Geometry and energy constrained projection extension

Abstract

BACKGROUND:

In clinical computed tomography (CT) applications, when a patient is obese or improperly positioned, the final tomographic scan is often partially truncated. Images directly reconstructed by the conventional reconstruction algorithms suffer from severe cupping and direct current bias artifacts. Moreover, the current methods for projection extension have limitations that preclude incorporation from clinical workflows, such as prohibitive computational time for iterative reconstruction, extra radiation dose, hardware modification, etc.

METHOD:

In this study, we first established a geometrical constraint and estimated the patient habitus using a modified scout configuration. Then, we established an energy constraint using the integral invariance of fan-beam projections. Two constraints were extracted from the existing CT scan process with minimal modification to the clinical workflows. Finally, we developed a novel dual-constraint based optimization model that can be rapidly solved for projection extrapolation and accurate local reconstruction.

RESULTS:

Both numerical phantom and realistic patient image simulations were performed, and the results confirmed the effectiveness of our proposed approach.

CONCLUSION:

We establish a dual-constraint-based optimization model and correspondingly develop an accurate extrapolation method for partially truncated projections. The proposed method can be readily integrated into the clinical workflow and efficiently solved by using a one-dimensional optimization algorithm. Moreover, it is robust for noisy cases with various truncations and can be further accelerated by GPU based parallel computing.

Keywords

Projection extension dual-constraint scout view integral invariance optimization model

1 Introduction

X-ray computed tomography (CT) has a profound impact and continuously increasing demand in the field of biomedical imaging [1]. Classic CT theory aims at exactly reconstructing an image from a complete projection set, i.e., the object is fully encompassed by the field of view (FOV) and all line integrals passing through the object are measured [2]. However in clinical practice, the ideal condition is not obeyed in multiple cases, e.g. when the patient is obese or improperly positioned [3, 4]. In such cases, the real projections extend beyond the measurable range of the detector [2]. Due to the data truncation, results directly reconstructed by the traditional methods suffer from severe artifacts, e.g. cupping or direct current (DC) bias [5, 6].

For this practical problem, a kind of popular methods can be employed to generalize projection extensions, of which the main purpose is to estimate the truncated projection data [7, 8]. The projection correction methods can be further subdivided into two categories based on the nature of prior knowledge incorporated in the extrapolation process: heuristic prior based methods and aided-measurement-based methods. The heuristic prior based methods employ some general assumptions (symmetric mirror scheme or water cylinder estimation) or models (Gaussian or square root function) to guide the extrapolation [3 , 9–11]. Due to lack of specific constraints from the object, those methods fail to achieve high accuracy when the truncation is severe [8]. The aided-measurement-based methods employ an additional low-dose CT scan, fluoroscopic X-ray projection, or a specific beam filter fixed in the hardware [12 –17]. Although they can provide certain specific prior knowledge of a scanned patient, the radiation dose is inevitably increased, extra procedures interrupt the current workflow, and/or the modification of hardware is expensiveand limiting.

Acquisition of scout views is a part of the routine clinical workflow to center the patient in the scanner, and the views are usually not utilized in the final reconstruction. However, the scout views can be employed to extract reliable prior knowledge for local reconstruction (e.g. geometrical information), which can be classified as an aided measurement-based method. In the field of high-resolution micro-CT, we demonstrated that few global scout views can provide highly accurate interior reconstruction [6]. More recently, Xia et al. reported a method to improve image quality for truncated volume-of-interest (VOI) imaging using anterior-posterior (AP) and medio-lateral (ML) scout views [8]. Their work was applied to C-arm based imaging and relied on an assumption that the scout views cover the entire object, while the final VOI scan is severely truncated.

When clinical CT is carried out on obese patients or the patient is improperly positioned, it is quite common that the final tomographic scan is partially truncated (i.e. a portion of the views are truncated whereas the rest are not) [8 , 18]. In such cases, the AP scout view is usually truncated, but the ML one is not. Aiming at this kind of specific scenario, we first move the patient table down to generate a non-truncated AP scout view. Then, by combining with the original ML scout view, a rough ellipse estimation of the patient habitus can be obtained and then employed as a geometry constraint to estimate the projection extrapolation boundary. However, comparing with a real patient habitus, the ellipse assumption is not accurate enough, which need to be relaxed or transformed. Thus, we further introduce a data consistency condition (DCC) as an energy constraint to improve the accuracy of projection estimation.

So far, various investigations about the DCCs for X-ray CT have been proposed. The DCCs reveal the projection redundancy under different scan protocol, such as parallel beam [19], fan-beam [20 –22] and cone-beam [23 –26]. The DCCs have been employed for hardening correction [27], motion artifact reduction [29] and truncated projection estimations [4 , 32].

In this work, we employ the integral invariance of fan-beam geometry to estimate the total energy of missing data in the truncated projections. The geometry and energy constraints are incorporated into an optimization model. Then, we develop the corresponding accurate extrapolation method. The proposed optimization model can be effectively solved by simple one-dimensional search methods. Because the solution process is independent of different views, it is also highly amenable for parallel computation which leads to a considerably short runtime.

The remainder of this paper is organized as follows. Section 2 describes the geometry and energy constraints, and the multidimensional optimization model. In section 3, both numerical phantom and realistic patient image simulations are performed to verify the effectiveness of the proposed methods. In the last section, we discuss some related issues and conclude the paper.

2 Method

In this section, we first present the data acquisition of a modified configuration allowing a non-truncated AP scout view and estimate the geometrical shape by using a slice-wise ellipse model. Then, we establish an energy constraint according to the integral invariance in fan-beam geometry. By incorporating the geometry and energy characteristics, we propose a multidomain-constraint-based optimization model for local reconstruction from partially truncated projections and develop a corresponding accurate extrapolation method to estimate missing projections.

2.1 Scout view based geometry constraint

In clinical CT, when a patient is very large or improperly positioned, the corresponding tomographic scan often suffers from partial truncations. The technician typically attempts to align the patient at the isocenter (patient centering might not be accurate and is typically improved through information gathered through the scouts). Figure 1A shows a normal scout configuration for such a case. The source-to-(assumed)-isocenter distance (SOD) is noted as R in the figure. For a patient, whose body habitus is not within the full field of view (FFOV), the truncation predominantly arises in the AP direction but not in the ML scout.

As is illustrated in Fig. 1B, we propose a modified scout acquisition. While the ML scout is kept in the original configuration, a new AP scout is acquired after moving the table down by a distance d, so that the current SOD is R_m = R + d. By moving the patient closer to the detector, the geometrical magnification factor is reduced, and a larger patient habitus can be covered by the X-ray beam. The modified AP scout configuration allows the acquisition of non-truncated AP projections up to an extended body habitus of length L_eff, where

Fig.1

Various scout configurations. (A) is a normal configuration where ML scout is non-truncated and AP scout is truncated. (B) is a modified scout configuration where ML scout is unchanged and change in SOD allows non-truncated AP scout.

$L_{eff} = \frac{R_{m}}{R} \cdot FFOV = \frac{R + d}{R} \cdot FFOV .$

It is obvious that the maximal L_eff depends on how far the table can move down in the gantry, which is actually controlled by the geometrical specifications of a specific clinical CT scanner. Here we investigate four mainstream CT scanners: GE LightSpeed Xtra (General Electric Co., Waukesha, USA), Philips Brilliance Big Bore (Koninklijke Philips N.V., Amsterdam, Netherlands), Siemens SOMATOM Sensation Open (24/40) (Siemens AG, Berlin, Germany) and Toshiba Aquilion LB (Toshiba Medical Systems, Otawara, Japan). Table 1 provides a list of relevant geometrical specifications retrieved from a report compiled by the ImPACT group, London, UK [33].

Table 1

Geometric specifications of four CT scanners* and the corresponding increased patient coverage L_eff

	GE LSX	Philips Br.	Siemens SOM.	Toshiba Aq.
Aperture (cm)	80	85	82	90
SOD (mm)	606	645	570	712
SDD (mm)	1062.5	1183	1040	1275
Maximal FFOV (cm)	50	50	50	50
Reconstruction grid	512 × 512	512 × 512	512 × 512	512 × 512
L_eff (cm)	62.37	61.63	63.15	60.53
increase (%)	24.75	23.26	26.32	21.07

^*The four CT scanners are GE LightSpeed Xtra (GE LSX), Philips Brilliance Big Bore (Philips Br.), Siemens SOMATOM Sensation Open (Siemens SOM.) and Toshiba Aquilion LB (Toshiba Aq.).

The calculation of L_eff is based on the following assumptions:

AP scouts are typically acquired at or near the actual isocenter;

Patient table can be moved down in the gantry by at least 150 mm, which appears to be a realistic estimation as the bore radius is greater than 400 mm for all the aforementioned scanners.

The resulting value of L_eff for a table displacement d = 150 mm is shown in Table 1 as well. In addition, we calculate the corresponding percentage increase (w.r.t. maximal FOV reported by the manufacturer) in patient habitus coverage. According to Table 1, the percentage increases 21– 26% . Thus, it is evident that the modified scout configuration allows a significant increase of FFOV in the AP scout view.

Next, we show how to combine the original ML scout and the modified AP scout to obtain the specific geometry knowledge of various patients. Here, we assume that the patient habitus can be approximately described by an ellipse at each slice. In our setup (see Fig. 2), we place the origin of our coordinate axes (0,0) at the scanner isocenter. The source-to-isocenter distance (SOD) is R and source-to-isocenter distance (SDD) is D. Thus, the source is at (0, R) when acquiring the AP view, and at (- R, 0) when acquiring the ML view. The patient’s body habitus in the slice represented in this diagram is estimated as an ellipse with a center (x₀, y₀) and major and minor axis lengths of a and b, respectively. As is illustrated in Fig. 2, the ML scout is non-truncated in the original acquisition configuration. However, the AP scout would be truncated in this setting. By lowering the patient table a distance d, the truncation is avoided.

Fig.2

Setup for the acquisition of ML scout in the original configuration and AP scout in a modified configuration.

The tangents to the ellipse in the original ML scout and the modified AP scout are marked by black dash lines. In practice, they can be determined by the scout view measurements. We present the equations of these four tangents as follows, $\begin{matrix} Line S^{AP} P_{l}^{AP} : y = k_{l}^{AP} x + R, k_{l}^{AP} = cot (α_{l}^{AP}), \\ Line S^{AP} P_{r}^{AP} : y = k_{r}^{AP} x + R, k_{r}^{AP} = - cot (α_{r}^{AP}), \\ Line S^{ML} P_{l}^{ML} : y = k_{l}^{ML} (x + R), k_{l}^{ML} = - tan (α_{l}^{ML}), \\ Line S^{ML} P_{r}^{ML} : y = k_{r}^{ML} (x + R), k_{r}^{ML} = tan (α_{r}^{ML}), \end{matrix}$

where P represents the intersection between each tangent and the detector array, k stands for the slope, and α is the corresponding fan angle between the tangent and the central X-ray. To simplify the following solution process, we employ a more concise description of each scout view, $Line S^{AP} P^{AP} : y = k^{AP} x + R,$ $Line S^{ML} P^{ML} : y = k^{ML} (x + R),$

where $k^{AP} = k_{l}^{AP} or k_{r}^{AP}, k^{ML} = k_{l}^{ML} or k_{r}^{ML}$ .

The equation of the ellipse in the two configurations are noted below, $Ellipse in modified AP scout : {(\frac{x - x_{0}}{a})}^{2} + {(\frac{y - (y_{0} - d)}{b})}^{2} = 1,$ $Ellipse in original ML scout : {(\frac{x - x_{0}}{a})}^{2} + {(\frac{y - y_{0}}{b})}^{2} = 1 .$

By respectively substituting the equation for y provided in (1a) and (1b) into (2a) and (2b), we can obtain two quadratic equations in x, $\begin{matrix} A (x - x_{0})^{2} + B (k^{AP} x + R - y_{0} + d)^{2} = 1, \\ A (x - x_{0})^{2} + B (k^{ML} (x + R) - y_{0})^{2} = 1, \end{matrix}$

where A = 1/a² and B = 1/b². Since the system of equation allows a unique solution for each tangent, we can apply the formula that the discriminant of the quadratic is zero, i.e., $\begin{matrix} B ({Ax}_{0}^{2} - 1) (k^{AP})^{2} + 2 {ABx}_{0} (R - y_{0} + d) k^{AP} + A (B (R - y_{0} + d)^{2} - 1) = 0, \\ B (A (x_{0} + R)^{2} - 1) {(k ML)}^{2} - 2 AB (x_{0} + R) y_{0} k^{ML} + A {({By}_{0}^{2} - 1)}^{2} = 0 . \end{matrix}$

Each discriminant is a quadratic equation in slope k^AP or k^ML, which can be further employed to describe the sum and product of the slopes in each scout view, $({Ax}_{0}^{2} - 1) \sum_{i \in {l, r}} k_{i}^{AP} + 2 {Ax}_{0} (R - y_{0} + d) = 0,$ $B ({Ax}_{0}^{2} - 1) \sum_{i \in {l, r}} k_{i}^{AP} - A (B {(R - y_{0} + d)}^{2} - 1) = 0,$ $(A (x_{0} + R)^{2} - 1) \sum_{i \in {l, r}} k_{i}^{ML} - 2 A (x_{0} + R) y_{0} = 0,$ $B (A (x_{0} + R)^{2} - 1) \sum_{i \in {l, r}} k_{i}^{ML} - A ({By}_{0}^{2} - 1) = 0 .$

With the known value of each slope, Equations (3a– (3d) contain four searched-for variables x₀, y₀, A, and B, which can be solved with numerical solvers. Thus, the parameters of the ellipse (i.e. an estimation of the patient body habitus) are yielded. By forward projecting the achieved ellipse, a prolongation boundary of truncated projections can be obtained. As a geometrical constraint, it can be further employed to restrict the projection extrapolation process.

2.2 Integral-invariance based energy constraint

To describe the integral invariance in fan-beam geometry, we introduce a definition of projection energy as follow.

Definition 2.1. For a specific specimen, the projection energy is a function of scan angle, which is a linear integral of the projection along the detector array direction under each scan view.

According to definition 2.1, for fan-beam geometry with equal angular detector array (Fig. 3), the projection energy of specimen f reads $E (θ) = \int_{γ \in α} P_{f, θ} (γ) d γ,$

where θ is the current scan angle, P_f,θ is the corresponding projection, α stands for the detector fan angle of the X-ray beam coverage, and γ ∈ α represents a scanned fan angle.

With a parallel-beam geometry, when the scanned object is completely inside the FOV, the total attenuation remains constant view by view, i.e., the projection energy share the same value for each scan angle [10]. Hence, this constancy can be treated as parallel-beam based energy conservation. Actually, it is just a special case of the following three properties: periodicity, continuity, and boundedness. In fan-beam CT scan, the principle of energy conservation does not hold anymore due to X-ray beam divergence. However, the aforementioned three properties are still satisfied. Moreover, it satisfies the property of integral invariance [21],

Fig.3

Illustration of fan-beam based CT scan.

$J (θ, θ + β) = \int_{γ \in α} \frac{P_{f, θ} (γ)}{cos (γ - β / 2)} d γ - \int_{γ \in α} \frac{P_{f, θ + β} (γ)}{cos (γ + β / 2)} d γ \equiv 0,$ where J (θ, θ + β) is a judgment function of two projections at angles θ and θ + β. Different from the original ratio based version [21], here we employ a difference based judgment function, which is more suitable for the projection extrapolation process. It is obvious that both subtrahend and subtractor of Equation (5) are cosine weighted version of projection energy in Equation (4).

In [21], integral invariance of projection data in fan-beam and cone-beam geometries are studied and applied to sense an object motion and detect a contrast bolus arrival. In this work, we extend their application into local reconstruction from partially truncated projections. In the CT problem of obese or improperly positioned patients, projections are truncated in a portion of scan views. This leads to a reduction of corresponding projection energy. However, other non-truncated projections contain the entire energy. According to the integral invariance, the non-truncated data can be employed to provide weighted estimations of the missing projections. Thus, the weighted energy of missing projection can be achieved in each truncated view and further employed as an energy constraint to refine the projection extrapolation process.

2.3 Multidomain-constraint-based optimization model

By employing the scout-view based shape estimation and the integral-invariance based energy property, we propose a multidomain-constraint-based optimization model, $\begin{matrix} min_{λ (θ_{T})} {∥ J (θ_{T}, θ) ∥}_{L_{2}}^{2}, θ_{T} \in Θ_{T}, θ \in Θ ∖ Θ_{T}, \\ s . t . P_{θ_{T}} (x) = {\begin{matrix} μ_{θ_{T}}^{l} ω_{θ_{T}}^{l} (x) exp (- \frac{1}{2} {(\frac{x - x_{T}^{l}}{λ_{θ_{T}} σ_{θ_{T}}^{l}})}^{2}), for (1 - λ_{θ_{T}}) x_{T}^{l} + λ_{θ_{T}} x_{θ_{T}}^{l} \leq x < x_{T}^{l} and x_{θ_{T}}^{l} < x_{T}^{l}, \\ μ_{θ_{T}}^{r} ω_{θ_{T}}^{r} (x) exp (- \frac{1}{2} {(\frac{x - x_{T}^{r}}{λ_{θ_{T}} σ_{θ_{T}}^{r}})}^{2}), for x_{T}^{r} < x \leq (1 - λ_{θ_{T}}) x_{T}^{r} + λ_{θ_{T}} x_{θ_{T}}^{r} and x_{T}^{r} < x_{θ_{T}}^{r}, \end{matrix} \end{matrix}$

where Θ is the set of all scan angles, Θ_T is the subset of Θ with all truncated cases, and P is the searched-for non-truncated projection. For a scanning view θ_T, the corresponding one-dimensional projection is represented as P_{θ
_T}, where a datum is indexed by its spatial position x. Particularly, the truncated positions at left and right are noted as $x_{T}^{l}$ and $x_{T}^{r}$ , and the left and right projection boundary positions of the estimated patient habitus are $x_{θ_{T}}^{l}$ and $x_{θ_{T}}^{r}$ , respectively. As is illustrated in Fig. 4, for each θ_T ∈ Θ_T, we extract the truncated projections and employ a Gaussian function to guide estimating the missing projections. The center of the Gaussian model is located at the truncated position and the maximal magnitude is the corresponding projection value ( $μ_{θ_{T}}^{l}$ and $μ_{θ_{T}}^{r}$ for left and right sides). Based on the estimated patient habitus, the initial guess of the standard deviation is set as 1/3 of the distance between the truncated position and the corresponding estimated boundary position, i.e., $σ_{θ_{T}}^{i} = | x_{θ_{T}}^{i} - x_{T}^{i} | / 3, i \in {l, r}$ . Thus, the extrapolation obeys a Gaussian distribution, and the estimated projection value at the boundary position is very small. To further ensure the compensated projections gradually decrease to 0, we employ a magnitude weighting function $ω_{θ_{T}}^{i} (x) = \sqrt{1 - {(\frac{x - x_{T}^{i}}{λ_{θ_{T}} (x_{θ_{T}}^{i} - x_{T}^{i})})}^{2}}, i \in {l, γ}$ , where λ_{θ
_T} is the searched-for minimizer of the proposed optimization model to adjust the standard deviation of the weighted Gaussian model so that the energy constraint can be satisfied. λ_{θ
_T} can be easily obtained by using one-dimensional optimization algorithms.

Fig.4

Multidomain-constraint-based projection extrapolation. The standard deviation of the employed Gaussian distribution is marked by an arrow.

The proposed optimization model employs two constraints extracted from different measurement domains. The energy constraint as the optimization objective function is in light of the total attenuation capability of the scanned object. However, the geometry constraint provides a rough shape estimation of the searched-for projection. The dual constraints work together to well control the extrapolation process. In Equation (6), the employment of λ_{θ
_T} minimizes the energy difference and modifies the standard deviation as $λ_{θ_{T}} σ_{θ_{T}}^{i}, i \in {l, r}$ . The update of the standard deviation is based on two considerations as follows. First, the λ_{θ
_T} renders the energy of extrapolated projection weighted coincide for each scan view. Second, modifying the geometry-constraint-based standard deviation can make the patient body habitus estimation more accurate. This is because the ellipse assumption is a rough approximation and it cannot perfectly describe various patient cross-section shapes. Thus, energy constraint can be effectively incorporated to further refine the inaccurate boundaries. The solving process for Equation (6) is illustrated in Fig. 4. A complete procedure of the proposed method is summarized in Algorithm 1. Although the proposed method is derived and implemented in fan-beam geometry, it can be easily extended to cone-beam geometry.

Algorithm 1. Solving Multidomain Regularization based Optimization Model
S1: Extracting non-truncated projections and calculating the weighted projection energy corresponding to each truncated scan angle;
S2: Estimating parameters of the ellipse model based on the original ML and modified AP scout configurations;
S3: Forward projecting the ellipse boundaries and calculating the center and standard deviation of Gaussian function;
S4: Determining the minimizer λ (θ_T) , θ_T ∈ Θ_T for model (6) by one-dimensional optimization algorithm and updating the standard deviation;
S5: Employing the Gaussian function based extrapolation to correct the truncated projections.

3 Results

To verify the effectiveness of the proposed method, numerical simulations are performed with both phantoms and patient images. In the phantom simulations, we employ both Shepp-Logan phantom and Forbild shoulder-included thorax phantom. The Shepp-Logan phantom is used to verify the estimation accuracy of ellipse parameters and to assess the quality of reconstructed images in both noise-free and noisy cases. The thorax phantom is to validate the robustness of the proposed method with various truncation degrees. In the realistic patient image simulations, we test the proposed algorithm on two different body parts: abdomen and thorax. For all the experiments, fan-beam geometry is assumed for simplicity. Because the work aims at achieving an accurate estimation of missing projections to eliminate the bias and truncated artifacts in the corresponding FBP-based results, global reconstructions from FBP are employed as reference images. As a comparison, we also implement an Edge-Gauss extrapolation method [34]. Meanwhile, to further demonstrate the superiority of our dual-constraint-based method, we modify the proposed method to a single-constraint-based version and employ it as a comparison as well. Particularly, we only use the geometry estimation and ignore the energy constraint, i.e., $P_{θ_{T} \in Θ_{T}} (x) = {\begin{matrix} μ_{θ_{T}}^{l} ω_{θ_{T}}^{l} (x) exp (- \frac{1}{2} {(\frac{x - x_{T}^{l}}{σ_{θ_{T}}^{l}})}^{2}), for x_{θ_{T}}^{l} \leq x < x_{T}^{l} and x_{θ_{T}}^{l} < x_{T}^{l}, \\ μ_{θ_{T}}^{r} ω_{θ_{T}}^{r} (x) exp (- \frac{1}{2} {(\frac{x - x_{T}^{r}}{σ_{θ_{T}}^{r}})}^{2}), for x_{T}^{r} < x \leq x_{θ_{T}}^{r} and x_{T}^{r} < x_{θ_{T}}^{r} . \end{matrix}$

This is a special case of Equation (6) with λ_{θ
_T} = 1. In the following, we call this modified method as energy constraint-free for short. Three quantitative measures are employed to evaluate the image quality for all the reconstructed results. They are peak signal-to-noise ratio (PSNR) [35], normal mean absolute deviation (NMAD) [36] and structural similarity (SSIM) [37].

3.1 Numerical phantom simulations

First, the proposed method is evaluated by phantom simulations with different noise levels, i.e., noise-free case and noisy case with 10⁵ emitted photons per X-ray path. A Shepp-Logan phantom is employed assuming a 512 × 512 image grid, and its pixel size is 1.405 × 1.405 mm². On this image grid, the major and minor axes of the major ellipse comprising the phantom are measured as a = 310.505 mm and b = 238.850 mm. When simulations are carried out, the major ellipse is typically centered at the isocenter of the simulated gantry. For the purpose of this paper (and in most real-life situations), this is not a realistic assumption because the patient is not always centered perfectly at the gantry isocenter. In the following test, we use a perturbation from isocenter (x₀ = 7.025 mm, y₀ = 11.240 mm), i.e. a 2.26% deviation w.r.t. a and a 4.74% deviation w.r.t b. While these values are arbitrarily selected, the accuracy of the proposed method is expected to hold for other values of perturbation from gantry isocenter (provided x₀ and y₀ are within reasonable bounds of patient centering error). The SOD is 595 mm. The detector consists of 920 channels and each is covered by a fan-beam X-ray of 0.0578°. The view number is set as 984. The reference projection is a global tomographic scan (see the first column of Fig. 5), where the truncated positions are marked by vertical lines. The corresponding reference reconstructions are shown in the first column of Fig. 6, where the truncated-FOV (TFOV), i.e., the area completely covered by the detector, is marked by a yellow circle. Obviously, the AP scout is truncated, and the ML scout is non-truncated.

Fig.5

Shepp-Logan phantom projections without/with noise. The first column images are reference projections. The second to fourth columns are the estimated projections by the Edge-Gauss method, energy constraint-free method and the proposed method, respectively. The top and bottom rows are noise-free and noisy projections, respectively.

Table 2

Estimated results of the ellipse parameters in the Shepp-Logan phantom experiments (unit: mm)

	a	b	x ₀	y ₀
Ground truth	310.505	238.850	7.025	11.240
Estimation of Noise-free Case	311.348	239.165	6.903	11.350
Estimation of Noisy Case	311.663	240.350	6.907	12.446

Fig.6

Same as Fig. 5 but the reconstructed images in a display window [0.15, 0.35].

By moving the detector down by 200 mm, a non-truncated AP scout is acquired. Based on the projection boundaries of the original ML and modified AP scout views, the unknowns x₀, y₀, a and b are calculated for both noise-free and noisy cases. The corresponding results are summarized in Table 2. The hulls of estimated ellipses are marked in the reference images by red curves (see the first column of Fig. 6).

For both noise-free and noisy cases, the estimated projections and corresponding reconstructions by the Edge-Gauss method, the energy constraint-free method, and the proposed method are shown in Figs. 5 and 6, respectively. To compare local details, a representative horizontal profile (marked with a green line) and vertical profile (marked with cyan line) are respectively extracted from each reconstructed image and plotted in Fig. 7. Here all the profiles are divided into two parts by the TFOV, where the boundary position is marked by a magenta dash line. The image quality measures calculated inside the TFOV are listed in Table 3.

Secondly, we employ the Forbild shoulder-included thorax phantom to verify the performance of the proposed method with respect to different patient sizes. The maximum shoulder breadth is fixed as 594.530 mm and 647.490 mm respectively for small and large size simulation. And the patient boundary is not satisfied with strict ellipse anymore. The emitted photon number per X-ray path is consistently 10⁵. The scan geometry is the same as the previous simulation. The perturbation from the isocenter is (x₀ = 16.860 mm, y₀ = 36.530 mm).

The reconstructed images by the Edge-Gauss method and the proposed method are shown in Fig. 8. Representative horizontal and vertical profiles along line-segments are also illustrated. The image quality measures calculated inside the TFOV are listed in Table 4.

Fig.7

Representative profiles along line-segments indicated in Fig. 6.

Table 3

PSNRs, NMADs, and SSIMs of the Shepp-Logan phantom simulations

	PSNR	NMAD	SSIM
Noise-free Projections
Edge-Gauss	34.31	0.0789	0.9788
Energy Constraint Free	26.45	0.2403	0.9430
Proposed Method	42.71	0.0273	0.9935
Noisy Projections
Edge-Gauss	26.88	0.2974	0.7837
Energy Constraint Free	24.18	0.3814	0.7708
Proposed Method	27.57	0.2780	0.7885

Fig.8

Forbild shoulder-included phantom reconstructions with various phantom size (small and large). The first column images are reference reconstructions. Results by the Edge-Gauss method and the proposed method are shown in the second and third columns, respectively. The display window is [0.9, 1.2]. Representative profiles along the line-segments are also illustrated.

3.2 Realistic patient image simulations

In the realistic patient image simulations, two cases are considered with different body parts, abdomen and thorax. The reference projections are shown in the first column of Fig. 9, where the truncated positions are marked by vertical lines. In real clinical applications, although the patient table also needs to be taken into consideration, as an experimental knowledge it can be easily removed from projections. Thus, in this simulation, we neglect the patient table and focus on the results inside the TFOV.

Table 4
PSNRs, NMADs, and SSIMs of the Forbild phantom simulations

PSNR NMAD SSIM

Small-size Projections

Edge-Gauss 39.74 0.0195 0.9982

Proposed Method 51.30 0.0059 0.9998

Large-size Projections

Edge-Gauss 39.17 0.0275 0.9979

Proposed Method 50.14 0.0051 0.9997

	PSNR	NMAD	SSIM
Small-size Projections
Edge-Gauss	39.74	0.0195	0.9982
Proposed Method	51.30	0.0059	0.9998
Large-size Projections
Edge-Gauss	39.17	0.0275	0.9979
Proposed Method	50.14	0.0051	0.9997

We reconstruct 512×512 images for the abdomen and the thorax experiments, of which the pixel sizes are 1.268 × 1.268 mm² and 1.308 × 1.308 mm², respectively. All the other configurations are the same as the noisy case of phantom simulation. The reference reconstructions are shown in the first column of Fig. 10, where the yellow circles indicate the TFOV and the red ellipses are the estimated patient hulls. The estimated projections and corresponding reconstructions by the Edge-Gauss method, the energy constraint-free method, and the proposed method are shown in the second to fourth columns of Figs. 9 and 10, respectively. A representative horizontal profile (marked with a green line) and vertical profile (marked with cyan line) are respectively extracted from each reconstructed image to magnify the local details (see Fig. 11). The magenta dash line suggests the boundary between inside and outside of the TFOV. The image quality measures inside the TFOV are calculated and listed in Table 5.

Fig.9

Realistic patient image projections. The top and bottom rows are for abdomen and thorax images, respectively. The first column images are the reference projections. The second to fourth columns are estimated projections by the Edge-Gauss method, the energy constraint-free method, and the proposed method, respectively.

Fig.10

Same as Fig. 9 but the reconstructed images in a display window [– 1000,1000] (unit: HU).

Fig.11

Representative profiles of the patient image simulation results along lines indicated in Fig. 10.

Table 5

PSNRs, NMADs, and SSIMs of the patient image simulation results

	PSNR	NMAD	SSIM
Abdomen
Edge-Gauss	35.04	0.0423	0.84275
Energy Constraint Free	34.81	0.0437	0.83802
Proposed Method	35.78	0.0400	0.84277
Thorax
Edge-Gauss	33.84	0.0661	0.86882
Energy Constraint Free	33.28	0.0701	0.85915
Proposed Method	36.85	0.0509	0.88819

3.3 Projection energy

Figure 12 plots the projection energy curves for both the Shepp-Logan phantom and patient image simulations. Extrapolated projections at a selected view (#250) are also shown in plots in Fig. 12 (centre and right).

Fig.12

Projection energy curves of the Shepp-Logan phantom simulations (upper two rows) and patient image simulations (lower two rows). The projection with view index 250 is extracted, and the corresponding compensated projections at left and right sides are respectively illustrated, where the truncated position is marked by a cyan dashed line.

4 Discussions

4.1 Numerical phantom simulations

First, the results in Table 2 demonstrate the feasibility of the proposed boundary estimation method. Moreover, the noisy experiment suggests the robustness against noise. The hulls of estimated ellipses shown in the first column of Fig. 6 suggest that the habitus estimation is very accurate when the object is exactly elliptical.

As are shown in Fig. 5, the extrapolated projections by the energy constraint-free method and the proposed method are much closer to the reference projection than the Edge-Gauss method. Moreover, they have more reasonable boundary shapes. From the reconstructed results shown in Fig. 6, it is obvious that the artifacts near the TFOV boundaries are effectively removed by the proposed method. By comparing the magnified profile plots in Fig. 7, we can see that the proposed method allows the profiles to be very close to the global fan-beam reconstruction within the TFOV. As compared with other methods, the proposed method has a much greater degree of fidelity w.r.t. global fan-beam reconstruction even slightly outside the TFOV. In addition, the image quality assessments listed in Table 3 consistently confirm the superiority of the proposed method.

For the experiments with different patient sizes, both the visualized comparisons in Fig. 8 and quantitative assessments in Table 4 demonstrate the robustness of the proposed method against various truncation degrees. Moreover, in this simulation, the patient habitus do not well meet the ellipse condition. However, the DCC as an energy constraint effectively remedies the deviation. The DC bias inside the TFOV is dramatically removed.

4.2 Realistic patient image simulations

Comparing the extrapolated projections in Fig. 9, we can see that the results by the proposed method contain the most reasonable boundaries. From the corresponding reconstructions in Fig. 10, it is obvious that the images by the Edge-Gauss method and the energy constraint-free method suffer from severe artificial boundaries outside the TFOV, which are generated by the inaccurate extrapolations. However, by combining geometrical and energy constraints, the proposed method allows a highly accurate reconstruction, and the artifacts are effectively reduced and even removed. As are shown in Fig. 11, the magnified profiles by the proposed method are much closer to the reference profiles both inside and outside the TFOV than other methods, which confirm the superiority of the proposed method. Moreover, the corresponding image quality measures in Table 5 suggest considerable improvement of the proposed method as well.

Table 6
Running times to solve the proposed optimization model (s)

Phantom Simulation Patient Image Simulation

Shepp-Logan Forbild Abdomen Thorax

4.66 4.68 4.65 4.68

Phantom Simulation	Patient Image Simulation
Shepp-Logan	Forbild	Abdomen	Thorax
4.66	4.68	4.65	4.68

4.3 Projection energy

The plots of projection energy on the left side of Fig. 12 show that the proposed method leads to near-identical projection energy at all views. The plots of extrapolated projections at view (#250) show that the proposed method more closely matches the original projection data while the Edge-Gauss and Energy-Constraint Free methods fail to do so in many cases.

5 Conclusions

For partially truncated projections, we propose a novel compensation method to achieve accurate extrapolations. This approach is based on the assumption of Gaussian function and aims at solving an optimization model with both geometrical and energy constraints. The solution leads to a one-dimensional search of the parameter λ_{θ
_T} to update the standard deviation of the Gaussian function. On one hand, the use of λ_{θ
_T} in the optimization model guides the solution such that the energy of extrapolated projection obeys the integral invariance of fan-beam geometry. Furthermore, modifying the geometry-constraint-based standard deviation improves the estimation accuracy of the patient body habitus. This is because in real applications the patient habitus often is not exactly elliptical, and the assumption of the ellipse is just a rough approximation and it fails to perfectly describe various patient shapes. Yet, even such rough estimates are powerful pieces of prior information that can be utilized to improve the final image reconstructions. Moreover, the energy constraint further refines the estimated boundaries and effectively incorporates more prior knowledge to guide the extrapolation process. The dual-constraints in different measurement domains work together and allow a highly accurate reconstruction from partially truncated projections. Moreover, the proposed approach is robust for noisy cases with various truncations and can be readily integrated into the clinical workflow as an individual module without requiring extra measurements or hardware modifications. Furthermore, the different pieces of the proposed method are modular as well. For instance, the ellipse estimation by modifying scout configuration can be used for many image reconstruction methods, e.g. [8] or some iterative algorithms.

The two employed constraints can be naturally achieved from the current workflow without extra measurements or hardware modification. Moreover, the proposed method can be readily employed as an individual module and integrated into existing CT workflows. Comparing with existing methods for local reconstruction from partially truncated projections, the proposed approach extracts sufficient prior knowledge from a specific patient without increasing radiation dose, and it can be easily utilized in clinical CT applications.

The main calculation time of the proposed method is to solve the optimization model. For each scan view, the solving method corresponds to a one-dimensional search process. And all the scan views are independent of each other, which is highly amenable to GPU based parallel computing. For all the aforementioned experiments, we use MATLAB GPU based matrix calculation to implement the solving process. The employed computer is single GPU of GeForce GTX 960M by NVIDIA (Santa Clara, CA, USA), 4 CPUs of Intel(R) Core(TM) i5-6300HQ @ 2.30 GHz, and 16GB RAM. The corresponding running times are summarized in Table 6. Comparing to the execution time of global FBP reconstruction 1.68 s, the average running time to solve the optimization problem is about 2.7 times of the FBP reconstruction. It is noticeable that the proposed optimization problem can be easily and fast solved.

Although our method is derived based on fan-beam geometry, it can be easily extended to cone-beam geometry. One possible and direct approach is to translate and scale the achieved ellipse in the central slice to other noncentral slices, and the coefficients of such geometric transformations can also be determined from the scout view images. Then, the DCCs in cone-beam geometry can be employed as energy constraint to control the projection extrapolation.

In summary, we establish a multidomain-constraint-based optimization model and correspondingly develop an accurate extrapolation method for partially truncated projections. The first constraint is based on the habitus estimation of the patient, named geometry constraint. It is calculated from dual scout view based measurements and it provides a rough boundary of the searched-for projection. The second constraint is based on the integral invariance of fan-beam CT, named energy constraint. It is in the measurement domain of X-ray attenuation and allows further refinement in the projection extrapolation. The proposed optimization model can be solved very efficiently using one-dimensional optimization algorithm. Because each optimization problem is independent for different scan views, it can be further accelerated by GPU based parallel computing. In addition, it is extremely robust against inevitable noise. Both numerical phantom and patient image simulations confirm the merits of the proposed approach.

Footnotes

Acknowledgments

This work was supported in part by the NSF/CBET CAREER award 1540898.

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Geometry and energy constrained projection extension

Abstract

BACKGROUND:

METHOD:

RESULTS:

CONCLUSION:

Keywords

1 Introduction

2 Method

2.1 Scout view based geometry constraint

3.1 Numerical phantom simulations

Table 4 PSNRs, NMADs, and SSIMs of the Forbild phantom simulations PSNR NMAD SSIM Small-size Projections Edge-Gauss 39.74 0.0195 0.9982 Proposed Method 51.30 0.0059 0.9998 Large-size Projections Edge-Gauss 39.17 0.0275 0.9979 Proposed Method 50.14 0.0051 0.9997

4.1 Numerical phantom simulations

4.2 Realistic patient image simulations

Table 6 Running times to solve the proposed optimization model (s) Phantom Simulation Patient Image Simulation Shepp-Logan Forbild Abdomen Thorax 4.66 4.68 4.65 4.68

5 Conclusions

Footnotes

Acknowledgments

References

Table 4
PSNRs, NMADs, and SSIMs of the Forbild phantom simulations

PSNR NMAD SSIM

Small-size Projections

Edge-Gauss 39.74 0.0195 0.9982

Proposed Method 51.30 0.0059 0.9998

Large-size Projections

Edge-Gauss 39.17 0.0275 0.9979

Proposed Method 50.14 0.0051 0.9997

Table 6
Running times to solve the proposed optimization model (s)

Phantom Simulation Patient Image Simulation

Shepp-Logan Forbild Abdomen Thorax

4.66 4.68 4.65 4.68