- Email: [email protected]
3D reconstruction based on compressed-sensing (CS)-based framework by using a dental panoramic detector
ARTICLE IN PRESS Physica Medica ■■ (2015) ■■–■■
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Physica Medica j o u r n a l h o m e p a g e : h t t p : / / w w w. p h y s i c a m e d i c a . c o m
Technical Notes
3D reconstruction based on compressed-sensing (CS)-based framework by using a dental panoramic detector U.K. Je, H.M. Cho, D.K. Hong, H.S. Cho *, Y.O. Park, C.K. Park, K.S. Kim, H.W. Lim, G.A. Kim, S.Y. Park, T.H. Woo, S.I. Cho Department of Radiation Convergence Engineering, iTOMO Research Group, Yonsei University, Wonju 220-710, Republic of Korea
A R T I C L E
I N F O
Article history: Received 17 July 2015 Received in revised form 22 September 2015 Accepted 24 September 2015 Available online Keywords: Dental panoramic detector Compressed-sensing Spiral scan Zigzag scan
A B S T R A C T
In this work, we propose a practical method that can combine the two functionalities of dental panoramic and cone-beam CT (CBCT) features in one by using a single panoramic detector. We implemented a CS-based reconstruction algorithm for the proposed method and performed a systematic simulation to demonstrate its viability for 3D dental X-ray imaging. We successfully reconstructed volumetric images of considerably high accuracy by using a panoramic detector having an active area of 198.4 mm × 6.4 mm and evaluated the reconstruction quality as a function of the pitch (p) and the angle step (Δθ). Our simulation results indicate that the CS-based reconstruction almost completely recovered the phantom structures, as in CBCT, for p ≤ 2.0 and θ ≤ 6° , indicating that it seems very promising for accurate image reconstruction even for large-pitch and few-view data. We expect the proposed method to be applicable to developing a cost-effective, volumetric dental X-ray imaging system. © 2015 Associazione Italiana di Fisica Medica. Published by Elsevier Ltd. All rights reserved.
Introduction Panoramic radiography, introduced in the late 1940s, made major progress in dentistry, providing dentists with a single comprehensive image of both the jaws and the maxillofacial structures [1,2]. Cone-beam computed tomography (CBCT), introduced in the mid1990s, also created a revolution, facilitating the transition of dental X-ray imaging from 2D to 3D images of sub-millimeter resolution with faster scanning time and lower dose than in conventional medical CT by employing a large-area flat-panel detector [3]. In addition, recently, handy two (or three)-in-one dental imaging systems that combine the panoramic and the CBCT (or/and cephalometric) features in one device have been launched in the global market, equipped with one panoramic detector and one flat-panel detector together. However, market prices for the combined systems are still high mainly due to the detector cost, thus posing an obstacle for the more widespread use of these systems, especially, in local dental clinics. In this work, we propose a practical method that can combine the two functionalities of dental panoramic and CBCT features in one by using a single panoramic detector. Here a linear-type panoramic detector is tilted by 90° from the orientation for panoramic
* Corresponding author. Department of Radiation Convergence Engineering, iTOMO Research Group, Yonsei University, Wonju 220-710, Republic of Korea. Tel.: +82 33 761 9660; fax: +82 33 761 9664. E-mail address: [email protected] (H.S. Cho).
imaging and rotates together with an X-ray tube around the rotational axis several times along a spiral (or zigzag) trajectory to cover the whole imaging volume during the projection data acquisition. Figure 1 shows the schematic illustrations of (a) a conventional dental CBCT geometry and (b) the proposed geometry. As indicated in Fig. 1(a), in the CBCT, there is a penalty for the large detector coverage because its exact reconstruction is not possible due to insufficient sampling of the imaging volume by the diverging X-ray beam. While the filtered-backprojection (FBP) algorithm has been widely used as an approximate method for the cone-beam image reconstruction, the resulting cone-beam artifacts increase with the imaging volume thickness [4]. Furthermore, the amount of scatter also increases with the detector size, reducing the image contrast. As a possible solution to the cone-beam related artifacts, a lineartype (e.g., 198.4 mm × 6.4 mm in our case) detector can be scanned helically in the axial direction to provide sufficient volumetric sampling. Figure 2 shows the schematic illustrations of the proposed scan geometries for volumetric dental X-ray imaging by using a linear-type panoramic detector: (a) a spiral scan geometry and (b) a zigzag scan geometry. Here the zigzag scan is devised to eliminate the use of the slip-ring technology in the spiral scan which allows continuous multiple rotations but increases the system complexity [5]. The proposed approach to volumetric dental X-ray imaging, compared to the CBCT, leads to imaging benefits such as alleviating cone-beam artifacts and reducing scatters, as well as decreasing system cost. Another advantage of the proposed method would be increased image quality in general, implying that one can go lower in exposure to reach the same image quality as a CBCT,
http://dx.doi.org/10.1016/j.ejmp.2015.09.005 1120-1797/© 2015 Associazione Italiana di Fisica Medica. Published by Elsevier Ltd. All rights reserved.
Please cite this article in press as: U.K. Je, et al., 3D reconstruction based on compressed-sensing (CS)-based framework by using a dental panoramic detector, Physica Medica (2015), doi: 10.1016/j.ejmp.2015.09.005
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Materials and methods The X-ray imaging procedure can be modeled approximately by a discrete linear system as follows:
Ax = b,
(1)
x = ( x1, x2, … , xN ) , T
b = ( b1, b2, … , bM ) , T
A = {aij }, i = 1, 2, … , M and j = 1, 2, … , N ,
(2)
where x is the original image vector to be reconstructed, b is the measured projection vector, A is the system matrix, relating x and b, N is the number of voxels, M is the total number of sampling points in the projection vector, and the superscript T is the transpose operator. In the CS framework, x is normally recovered as an optimal solution, x*, for the convex optimization problem described in Eq. (3) by minimizing the following objective function, f(x), assuming that most components of derivative images are negligibly small:
x* = arg min f ( x ) , x∈Q
N 1 2 f ( x ) = Ax-b 2 + α ∑ Di x 2, 2 i =1
where Q is the set of feasible x, 1 2 Ax-b
(3) 2 2
is the fidelity term,
N
∑ Dx i
Figure 1. Schematic illustrations of (a) a conventional dental CBCT geometry and (b) the proposed geometry. In the proposed geometry, a linear-type panoramic detector is tilted by 90° from the orientation for panoramic imaging and rotates together with an X-ray tube around the rotational axis several times along a spiral (or zigzag) trajectory to cover the whole imaging volume during the projection data acquisition.
because for a given field-of-view (FOV) size and exposure settings, scatter will be reduced in the proposed geometry due to the use of a thin detector. We employed a compressed-sensing (CS)-based framework, rather than the common FBP-based framework, for more accurate image reconstruction. Here the CS is a state-of-the-art mathematical scheme for solving the inverse problems, which exploits the sparsity of the image with substantially high accuracy [6]. We implemented a CS-based algorithm for the proposed method and performed a systematic simulation to demonstrate its viability for 3D dental X-ray imaging. In the following sections, we briefly describe the implementation of the CS-based reconstruction algorithm for the proposed method and present the simulation results.
i =1
2
is the sparsifying term, α is the parameter that balances
the two terms and is chosen so that signal-to-noise ratio is maximized (e.g., α = 0.1 was used in the simulation), and Di is the forward difference approximation to the gradient at voxel i. The convex optimization problem described in Eq. (3) can be solved approximately, but efficiently, by using the accelerated gradient-projection Barzilai– Borwein (GPBB) formulation [6]. Details on the mathematical descriptions of the CS reconstruction framework can be found in our previous paper [7]. Figure 3 shows the simplified flowchart of the CS-based reconstruction framework for the proposed method, and it is briefly described as follows. Firstly, the measured projections bm are acquired from the imaging system. The initial guess x(0) is then assumed as the image reconstructed by simple backprojection (BP), followed by forward projection (FP) to obtain the computed projections bc. By using the bm, the bc, and the current updated image x(k), the objective function f(x) is computed to determine the step size Δx(k) for the next updated image x(k+1). The image x is successively updated until the mismatch between the current and the updated images converges to a specified tolerance ε. During the iterative procedure,
Figure 2. Schematic illustrations of the proposed scan geometries for volumetric dental X-ray imaging by using a linear-type panoramic detector: (a) a spiral scan geometry and (b) a zigzag scan geometry.
Please cite this article in press as: U.K. Je, et al., 3D reconstruction based on compressed-sensing (CS)-based framework by using a dental panoramic detector, Physica Medica (2015), doi: 10.1016/j.ejmp.2015.09.005
ARTICLE IN PRESS U.K. Je et al./Physica Medica ■■ (2015) ■■–■■
3
Figure 3. Simplified flowchart of the CS-based reconstruction framework for the proposed method. Firstly, the measured projections bm are acquired from the imaging system. The initial guess x(0) is then assumed as the image reconstructed by simple BP, followed by FP to obtain the computed projections bc. By using the bm, the bc, and the current updated image x(k), the objective function f(x) is computed to determine the step size Δx(k) for the next updated image x(k+1). The image x is successively updated until the mismatch between the current and the updated images converges to a specified tolerance ε.
as indicated in Fig. 3, each iteration loop requires one FP and one BP which are performed by using the prepared system matrix A. We used a modified version of the distance-driven method [8], a current state-of-the-art scheme, for building the proposed system’s matrix. Based upon the above descriptions, we implemented a CSbased reconstruction algorithm for the proposed method and performed a systematic simulation to investigate the imaging characteristics. The number of iterations in the CS-based algorithm was set to 150 because the convergence rate of the algorithm seemed to be saturated after about 100 iterations. We also implemented a common FBP-based algorithm for comparison. A numerical mouth phantom (220 × 220 × 220) was derived by us from a 3D reconstructed image from a CBCT scan of a patient head. A lower part (220 × 220 × 60) of the phantom was used in the simulation for saving the reconstruction time. A detector dimension of 496 (width) × 16 (height) was used to imitate a linear-type panoramic detector. A voxel size of 0.4 mm and a pixel size of 0.4 mm were used in the reconstruction. Detailed reconstruction parameters used in the simulation are listed in Table 1. We investigated the reconstruction quality as a function of the pitch (p) and the angle step (Δθ). Here the pitch
Table 1 Reconstruction parameters used in the simulation. Parameter
Dimension
Source-to-detector distance (SDD) Source -to-objector distance (SOD) Pitch (p) Angle step (Δθ) Pixel size Voxel size Detector dimension (W × H) Voxel dimension Reconstruction algorithm Phantom
650 mm 450 mm 0.5, 1.0, 1.5, 2.0, 4.0, 6.0 3°, 6°, 12°, 24° 0.4 mm 0.4 mm 496 × 16 220 × 220 × 60 FBP, CS 3D numerical mouth phantom
is defined as the source-detector feed per rotation in the units of the nominal slice thickness.
Results and discussion Figure 4 shows the reconstructed (a) transverse and (b) coronal images of the mouth phantom by using the FBP- and the CS-based algorithms for selected pitches of p = 0.5, 1.0, 1.5, 2.0, 4.0, and 6.0 and a fixed angle step of Δθ = 3° (i.e., 120 projections per rotation). As indicated in Fig. 4, the CS-based reconstruction almost completely recovered the phantom structures for a pitch of p ≤ 2.0 , as in CBCT [9], while the FBP-based reconstruction seriously suffered from streaking artifacts for all the pitches, even for p = 0.5, due to the insufficient angular samplings. These results indicate that the CS-based reconstruction is promising for accurate image reconstruction even for large-pitch projection data in the spiral scan. Figure 5 shows the reconstructed transverse images of the mouth phantom by using the FBP- and the CS-based algorithms for selected angle steps of Δθ = 3°, 6°, 12°, and 24° and a fixed pitch of p = 1. As indicated in Fig. 5, the CS-reconstructed images did not seriously suffer from the image artifacts due to insufficient angular samplings, even for Δθ = 6°, while the FBP-reconstructed images seriously did, depending on the angle step. These results indicate that the CS-based reconstruction is also promising even for few-view projection data in the spiral scan. Figure 6 shows the reconstructed transverse (top) and coronal (bottom) images of the mouth phantom for the zigzag scan with a pitch of p = 1 and an angle step of Δθ = 3°, by using the FBP- and the CS-based algorithms. The image characteristics seemed very similar to those for the spiral scan, indicating the effectiveness of the zigzag scan for volumetric image reconstruction. Two main concerns that should be addressed in the proposed method for clinical applicability are the scan time and the reconstruction time; higher scan and reconstruction times than clinically acceptable levels would certainly be unacceptable in a clinical setting.
Please cite this article in press as: U.K. Je, et al., 3D reconstruction based on compressed-sensing (CS)-based framework by using a dental panoramic detector, Physica Medica (2015), doi: 10.1016/j.ejmp.2015.09.005
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Figure 5. Reconstructed transverse images of the mouth phantom by using the FBPand the CS-based algorithms for selected angle steps of Δθ = 3°, 6°, 12°, and 24° and a fixed pitch of p = 1. These results indicate that the CS-based reconstruction is also promising even for few-view projection data in the spiral scan.
Figure 6. Reconstructed transverse (top) and coronal (bottom) images of the mouth phantom for the zigzag scan with a pitch of p = 1 and an angle step of Δθ = 3°, by using the FBP- and the CS-based algorithms. The image characteristics seemed very similar to those for the spiral scan, indicating the effectiveness of the zigzag scan for volumetric image reconstruction.
Table 2 Estimation of the scan time needed for different FOVs at selected pitch settings. FOV (diameter × height)
Single jaw (80 mm × 60 mm) Both jaws (120 mm × 90 mm) Maxillofacial (160 mm × 140 mm)
Figure 4. Reconstructed (a) transverse and (b) coronal images of the mouth phantom by using the FBP- and the CS-based algorithms for selected pitches of p = 0.5, 1.0, 1.5, 2.0, 4.0, and 6.0 and a fixed angle step of Δθ = 3°. These results indicate that the CS-based reconstruction is promising even for large-pitch projection data in the spiral scan.
We estimated the scan time needed for a single jaw (e.g., FOV (diameter × height) ~ 80 mm × 60 mm), both jaws (120 mm × 90 mm), and entire maxillofacial area (160 mm × 140 mm) [10] at selected pitch settings of p = 1, 1.5, and 2.0. In this estimation, we assumed that the detector height is 6.4 mm, the angle step is Δθ = 3°, and the detector’s frame rate is 80 fps (available in our CMOS-type panoramic detector). Table 2 shows the resultant estimation of the scan time for given conditions. As indicated in Table 2, the scan times needed for both jaws at p = 1, 1.5, and 2.0 are estimated less than about 22, 15, and 11 seconds, respectively. Also note that the scan time for entire maxillofacial area at p = 2.0 are less than about 17 seconds, comparable to that in dental CBCT. However, for a common detector having a moderate frame rate, the scan time would be about two or three times larger than this estimation. One possible approach to reduce the scan time could be the use of a wider panoramic detector, which allows a larger pitch without sacrificing image performance.
Pitch (p) p=1
p = 1.5
p = 2.0
14.1 s 21.1 s 32.8 s
9.4 s 14.1 s 21.9 s
7.0 s 10.5 s 16.4 s
The other concern is the reconstruction time associated with iterative techniques in which, as indicated in Fig. 3, one FP and one BP are required for each iteration loop; the former is approximately a discretized evaluation of the Radon transform and the latter is the adjoint operation of the FP. These operations are the primary computational bottleneck in the iterative scheme, especially in 3D image reconstruction, and the use of the graphics processing unit (GPU) has been frequently considered for accelerating them. Although the implemented algorithm for this work has not yet been accelerated by the GPU, the reconstruction time for the given test conditions (i.e., FOV = 220 × 220 × 60 voxels, voxel size = 0.4 mm) was less than 30 minutes on a normal workstation, in its present state well beyond what current CBCTs (having large FOVs with small voxel sizes) offer. One possibility to speed it up could be adopting the FBP framework into the CS-based framework to provide a more feasible initial gauss (rather than a simple BP) and a more well-converged next updated image. We expect that it could improve the convergence rate of the present iterative algorithm considerably. Conclusion In this work, as a continuation of our dental imaging R&D, we propose a practical method that can combine two functionalities
Please cite this article in press as: U.K. Je, et al., 3D reconstruction based on compressed-sensing (CS)-based framework by using a dental panoramic detector, Physica Medica (2015), doi: 10.1016/j.ejmp.2015.09.005
ARTICLE IN PRESS U.K. Je et al./Physica Medica ■■ (2015) ■■–■■
of dental panoramic and CBCT features in one with a single panoramic detector intended for integration into local panoramic systems. Here a linear-type panoramic detector is tilted by 90° from the orientation for panoramic imaging and rotates together with an X-ray tube around the rotational axis several times along a spiral (or zigzag) trajectory to cover the whole imaging volume during the projection data acquisition. We employed a CS-based framework, rather than the common FBP-based framework, for more accurate image reconstruction. Here the CS is a state-of-the-art mathematical scheme for solving the inverse problems, which exploits the sparsity of the image with substantially high accuracy. We successfully reconstructed volumetric images of considerably high accuracy by using a panoramic detector having an active area of 198.4 mm × 6.4 mm and evaluated the reconstruction quality as a function of the pitch (p) and the angle step (Δθ). According to our simulation results, the CS-based reconstruction almost completely recovered the phantom structures, as in CBCT, for p ≤ 2.0 and θ ≤ 6° , demonstrating its viability for 3D dental X-ray imaging. The reconstruction characteristics for the zigzag scan were similar to those of the spiral scan. We expect the proposed method to be applicable to developing a cost-effective, volumetric dental X-ray imaging system. Acknowledgment This research was supported by the Radiation Technology Development Program of the National Research Foundation (NRF)
5
funded by the Korea Ministry of Science, ICT & Future Planning under contract no. 2015-51-0284. References [1] Friedlander AH. Identification of stroke-prone patients by panoramic and cervical spine radiography. Dentomaxillofac Radiol 1995;24(3):160–4. [2] Friedlander AH, Lande A. Panoramic radiographic identification of carotid arterial plaques. Oral Surg Oral Med Oral Pathol 1981;52:102–4. [3] Ning R, Wang X, Conover DL, Tang X. Image-intensifier-based volume tomographic angiography imaging system. Proc SPIE 1997;3032:238. [4] Hamberg LM, Hunter GH, Halpern EF, Hoop B, Gazelle GS, Wolf GL. Quantitative high-resolution measurement of cerebrovascular physiology with slip-ring CT. AJNR Am J Neuroradiol 1996;17(4):639–50. [5] Choi K, Wang J, Zhu L, Suh TS, Boyd S, Xing L. Compressed sensing based cone-beam computed tomography reconstruction with a first-order method. Med Phys 2010;37:5113. [6] Park J, Song BY, Kim JS, Park SH, Kim HK, Liu S, et al. Fast compressed sensingbased CBCT reconstruction using Barzilai-Borwein formulation for application to on-line IGRT. Med Phys 2012;39(3):1207–17. [7] Oh JE, Cho HS, Hong DK, Lee MS, Park YO, Je UK, et al. Compressed-sensing (CS)-based micro-DTS reconstruction for applications of fast, low-dose X-ray imaging. J Korean Phys Soc 2012;61(7):1120–4. [8] De Man B, Basu S. Distance-driven projection and backprojection in three dimensions. Phys Med Biol 2004;49:2463. [9] Je UK, Lee MS, Cho HS, Hong DK, Park YO, Park CK, et al. Simulation and experimental studies of three-dimensional (3D) image reconstruction from insufficient sampling data based on compressed-sensing theory for potential applications to dental cone-beam CT. Nucl Instr Meth A 2015; 784:550. [10] http://www.vatechkorea.com/products/zenith3d.asp, Vatech Corp., on-line catalog.
Please cite this article in press as: U.K. Je, et al., 3D reconstruction based on compressed-sensing (CS)-based framework by using a dental panoramic detector, Physica Medica (2015), doi: 10.1016/j.ejmp.2015.09.005
Contents lists available at ScienceDirect
Physica Medica j o u r n a l h o m e p a g e : h t t p : / / w w w. p h y s i c a m e d i c a . c o m
Technical Notes
3D reconstruction based on compressed-sensing (CS)-based framework by using a dental panoramic detector U.K. Je, H.M. Cho, D.K. Hong, H.S. Cho *, Y.O. Park, C.K. Park, K.S. Kim, H.W. Lim, G.A. Kim, S.Y. Park, T.H. Woo, S.I. Cho Department of Radiation Convergence Engineering, iTOMO Research Group, Yonsei University, Wonju 220-710, Republic of Korea
A R T I C L E
I N F O
Article history: Received 17 July 2015 Received in revised form 22 September 2015 Accepted 24 September 2015 Available online Keywords: Dental panoramic detector Compressed-sensing Spiral scan Zigzag scan
A B S T R A C T
In this work, we propose a practical method that can combine the two functionalities of dental panoramic and cone-beam CT (CBCT) features in one by using a single panoramic detector. We implemented a CS-based reconstruction algorithm for the proposed method and performed a systematic simulation to demonstrate its viability for 3D dental X-ray imaging. We successfully reconstructed volumetric images of considerably high accuracy by using a panoramic detector having an active area of 198.4 mm × 6.4 mm and evaluated the reconstruction quality as a function of the pitch (p) and the angle step (Δθ). Our simulation results indicate that the CS-based reconstruction almost completely recovered the phantom structures, as in CBCT, for p ≤ 2.0 and θ ≤ 6° , indicating that it seems very promising for accurate image reconstruction even for large-pitch and few-view data. We expect the proposed method to be applicable to developing a cost-effective, volumetric dental X-ray imaging system. © 2015 Associazione Italiana di Fisica Medica. Published by Elsevier Ltd. All rights reserved.
Introduction Panoramic radiography, introduced in the late 1940s, made major progress in dentistry, providing dentists with a single comprehensive image of both the jaws and the maxillofacial structures [1,2]. Cone-beam computed tomography (CBCT), introduced in the mid1990s, also created a revolution, facilitating the transition of dental X-ray imaging from 2D to 3D images of sub-millimeter resolution with faster scanning time and lower dose than in conventional medical CT by employing a large-area flat-panel detector [3]. In addition, recently, handy two (or three)-in-one dental imaging systems that combine the panoramic and the CBCT (or/and cephalometric) features in one device have been launched in the global market, equipped with one panoramic detector and one flat-panel detector together. However, market prices for the combined systems are still high mainly due to the detector cost, thus posing an obstacle for the more widespread use of these systems, especially, in local dental clinics. In this work, we propose a practical method that can combine the two functionalities of dental panoramic and CBCT features in one by using a single panoramic detector. Here a linear-type panoramic detector is tilted by 90° from the orientation for panoramic
* Corresponding author. Department of Radiation Convergence Engineering, iTOMO Research Group, Yonsei University, Wonju 220-710, Republic of Korea. Tel.: +82 33 761 9660; fax: +82 33 761 9664. E-mail address: [email protected] (H.S. Cho).
imaging and rotates together with an X-ray tube around the rotational axis several times along a spiral (or zigzag) trajectory to cover the whole imaging volume during the projection data acquisition. Figure 1 shows the schematic illustrations of (a) a conventional dental CBCT geometry and (b) the proposed geometry. As indicated in Fig. 1(a), in the CBCT, there is a penalty for the large detector coverage because its exact reconstruction is not possible due to insufficient sampling of the imaging volume by the diverging X-ray beam. While the filtered-backprojection (FBP) algorithm has been widely used as an approximate method for the cone-beam image reconstruction, the resulting cone-beam artifacts increase with the imaging volume thickness [4]. Furthermore, the amount of scatter also increases with the detector size, reducing the image contrast. As a possible solution to the cone-beam related artifacts, a lineartype (e.g., 198.4 mm × 6.4 mm in our case) detector can be scanned helically in the axial direction to provide sufficient volumetric sampling. Figure 2 shows the schematic illustrations of the proposed scan geometries for volumetric dental X-ray imaging by using a linear-type panoramic detector: (a) a spiral scan geometry and (b) a zigzag scan geometry. Here the zigzag scan is devised to eliminate the use of the slip-ring technology in the spiral scan which allows continuous multiple rotations but increases the system complexity [5]. The proposed approach to volumetric dental X-ray imaging, compared to the CBCT, leads to imaging benefits such as alleviating cone-beam artifacts and reducing scatters, as well as decreasing system cost. Another advantage of the proposed method would be increased image quality in general, implying that one can go lower in exposure to reach the same image quality as a CBCT,
http://dx.doi.org/10.1016/j.ejmp.2015.09.005 1120-1797/© 2015 Associazione Italiana di Fisica Medica. Published by Elsevier Ltd. All rights reserved.
Please cite this article in press as: U.K. Je, et al., 3D reconstruction based on compressed-sensing (CS)-based framework by using a dental panoramic detector, Physica Medica (2015), doi: 10.1016/j.ejmp.2015.09.005
ARTICLE IN PRESS 2
U.K. Je et al./Physica Medica ■■ (2015) ■■–■■
Materials and methods The X-ray imaging procedure can be modeled approximately by a discrete linear system as follows:
Ax = b,
(1)
x = ( x1, x2, … , xN ) , T
b = ( b1, b2, … , bM ) , T
A = {aij }, i = 1, 2, … , M and j = 1, 2, … , N ,
(2)
where x is the original image vector to be reconstructed, b is the measured projection vector, A is the system matrix, relating x and b, N is the number of voxels, M is the total number of sampling points in the projection vector, and the superscript T is the transpose operator. In the CS framework, x is normally recovered as an optimal solution, x*, for the convex optimization problem described in Eq. (3) by minimizing the following objective function, f(x), assuming that most components of derivative images are negligibly small:
x* = arg min f ( x ) , x∈Q
N 1 2 f ( x ) = Ax-b 2 + α ∑ Di x 2, 2 i =1
where Q is the set of feasible x, 1 2 Ax-b
(3) 2 2
is the fidelity term,
N
∑ Dx i
Figure 1. Schematic illustrations of (a) a conventional dental CBCT geometry and (b) the proposed geometry. In the proposed geometry, a linear-type panoramic detector is tilted by 90° from the orientation for panoramic imaging and rotates together with an X-ray tube around the rotational axis several times along a spiral (or zigzag) trajectory to cover the whole imaging volume during the projection data acquisition.
because for a given field-of-view (FOV) size and exposure settings, scatter will be reduced in the proposed geometry due to the use of a thin detector. We employed a compressed-sensing (CS)-based framework, rather than the common FBP-based framework, for more accurate image reconstruction. Here the CS is a state-of-the-art mathematical scheme for solving the inverse problems, which exploits the sparsity of the image with substantially high accuracy [6]. We implemented a CS-based algorithm for the proposed method and performed a systematic simulation to demonstrate its viability for 3D dental X-ray imaging. In the following sections, we briefly describe the implementation of the CS-based reconstruction algorithm for the proposed method and present the simulation results.
i =1
2
is the sparsifying term, α is the parameter that balances
the two terms and is chosen so that signal-to-noise ratio is maximized (e.g., α = 0.1 was used in the simulation), and Di is the forward difference approximation to the gradient at voxel i. The convex optimization problem described in Eq. (3) can be solved approximately, but efficiently, by using the accelerated gradient-projection Barzilai– Borwein (GPBB) formulation [6]. Details on the mathematical descriptions of the CS reconstruction framework can be found in our previous paper [7]. Figure 3 shows the simplified flowchart of the CS-based reconstruction framework for the proposed method, and it is briefly described as follows. Firstly, the measured projections bm are acquired from the imaging system. The initial guess x(0) is then assumed as the image reconstructed by simple backprojection (BP), followed by forward projection (FP) to obtain the computed projections bc. By using the bm, the bc, and the current updated image x(k), the objective function f(x) is computed to determine the step size Δx(k) for the next updated image x(k+1). The image x is successively updated until the mismatch between the current and the updated images converges to a specified tolerance ε. During the iterative procedure,
Figure 2. Schematic illustrations of the proposed scan geometries for volumetric dental X-ray imaging by using a linear-type panoramic detector: (a) a spiral scan geometry and (b) a zigzag scan geometry.
Please cite this article in press as: U.K. Je, et al., 3D reconstruction based on compressed-sensing (CS)-based framework by using a dental panoramic detector, Physica Medica (2015), doi: 10.1016/j.ejmp.2015.09.005
ARTICLE IN PRESS U.K. Je et al./Physica Medica ■■ (2015) ■■–■■
3
Figure 3. Simplified flowchart of the CS-based reconstruction framework for the proposed method. Firstly, the measured projections bm are acquired from the imaging system. The initial guess x(0) is then assumed as the image reconstructed by simple BP, followed by FP to obtain the computed projections bc. By using the bm, the bc, and the current updated image x(k), the objective function f(x) is computed to determine the step size Δx(k) for the next updated image x(k+1). The image x is successively updated until the mismatch between the current and the updated images converges to a specified tolerance ε.
as indicated in Fig. 3, each iteration loop requires one FP and one BP which are performed by using the prepared system matrix A. We used a modified version of the distance-driven method [8], a current state-of-the-art scheme, for building the proposed system’s matrix. Based upon the above descriptions, we implemented a CSbased reconstruction algorithm for the proposed method and performed a systematic simulation to investigate the imaging characteristics. The number of iterations in the CS-based algorithm was set to 150 because the convergence rate of the algorithm seemed to be saturated after about 100 iterations. We also implemented a common FBP-based algorithm for comparison. A numerical mouth phantom (220 × 220 × 220) was derived by us from a 3D reconstructed image from a CBCT scan of a patient head. A lower part (220 × 220 × 60) of the phantom was used in the simulation for saving the reconstruction time. A detector dimension of 496 (width) × 16 (height) was used to imitate a linear-type panoramic detector. A voxel size of 0.4 mm and a pixel size of 0.4 mm were used in the reconstruction. Detailed reconstruction parameters used in the simulation are listed in Table 1. We investigated the reconstruction quality as a function of the pitch (p) and the angle step (Δθ). Here the pitch
Table 1 Reconstruction parameters used in the simulation. Parameter
Dimension
Source-to-detector distance (SDD) Source -to-objector distance (SOD) Pitch (p) Angle step (Δθ) Pixel size Voxel size Detector dimension (W × H) Voxel dimension Reconstruction algorithm Phantom
650 mm 450 mm 0.5, 1.0, 1.5, 2.0, 4.0, 6.0 3°, 6°, 12°, 24° 0.4 mm 0.4 mm 496 × 16 220 × 220 × 60 FBP, CS 3D numerical mouth phantom
is defined as the source-detector feed per rotation in the units of the nominal slice thickness.
Results and discussion Figure 4 shows the reconstructed (a) transverse and (b) coronal images of the mouth phantom by using the FBP- and the CS-based algorithms for selected pitches of p = 0.5, 1.0, 1.5, 2.0, 4.0, and 6.0 and a fixed angle step of Δθ = 3° (i.e., 120 projections per rotation). As indicated in Fig. 4, the CS-based reconstruction almost completely recovered the phantom structures for a pitch of p ≤ 2.0 , as in CBCT [9], while the FBP-based reconstruction seriously suffered from streaking artifacts for all the pitches, even for p = 0.5, due to the insufficient angular samplings. These results indicate that the CS-based reconstruction is promising for accurate image reconstruction even for large-pitch projection data in the spiral scan. Figure 5 shows the reconstructed transverse images of the mouth phantom by using the FBP- and the CS-based algorithms for selected angle steps of Δθ = 3°, 6°, 12°, and 24° and a fixed pitch of p = 1. As indicated in Fig. 5, the CS-reconstructed images did not seriously suffer from the image artifacts due to insufficient angular samplings, even for Δθ = 6°, while the FBP-reconstructed images seriously did, depending on the angle step. These results indicate that the CS-based reconstruction is also promising even for few-view projection data in the spiral scan. Figure 6 shows the reconstructed transverse (top) and coronal (bottom) images of the mouth phantom for the zigzag scan with a pitch of p = 1 and an angle step of Δθ = 3°, by using the FBP- and the CS-based algorithms. The image characteristics seemed very similar to those for the spiral scan, indicating the effectiveness of the zigzag scan for volumetric image reconstruction. Two main concerns that should be addressed in the proposed method for clinical applicability are the scan time and the reconstruction time; higher scan and reconstruction times than clinically acceptable levels would certainly be unacceptable in a clinical setting.
Please cite this article in press as: U.K. Je, et al., 3D reconstruction based on compressed-sensing (CS)-based framework by using a dental panoramic detector, Physica Medica (2015), doi: 10.1016/j.ejmp.2015.09.005
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Figure 5. Reconstructed transverse images of the mouth phantom by using the FBPand the CS-based algorithms for selected angle steps of Δθ = 3°, 6°, 12°, and 24° and a fixed pitch of p = 1. These results indicate that the CS-based reconstruction is also promising even for few-view projection data in the spiral scan.
Figure 6. Reconstructed transverse (top) and coronal (bottom) images of the mouth phantom for the zigzag scan with a pitch of p = 1 and an angle step of Δθ = 3°, by using the FBP- and the CS-based algorithms. The image characteristics seemed very similar to those for the spiral scan, indicating the effectiveness of the zigzag scan for volumetric image reconstruction.
Table 2 Estimation of the scan time needed for different FOVs at selected pitch settings. FOV (diameter × height)
Single jaw (80 mm × 60 mm) Both jaws (120 mm × 90 mm) Maxillofacial (160 mm × 140 mm)
Figure 4. Reconstructed (a) transverse and (b) coronal images of the mouth phantom by using the FBP- and the CS-based algorithms for selected pitches of p = 0.5, 1.0, 1.5, 2.0, 4.0, and 6.0 and a fixed angle step of Δθ = 3°. These results indicate that the CS-based reconstruction is promising even for large-pitch projection data in the spiral scan.
We estimated the scan time needed for a single jaw (e.g., FOV (diameter × height) ~ 80 mm × 60 mm), both jaws (120 mm × 90 mm), and entire maxillofacial area (160 mm × 140 mm) [10] at selected pitch settings of p = 1, 1.5, and 2.0. In this estimation, we assumed that the detector height is 6.4 mm, the angle step is Δθ = 3°, and the detector’s frame rate is 80 fps (available in our CMOS-type panoramic detector). Table 2 shows the resultant estimation of the scan time for given conditions. As indicated in Table 2, the scan times needed for both jaws at p = 1, 1.5, and 2.0 are estimated less than about 22, 15, and 11 seconds, respectively. Also note that the scan time for entire maxillofacial area at p = 2.0 are less than about 17 seconds, comparable to that in dental CBCT. However, for a common detector having a moderate frame rate, the scan time would be about two or three times larger than this estimation. One possible approach to reduce the scan time could be the use of a wider panoramic detector, which allows a larger pitch without sacrificing image performance.
Pitch (p) p=1
p = 1.5
p = 2.0
14.1 s 21.1 s 32.8 s
9.4 s 14.1 s 21.9 s
7.0 s 10.5 s 16.4 s
The other concern is the reconstruction time associated with iterative techniques in which, as indicated in Fig. 3, one FP and one BP are required for each iteration loop; the former is approximately a discretized evaluation of the Radon transform and the latter is the adjoint operation of the FP. These operations are the primary computational bottleneck in the iterative scheme, especially in 3D image reconstruction, and the use of the graphics processing unit (GPU) has been frequently considered for accelerating them. Although the implemented algorithm for this work has not yet been accelerated by the GPU, the reconstruction time for the given test conditions (i.e., FOV = 220 × 220 × 60 voxels, voxel size = 0.4 mm) was less than 30 minutes on a normal workstation, in its present state well beyond what current CBCTs (having large FOVs with small voxel sizes) offer. One possibility to speed it up could be adopting the FBP framework into the CS-based framework to provide a more feasible initial gauss (rather than a simple BP) and a more well-converged next updated image. We expect that it could improve the convergence rate of the present iterative algorithm considerably. Conclusion In this work, as a continuation of our dental imaging R&D, we propose a practical method that can combine two functionalities
Please cite this article in press as: U.K. Je, et al., 3D reconstruction based on compressed-sensing (CS)-based framework by using a dental panoramic detector, Physica Medica (2015), doi: 10.1016/j.ejmp.2015.09.005
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of dental panoramic and CBCT features in one with a single panoramic detector intended for integration into local panoramic systems. Here a linear-type panoramic detector is tilted by 90° from the orientation for panoramic imaging and rotates together with an X-ray tube around the rotational axis several times along a spiral (or zigzag) trajectory to cover the whole imaging volume during the projection data acquisition. We employed a CS-based framework, rather than the common FBP-based framework, for more accurate image reconstruction. Here the CS is a state-of-the-art mathematical scheme for solving the inverse problems, which exploits the sparsity of the image with substantially high accuracy. We successfully reconstructed volumetric images of considerably high accuracy by using a panoramic detector having an active area of 198.4 mm × 6.4 mm and evaluated the reconstruction quality as a function of the pitch (p) and the angle step (Δθ). According to our simulation results, the CS-based reconstruction almost completely recovered the phantom structures, as in CBCT, for p ≤ 2.0 and θ ≤ 6° , demonstrating its viability for 3D dental X-ray imaging. The reconstruction characteristics for the zigzag scan were similar to those of the spiral scan. We expect the proposed method to be applicable to developing a cost-effective, volumetric dental X-ray imaging system. Acknowledgment This research was supported by the Radiation Technology Development Program of the National Research Foundation (NRF)
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Please cite this article in press as: U.K. Je, et al., 3D reconstruction based on compressed-sensing (CS)-based framework by using a dental panoramic detector, Physica Medica (2015), doi: 10.1016/j.ejmp.2015.09.005