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Dieser Artikel geht auf die Herausforderungen der geometrischen Kalibrierung in Twin-Roboter-CT-Systemen ein, die zunehmend aufgrund ihrer Flexibilität und Fähigkeit, große Objekte zu handhaben, eingesetzt werden. Herkömmliche Offline-Kalibrierungsmethoden erfordern zusätzliche Scans und manuelle Eingriffe, was sie für bestimmte Anwendungen zeitaufwändig und unpraktisch macht. Der vorgeschlagene Online-Kalibrierungsalgorithmus adressiert diese Beschränkungen, indem er Kugeln verwendet, die an der Probe befestigt sind, und einen iterativen Optimierungsprozess anwendet, um geometrische Informationen zu verfeinern. Die Methode wird durch Simulationen und Experimente in der realen Welt ausgewertet und zeigt signifikante Verbesserungen bei der Qualität der CT-Rekonstruktion und der geometrischen Genauigkeit. Zu den wichtigsten Erkenntnissen zählen die Robustheit des Algorithmus über verschiedene CT-Verläufe hinweg, seine Fähigkeit, erhebliche geometrische Abweichungen auszugleichen, und sein Potenzial, bei Rekonstruktionen Metallartefakte zu reduzieren. Der Artikel diskutiert auch die Vorteile der Methode gegenüber bestehenden Kalibriertechniken, ihre Eignung für kostengünstige Systeme und zukünftige Forschungsrichtungen, um ihre Anwendbarkeit und Leistung zu verbessern.
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Abstract
Robotic CT systems offer several advantages over conventional systems due to their high flexibility. They can perform almost any CT trajectory and are particularly well suited for region-of-interest (ROI) scans of objects that exceed the size limitations of conventional CT systems. However, robot-based manipulators have a significantly lower absolute positioning accuracy compared to conventional manipulators, necessitating additional calibration methods to refine the geometric information about the spatial position and orientation of the X-ray source and detector for each projection for higher resolution reconstructions. We propose a geometric calibration method for CT systems with twelve degrees of freedom that does not require additional calibration scans. The method is easy to use, computationally efficient, and supports continuous CT trajectories. It utilises spheres with unknown positions that are attached to the specimen. The calibration process is divided into two stages. First, the spatial positions of the spheres are estimated using the initial geometric information and the acquired projections. Second, these estimates serve as input to an iterative optimisation that calibrates each projection individually. The applicability of the proposed method is demonstrated through simulations and real-world scans using a twin robotic CT system. Both quantitative and qualitative evaluations show a significant improvement in scan quality, comparable to results obtained via offline calibration. Moreover, evaluations on simulated data confirm the method’s robustness even for systems with positioning errors in the millimetre range. This novel online calibration technique is computationally efficient, compatible with highly flexible CT systems, and holds promise for enabling future mobile CT applications.
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1 Introduction
Twin robotic CT systems, where industrial robots control both the X-ray source and detector, offer several advantages over conventional CT systems. They can follow nearly arbitrary CT trajectories, perform region of interest (ROI) scans on objects too big for conventional CT systems and are suitable for mobile applications on non-mobile objects. One of the reasons for the low prevalence of these systems is attributed to geometric calibration problems. To enable accurate CT reconstruction, the geometric information, i.e. the pose (position and orientation) of the X-ray source and the detector with regard to a coordinate system in which the measurement object has a static pose, must be known for each X-ray projection. Industrial robots have a low absolute positional accuracy compared to conventional manipulator systems, which, when using uncorrected geometric information in reconstruction, can lead to artefacts such as double contours and blurring in the reconstructed volume. In the following, the geometric information output by the CT system is referred to as the inaccurate geometric information.
Offline calibration methods, as described in [1‐5], leverage the high repeatability of industrial robots. The CT trajectory of the CT scan is repeated once more, measuring a calibration body composed of easily penetrable material and several spheres. Spheres are used as markers, because they are rotation-invariant and easy to detect on projections. Geometric information of the positions of the spheres is either determined in advance, e.g. by tactile measurement or another metrological CT scan, or measured during the calibration scan [6, 7]. Optimisation methods, e.g. in [8] or methods from computer vision, e.g. the direct linear transformation (DLT) [9] then compute the geometric information of each individual projection relative to the calibration body’s coordinate system. These methods require a second scan, and the calibration body must be positioned exactly at the same location as the ROI, which makes it unusable for non-mobile objects and in situ measurements.
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Online calibration methods solely rely on the data from the scan of the relevant region of interest of the specimen. The use of a calibration body around a measurement object, e.g. [10, 11], would eliminate the need for an additional calibration scan. For objects that fit inside standard calibration bodies, this approach is indeed practical. However, these objects can usually also be measured using conventional CT systems. For larger objects this requires the custom manufacturing of such a calibration body specific to the specimen.
3D-2D registration methods use forward projections of CT reconstructions previously acquired on a conventional CT system or CAD-models to compare them with the acquired projections [12‐16]. Their algorithms minimise an objective function that compares the forward projections and actual projections while taking all geometric parameters into account. These methods are computationally expensive and again need a prior scan or CAD model of the specimen.
Table 1
Overview of CT calibration methods with their basic idea, advantages, disadvantages, and references
Spheres with unknown position are attached; geometry is optimized based on their projections.
Low computational effort; no second scan; suitable for arbitrary trajectories.
Requires sphere visibility in projections; stability depends on object properties and marker placement.
Other marker-free online calibration methods that do not rely on prior information, as described in [17‐20], are based on data consistency conditions. This requires the absence of significant artefacts and that the entire specimen is fully captured within the field of view of all projections. These constraints limit the applicability of these methods in general robotic CT calibration scenarios, where scans of large objects, heterogeneous materials and partial object scans are often necessary. In recent years, [21, 22] the limitation of full object coverage could be dropped, but with the disadvantage of high computational costs.
Methods using artificial intelligence have shown promising results in reconstructing images directly from uncalibrated CT scans, potentially reducing the need for explicit calibration procedures [23, 24]. However, these approaches still come with high computational costs and can benefit from calibrated data, which may improve reconstruction accuracy and robustness, especially in challenging imaging scenarios.
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Marker-based online calibration methods apply markers with unknown positions to the specimen. These methods begin with educated guesses of the sphere 3D positions, followed by an optimisation of the geometric information using this information. This approach only uses the inaccurate geometric information and the centre points of the projected spheres, hereafter referred to as blobs, which results in low computational effort. Bossema et al. [25] developed such a calibration method for CT systems with stationary X-ray focal spot- and detector position and a rotary table in order to enable 3D CT scanning with in-house 2D X-ray equipment from museums. Ma et al. [26] proposed a fully automatic online geometric calibration for non-circular CT trajectories on six degrees of freedom C-arm CT systems. For improved clarity, the preceding content has been concisely summarized in Table 1.
In this contribution, the method proposed by Ma et al. [26] is adapted and simplified to enable its use in a twin robotic CT system featuring twelve degrees of freedom. The algorithm is designed to refine the pose of the detector and the position of the X-ray source, resulting in nine optimisation parameters. It requires no additional calibration scans or human intervention. Its implementation is computationally efficient, requires only a short runtime and can also be used for the calibration of conventional CT systems. Importantly, such calibration is only necessary when the absolute positioning accuracy of the robotic system is insufficient to ensure reconstruction quality. The accuracy and robustness of the approach are evaluated using both simulated and experimental data.
In Section 2, the methodology of the calibration algorithm is described. Experiments and the used twin robotic CT system are presented in Section 3. Results are shown in Section 4 and discussed in Section 5. In Section 6, a conclusion is presented.
2 Methods
The general concept of the proposed method is as follows: First, spheres with unknown position are applied to the specimen. The spheres should be evenly distributed all over the ROI and experiments have shown that their distance between each other should not fall below the bound of ten times the diameter of the spheres. Experimental results on the twin robotic CT system indicate that the algorithm yields stable and robust calibration results when at least six spheres are reliably detected in every projection. Consequently, the number of spheres should be chosen based on the attenuation characteristics and geometry of the specimen, in order to maintain sufficient visibility of the spheres across all projections. Second, the blobs on the projections are detected and their centre points are determined. Next, each blob is assigned to its corresponding sphere by clustering them to, in the following called, traces. The inaccurate geometric information and the blob centre positions are then combined for a precise estimate of each sphere 3D position. The estimated sphere positions, the blob positions and the inaccurate geometric information then serve as input for an iterative optimisation minimizing the reprojection error with the Levenberg-Marquardt algorithm. The software was developed in Python and uses the implementation of the optimiser from the jaxopt package [27]. The individual steps are described in more detail in the following sections.
2.1 Initial Sphere Localisation
The blobs on the projections normalised to 8-bit unsigned integers are detected with the SimpleBlobDetector class from OpenCV [28]. Blobs that are closer than 1.5 times their diameter to each other are disregarded to later assure a correct assignment to their corresponding sphere. Once all projections have been loaded and the blobs on the projections were detected, the traces are formed. A trace is defined as a collection of blobs, each of them on a different projection, but assigned to the same sphere. The general concept on how to form the traces was first described in [26]. A short explanation on how this works is given in the following, referring to the original paper for further information.
The process begins with the first projection, identifying the blob centres within it. Next, for each blob, the area surrounding its position is searched in the subsequent projection for the nearest blob centre. If the distance between them falls below a defined threshold, the blobs are linked to form a trace. If the distance is above the threshold, the process stops for this trace. Blobs already assigned to a trace are deleted from the list of blobs to avoid blobs appearing several times in different traces and to reduce computing time. This is iterated over all projections. The threshold depends on the amount of projections acquired and needs to be experimentally determined. It is important to note that this method is applicable exclusively to continuous acquisition trajectories and yields more reliable results when the angular increments between successive projections are small. However, this condition is usually fulfilled by the scan trajectories of robotic CT systems. For all performed experiments in this paper, 30 pixels were chosen as the threshold.
This process can result in more traces than spheres applied to the specimen. This can result from different circumstances. Blobs can vanish behind denser materials or can come too close to other blobs for example. Then, a trace ends and a new trace starts as soon as the blob reappears on a different projection.
The next step of the method merges the traces that were interrupted, but are belonging to the same sphere. This starts by estimating the 3D positions of the corresponding spheres of all traces, using the DLT reconstruction. For each trace, one blob is visible on a number of projections. For simplification the method is explained for one sphere that can be seen on two projections.
Let \(X\in \mathbb {R}^3\times \lbrace 1\rbrace \) be the centre point of the sphere that is to be estimated in projective coordinates, \(P\in \mathbb {R}^{3\times 4}\) the projection matrix of a scene and \((u, v)\in \mathbb {R}^2\) the blob centre point on the image. A projection matrix in this case describes the projection of 3D points in the world coordinate system to 2D points on the detector. For a detailed explanation see e.g. [9].
$$\begin{aligned} \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} \times PX&= \begin{bmatrix} v\cdot p_3X - p_2X \\ p_1X - u\cdot p_3X \\ u\cdot p_2X - p_1X\cdot v \end{bmatrix} = 0. \end{aligned}$$
(1)
Now, let \(P'\in \mathbb {R}^{3\times 4}\) be the projection matrix of the second projection, \(X\in \mathbb {R}^3\times \lbrace 1\rbrace \) again the centre point of the sphere that is about to be estimated and \((u', v')\in \mathbb {R}^2\) the blob centre point on the second image. Combining the first two equations of Eq. 1 for both results in the following equation
The equation \(A\cdot X = 0\) can be solved using singular value decomposition and yields the estimated 3D position of the sphere forming this trace. This can simply be extended to any number of projections by adding more rows to the matrix A.
Once an estimated 3D position for each trace is determined, two traces are merged, if their estimated sphere positions are closer to each other than a defined threshold. This threshold depends on the distances between the applied spheres. To achieve reliable results, the spheres should be placed with a reasonable distance, depending on the magnification and diameter of the spheres. From experience, a reasonable threshold is one third of the smallest distance between the spheres. For the experiments at the twin robotic CT system, spheres were applied with a distance of at least 3 cm onto the specimen and a threshold of 1 cm was used.
After merging traces, the position of the assigned sphere of the new trace is again estimated and this process is repeated until the amount of traces equals the amount of spheres applied to the specimen.
This process ends with an estimate for the 3D positions of all spheres and an assignment of most of the blobs to their corresponding sphere.
2.2 Iterative Geometric Refinement using the Levenberg-Marquardt Optimiser
The information obtained by the method from Section 2.1 is used to refine the geometric information of every projection utilizing the Levenberg-Marquardt optimiser.
Let \(P(s, d_c, d_q)\in \mathbb {R}^{3\times 4}\) be the projection matrix of the geometry described by the X-ray source focal spot position, \(s\in \mathbb {R}^3\), the detector centre position, \(d_c\in \mathbb {R}^3\), and the detector orientation quaternion, \(d_q\in \mathbb {R}^4\).
Let \(X_1,\ldots , X_n\in \mathbb {R}^4\) be the sphere 3D positions associated to the measured blobs \(b_1, \ldots , b_n\in \mathbb {R}^3\) on the projection and \(\tilde{b}_1, \ldots , \tilde{b}_n\) the sphere positions projected onto the detector by \(P(s + \bigtriangleup s, d_c + \bigtriangleup d_c, d_q + \bigtriangleup d_q)\). The geometric information of a projection is refined by minimising the following residual function
where \(c>0\) is a spatial constraint, \(\alpha >0\) an angle constraint, \(k>0\) a punishment factor and \(\left\Vert {\cdot } \right\Vert _e\) defines the norm of the euler angles of a quaternion. These constants should be chosen in the order of magnitude of the expected errors in position, respectively rotation of the underlying CT system. In case of the twin robotic CT system at the Deggendorf Institute of Technology, the constants were set to \(c=2\,{\text {m}\text {m}}\) and \(\alpha =2^\circ \). The factor \(k>0\) should be a huge positive number, here it was set to 1000. After refining the geometric information, the 3D positions of the sphere for all traces are re-estimated with the new geometric information and the entire process is repeated until each 3D position changes less than a threshold value, which in this implementation was set to \(50\,{\upmu \text {m}}\).
For this article, the Levenberg Marquardt implementation in jaxopt [27] with Madsen-Nielsen stop criterion, conjugate gradient solver and 100 iterations maximum were used. The optimisation was not parallelised, but vectorised with the vmap functionality from jax [29]. GPU support was not available as the software was running on a Windows operating system.
Fig. 1
Experimental setup at the (a) twin robotic CT system at the Deggendorf Institute of Technology measuring (b) the calibration body used for offline calibration
Experiments were done both on simulated and real CT data. The method is evaluated on two different CT trajectories, a circular CT trajectory and a saw tooth CT trajectory. A visualisation of the latter can be seen in Fig. 2(b). The simulations were carried out using the software aRTist [30] and the real experiments were performed on the twin robotic CT system at Deggendorf Institute of Technology. It consists of two robot arms (Kuka KR120 R2900) mounted on linear axes to move the X-ray source and the detector and an additional rotary table for the measurement object (Fig. 1). The calibration algorithm was run on a laptop with an Intel Core i7-13700H CPU and one NVIDIA RTX 2000 Ada Generation Laptop GPU. The reconstruction of volumetric data was performed using the in-house developed software thyGraphics, which implements the SART [31] and FDK [32] algorithms. No pre-processing or post-processing was applied. All reconstructions were conducted using textbook-standard implementations. For FDK, the standard ramp filter was used.
3.1 Simulations
A simulation of a CT scan of a smurf as specimen consisting of high-density polyethylen, surrounded by a spiral of 15 iron spheres with a radius of \(1.5\,{\text {m}\text {m}}\), was performed. A picture of the scan scene is shown in Fig. 2(a). A circular CT trajectory and a saw tooth CT trajectory, displayed in Fig. 2(b), were applied.
The results from the simulations were used to evaluate the quality of the first estimate of the sphere positions under the influence of erroneous data of varying magnitude in order to estimate the maximum deviations of the geometric information of the manipulators. The quality of the CT reconstructions before and after geometric calibration is analysed using the mean squared error (MSE) in comparison to the ground truth data and by a qualitative inspection.
Table 2
Used scan parameters for the simulations and the twin robotic CT system
Smurf
Smurf
Bicycle helmet
Calibration body
Calibration body
Scan trajectory
circular
saw tooth
saw tooth
circular
saw tooth
FOD / mm
1000
1000
1100
1000
1000
FDD / mm
1500
1500
1600
1500
1500
Amplitude / mm
-
150
150
-
150
Projections
1000
1000
2400
3000
3000
X-ray tube
X-ray tube voltage / kV
150
150
225
220
220
X-ray tube current / µA
1000
1000
2400
2000
2000
Detector
Size / Pixels
\(3072\times 3072\)
\(3072\times 3072\)
\(3072\times 3072\)
\(3072\times 3072\)
\(3072\times 3072\)
Pixel pitch / mm
0.139
0.139
0.139
0.139
0.139
Integration time / s
-
-
0.3
0.25
0.25
Reconstruction
Algorithm
FBP
SART
SART
FBP
SART
Number of Iterations
-
3
5
-
10
Relaxation Factor
-
0.5
0.5
-
0.5
Voxel side length / \(\upmu \)m
370
370
400
137
184
Fig. 3
Experimental setup at the twin robotic CT at the Deggendorf Institute of Technology. (a) and (b): bicycle helmet applied with a total of 15 spheres. (c) X-ray projection of the bicycle helmet with detected blob centre points
The projections of the CT scan were taken once and the geo-metric information were subsequently adjusted to simulate the influence of different error magnitudes. Therefore, a normally distributed spatial error with a mean magnitude of \(0.25\,{\text {m}\text {m}},\)\(\,0.5\,{\text {m}\text {m}},\,1.0\,{\text {m}\text {m}},\,2.0\,{\text {m}\text {m}},\,3.0\,{\text {m}\text {m}},\,4.0\,{\text {m}\text {m}}\) and \(5.0\,{\text {m}\text {m}}\) and a standard deviation of \(0.125\,{\text {m}\text {m}},\,0.25\,{\text {m}\text {m}},\)\(\,0.5\,{\text {m}\text {m}},\,1.0\,{\text {m}\text {m}},\,1.5\,{\text {m}\text {m}},\,2.0\,{\text {m}\text {m}}\) and \(2.5\,{\text {m}\text {m}}\) was assigned to the 3D positions of the focal spot and the detector centre.
All relevant parameters of the simulated scans are listed in Table 2.
3.2 Experiments with the Twin Robotic CT System
Two different experiments were performed at the twin robotic CT system, shown in Fig. 1(a), to evaluate the presented calibration method. A scan with a bicycle helmet, shown in Fig. 3, using the saw tooth CT trajectory was performed. A total of 15 iron spheres with a radius of \(1.5\,{\text {m}\text {m}}\) were attached all over the helmet. The results will be evaluated qualitatively.
Fig. 4
Box plots of the errors of the first estimate of the sphere centres for different positional errors added to (a) a circular CT trajectory and (b) the saw tooth CT trajectory shown in Fig. 2(b)
In addition, the calibration body shown in Fig. 1(b), which is used in RoboCT for offline calibration, was measured with a circular- and a saw tooth CT trajectory. This specimen has the advantage that the measurement results can be quantitatively evaluated against the 3D coordinates of the sphere centre points, obtained with a metrological CT system with higher accuracy (Werth Tomoscope HV 500 with maximum permissible error of \(\text {EMPE} =\pm (4.5+L/75)\,{\upmu \text {m}}\), where L is the measured length in millimeters) in advance. All sphere centre points were determined in the reconstructed volumes with the software VGStudio Max 3.4. All possible distances that can be built between pairs of sphere centres were calculated. This results in 861 distances, as the calibration body is equipped with 42 iron spheres with a radius of \(1.5\,{\text {m}\text {m}}\). A more detailed description of this analysis can be found in [33]. All relevant parameters of the performed scans are listed in Table 2.
Fig. 5
Mean squared error (MSE) of CT reconstructions of a smurf phantom compared to the reconstruction with the ground truth geometric information of the simulation under the different mean positional errors before and after geometric calibration
The results from the simulations were used to evaluate the quality of the first estimate of the sphere positions under the influence of erroneous data and to compare the quality of the CT reconstruction using the MSE before and after geometric calibration with the ground truth. Visual results are also presented.
For both CT trajectories and all error magnitudes, the algorithm reduced the amount of traces to the amount of spheres, 15, positioned around the smurf automatically. Figure 4 shows that the error of estimated sphere positions increases with the magnitude of the added positional error. The experiments show similar results for the circular- and the saw tooth CT trajectory.
Figure 5 shows the MSE of the CT reconstructions of the smurf compared to the reconstruction with the ground truth geometric information of the simulation under the different mean positional errors before and after geometric calibration. The x-axis represents the mean positional error (ranging from \(0.25\,{\text {m}\text {m}}\) to \(5.0\,{\text {m}\text {m}}\)), while the y-axis shows the MSE scaled by \(10^{-4}\).
The data shows a clear trend. As positional error increases, the MSE increases in all cases, which is expected due to accumulating geometric inconsistencies.
Importantly, geometric calibration consistently reduces the reconstruction error for both scan types. For the saw tooth trajectory, calibration yields a significantly lower MSE at all error levels, particularly for larger displacements (e.g., halving the error at \(2.0-5.0\,{\text {m}\text {m}}\)). The circular trajectory shows a similar but slightly less pronounced improvement after calibration, though still substantial. The results for the circular scan with \(3\,{\text {m}\text {m}}\) error are an exception. They show a barely recognisable reduction in the MSE.
Figure 6 shows slices of the reconstructed volumes of the smurf scanned with a saw tooth CT trajectory before and after geometric calibration. All slices of the calibrated scans show clear improvements to the uncalibrated ones. The quality of the results decreases with the magnitude of the error, but still delivers significant improvements even for the highest errors. The amount of optimisation iterations increased from two iterations for the errors of magnitude \(0.25\,{\text {m}\text {m}}\) and \(0.5\,{\text {m}\text {m}}\) to four iterations for errors of magnitude \(1.0\,{\text {m}\text {m}}\), nine iterations for errors of magnitude \(2.0\,{\text {m}\text {m}}\), 12 iterations for errors of magnitude \(3.0\,{\text {m}\text {m}}\), 14 iterations for errors of magnitude \(4.0\,{\text {m}\text {m}}\) and 17 iterations for errors of magnitude \(5.0\,{\text {m}\text {m}}\).
Fig. 6
Slices of the reconstruction of the smurf simulation. (a) - (d) Reconstructions of the uncalibrated CT scan with different mean positioning errors. (e) - (h) Reconstructions of the above CT scans after geometric calibration and the required amount of optimisation iterations
Slices of the reconstruction of the scan of a bicycle helmet with a saw tooth CT trajectory. (a)-(c) show the results of the uncalibrated scans. (d)-(f) show the result of the calibrated scan after three iterations
A visual examination of different slices and scans is performed in Section 4.2.1. The results of the measured sphere distance errors for the spheres in the calibration body are presented in Section 4.2.2.
Fig. 8
Slices of the reconstruction of a scan of the calibration body, shown in Fig. 1(b), measured with a saw tooth CT trajectory. (a) Uncalibrated scan. (b) Calibrated with the proposed method. (c) DLT calibrated
4.2.1 Visual Results from the Twin Robotic CT System
Figure 7 shows slices of the reconstructed volumes of the bicycle helmet measured with a saw tooth CT trajectory before and after calibration. Blurring has been significantly reduced and the overall image quality has improved considerably.
Figure 8 shows slices of the reconstructed volumes of the calibration body with uncalibrated-, calibrated with the proposed method- and DLT calibrated geometric information. The uncalibrated scan shows strong double contours that vanish completely after geometric calibration. Results calibrated by the presented method and by the DLT calibration hardly differ.
Fig. 9
Sphere distance errors \(\bigtriangleup L\) and the line through the origin fitted with the least squares method through all data points for (a) circular CT trajectory and (b) saw tooth CT trajectory
The measured sphere distance errors \(\bigtriangleup L\) of all sphere combinations and both types of CT trajectory are shown in Fig. 9. The diagrams show the result of the measurement of the twin robotic CT system minus the measurement of the metrological CT system. The diagrams show a linear trend, which indicates a scaling error. The line through the origin fitted with the least squares method through all data points has a slope of \(0.14\,{\upmu \text {m}}\,{\text {m}\text {m}}^{-1}\) for the circular CT trajectory and \(0.3\,{\upmu \text {m}}\,{\text {m}\text {m}}^{-1}\) for the saw tooth CT trajectory.
5 Discussion
The presented algorithm represents a well-functioning alternative to offline calibration for flexible CT systems. The algorithm only uses spheres made of highly attenuating material attached to the specimen and does not require a second calibration scan to correct the geometric information. Compared to offline calibration methods this halves the scanning time.
The overall runtime is comparable to the benchmark DLT offline calibration method [1]. The most time-consuming part of the algorithm is the detection of the blobs and their centers. This step is not yet parallelised and can therefore only process six images per second on the computer used. Creating the traces and merging them until the amount of traces equals the amount of applied spheres took between 20 to 30 s in all described experiments. The subsequent optimisation of the geometric information required roughly 10 s per iteration in all presented experiments.
The algorithm is fully automated and does not require human-assisted assignment of the blobs to the spheres, as is required by most offline calibration methods [1, 2, 8].
It can be used for all continuous CT trajectories and yields comparable results to the benchmark algorithm in terms of length measurement deviations. The line through the origin fitted with the least squares method through all length measurement deviations from the saw tooth measurement shown in Fig. 9 has a slope of \(0.3\,{\upmu \text {m}}\,{\text {m}\text {m}}^{-1}\), which was also the result of the same evaluation performed in [33] for the benchmark calibration.
The results in Fig. 4 demonstrate the effectiveness of the proposed geometric calibration method. Across almost all perturbation magnitudes, a clear reduction in reconstruction errors is observed after geometric calibration, with particularly strong improvements for the saw tooth CT trajectory. In some cases, the MSE is reduced by more than 50%, highlighting the method’s ability to compensate for substantial geometric deviations. An exception occurs for the circular scan with a \(3\,{\text {m}\text {m}}\) perturbation. While the calibration results in only a marginal improvement in the MSE for this case, qualitative inspection of the reconstructed slices in Fig. 6 reveals a noticeable enhancement in structural consistency and image quality. This suggests that the limited improvement in the MSE may be due to a slight shift in the reconstructed volume position rather than a failure of the calibration itself. These findings indicate that the proposed method is well suited for flexible CT systems where high reconstruction accuracy must be maintained despite mechanical inaccuracies.
The effectiveness of the proposed method is closely linked to the geometric conditions of the acquisition setup. As the desired spatial resolution increases, the system becomes more sensitive to deviations in the positioning of the X-ray source, -detector, and object. In such cases, even small inaccuracies can lead to measurable reconstruction errors. The proposed online calibration algorithm compensates for these deviations, which significantly reduces the dependency on mechanically precise motion systems. This is particularly advantageous in flexible or robotic CT systems, where absolute positioning accuracy is inherently limited. Nevertheless, for applications requiring extremely high spatial resolution, the limits of geometric correction must be considered, and the method may need to be complemented by improved hardware accuracy.
CT scans of larger objects that require an extension of the measuring field can also be calibrated by the proposed method, which would require the production of a larger calibration body with the benchmark method.
The results of the evaluation of the quality of the first estimate of the sphere positions under the influence of erroneous data showed error magnitudes under \(50\,{\upmu \text {m}}\) for positioning errors up to \(1.5\,{\text {m}\text {m}}\). The choice of CT trajectory hardly influenced the results. The magnitude of the errors allows the conclusion that the method can also be used on low-cost systems with low absolute accuracy. The corresponding experiments are being conducted within the framework of the BrückenCT project (FE 88.0207/2024/AV10), funded by Bundesanstalt für Straßen- und Verkehrswesen (BASt, Federal Highway Research Institute).
A detailed investigation into the influence of the number and spatial distribution of spheres-particularly under challenging imaging conditions such as high noise levels or partial occlusion - has not yet been conducted and is planned for future work. In this context, we will also investigate how blob detection on the projections can be optimised for strongly attenuating specimens. Reliable detection of blobs and their centres is fundamental to the proposed calibration approach and currently limits its applicability primarily to objects with low attenuation.
The attached spheres also generate metal artefacts that influence the quality of the reconstruction. As the positions of the spheres, their size and the projection geometries are determined by the method, the attenuations of the spheres could be subtracted from the projections, which should lead to a significant reduction in metal artefacts. Studies on this are also planned for the future.
In addition, improvements are planned regarding the assignment of blobs to their corresponding spheres for non-continuous CT trajectories and the optimisation of the implementation to reduce the runtime.
6 Conclusion
The proposed online calibration algorithm offers a fully automated and efficient alternative to traditional offline calibration methods for flexible CT systems. By eliminating the need for a second calibration scan and manual assignment of the spheres, the method significantly reduces scanning time and user interaction while maintaining a comparable level of geometric accuracy.
The algorithm proved to be robust across various continuous CT trajectories and tolerant to considerable perturbations in the inaccurate geometric information of the CT system. This suggests its suitability for a broad range of applications, including low-cost systems with limited absolute positioning accuracy as well as conventional CT systems. The method is particularly effective for trajectories with small angular increments between projections, as these conditions ensure a reliable assignment between the detected blobs and the corresponding spheres.
Overall, the presented approach marks a substantial step towards practical CT calibration and will enable mobile CT applications in the future.
Declarations
Competing Interests
Yiqun Ma is an inventor on a pending patent application related to the method described in this work.
Ethics and Consent to Participate
Not applicable.
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