High Technology Research and Development Program of China (2007AA02Z452), the National Basic Research Program of China (2010CB834300), and NIH/NIBIB (EB002667, EB004287, and EB007288). The work of A. Katsevich was supported in part by the National Science Foundation under Grant DMS-0806304.

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

Let us fix a point *x* Ω and find its three associated inter-PI lines as shown in Fig. 3. Then, we can select a source position
$s\in ({s}_{j}^{s},{s}_{j}^{e})$, *j* {1, 2, 3} and determine how the inter-PI lines project onto the corresponding detector plane. For simplicity, here and below the projection of *y _{j}*(

*Lemma 1*: On a detector plane, the slopes of the projected inter-PI lines
$\overline{{\widehat{s}}_{j}^{s}{\widehat{s}}_{j\text{mod}3+1}^{e}}$ and
$\overline{{\widehat{s}}_{(j+1)\text{mod}3+1}^{s}{\widehat{s}}_{j}^{e}}$ are always positive, and that of the inter-PI line
$\overline{{\widehat{s}}_{j\text{mod}3+1}^{s}{\widehat{s}}_{(j+1)\phantom{\rule{0.2em}{0ex}}\text{mod}3+1}^{e}}$, *j* {1, 2, 3} is always negative.

*Proof*: Without loss of generality, suppose the source position we select is *y*_{1}(*s*_{0}),
${s}_{0}\in \left({s}_{1}^{s},{s}_{1}^{e}\right)$. By construction,
${s}_{2}^{e}-{s}_{1}^{s}<2\pi $,
${s}_{1}^{e}-{s}_{3}^{s}<2\pi $, and
${s}_{3}^{e}-{s}_{2}^{s}>0$. Hence, the projections of
${s}_{1}^{s}$, and
${s}_{3}^{e}$ and
${s}_{3}^{e}$ are always to the left of those of
${s}_{2}^{e}$,
${s}_{2}^{s}$, and
${s}_{1}^{e}$ respectively (Fig. 18). When
$s\in \left({s}_{1}^{s},{s}_{1}^{e}\right)$ changes, the point *x^*, i.e., the projection of *x* onto the detector plane, moves inside the region G := *G*_{1} *G*_{2} *G*_{3}. Clearly, *x^* reaches its highest (resp., lowest) position in the vertical direction when *x^* is at the intersection of Γ* _{l}* and Γ

Projected inter-PI lines
$\overline{{\widehat{s}}_{1}^{s}{\widehat{s}}_{2}^{e}}$,
$\overline{{\widehat{s}}_{2}^{s}{\widehat{s}}_{3}^{e}}$ and
$\overline{{\widehat{s}}_{3}^{s}{\widehat{s}}_{1}^{e}}$ on the detector plane. Thick curve segments denote the inter-PI arcs.

As follows from the definition in [14], the inter-PI line $\overline{{s}_{2}^{s}{s}_{3}^{e}}$ satisfies:

$$\{\begin{array}{l}{x}_{1}=Rt\text{cos}{s}_{2}^{s}+R(1-t)\text{cos}{s}_{3}^{e}\\ {x}_{2}=Rt\text{sin}{s}_{2}^{s}+R(1-t)\text{sin}{s}_{3}^{e}\\ {x}_{3}=\frac{h}{2\pi}t\left({s}_{2}^{s}-\frac{2\pi}{3}\right)+\frac{h}{2\pi}\left(1-t\right)\left({s}_{3}^{e}-\frac{4\pi}{3}\right)\end{array}$$

(17)

where *G*_{2} and
${s}_{2}^{s}$,
${s}_{3}^{e}\in \left({s}_{0},{s}_{0}+2\pi \right)$. Let

$${x}_{1}={r}_{0}\text{cos}{\mu}_{0},\phantom{\rule{0.2em}{0ex}}{x}_{2}={r}_{0}\text{sin}{\mu}_{0}$$

(18)

where *r*_{0} [0, 0.495*R*] and *μ*_{0} [0, 2*π*]. We have

$$\text{cos}\frac{{s}_{3}^{e}-{s}_{2}^{s}}{2}=\frac{{r}_{0}\text{sin}\left({\mu}_{0}-{s}_{2}^{s}\right)}{\sqrt{{R}^{2}+{r}_{0}^{2}-2R{r}_{0}\text{cos}\left({\mu}_{0}-{s}_{2}^{s}\right)}}.$$

(19)

Then, (19) can be rewritten as

$$\text{cos}\frac{{s}_{3}^{e}-{s}_{2}^{s}}{2}=\frac{\text{sin}\left({\mu}_{0}-{s}_{2}^{s}\right)}{\sqrt{{\left(\frac{R}{{r}_{0}}+\text{cos}\left({\mu}_{0}-{s}_{2}^{s}\right)\right)}^{2}}+{\text{sin}}^{2}\left({\mu}_{0}-{s}_{2}^{s}\right)}.$$

(20)

When
${\mu}_{0}-{s}_{2}^{s}$ is fixed and *r*_{0} is reduced,
$\left|\text{cos}(1)/(2)({s}_{3}^{e}-{s}_{2}^{s})\right|$ decreases. Therefore, the right side of (20) reaches its maximum or minimum when *r*_{0} is maximized, i.e., at *r*_{0} = 0.495*R*. Those maximum and minimum values can be numerically calculated [Fig. 19(a)], and we have

$$-0.4949\le \text{cos}\frac{1}{2}\left({s}_{3}^{e}-{s}_{2}^{e}\right)\le 0.4949$$

or

$$2.1062\le {s}_{3}^{e}-{s}_{2}^{s}\le \mathrm{4.1770.}$$

(21)

Next we prove that $0\le {s}_{2}^{s}-{s}_{0}\le 4.1773$ implies

$${\upsilon}_{1}\left({s}_{3}^{e}-{s}_{0}\right){\upsilon}_{2}\left({s}_{2}^{s}-{s}_{0}\right)>0$$

(22)

where *υ*_{1}(*s*) = (Dh)/(2*πR*)(s – 4*π*/3)/(1 – cos *s*) and *υ*_{2}(*s*) = (Dh)/(2*πR*)(*s*–2*π*/3)/(l–cos *s*). Fig. 19(b) shows the function Φ(*s*) = 1 – cos *s* – (*s* – 4/3*π*)sin s in the range *s* (0,2*π*), demonstrating that *υ̇*_{1}(*s*) is always positive. Hence
${\upsilon}_{1}({s}_{3}^{e}-{s}_{0})$ is monotonically increasing. Let us fix
${s}_{3}^{e}={s}_{2}^{s}+2.1062$ and plot the function
$\Psi ({s}_{2}^{s})={\upsilon}_{1}({s}_{3}^{e}-{s}_{0})-{\upsilon}_{2}({s}_{2}^{s}-{s}_{0})$ [Fig. 19(c)]. Clearly, this function is always positive in the range
$0\le {s}_{2}^{s}-{s}_{0}\le 4.1773$. Note that *s* = 4.1773 + *s*_{0} is the intersection of Γ* _{l}* and Γ

Equation (22) indicates that ${\widehat{s}}_{2}^{s}$ is always lower than ${\widehat{s}}_{3}^{e}$ in the vertical direction. Since ${\widehat{s}}_{2}^{s}$ is to the right of ${\widehat{s}}_{3}^{e}$, the slope of the inter-PI line $\overline{{\widehat{s}}_{2}^{s}{\widehat{s}}_{3}^{e}}$ is always negative. Due to symmetry, the other two cases $s\in ({s}_{2}^{s},{s}_{2}^{e})$ and $s\in ({s}_{3}^{s},{s}_{3}^{e})$ can be handled similarly. This finishes the proof.

*Lemma 2*: A *T*-curve cannot be tangent to an *A*-curve at an interior point of *T*.

*Proof*: The interior point of *T*-curve is any point of the *T*-curve except an endpoint. It has been proved in [17] that a *T*-curve is smooth everywhere, except possibly at a cusp. Let
${s}_{i}^{c}$, *i* {1, 2,3} be the point where the cusp occurs. Suppose a *T*-curve is tangent to an *A*-curve at *α*(*s*). If
$s={s}_{i}^{c}\in ({s}_{i}^{s},{s}_{i}^{e})$, where
${s}_{i}^{s}$,
${s}_{i}^{e}$ are the endpoints of the *T*-curve, then the osculating plane Π* _{c}*(

*Lemma 3*: Case 1, 2, 4, 6, 7 do not occur for *r <* 0.495*R* and Case 5 does not occur for *r <* 0.265*R*.

*Proof*: Cases 1 and 2 are impossible because they mean that there will be a plane containing three inter-PI lines or tangent to three inter-PI arcs.

In Case 4, there will be one plane Π containing one inter-PI line and tangent to two inter-PI arcs. Assume this inter-PI line is
$\overline{{s}_{1}^{s}{s}_{2}^{e}}$, let us pick a point
$s={s}_{1}^{s}$ on *y _{i}*(

In Case 5, there will be one plane containing one inter-PI line and tangent to one inter-PI arc at the inflection point. Thus, the inter-PI line on the detector overlaps the inflection line, i.e.,
$\overline{{\widehat{s}}_{2}^{s}{\widehat{s}}_{3}^{e}}$ overlaps *L _{it}* when,
${s}_{2}^{s}={s}_{u}$ and
${s}_{3}^{e}=2{s}_{u}$, where

In Case 6, a *T*-curve will intersect one *A*-curve twice before meeting a cusp. Suppose this takes place at inter-PI arc
$\widehat{{s}_{1}^{s}{s}_{1}^{e}.}$. Let us pick a point
$s={s}_{1}^{s}$ on *y*_{1}(*s*) and see what happens on the detector plane when *s* moves. By construction, the plane II containing _{1}(*s*) and *x* intersects the detector plane at the line *L* which is parallel to the helix tangent across *x^*. At
$s={s}_{1}^{s}$, *x^* is on Γ^{+1} and Π contains inter-PI line
$\overline{{\widehat{s}}_{1}^{s}{\widehat{s}}_{2}^{c}}$. As *s* moves along *y*_{1}(*s*), *x^* moves downwards. Notice that *L* is parallel to the asymptote of the Tam–Dannielson window, so it will not intersect Γ^{−2} provided that *x^* moves across the asymptote, at where the cusp occurs. Hence, *n* will not contain the inter-PI line
$\overline{{\widehat{s}}_{1}^{s}{\widehat{s}}_{2}^{e}}$ and Case 6 is impossible.

By Lemma 2, Case 7 is impossible. This finishes the proof. Let *G*_{21} be the area bounded by *L _{it}*,
${\Gamma}_{l}^{\text{'}}$ and the line

*Lemma 4*: The inflection point *s^ _{u}* (

*Proof*: By Lemma 3, any point in the area *r* < 0.265*R* has three cusps in the diagram. Note that there is one IP in each inter-PI arc within *D*_{4}. Since all three cusps are in *D*_{4}, an osculating plane of one inter-PI arc intersects two other inter-PI arcs exactly once at one point. Assume that this osculating plane Π* _{c}* contains

*Lemma 5*: An *L* -curve never intersects an *A* -curve for *r* < 0.265*R*.

*Proof*: If an *L*- and *A* -curves intersect, an *L* -line through *x^* overlaps the inter-PI line. Let us pick one point on the inter-PI arc
$\stackrel{\u2040}{{s}_{3}^{s}{s}_{3}^{e}}$ and consider the slope of the inter-PI line on the corresponding detector plane. By construction, an *L*-curve always starts from a cusp of a *T*-curve and ends on a cusp of another *T*-curve. For the osculating plane Π* _{C}*, its intersection with the detector plane is the line tangent to Γ

Relationship among the inter-PI line *L*_{a}, *L*-line *L*_{l} and inflection line *L*_{it}(*L*_{ib}). (a) *x^* in *G*_{2} and above *s^ *_{u}, and (b) *x^* in *G*_{2} and below *s^*_{d}.

If *x^* is in *G _{2}* and above

For the point on other inter-PI arcs, the situation is the same. This finishes the proof.

The angles in the detector plane are not equivalent to that in the spherical coordinate system because the detector plane is parallel to the *x*_{3}-axis while the plane in spherical coordinates is perpendicular to *y _{i}*(

Let *S* = (0, 0, *R*) denote the source position, *x* one point in the sphere centered at *o* with radius *r*, *A* = (*d*, 0,0) the projection of *x*, *B* and *C* two points on the detector plane, and *θ* the angle between lines *AB* and *AC*. The plane perpendicular to the line *SA* intersects the lines *SB* and *SC* at *P*_{1} and *P*_{2}, respectively. The angle between lines *AP*_{1} and *AP*_{2} is what we need (see Fig. 23).

By construction

$$\{\begin{array}{c}\left({P}_{1}-A\right)\cdot \left(S-A\right)=0\\ \left({P}_{1}-S\right)={c}_{1}\left({P}_{1}-B\right)\end{array}$$

(23)

$$\{\begin{array}{c}\left({P}_{2}-A\right)\cdot \left(S-A\right)=0\\ \left({P}_{2}-S\right)={c}_{2}\left({P}_{2}-C\right)\end{array}$$

(24)

where *B* = (*d* + *d* cos *α*, *d* sin *α*,0), *C* = (*d + d cos*(*α + θ*), *d* sin(*α* + *θ*), 0), *c*_{1} and *c*_{2} are two nonzero constants. Solving (23) and (24), we have

$$\begin{array}{l}{P}_{1}=(\frac{d\left({R}^{2}+{d}^{2}\right)\left(1+\text{cos}\alpha \right)}{{R}^{2}+{d}^{2}\left(1+\text{cos}\alpha \right)},\frac{d\left({R}^{2}+{d}^{2}\right)\text{sin}\alpha}{{R}^{2}+{d}^{2}\left(1+\text{cos}\alpha \right)},\frac{{d}^{2}R\text{cos}\alpha}{{R}^{2}+{d}^{2}\left(1+\text{cos}\alpha \right)})\end{array}$$

(25)

$$\begin{array}{l}{P}_{2}=(\frac{d\left({R}^{2}+{d}^{2}\right)\left(1+\text{cos}\left(\alpha +\theta \right)\right)}{{R}^{2}+{d}^{2}\left(1+\text{cos}\left(\alpha +\theta \right)\right)},\frac{d\left({R}^{2}+{d}^{2}\right)\text{sin}\left(\alpha +\theta \right)}{{R}^{2}+{d}^{2}\left(1+\text{cos}\left(\alpha +\theta \right)\right)},\frac{{d}^{2}R\text{cos}\left(\alpha +\theta \right)}{{R}^{2}+{d}^{2}\left(1+\text{cos}\left(\alpha +\theta \right)\right)})\end{array}$$

(26)

$$\begin{array}{l}\text{cos}\phi =\frac{\left({P}_{1}-A\right)\cdot \left({P}_{2}-A\right)}{\sqrt{\left({P}_{1}-A\right)}\sqrt{\left({P}_{2}-A\right)}}\\ \phantom{\rule{2em}{0ex}}=\frac{\text{cos}\alpha \text{cos}\left(\alpha +\theta \right)+\left(1+{k}^{2}\right)\text{sin}\alpha \text{sin}\left(\alpha +\theta \right)}{\sqrt{1+{k}^{2}{\text{sin}}^{2}\alpha}\sqrt{1+{k}^{2}{\text{sin}}^{2}\left(\alpha +\theta \right)}}.\end{array}$$

(27)

where *k* = *d/R*.

Computing the derivative of (27) we find that reaches its maximum at *α* = −*θ*/*2* or *α* = −*θ*/*2* + *π*. In this case, (27) becomes

$$\text{cos}\phi =\frac{{\text{cos}}^{2}\frac{\theta}{2}-\left(1+{k}^{2}\right){\text{sin}}^{2}\frac{\theta}{2}}{{\text{cos}}^{2}\frac{\theta}{2}+\left(1+{k}^{2}\right){\text{sin}}^{2}\frac{\theta}{2}}.$$

(28)

For a fixed *θ*, cos is monotonically decreasing while *k* is increased. Since *r <* 0.495 *R*, *k* reaches its maximum
$1/\sqrt{3}$ when the line *OX* is perpendicular to the line *SA* and its minimum 0 when the line *OX* is parallel to the line *OS*. Hence,

$$\theta <\phi <\text{arccos}\frac{3-4{\text{tan}}^{2}\frac{\theta}{2}}{3+4{\text{tan}}^{2}\frac{\theta}{2}}$$

or

$$\theta <\phi <2\text{arctan}\left(\frac{2}{\sqrt{3}}\text{tan}\frac{\theta}{2}\right).$$

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Yang Lu, Department of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (Email: nc.ude.utjs@gnayvl)

Alexander Katsevich, Department of Mathematics, University of Central Florida, Orlando, FL 32816 USA (Email: ude.fcu.liam@ivestaka)

Jun Zhao, Department of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (Email: nc.ude.utjs@oahznuj).

Hengyong Yu, SBES Division & ICTAS Center for Biomedical Imaging, Virginia Tech, Blacksburg, VA 24061 USA (Email: gro.eeei@uy-gnoygneh)

Ge Wang, SBES Division & ICTAS Center for Biomedical Imaging, Virginia Tech, Blacksburg, VA 24061 USA (Email: gro.eeei@gnaw-eg).

1. Wang G, Crawford CR, Kalender WA. Guest Editorial: Multirow detector and cone-beam spiral/helical CT. IEEE Trans Med Imag. 2000 Sep;19(9):817–821. [PubMed]

2. Flohr TG, McCollough CH, Bruder H, Petersilka M, Gruber K, Süß C, Grasruck M, Stierstorfer K, Krauss B, Raupach R. First performance evaluation of a dual-source CT (DSCT) system. Eur Radiol. 2006;16:256–268. [PubMed]

3. Kachelrieß M, Knaup M, Kalender WA. Multithreaded cardiac CT. Med Phys. 2006;33:2435–2447. [PubMed]

4. Vannier M, Wang G. Spiral CT refines imaging of temporal bone disorders. Diagnostic Imag. 1993;15:116–121. [PubMed]

5. Wang G, Zhao S, Yu H, Miller C, Abbas P, Gantz B, Lee S, Rubinstein J. Design, analysis and simulation for development of the first clinical micro-CT scanner1. Academic Radiol. 2005;12:511–525. [PubMed]

6. Wang G, Lin TH, Cheng PC, Shinozaki DM. A general cone-beam reconstruction algorithm. IEEE Trans Med Imag. 1993 Sep;12(3):486–496. [PubMed]

7. Katsevich A. Theoretically exact filtered backprojection-type inversion algorithm for spiral CT. SIAM J Appl Math. 2002;62:2012–2026.

8. Orlov SS. Theory of three-dimensional reconstruction. 1. Conditions of a complete set of projections. Sov Phys Crystallogr. 1975;20:312–314.

9. Gel'fand I, Graev M. Crofton's function and inversion formulas in real integral geometry. Functional Anal Its Appl. 1991;25:1–5.

10. Rullgård H. An explicit inversion formula for the exponential Radon transform using data from 180. Arkiv för Matematik. 2004;42:353–362.

11. Wang G, Ye Y, Yu H. Approximate and exact cone-beam reconstruction with standard and non-standard spiral scanning. Phys Med Biol. 2007;52:1–13. [PubMed]

12. Zhao J, Jiang M, Zhuang T, Wang G. An exact reconstruction algorithm for triple-source helical cone-beam CT. J X-ray Sci Technol. 2006;14:191–206.

13. Zhao J, Jin Y, Lu Y, Wang G. A filtered backprojection algorithm for triple-source helical cone-beam CT. IEEE Trans Med Imag. 2009 Mar;28(3):384–393. [PMC free article] [PubMed]

14. Zhao J, Jiang M, Zhuang T, Wang G. Minimum detection window and inter-helix PI-line with triple-source helical cone-beam scanning. J X-ray Sci Technol. 2006;14:95–107.

15. Noo F, Pack J, Heuscher D. Exact helical reconstruction using native cone-beam geometries. Phys Med Biol. 2003;48:3787–3818. [PubMed]

16. Katsevich A. A general scheme for constructing inversion algorithms for cone-beam CT. Int J Math Math Sci. 2003;21:1305–1321.

17. Katsevich A. 3PI algorithms for helical computer tomography. Adv Appl Math. 2006;36:213–250.

18. Kapralov M, Katsevich A. A one-PI algorithm for helical trajectories that violate the convexity condition. Inverse Problems. 2006;22:2123–2143.

19. Kapralov M, Katsevich A. On the study of 1PI algorithms for a general class of curves. SIAM J Imag Sci. 2008;1:418–459.

20. Yu H, Wang G. Studies on implementation of the Katsevich algorithm for spiral cone-beam CT. J X-ray Sci Technol. 2004;12:97–116.

21. Ye Y, Yu H, Wei Y, Wang G. A general local reconstruction approach based on a truncated Hilbert transform. Int J Biomed Imag. 2007;2007:63634. [PMC free article] [PubMed]

22. Kudo H, Courdurier M, Noo F, Defrise M. Tiny a priori knowledge solves the interior problem in computed tomography. Phys Med Biol. 2008;53:2207–2231. [PubMed]

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |