The method aims to maximize the regularity of ball markers in the calibration phantom and also in its projection image

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images. A standard calibration process usually involves the illumination of an accurate calibration phantom with known coordinates of ball markers using the imaging system, followed by calculation of geometrical parameters by minimizing the errors between reprojected projection of ball markers and its acquired projection image. Although many attempts have been made to estimate the geometrical parameters, little attention has been paid to the optimal structure of calibration phantom. Inspired by the assumption that the larger the regularity of ball markers in the calibration phantom is, the more the stable is, and the better accuracy of estimated geometric parameters is, we propose a method to design phantom that maximize the accuracy of calibration process and mitigate the contribution of errors in indicating the ball centers. The method aims to maximize the regularity of ball markers in the calibration phantom and also in its projection image. The proposed method is applied to different phantom designs with the standard cylindrical holder and is proven to provide more accurate results than the traditional designs. The method can be applied to design scanner-dependent calibration phantoms and potentially free manufacturers and practitioners from manually searching work.© 2022 Society of Photo-Optical Instrumentation Engineers (SPIE)[DOI:10.1117/1.OE.61.2.023104]

Keywords:imaging systems; computed tomography; geometrical calibration; optimal phantom design.

Paper 20210886 received Aug. 12, 2021; accepted for publication Feb. 8, 2022; published online Feb. 25, 2022.

1 Introduction

Acquiring geometrical parameters is an important step in perfecting a computed tomography (CT) imaging system. Accurate values for geometric parameters will ensure the correctness of the reconstruction process that brings back the underlying image since minor errors in geometric parameters will subsequently lead to incorrect results, even major artifacts in reconstructed images. This is more important in the low-dose CT imaging era where the iterative methods are used to reconstruct the image.¹In that case, the iterative methods might propagate the incorrect values of geometric parameters into the reconstructed images.

In general practice, the geometric parameters are usually estimated via a calibration process.

Geometric calibration methods are classified into two groups: online and offline methods (see Ref. 2 for extensive review). Online methods are carried indirectly via 2D–3D registration processing that aims to minimize the errors between acquired projection image and the reprojected projection.^3–6Though these methods do not require a calibration phantom, they usually involve computationally expensive and heuristic registration that might provide less accurate values than the offline methods. On the other hand, the offline methods work by illuminating a predefined calibration phantom having steel ball markers, followed by computation to find the geometric parameters. The computation consists of two steps: (i) extracting the ball markers location and then center of ball markers in projection image and (ii) solving the equations that relate the location of center of balls in the projection image and its position in predefined calibration phantom to find the values of geometric parameters. The offline methods have been shown to provide more accurate results than the online methods, with the condition of having

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highly accurate coordinates of balls markers in the calibration phantom and also highly accurate extracted center of balls markers in the projection image.^7–14

Although the process of offline calibration is well-defined and investigated, little attention has been paid to the structure of calibration phantom. From an industrial point of view, an easy- to-manufactured and assembled phantom is a cylindrical plastic phantom with ball markers located in two circular patterns as in Ref.13. Another phantom is helical-shape phantom with ball markers positioned on a helix made of plastic or glass holder.⁷However, a specific and systematic method to distribute the balls along the helix holder is yet to be available. This leaves the manufacturers with a difficult task in calibrating the CT systems (for both CBCT and C-arm systems) and might eventually end-up with suboptimal design. This is particularly more difficult when the existence of errors in measuring the ball markers center in the projection image and the errors in manufacturing the calibration phantoms are unavoidable.

Efforts have been made to create a guideline on how to make an efficient calibration phantom.^8,15–20 Claus¹⁵ presented a detailed analysis of critical criteria related to the design of an artifact for CT geometry calibration. Then they proposed a reference object with spheres mounted on a cylindrical frame arranged in rotationally shifted four quarter-helix arrays. This design helps to reduce possible overlaps between projected ball markers and create a uniform spatial distribution of the spheres. Zechner et al.⁸proposed a ball bearing phantom applicable for nine degrees of freedom calibration of a cone-beam computed tomography (CBCT) scanner. To ensure high accuracy of marker detection, three different marker distributions on the phantom cylinder surface were studied. Those included: (i) random distribution where the positions of markers are randomized; (ii) quasi-random distribution where ball positions were generated randomly with the condition of having a distance larger than a fixed threshold on the detector plane;

and (iii) distribution of balls followed a helical shape based on the Golden angle distribution. The best method is the compromise between uniform distribution and a high packing fraction of balls. The effectiveness of the investigated method was judged based on the uniformity of the distribution over the detector area. The result indicated that the golden ratio ball bearing phantom with the uniform distribution of ball markers over the entire detector in the projection image combined with the highest possible packing fraction of markers distributed over the entire detector area helps to achieve highest accuracy of geometrical parameters. More recently, in Refs.16 and17, modular phantom constructed from LEGO brick was used. The balls were positioned in which their projection trajectories cover as much of the detector’s regularity as possible (the point pattern of projection of ball markers in the projection image is highly regular) and there are no overlaps of the balls in the projection images. Similarly, in Ref.9, the geometrical measurement procedure was carried via a calibration phantom that was designed to reduce sphere overlaps in the projected image and broadening the distribution of projected sphere across the detector area. However, there is no systematic method in designing calibration phantoms to sat- isfy those requirements. That leaves the manufacturers and practitioners rough ideas on a good calibration phantom, but not a method to make that phantom optimally.

In CBCT imaging systems, from a geometrical point of view, the wider the view of the imaging object in the projection image is, the better the estimation of the distance between the focal point to the detector is, and the more accurate the estimation of geometric parameters is.

In this work, inspired by that principle, we propose a method to distribute the balls in the calibration phantom so that it maximizes the accuracy of geometric parameters estimation. It is done by maximizing the regularity of ball markers in the phantom (the point pattern of ball markers is highly regular in the phantom domain) and the regularity of projection of ball markers in the projection image. In addition to providing accurate values to geometrical parameters, the method is expected to ease the burden on the accurate estimation of ball centers in the projection images.

The proposed method is then applied to popular calibration phantom designs with cylindrical plastic holder, which is usually used in calibrating circular trajectory CBCT systems and proven to work in a standard CBCT imaging system where it helps to robustly estimate the geometrical parameters.

The remainder of this paper is organized as follows. Section2presents the imaging geometry and an offline calibration method which can estimate the geometrical parameters per projection view. Section3presents our method to optimize the distribution of ball markers in the calibration

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phantom and its application to the designing of calibration phantom with cylindrical plastic holder for a standard CBCT system. Section 4 presents our experimental studies. Section5 concludes.

2 Calibration Method

2.1 Image Model in CBCT Systems

As in a general pinhole camera model, in x-ray imaging, the relation between a pointðx; y; zÞin 3D dimension and its projected pointðu; vÞin 2D detector is modeled via a mapping as follows (see Fig. 1):

EQ-TARGET;temp:intralink-;e001;116;387

½uω; vω;ω^T¼P½x; y; z;1^T; (1) whereP is projection matrix andω is the weighting factor.

The projection matrixPis composed from geometric parameters as follows:

EQ-TARGET;temp:intralink-;e002;116;334P¼K½Rjt; (2)

whereKis the intrinsic matrix,Ris the rotation matrix, andtis the translation matrix.¹⁹ The intrinsic matrix is formed by intrinsic geometric information about the imaging system and has the following form:

K¼ 2 64

f pu

f putanα u0

0 _p ^f

vsinα v0

0 0 1

3

75; (3)

wherefis the distance from source to the detector;p_u; p_vare the width and height of detector bin, respectively;αis the angle formed by two axes of detector bin; andu0; v0is the coordinate of the intersecting point between central ray and the detector. Meanwhile, the rotation matrixR is formed by R¼R_zR_yR_x, where R_x;R_y;R_z are the rotation matrix in x; y, and z axes, respectively.

2.2 Phantom-Based Calibration Method

In phantom-based calibration methods, the measurement of projection matrixPfor each projection view is usually carried via the acquisition of several pairs ofðu_i; v_iÞandðx_i; y_i; z_iÞwith i¼1;2; : : : ; N. To obtain these pairs, a calibration phantom is used. The phantom has several steel-made ball markers centered at known positionsðx_i; y_i; z_iÞwith i¼1;2; : : : ; N.

Fig. 1 Illustration of the 3D to 2D projection in a standard CBCT system.

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The phantom is then illuminated in the real CBCT scanner, then the centroids of the ellipses formed by projected area of ball markericentered atðx_i; y_i; z_iÞare the associatedðu_i; v_iÞ. The computation ofðu_i; v_iÞis done via the following steps:^19,20(i) remove the noise in the projection image; (ii) find edges of the ellipse; and (iii) use a curve-fitting algorithm to find the ellipse centers. (The computation can also be done via a deep learning method such as Beadnet as done in Ref.17, given the availability of simulated dataset containing x-ray bead projections from different geometry configurations.)

To estimate the projection matrix for a projection view, we need two lists sorted in the same order: ball markers positionðx_i; y_i; z_iÞand its associated projected pointðu_i; v_iÞin the detector, which are also known as point correspondences. The correspondence between the ball marker and its projection can be achieved using: (i) the horizontal information of the projected point ðu_i; v_iÞif the phantom is the equally distributed helical phantom (see Sec.4) and (ii) the angle of the line connecting projected point to the virtual center of the ball centers and horizontal axis if the phantom is the circular pattern phantom.²¹For other types of ball distribution, finding the correspondence is a challenging problem and needs a phantom-oriented method to solve.

Having the ball markers positionðx_i; y_i; z_iÞand its associated projected pointðu_i; v_iÞin the detector withi¼1; : : : ; N, the geometric projection matrixPis estimated by solving the linear system of Eq. (1), which can be described in a clarified form as follows:¹⁹

EQ-TARGET;temp:intralink-;e004;116;520A_ip¼0; (4)

withi¼1; : : : ; N,A_i is a2×12 matrix.

A_i¼

x_i y_i z_i 1 0 0 0 0 −u_ix_i −u_iy_i −u_iz_i −u_i

0 0 0 0 x_i y_i z_i 1 −v_ix_i −v_iy_i −v_iz_i −v_i

; (5)

and pis a column vector constructed by 12 elements of the projection matrixP:

p¼ ðP11; P12; P13; P14; P21; P22; P23; P24; P31; P32; P33; P34Þ^T (6) HavingNcalibration points, the vectorpcan be found by solving the following equation:

EQ-TARGET;temp:intralink-;e007;116;375Ap¼0; (7)

whereAis a2N×12matrix formed by stacking A_i, withi¼1; : : : ; N.

Equation (7) can be solved via the singular value decomposition technique or a least square optimization technique.²² However, some issues need to be addressed to accurately findp.

Theoretically, the accuracy of measurement for ðx_i; y_i; z_iÞ can be achieved by precision manufacturing. However, it is not the case in practice due to the dependence on the quality of ball markers not only the perfect shape but also the density of the ball marker. Therefore, the errors in determiningðx_i; y_i; z_iÞare unavoidable. In addition, the accuracy ofðu_i; v_iÞis difficult to maintain due to the following reasons. First of all, it is the centroid of the ellipse formed by the projected area of the ball markericentered atðx_i; y_i; z_iÞ(since the projection of a ball of uniform density onto a flat detector forms an ellipse). And the centroid of the intensity value is different from the ellipse center and is also different from the center projection of the ball center onto the projection.²³In ideal situation, center projection (projected location of the ball center in the detector) should be used asðu_i; v_iÞ. However, due to the difficulties in accurately identify it, the centroid of the intensity value is usually used. Nevertheless, all these values are noise sen- sitive and heavily dependent on the accuracy of phantom manufacturing.

3 Optimal Phantom Design

3.1 Method for Optimal Phantom Design

Previous works consider improving the accuracy of solutionpto Eq. (7) by employing different techniques to accurately determineðu_i; v_iÞ. In this work, we assume that the errors in determining ðu_i; v_iÞ are inevitable. Furthermore, the errors in placing ball markers in the phantom

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ðx_i; y_i; z_iÞare also inevitable. To overcome those issues, we aim at having a good ball markers placement so that it will generate a stable result forp. To do so, we introduce a method to design phantom as follows.

From a geometrical point of view, as shown in Fig.2, the wider view of the object in the detector is, the more accurate the estimation of geometric parameters is. With that intuition in mind, we propose a method to place ball marker in the volume image effectively. The proposed method is defined in both object (phantom) domain and projection image domain.

Our method is to maximize the spatial randomness of the ball markers in the object domain. It also aims to maximize spatial randomness of the projection of ball markers in the projection image. It is equivalent to have a highly regular distribution (or regularity) in both object domain and projection image domain.

In the object domain, for each ballicentered atðx_i; y_i; z_iÞ, we definer^obj_i is the nearest neighbor distance:

r^obj_i ¼min_j¼1_{;: : : ;N}_;_j≠ifd^obj_ij g; (8)

where dôbj_ij ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx_i−x_jÞ²þ ðy_i−y_jÞ²þ ðz_i−z_jÞ² q

, then the mean nearest neighbor for all points in the object domain is defined as follows:

EQ-TARGET;temp:intralink-;e009;116;360^ r^obj¼ 1

N X^N

i¼1

r^obj_i : (9)

In the projection domain, for a particular projection viewυ, we define ther^ðυÞ_i is the nearest neighbor distance fori’th ball center in the projection image:

r^ðυÞ_i ¼min_j¼1_{;: : : ;}N¯;j≠ifd^ðυÞ_ij g; (10)

whered^ðυÞ_ij ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu^ðυÞ_i −u^ðυÞ_j Þ²þ ðv^ðυÞ_i −v^ðυÞ_j Þ² q

, then the mean nearest neighbor for all points in this projection viewυis defined as follows:

EQ-TARGET;temp:intralink-;e011;116;214^r^ðυÞ¼ 1 N¯

X^N^¯

i¼1

r^ðυÞ_i ; (11)

whereN¯ is the number of balls marker viewable in the projection image (detector) (N¯ ≤N).

Then the mean nearest neighbor for all view is

EQ-TARGET;temp:intralink-;e012;116;138^ r^proj¼ 1

V X^V

υ¼1

r^^ðυÞ; (12)

whereV is the number of projection views in the CBCT system. Note that the calculation for Eq. (12) is not limited to any particular CT trajectory sincer^^ðυÞ is defined for each view.

Fig. 2 Illustration on the contribution of viewing condition on the accuracy of calibration where the wider view about the calibration phantom the higher accuracy of geometrical parameters:

(a) narrower view and (b) wider view.

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The connection between the mean nearest neighbor for all points in the object domainr^^obj and the regularity in the phantom domain is that, increasing r^^obj will lead to highly regular (or higher regularity) distribution of ball markers in the phantom domain. Meanwhile, increasing r^^prj will lead to highly regular (or higher regularity) distribution of projection of ball markers in the projection image.

Thefore, we propose to use the following combined objective function:

EQ-TARGET;temp:intralink-;e013;116;661^

r¼r^^objþγr^^proj; (13)

whereγis the hyperparameter which weights the contribution from spatial randomness in the projection domain.

When we considerr^¼r^^proj, having large value ofr^in Eq. (13) is a good indicator for evaluating the projection image of calibration phantom. (And this is the same as in Ref.8.) We can also include constraints to eachr^ðυÞ_i such as the lower threshold or the percentage of the ball havingr^ðυÞ_i greater than a given threshold. (These additional constraints can be included in the optimization process using the randomized algorithms as shown later in Sec.3.2.)

In order to apply this method to calibration phantom design whenγ≠0, we need to have prior geometrical information about the imaging system to be calibrated. The prior information (especially geometric parameters forming the intrinsic matrix and translation matrix) might come from physical measurement using other sensors equipped within the CBCT scanner. Then the later calibration process is used to finely determine the geometrical values in a narrow range around the previously measured values.

With that initialized values for geometric parameter, we can have an approximated projection matrix Pfrom which the calibration phantom can be used to fine tune.

Given the projection matrixPand by denotingH¼ 2 66 66 64

x1 y1 z1

x2 y2 z2

x3 y3 z3

: : : : : : : : : x_N y_N z_N

3 77 77 75

, whereðx_i; y_i; z_iÞis

the center of ball markeriwithi¼1; : : : ; N, then our goal (to have optimal ball marker placement) is to solve the following optimization problem:

H^ ¼arg max

H

r :^ (14)

From a theoretical point of view,^24,25the spatial randomness for N points in an areaA is described by the Poisson process, where the probability density function for nearest neighbor distance dis

pðdÞ ¼2πδde^−πδd²; (15)

withδ¼^N_Ais the point density which can be modeled as the mean number of points per unit area.

The expectation of this distributionr_E¼¹₂ ffiffiffi

AN

q

gives the average distance between nearest neigh- bors for a random process. Then the nearest neighbor statisticR¼r^_υ∕r_Eprovides a quantitative summary of the spacing between stimulus elements. As illustrated in Ref.25, asRincreases, the point pattern stimuli change from highly clustered through random to highly regular. Inspired by that, in this work, we maximizer^¼r^^objin Eq. (13) to encourage ball markers distribution so that they form a highly regular pattern in the object domain. Meanwhile, we maximizer^¼^r^projto encourage ball markers distribution so that its projection has wide coverage and a highly regular pattern. The constraint of having no overlap between points might also be included, though not required. Note that we can also add the constraint to have as large a number of visible balls as possible to the objective function in Eq. (11) to perfect the ball distribution.

In this work, to model in a realistic and traditional condition, we consider a specific configuration where the ball markers are distributed along a standard cylindrical holder to make the calibration phantom. This kind of phantom is chosen since it is usually used in current practice.¹³

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SQP method can be trapped in local solution that is far from the global (optimal) solution.

To overcome that issue of SGP method, in this work, we use the evoluationary algorithm to solve the optimization problem in Eq. (14). In particular, we use the covariance matrix adapta- tion-evolutionary strategy (CMA-ES).²⁷The CMA-ES is a stochastic search algorithm known for robustness and efficiency in a rugged search landscape such as with nonlinear, nonquadratic, nonconvex functions, or nonsmooth, discontinuous, multimodal, and/or noisy function. It is a proven, reliable, and highly competitive evolutionary algorithm for local optimization and global optimization.

The CMA-ES algorithm works by randomly generating candidate sample points around the current estimate based on a multivariate normal distribution with a certain mean and a covariance matrix. The number of sample points in one step (generation) is called the “population size” (denoted λ). The objective function (also known as fitness or loss function), which is r^_υ in Eq. (13) in our case, is evaluated at each sample point, and based on the function values at all sample points in one step, the algorithm adapts the mean and the covariance matrix of the sample distribution for the next step in such a way that the distribution aligns with the gradient direction of the objective function. The choice of population sizeλdepends on the preference of convergence speed or robustness, where a smallerλleads to faster convergence, and a largerλhelps avoid local optima. (More details about the CMA-ES algorithm can be found in Ref.27.)

In this work, we use the MATLAB implementation of CMA-ES algorithm (available in Ref.28) with modifications to define the objective function in Eq. (13).

Note that the use of CMA-ES algorithm also helps us to freely include the other constraints such as having the lower threshold forr^ðυÞ_i or set the percentage of the ball havingr^ðυÞ_i greater than a given threshold.

4 Simulation Studies

To evaluate the performance of the proposed method, we performed the experiment with a soft- ware modeled CBCT scanner having the following setting. The detector of CBCT system had a size of128 mm×130 mm. The detector bin size was0.1 mm×0.1 mmthat makes the detector resolution of1280×1300. The distance from source to detector was set to f¼620 mm. The distance from source to isocenter was set to 418 mm. The angle formed by two axes of detector bin was set toα¼90deg. From these geometric parameters, the projection matrix was generated according to Eq. (2). This projection matrix was considered as the ground truth (underlying values to be searched for) in our experiment studies, and we denoted it asPgroundtruth.

To search for the projection matrix, we used six different design approaches to distribute the ball markers along the cylindrical holder. The three conventional design approaches were: two circular patterns phantom (P1) where the ball markers were distributed in usually used two circular patterns,¹³equal helical phantom (P2) where the ball markers were distributed evenly along a helix, and golden ratio phantom (P3) where we distributed the ball along a helix using the golden ratio criterion as in Ref.8. The three modified approaches by applying our method were:

modififed two circular pattern phantom (P4), phantom domain oriented design (P5), and cross domain oriented design (P6).

In particular, the phantomP4was made by constrainting the markers in two circular patterns asP1, but varying the two circular patterns along theyaxis so thatr^¼r^^projreaches its maximum value. Meanwhile, the phantomP5 was made by distributing the ball markers along the cylindrical holder so thatr^¼r^^objreaches its maximum value. Finally, the phantomP6was made by distributing the ball markers along the cylindrical holder so thatr^¼r^^objþr^^prjreaches its maximum value with prior information about projection matrixP.

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To have a fair comparison, we restricted the design in the phantom having length of 63 mm, the inner radius is of 23 mm, and outer radius is of 27 mm (see Fig.3). There were 24 metal ball markers in each phantom. The ball markers had the diameter of 1.5 mm. The pitch value of the helix in phantomsP2andP3was 30 mm. That value was chosen so that 24 metal balls are well- distributed in two complete helix turns. Note that our proposed method, whenγ≠0, used an approximated projection matrix derived from physical measurement for basic parameters. This was consistent to current practice where the manufacturers use additional sensors to approxi- mately estimate the geometric parameters. The calibration phantom was later use to fine-tuning the value of geometric parameters.

Table1summarizes the coordinates of ball markers in our six designs. Note that the coordinates were shown in that the center of phantom was at the origin in the object coordinate.

With the six calibration phantoms, we performed forward projection to get the corresponding projection images in the detector. To simplify the analysis, we performed the experiments with one projection view where all three Euler angles (that forms the rotation matrixR) were set to 0.

Figure3shows the resulting projection images where the distribution of ball markers inP1and P4followed two circular patterns, but inP4, the circular patterns were pushed to the two sides of the cylinder so that they have widest view and largest coverage area in the detector. Meanwhile for the phantom designP2 andP3, its projection images follow the expected pattern of a helix (P2) and golden ratio distribution (P3) of angle 137.5 deg. For the phantom designP5, the objection function wasr^¼r^^obj. Upon having the solution, we found that for that solution, the ball markers were positioned equally in four“virtual”circular patterns in the phantom domain. And finally, for the phantom designP6, the ball markers were distributed so that its projection has regular pattern (with highest measured regularity score). Note that forP6, the objection function wasr^¼^r^objþγr^^projwith γ¼1. One can change the value ofγ to adjust the contribution of constraint in the projection domain to the overall objective function.

To evaluate the performance of projection matrix acquired using different calibration phantoms, we used an evaluating phantom which has the length of 73 mm, the inner radius of 31 mm, and outer radius of 33 mm as the calibration phantoms. The evaluating phantom consists of Fig. 3 Projection images of six calibration phantoms: (a)P₁, (b)P₂, (c)P₃, (d)P₄, (e)P₅, and (f)P6. Since the ball markers are of the same size, in the projection images, the smaller the ball marker is, the closer to the detector is. The rectangle around (a) indicates the imaging range of the detector.

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Table1Coordinatesofballmarkersinoursixcalibrationphantomdesigns:twocircularpatternsphantom(P1),equallydistributedhelicalphantom( phantom(P3),modifiedtwocircularpatternsphantom(P4),phantomdomainorienteddesign(P5),andcrossdomainorienteddesign(P6). Ballmarker

xyzxyzxy Phantom1(P1)Phantom2(P2)Phantom 121.651−15.00012.50024.148−28.7506.4709.365−28.764 221.65115.00012.50017.678−26.25017.678−22.470−26.255 312.500−15.00021.6516.470−23.75024.14823.908−23.764 412.50015.00021.651−6.470−21.25024.148−12.876−21.255 50.000−15.00025.000−17.678−18.75017.678−5.198−18.764 60.00015.00025.000−24.148−16.2506.47020.225−16.255 7−12.500−15.00021.651−24.148−13.750−6.470−24.814−13.764 8−12.50015.00021.651−17.678−11.250−17.67816.401−11.255 9−21.651−15.00012.500−6.470−8.750−24.1480.872− 10−21.65115.00012.5006.470−6.250−24.148−17.366− 11−25.000−15.0000.00017.678−3.750−17.67824.966− 12−25.00015.0000.00024.148−1.250−6.470−19.429− 13−21.651−15.000−12.50024.1481.2506.4703.4791.236 14−21.65115.000−12.50017.6783.75017.67813.9803.745 15−12.500−15.000−21.6516.4706.25024.148−24.3596.236 16−12.50015.000−21.651−6.4708.75024.14821.8658.745 170.000−15.000−25.000−17.67811.25017.678−7.72511.23 180.00015.000−25.000−24.14813.7506.470−10.16813.74 1912.500−15.000−21.651−24.14816.250−6.47023.01316.23 2012.50015.000−21.651−17.67818.750−17.678−23.63818.74 2121.651−15.000−12.500−6.47021.250−24.14811.73721.23 2221.65115.000−12.5006.47023.750−24.1486.04823.74 2325.000−15.0000.00017.67826.250−17.678−20.96726.23 2425.00015.0000.00024.14828.750−6.47024.69228.74

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Table1(Continued). Ballmarker

xyzxyzxyz Phantom4(P4)Phantom5(P5)Phantom6(P6) 121.651−30.00012.50013.74830.00020.88024.995−16.3640.495 221.65130.00012.50010.01130.000−22.908−24.574−30.000−4.595 312.500−30.00021.6512.1269.57224.909−24.998−16.4570.348 412.50030.00021.651−4.096−30.000−24.662−12.243−9.053−21.797 50.000−30.00025.000−24.909−9.5722.126−24.99712.339−0.355 60.00030.00025.00022.908−30.00010.011−11.9954.110−21.935 7−12.500−30.00021.651−2.1269.572−24.909−24.852−2.365−2.719 8−12.50030.00021.651−13.74830.000−20.88010.976−23.838−22.462 9−21.651−30.00012.500−20.880−30.00013.748−6.81630.000−24.053 10−21.65130.00012.500−10.01130.00022.90811.78015.397−22.051 11−25.000−30.0000.00010.011−10.855−22.90824.98511.354−0.866 12−25.00030.0000.00024.909−9.572−2.126−0.09110.85425.000 13−21.651−30.000−12.5004.096−30.00024.662−24.84330.000−2.799 14−21.65130.000−12.500−24.66230.0004.09625.00030.0000.000 15−12.500−30.000−21.651−19.7249.57215.361−0.226−18.58624.999 16−12.50030.000−21.65115.361−9.57219.72412.6062.89321.589 170.000−30.000−25.000−10.011−10.85522.908−10.97117.421−22.464 180.00030.000−25.00024.66230.000−4.09624.527−2.441−4.839 1912.500−30.000−21.651−22.908−30.000−10.0116.82130.00024.052 2012.50030.000−21.65120.880−30.000−13.74812.732−11.69821.515 2121.651−30.000−12.50022.90810.85510.011−1.071−30.000−24.977 2221.65130.000−12.50019.7249.572−15.36124.330−30.000−5.748 2325.000−30.0000.000−22.90810.855−10.011−12.118−22.215−21.867 2425.00030.0000.000−15.361−9.572−19.724−0.323−3.429−24.998

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N_E¼144ball markers distributed in 12 circular patterns as shown in Fig.4(a). The ball markers had the diameter of 1.5 mm. Four regions of interest (ROIs) [as shown in Fig.4(b)] in the projection image were selected. The ROI #1 and #2 cover the ball markers close to the x-ray source and the ROI #3 and #4 cover the ball markers close to the detector. These ROIs cover different regions of the projection image and thereby have different sensitivities to the geometrical parameters.

Since it was hard to evaluate (and compare) the performance of the calibration phantom designs (P1; P2; P3; P4; P5; P6) in providing accurate geometrical information, we performed the numerical simulations as follows. For each phantomP_k, we added the Gaussian noise to the projected positions of the ball centers. (In practice, this must be done via the computation of centroids of ellipse, however, we performed a proof-of-concept to our phantom design by ignoring that procedure.) This is to simulate the errors associated in measuring centroids that are unavoidable in practice.¹¹We also added the Gaussian noise to the ball markers’centers in the object (phantom) domain. Although the means of Gaussian distribution were 0, the standard deviation was 0.075 for phantom domain and 0.25 for projection domain. We then use the noise contaminated ball marker centers and its noise contaminated projection to estimate the projection matrix. [We denoted the result as PðP_kÞ.] Since the direct matrix–matrix comparison would make no sense, so do the geometric parameters,²¹ the estimated projection matrix was then be used to perform forward projection of the evaluating phantom. We then denoted the resulting projected ball centers (of evaluating phantom) in the projection image asðu_i; v_iÞðPðP_kÞÞwith i¼1; : : : ; N_E. The Euclidean distance between theðu_i; v_iÞðPgroundtruthÞandðu_i; v_iÞðPðP_kÞÞwas measured. Then the average distance for all ball markers in the ROIris defined as

E^r_P

k¼ 1

N_r X

i∈ROIðrÞ

dððu_i; v_iÞðPgroundtruthÞ;ðu_i; v_iÞðPðP_kÞÞÞ; (16)

withdððu_i; v_iÞ;ðu_i⁰; v_i⁰ÞÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu_i⁰−u_iÞ²þ ðv_i⁰−v_iÞ²

p ,N_ris the number of ball markers in the ROIr.

E^r_P_k was then used as an indicator where the smaller the value is, the better the projection matrix estimated by the phantom design P_k in ROI r is. We repeated the above calculation over 50 independent noise trials and calculate the bias and standard deviation for allE^r_P_k with r¼1; : : : ;4 and k¼1; : : : ;6.

When there was no noise in both ball marker centers in the phantom domain and in the projection domain, all calibration phantomsP_k(withk¼1; : : : ;6) obtain results that are exactly the same as the ground truth projection matrix. However, this is not the case in practice.

Fig. 4 (a) Projection image of evaluating phantom and (b) dashed rectangles denote ROIs used in quantitative studies.

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We measured the bias and standard deviation forE^r_P_kwithr¼1; : : : ;4andk¼1; : : : ;6in three conditions: (i) when noise was added to ball marker centers in projection image only;

(ii) when noise was added to ball marker centers in phantom domain only; and (iii) when noise was added to ball marker centers in both projection image domain and phantom domain. The measuring results were summarized in Tables2–4. Note that, in the result for each ROI (each row in Tables2–4), the bold-faced cell indicated the one having smallest error for that ROI.

As shown in Table2, in the ideal case, when the calibration phantoms were precisely manufactured and the ball markers centers were perfectly measured, even when there were errors in the measuring of ball marker centers in projection image, there were almost no differences between calibration phantoms.

When there were errors in ball marker centers in phantom domain, different calibration phantoms started working differently. As shown in Table3, the optimal phantom designs (P4; P5; P6) work better than traditional phantom designs (P1; P2; P3) in all ROIs. In particular, the cross- domain oriented designP6 results almost smallest errors in three over four ROIs.

Finally, when the errors were introduced to ball marker centers in both phantom domain and projection domain, the calibration phantom designed by applying our method works efficiently with smallest errors in all ROIs. Once again, the calibration phantoms P6 works best (see Table4).

Table 2 Mean and standard deviation of error for 50 noise realizations when the noise was only added to projection image.

ROI P1 P2 P3 P4 P5 P6

1 0.99 ± 0.28 1.00 ± 0.29 1.05 ± 0.20 1.00 ± 0.14 1.01 ± 0.20 1.00 ± 0.19 2 0.99 ± 0.25 1.00 ± 0.31 1.05 ± 0.22 1.00 ± 0.15 1.00 ± 0.20 1.00 ± 0.20 3 1.00 ± 0.26 1.02 ± 0.22 1.01 ± 0.22 1.00 ± 0.14 0.99 ± 0.13 1.00 ± 0.20 4 0.99 ± 0.26 1.03 ± 0.23 1.01 ± 0.20 1.01 ± 0.13 1.00 ± 0.13 1.01 ± 0.20

Table 3 Mean and standard deviation of error for 50 noise realizations when the noise was only added to phantom domain.

1 1.90 ± 0.93 1.93 ± 1.15 1.51 ± 0.76 1.33 ± 0.56 1.50 ± 0.82 1.26 ± 0.64 2 1.85 ± 1.01 1.88 ± 1.25 1.55 ± 0.90 1.39 ± 0.57 1.59 ± 0.77 1.23 ± 0.68 3 1.59 ± 0.83 1.74 ± 1.08 1.41 ± 0.74 1.27 ± 0.57 1.20 ± 0.64 1.31 ± 0.76 4 1.54 ± 0.82 1.80 ± 1.14 1.43 ± 0.70 1.25 ± 0.55 1.22 ± 0.65 1.22 ± 0.72

Table 4 Mean and standard deviation of error for 50 noise realizations when the noise was added to both phantom domain and projection domain.

1 1.40 ± 0.90 1.78 ± 1.11 1.46 ± 0.88 1.39 ± 0.61 1.46 ± 0.75 1.23 ± 0.66 2 1.38 ± 0.80 1.90 ± 1.14 1.48 ± 0.92 1.36 ± 0.64 1.45 ± 0.73 1.22 ± 0.62 3 1.40 ± 0.77 1.47 ± 0.80 1.28 ± 0.60 1.19 ± 0.50 1.12 ± 0.55 1.24 ± 0.83 4 1.45 ± 0.80 1.50 ± 0.84 1.21 ± 0.64 1.20 ± 0.50 1.15 ± 0.61 1.20 ± 0.80

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in our experiments while the cross-domain oriented designP6works best. For that phantom, the hyperparameterγ¼1means we set the contribution from projection domain (^r^prj) equal to the contribution from the phantom domain. In practice, this value needs to be fine-tuning according to the distance from source to the isocenter, the distance from isocenter to the detector and the phantom size.

From a computation point of view, due to its well-rooted design, the calibration phantoms P1; P2; P4, andP5, are easier to use since the methods to find the correspondence between ball markers positionðx_i; y_i; z_iÞand its associated projected pointðu_i; v_iÞin the detector are well- established. On the other hand, for the complex calibration phantomsP3 and P6, finding the correspondence is a challenging task and requires substantial consideration when using in practice.

Finally, note that, since it is impossible to equally model the imperfects (of ball marker place- ments and measurements of projection of ball marker center in the projection image) for different phantom designs, we did not include the comparison using the reconstructed images.

5 Conclusions

We have proposed and validated a method to design calibration for CBCT systems. The method works by promoting the regularity of the ball markers in phantom domain and/or projection domain. The method has been numerically modeled and can incorporate prior information about the imaging system into construction of calibration phantom. Via numerical experiments, the proposed method has been shown to maximize the accuracy of calibration process and mitigate the contribution of measurement errors in indicating the ball centers in both phantom domain and projection domain. It can also be used to improve popular phantom designs further. To simplify the analysis, in the experiment we tried with a simple one projection view configuration.

However, our method can be applied to general cases such as circular trajectory or other trajectories since the formulation in Eq. (12) can be generalized to such cases. Since our method introduces a hyperparameterγ, which weights the contribution from spatial randomness in the projection domain, fine-tuning this parameter needs to be considered in the future. Here the proposed method helps to improve to accuracy of calibration process and assist manufacturer to calibrate the imaging system easier.

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Van-Giang Nguyen received his BS degree in informatics from Le Quy Don Technical University, Hanoi, Vietnam, in 2005 and his MS and PhD degrees in 2009 and 2012, respectively, in electronic engineering from Paichai University, Daejeon, Republic of Korea. He is a lecturer at Le Quy Don Technical University. His current research interests include image processing, computer vision, and their applications to medical imaging and medical physics.