A synergistic framework for geometric calibration and reconstruction in dual‐arm robotic cone‐beam computed tomography

Abstract

Background

Dual‐arm robotic cone‐beam computed tomography (CBCT) systems are susceptible to geometric instability due to their decoupled kinematics, which can limit their practical application.

Purpose

This study aims to develop and evaluate a synergistic framework for improving geometric calibration and reconstruction consistency in flexible dual‐arm robotic CBCT platforms.

Methods

We introduce the Synergistic Calibration and Reconstruction (SyCaR) framework. It uses a two‐stage geometric calibration strategy that combines real‐time external metrology for projection‐wise pose estimation with an optional reference‐free projection‐consistency refinement for residual geometric correction. For reconstruction, we developed a Pose‐Driven Feldkamp‐Davis‐Kress (PD‐FDK) algorithm that operates directly on per‐projection pose data to account for detector misalignments and non‐ideal source‐detector trajectory deviations.

Results

Physical phantom experiments showed that encoder geometry produced severe streaking artifacts and structural inconsistency, whereas motion‐capture geometry restored clearer phantom structures. In the numerical study, PD‐FDK achieved the best PSNR and RMSE among the evaluated analytical methods, improving from 28.99 dB and 0.036 with standard FDK to 32.72 dB and 0.023. Across numerical and physical experiments, PD‐FDK reduced geometry‐related artifacts and showed consistent performance among FDK type methods under the tested conditions.

Conclusions

The proposed SyCaR framework provides a feasible computational approach for mitigating geometric instability in dual‐arm robotic CBCT. The combination of geometric calibration and PD‐FDK reconstruction improved reconstruction consistency and reduced geometry‐related artifacts under the evaluated numerical and physical settings, while retaining the computational efficiency of analytical reconstruction. This work supports the further development of flexible dual‐arm robotic CBCT for biomedical imaging applications, although broader validation with more diverse trajectories, objects, and acquisition conditions remains necessary.

1. INTRODUCTION

Cone‐beam computed tomography (CBCT) is widely used for noninvasive visualization in scientific, industrial, and clinical applications. In industry, conventional turntable‐based CT is a mature approach for nondestructive testing and dimensional inspection in fields such as aerospace, precision manufacturing, and microelectronics. ¹ In medicine, mobile C‐arm systems provide flexible intraoperative 3D imaging for image‐guided procedures, including orthopedics and interventional radiology. ² , ³ However, both architectures share a fundamental limitation rooted in their kinematically coupled designs, where the X‐ray source and detector are linked by a rigid structure. This mechanical coupling, while ensuring stability, severely constrains the geometric freedom of the imaging chain, ² restricting the orientation of the scan plane and preventing access to spatially confined environments. Robotic CT (RoboCT) systems have therefore been investigated as a flexible alternative that uses robotic manipulators to expand trajectory design freedom and improve access to regions that are difficult to image with monolithic source‐detector assemblies. ⁴ , ⁵ , ⁶ , ⁷

Among the various RoboCT configurations that embrace this kinematic decoupling, the dual‐arm architecture represents a particularly flexible configuration. However, the freedom gained by breaking the rigid mechanical link introduces a distinct geometric challenge: the relative source‐detector geometry is no longer enforced by a rigid gantry, but must be determined projection by projection from the coordinated poses of two independently actuated robotic arms. ⁷ Although industrial robots generally provide high repeatability, flexible robotic CT requires accurate absolute geometry at each projection; millimeter‐level absolute positioning errors can remain even when repeatability is much better. ⁸ , ⁹ , ¹⁰ In a dual‐arm CBCT system, these absolute pose errors, together with inter‐arm coordination errors, can propagate into projection‐wise geometric inconsistencies, especially when scan trajectories are modified on demand rather than repeatedly executed as pre‐calibrated paths. These inconsistencies lead to two interconnected challenges: (1) achieving precise geometric calibration to measure or estimate the true, time‐varying pose of the imaging components; and (2) performing robust 3D reconstruction from data acquired along geometrically deviated scan paths. Even when commanded to follow a standard circular trajectory, the executed path can exhibit non‐negligible projection‐dependent deviations from the nominal geometry, which may degrade conventional reconstruction algorithms. The central challenge is therefore to recover and incorporate the actual projection geometry during image formation, so that flexible trajectories can be used while maintaining reconstruction consistency.

While substantial research has addressed geometric inaccuracies, existing solutions from the distinct domains of medical imaging and robotics do not directly resolve the unique challenges of dual‐arm robotic CT. Within the imaging domain, conventional CT geometric calibration techniques provide an important foundation for correcting systematic geometric errors and have been extensively reviewed in the context of X‐ray CT metrology. ¹¹ Offline strategies using dedicated calibration phantoms are commonly used to correct static or repeatable biases in mechanically stable systems. However, their direct use in flexible RoboCT is limited by trajectory‐specific calibration requirements and by the assumption that the calibrated geometry remains transferable to later scans. Leveraging the robot's high repeatability to pre‐calibrate a fixed trajectory can be effective when the same trajectory is repeatedly executed, ¹² , ¹³ , ¹⁴ , ¹⁵ but it reduces the flexibility of modifying scan paths on demand because substantially different trajectories may require separate calibration or validation. Alternatively, per‐projection calibration can support nonstandard or trajectory‐varying acquisitions, but it often relies on calibration features or fiducials visible in individual projections. ¹⁶ , ¹⁷ , ¹⁸ This introduces practical constraints when the calibration object must be scanned together with the sample, as the phantom or fiducials may occupy the imaging volume, reduce the available field of view, occlude the object, or introduce additional artifacts.

Online or self‐calibration approaches provide another route for estimating residual geometric errors without relying solely on a pre‐calibrated trajectory or a simultaneously scanned calibration phantom. These methods infer geometric parameters using model‐based optimization. ¹⁹ , ²⁰ They generally fall into two broad categories: methods that optimize reconstructed image‐quality metrics, such as entropy or sharpness, ²¹ , ²² , ²³ , ²⁴ and methods that exploit projection‐domain information. The latter include 3D–2D registration methods that maximize the similarity between acquired projections and digitally reconstructed radiographs generated from a 3D reconstruction, ¹⁹ , ²⁵ , ²⁶ , ²⁷ as well as reference‐free robot CT geometry estimation methods based on projection features or consistency constraints. ²⁰ While valuable for compensating residual geometric drifts or pose inconsistencies, these methods can be limited as stand‐alone solutions for dual‐arm robotic CT. Their iterative optimization can be computationally intensive, ¹⁹ , ²⁷ and their capture range may be insufficient for large projection‐wise deviations without an accurate initial pose estimate. This motivates a coarse‐to‐fine strategy in which external metrology first provides projection‐wise pose initialization, followed by optional reference‐free projection‐consistency refinement for residual geometric correction.

External tracking systems provide a natural source of such projection‐wise initialization by delivering independent six degree‐of‐freedom (6‐DoF) pose measurements of tracked rigid bodies. In robotics, they are widely used to provide external pose measurements and mitigate the limited absolute accuracy of robotic manipulators in high‐precision tasks. ²⁸ , ²⁹ , ³⁰ , ³¹ , ³² , ³³ In medical imaging, they have also been used to track and compensate for involuntary patient motion. ³⁴ , ³⁵ , ³⁶ While these studies demonstrate the capability of external tracking to measure dynamic targets or end‐effector motion, its integration into RoboCT geometric calibration remains less explored. Here, external tracking is used not primarily to track the imaged object, but to estimate the projection‐wise poses of the imaging components themselves, thereby providing the geometry required for reconstruction and for optional residual refinement.

Regarding reconstruction, the choice of algorithm involves a practical trade‐off between robustness and computational efficiency. Iterative reconstruction methods, for instance, are inherently flexible. They typically formulate the reconstruction problem as solving a large system of linear equations, which allows them to accommodate arbitrary scan geometries. However, despite being theoretically well‐suited for deviated robotic paths, their practical application is hindered by substantial computational costs and the need to carefully tune regularization hyperparameters, ³⁷ , ³⁸ , ³⁹ , ⁴⁰ which can limit their suitability for rapid feedback or repeated geometric refinement.

Deep learning methods have emerged as an important approach in medical imaging, ⁴¹ , ⁴² , ⁴³ demonstrating strong performance in many reconstruction tasks. However, their application to this problem faces a practical challenge: a reliance on large, high‐quality training datasets. Such datasets are often unavailable for novel hardware configurations like the dual‐arm robotic systems central to this work, which may limit model generalizability across trajectories, system geometries, and calibration states.

Analytical methods, particularly the Feldkamp–Davis–Kress (FDK) algorithm, ⁴⁴ therefore remain attractive because of their computational efficiency and well‐understood implementation. However, standard FDK assumes an idealized circular geometry. While extensions have been developed to correct specific geometric deviations, ⁴⁵ , ⁴⁶ , ⁴⁷ , ⁴⁸ such as detector tilt or a displaced center‐of‐rotation, they were not primarily designed for projection‐wise, coupled source‐detector pose variations in dual‐arm robotic CT. Therefore, once projection‐wise geometry is estimated through external tracking or optional online refinement, the reconstruction algorithm must incorporate this nonideal geometry explicitly rather than relying on a fixed circular‐trajectory assumption. Together with the calibration challenges described above, this motivates a unified computational framework that can establish an accurate projection‐wise geometric model and perform computationally efficient, geometrically consistent reconstruction from nonideal robotic scan paths.

To address these challenges, this paper introduces the Synergistic Calibration and Reconstruction (SyCaR) framework, a systematic methodology designed to improve geometric and reconstruction consistency on dual‐arm robotic CBCT systems. The framework consists of three main components. First, we propose a geometric parameterization pipeline that translates raw robotic and motion capture poses into a projection‐wise geometric model required for CT reconstruction. This step links the kinematic representations used in robotics and motion capture with the geometric inputs required by CT reconstruction algorithms. Second, we formulate a two‐stage geometric calibration strategy that leverages this pipeline. It combines robotics‐domain external metrology as an initial projection‐wise geometric estimate with an optional reference‐free projection‐consistency refinement for residual geometric correction. Third, we implement a Pose‐Driven FDK (PD‐FDK) algorithm. By operating directly on the projection‐wise pose estimates from our calibration strategy, it accounts for detector pose misalignments and non‐ideal source‐detector trajectories within an analytical reconstruction workflow, enabling computationally efficient reconstruction under the evaluated robotic scan conditions.

The remainder of this paper is organized as follows. Section II describes the dual‐arm robotic CBCT system, the geometric calibration procedure, the PD‐FDK reconstruction algorithm, and the experimental protocols used for validation. Section III reports the reconstruction and calibration results. Section IV discusses the scope, limitations, and conclusions of the proposed framework.

2. MATERIALS AND METHODS

This section details the proposed SyCaR framework, a computational pipeline designed to improve geometric consistency and reconstruction robustness on a custom‐built dual‐arm robotic CBCT system, which is depicted in Figure 1. An overview of the complete imaging, calibration, and reconstruction workflow is provided in Figure 2. The system's kinematically decoupled design, while offering increased scanning flexibility, and introduces projection‐wise geometric deviations that are difficult to handle with conventional fixed‐geometry reconstruction algorithms. Our framework addresses this challenge through two core computational stages: (1) a real‐time geometry estimation and refinement process to establish an projection‐wise geometric model, and (2) our PD‐FDK algorithm for computationally efficient reconstruction using the estimated scan geometry. The validation protocols are described at the end of this section.

FIGURE 1.

The custom‐built dual‐arm robotic CBCT system. The setup features two industrial robotic arms manipulating an X‐ray source and a flat‐panel detector. An optical motion capture system provides real‐time, external pose measurements of the imaging components.

FIGURE 2.

Overview of the proposed imaging and reconstruction pipeline. The workflow consists of three stages: (a) Data acquisition, where two robotic arms maneuver an X‐ray source and detector while their real‐time poses are tracked. This stage provides the necessary inputs: projection images ($p_i$) and raw pose data ($T_{M{S}_i}^W,T_{M{D}_i}^W$) for the computational framework. (b) and (c) constitute the computational SyCaR framework, which processes the acquired data. This begins with (b) a two‐stage geometric calibration to estimate the projection‐wise imaging geometry. (c) Finally, the PD‐FDK algorithm incorporates the estimated geometry into analytical reconstruction to account for geometric deviations and reconstruct the final volume.

Because the proposed calibration procedure relies on projection‐wise pose measurements from the physical system, the main hardware components are summarized here. The dual‐arm platform consists of two UR20 collaborative robotic arms (Universal Robots), each with a manufacturer‐specified pose repeatability of $\pm 0.1$ mm. Since the absolute positioning accuracy of an industrial robot can be affected by kinematic calibration, payload, and workspace configuration, the robot encoder poses are not used as the final CT geometry in this work. Instead, an eight‐camera FZMotion Swift 30A optical motion capture system (LUSTER) provides external pose measurements of rigid marker bodies attached to the source and detector assemblies. The X‐ray imaging chain consists of an IMD MONOBLOCK E‐40R HF X22C X‐ray source and a Varex Imaging 4030DX flat‐panel detector.

2.1. Real‐time geometry estimation and refinement

The foundational step of our framework is to estimate the time‐varying imaging geometry of the dual‐arm system. To achieve this, we employ a multi‐camera optical motion capture system that tracks the real‐time 6‐DoF poses of rigid marker bodies attached to the X‐ray source and detector assemblies. This external pose measurement approach motivates our choice of a general, vector‐based geometric model, consistent with the ASTRA toolbox. ⁴⁹ For each projection $i$, this model explicitly defines the source position $s_i\in {\mathbb{R}}^3$, the detector center $d_i\in {\mathbb{R}}^3$, and two orthogonal unit vectors, defining the directions of the detector's rows ($u_i$) and columns ($v_i$), which together establish the orientation of the detector plane. This parameterization is particularly suitable for two reasons. Firstly, it natively aligns with the pose data supplied by the motion capture system, after calibrated transformations from the marker bodies to the corresponding imaging components. Secondly, it does not presume a fixed isocenter, making it suitable for representing the flexible trajectories of our dual‐arm robotic system.

To convert the tracked marker‐body poses into the vector‐based CT geometry, two fixed transformations between the marker‐body and the corresponding imaging component must first be determined. This mapping, illustrated in Figure 3, relates the source‐side marker body to the X‐ray focal spot and the detector‐side marker body to the detector center. The specific transformations are:

• ${\mathbf{T}}_S^{MS}$: the fixed transform from the X‐ray focal spot to the source‐side marker‐body.

• $T_D^{MD}$: the fixed transform from the detector center to the detector‐side marker‐body.

FIGURE 3.

Illustration of the fixed, pre‐calibrated transformation from a tracked marker rigid body to an imaging component. The inset details this concept for the X‐ray source, where the world pose of the focal spot ($T_{S_i}^W$), graphically represented by its position for clarity, is derived from the world pose of the tracked marker body ($T_{M{S}_i}^W$) via the pre‐calibrated, fixed transform $T_S^{MS}$. An analogous transformation, $T_D^{MD}$, is applied for the detector.

With these pre‐calibrated transforms established, we can determine the projection‐wise poses of the imaging components during a scan. For each projection $i$, the motion capture system provides the dynamic 6‐DoF poses of the marker bodies relative to the world coordinate system $W$, denoted as $T_{M{S}_i}^W$ and $T_{M{D}_i}^W$. By composing these dynamic poses with the corresponding static transforms, we obtain the poses of the X‐ray source and detector expressed in the world coordinate system as:

$$\begin{array}{cc}\hfill T_{S_i}^W& =T_{M{S}_i}^W·T_S^{MS},\hfill \end{array}$$ (1)

$$\begin{array}{cc}\hfill T_{D_i}^W& =T_{M{D}_i}^W·T_D^{MD}.\hfill \end{array}$$ (2)

We define a reconstruction coordinate system $\mathcal{R}$ to ensure all geometric data is represented in a consistent and stable coordinate system for reconstruction. To ensure the reconstruction is independent of the object's position, we define the origin of $\mathcal{R}$ in a data‐driven manner as the effective isocenter of the scan, computed directly from the motion capture data. While our dual‐arm robotic system mechanically allows flexible source‐detector motion, the present study employs a circular trajectory to evaluate the basic geometric calibration and reconstruction pipeline. For this circular scan configuration, the origin is specifically computed as the midpoint between the centers of the best‐fit circular paths of the source and detector trajectories.

The process begins by extracting the 3D position vectors of the source $s_i^W$ and detector $d_i^W$ from their corresponding world poses $T_{S_i}^W$, $T_{D_i}^W$ for each projection $i$. The position vector is the translational component of the poses:

$$\begin{array}{cc}\hfill s_i^W& =\text{transl}(T_{S_i}^W),\hfill \end{array}$$ (3)

$$\begin{array}{cc}\hfill d_i^W& =\text{transl}(T_{D_i}^W).\hfill \end{array}$$ (4)

To determine the best‐fit circular path for each trajectory, we first identify its principal plane using Principal Component Analysis (PCA) on the point clouds $\{s_i^W\}$ and $\{d_i^W\}$. For each point cloud, the normal vector of its plane ($n_s$, $n_d$) corresponds to the direction of minimum variance. This is found by identifying the eigenvector of the covariance matrix with the smallest eigenvalue, which is mathematically equivalent to solving the following minimization problem:

$$\mathrm{n}=arg\underset{\parallel \stackrel{\sim }{n}{\parallel }_2=1}{\min }\sum _{i=1}^N{\left({\stackrel{\sim }{n}}^{\top }(p_i-\overline{p})\right)}^2,$$ (5)

where, $\{p_i\}$ is the point cloud and $\overline{\mathbf{p}}$ is its centroid.

The 3D trajectory points are then orthogonally projected onto their respective best‐fit planes. A least‐squares circle fitting is subsequently performed on these projected points within each plane's local 2D coordinate system. This yields the 2D coordinates of the centers, which are then transformed back into the 3D world coordinate system to obtain the source trajectory's center $o_s^W$ and the detector's $o_d^W$. The origin of the reconstruction coordinate system $o_{\mathcal{R}}$ is then defined as the midpoint of these two 3D centers:

$$o_{\mathcal{R}}^W=\frac{1}{2}(o_s^W+o_d^W).$$ (6)

With the origin $o_{\mathcal{R}}^W$ determined, the transformation from the world coordinate system $W$ to the reconstruction coordinate system $\mathcal{R}$ is defined as a pure translation that centers the geometry at the effective isocenter, while provisionally keeping the axes aligned with the world system. Any remaining orientation difference between the principal plane of the scan trajectory and the canonical planes of the reconstruction coordinate system is handled in the subsequent PD‐FDK reconstruction procedure.

The transformation from world to reconstruction coordinates is therefore:

$${\mathbf{T}}_W^{\mathcal{R}}=\left[\begin{array}{cc}{\mathbf{I}}_3& -{\mathbf{o}}_{\mathcal{R}}^W\\ {\mathbf{0}}^{\top }& 1\end{array}\right],$$ (7)

Applying this transformation yields the tracking‐based pose of each imaging component in the reconstruction coordinate system, $T_{S_i}^{\mathcal{R}}=T_W^{\mathcal{R}}·T_{S_i}^W$ and $T_{D_i}^{\mathcal{R}}=T_W^{\mathcal{R}}·T_{D_i}^W$. From these pose matrices, the complete set of geometric parameters required for reconstruction is extracted. Specifically, the source position $s_i$ and detector center $d_i$ are obtained as the translational components of $T_{S_i}^{\mathcal{R}}$ and $T_{D_i}^{\mathcal{R}}$, respectively. The detector orientation vectors, namely the row direction $u_i$ and column direction $v_i$, are extracted from the rotation matrix $R_{D_i}^{\mathcal{R}}$, which is the upper‐left $3\times 3$ submatrix of $T_{D_i}^{\mathcal{R}}$. In our detector‐frame definition, the first and second local axes correspond to the detector row and column directions after applying the pre‐calibrated marker‐to‐detector transformation. Therefore, $u_i$ and $v_i$ are given by the first and second columns of $R_{D_i}^{\mathcal{R}}$. This process yields an initial tracking‐based geometric parameter set ${\mathit{\omega }}_i^{(0)}=\{s_i,d_i,u_i,v_i\}$ for each projection $i$.

Although the tracking‐based geometry provides a projection‐wise external estimate of the source and detector poses, small residual geometric errors may remain due to calibration uncertainty of the fixed marker‐to‐component transformations and residual pose‐measurement uncertainty during robotic motion. We therefore implemented a reference‐free projection‐consistency refinement adapted from Kang et al. ²⁰ as an optional diagnostic module. Initialized by the tracking‐based geometry ${\mathit{\omega }}_i^{(0)}$, this refinement uses the epipolar consistency condition (ECC) to search only small perturbations of the source position, detector center, and detector orientation within a bounded interval. Because the tracking‐based geometry provides the primary projection‐wise pose estimate and the ECC search is limited to local residual perturbations, this module is treated as an optional residual‐correction and diagnostic step rather than a replacement for external metrology.

2.2. PD‐FDK reconstruction algorithm

Given the precisely calibrated geometry from the previous stage, we develop our PD‐FDK algorithm to compensate for the system's geometric non‐idealities. The algorithm consists of two main stages: (1) correction for detector orientation misalignments (pitch and roll), and (2) generalized reconstruction for complex source‐detector trajectory deviations. Detector yaw is not corrected by a separate projection‐resampling step; instead, it is retained in the projection‐wise vector geometry and incorporated during the generalized reconstruction stage.

The first stage corrects detector pose misalignments by virtually resampling each projection onto a corrected plane, as illustrated in Figure 4(a) and (b). To achieve this within our native vector‐based geometry, the procedure begins by establishing a stable reference axis for the entire scan trajectory. We estimate this as the effective rotation axis by applying the same PCA‐based method as described by (5) to the source trajectory positions $\{s_i\}$. The resulting normal vector, $n_{\text{rot}}$, provides a stable reference for correcting pitch and roll misalignments.

FIGURE 4.

The two‐stage strategy for geometric deviations integrated into our PD‐FDK algorithm. Stage 1: Detector Pose Correction. We first virtually resample the projection data onto a corrected plane to mitigate the detector‐orientation misalignments shown in (a) and (b). This process aligns the virtual detector's column vectors parallel to the effective rotation axis $n_{\text{rot}}$. The remaining detector yaw, corresponding to horizontal detector tilt, is retained in the projection‐wise vector geometry for stage 2 reconstruction. Stage 2: Generalized Reconstruction. A generalized FDK reconstruction is then applied. (c) This 2D schematic illustrates the fan‐beam geometry within the reference mid‐plane ( ${\mathcal{M}}_i$ ), viewed along the rotation axis $n_{\text{rot}}$. This geometric construction is used to handle asymmetric trajectory deviations where the source‐to‐origin line is not orthogonal to the detector.

To correct pitch misalignment, we first find the component of $n_{\text{rot}}$ that is orthogonal to the detector's row vector $u_i$, and compute the signed angle ${\theta }_{\text{pitch}}^{(i)}$ between this vector and the detector's column vector $v_i$:

$$\begin{array}{cc}\hfill n_{\text{proj}}^{(i)}& =n_{\text{rot}}-(n_{\text{rot}}^{\top }{u}_i)·u_i,\hfill \end{array}$$ (8)

$$\begin{array}{cc}\hfill {\theta }_{\text{pitch}}^{(i)}& =atan2\left(u_i^{\top }\left(v_i\times n_{\text{proj}}^{(i)}\right),v_i^{\top }{n}_{\text{proj}}^{(i)}\right).\hfill \end{array}$$ (9)

We then correct this deviation by rotating the detector about its central row axis, which passes through the detector center $d_i$ and points along direction $u_i$:

$$T_{\text{pitch}}^{(i)}=T(d_i)R(u_i,{\theta }_{\text{pitch}}^{(i)})T(-d_i)$$ (10)

where, $T(·)$ denotes a homogeneous translation. The rotation matrix $R(u_i,\theta )$ is computed using Rodrigues' rotation formula:

$$R(u_i,\theta )=\mathbf{I}+\sin (\theta ){[u_i]}_{\times }+(1-\cos (\theta )){[u_i]}_{\times }^2$$ (11)

Here, $\mathbf{I}$ is the identity matrix and ${[u_i]}_{\times }$ is the skew‐symmetric matrix of $u_i$.

After applying the pitch correction, the detector's column vector is updated to ${\stackrel{\sim }{\mathbf{v}}}_i$. The axis for the roll correction is the vector normal to the updated detector plane, calculated as ${\stackrel{\sim }{n}}_{\text{det}}^{(i)}=u_i\times {\stackrel{\sim }{v}}_i$. The residual roll angle ${\theta }_{\text{roll}}^{(i)}$ is then computed as the signed angle:

$${\theta }_{\text{roll}}^{(i)}=atan2\left({\left({\stackrel{\sim }{n}}_{\text{det}}^{(i)}\right)}^{\top }\left({\stackrel{\sim }{\mathbf{v}}}_i\times n_{\text{rot}}\right),{\stackrel{\sim }{\mathbf{v}}}_i^{\top }{n}_{\text{rot}}\right).$$ (12)

This roll misalignment is corrected by a rotation around the detector's normal vector ${\stackrel{\sim }{\mathbf{n}}}_{\text{det}}^{(i)}$:

$$T_{\text{roll}}^{(i)}=T(d_i)R({\stackrel{\sim }{n}}_{\text{det}}^{(i)},{\theta }_{\text{roll}}^{(i)})T(-d_i).$$ (13)

The complete detector pose correction for the $i$‐th projection is the composition of the pitch and roll corrections:

$$T_{\text{det}}^{(i)}=T_{\text{roll}}^{(i)}·T_{\text{pitch}}^{(i)}.$$ (14)

With the full corrective transformation defined, a new corrected projection image $P_i^{\prime \prime }$ is generated by virtually resampling the original projection image $P_i$. For each pixel on the corrected detector plane, we trace the ray from the X‐ray source through its 3D position and intersect this ray with the original detector plane. The resulting point is then converted to the corresponding coordinate in the original projection image. This source‐induced mapping between the corrected and original detector planes defines a projective transformation, which is implemented by homography‐based warping with bilinear interpolation. This warping process effectively removes artifacts caused by detector pitch and roll.

After correcting for detector pose misalignments, the second stage addresses remaining trajectory deviations such as asymmetric projection geometries and nonuniform angular sampling, as illustrated in Figure 4(c). To compensate for these compounded issues, we adapt the generalized FDK framework of. ⁴⁷ As their framework is formulated for a parameterized geometry (e.g., SOD, SDD) incompatible with our system, we reformulate it to operate directly on the outputs of our preceding correction stage: the resampled projection data $P_i^{\prime \prime }$ and the corresponding corrected orientation vectors and , while retaining the original source position $s_i$ and detector center $d_i$.

To implement the core fan‐beam filtering step, we first define a unique reference mid‐plane, ${\mathcal{M}}_i$, for each projection $i$. This plane is mathematically defined as the plane that contains the source position $s_i$ and is orthogonal to the effective rotation axis $n_{\text{rot}}$. The filtering geometry is first constructed on this reference mid‐plane and then extended row‐wise to the full cone‐beam projection.

Our immediate goal is to identify the detector row that lies most closely within this now well‐defined reference mid‐plane ${\mathcal{M}}_i$. We begin by defining the 3D position of any point along the detector's vertical midline as:

(15)

where, $d_i$, $j_c$, ${\mathrm{\Delta }}_v$, and represent the detector center, central row index, vertical pixel spacing, and the corrected detector column direction, respectively.

The index $j_0$ of this reference row is then found by minimizing the orthogonal distance of the points $p_{i,j}$ to the plane ${\mathcal{M}}_i$. This minimization is computed using the plane's known properties:

$$j_0=arg\underset{j}{\min }\left|{(p_{i,j}-s_i)}^{\top }{n}_{\text{rot}}\right|.$$ (16)

This reference detector row, which we denote as the line $L_i$, provides the spatial domain for the 1D ramp filtering in the reference mid‐plane and is formally defined as:

(17)

Performing ramp filtering along this reference detector row $L_i$ requires defining a local coordinate origin. Following the geometry proposed by, ⁴⁷ this origin is chosen as the projection of the effective rotation axis onto $L_i$. Specifically, we denote by $o_i$ the intersection between the reference mid‐plane ${\mathcal{M}}_i$ and the effective rotation axis passing through the reconstruction origin along $n_{\text{rot}}$. The filtering origin ${\mathbf{h}}_i$ is then obtained by projecting ${\mathbf{o}}_i$ onto $L_i$:

(18)

With the filtering coordinate system established, the next step is to compute the necessary weighting factors. Instead of relying on fixed global parameters like SOD or SDD, we dynamically compute these values for each projection directly from our calibrated vector geometry, providing a direct, vector‐based translation of the parameters used by, ⁴⁷ The two critical distances for fan‐beam scaling are the source‐to‐rotation‐axis distance $R_i$ and the effective source‐to‐detector distance $D_i$:

(19)

$$\begin{array}{cc}\hfill R_i& =\parallel s_i-o_i\parallel .\hfill \end{array}$$ (20)

Additionally, a key component of the pre‐weighting is the cosine of the angle ${\alpha }_u$ between the central ray (from $s_i$ to $o_i$) and the ray through a given point $p_u$ on the detector line $L_i$:

$$\cos {\alpha }_u=\frac{{(p_u-{\mathbf{s}}_i)}^{\top }(o_i-s_i)}{\parallel p_u-s_i\parallel ·\parallel o_i-s_i\parallel }.$$ (21)

The data from the reference row of the corrected projection $P_i^{\prime \prime }(u,j_0)$ is first pre‐weighted:

$$g_i(u)=W_i(u)P_i^{\prime \prime }(u,j_0),W_i(u)=\cos {\alpha }_u\frac{R_i}{D_i}.$$ (22)

This pre‐weighted 1D projection $g_i(u)$ is then convolved with a ramp filter $h(u)$ to produce the filtered projection ${\stackrel{\sim }{g}}_i(u)$:

$${\stackrel{\sim }{g}}_i(u)=g_i(u)\ast h(u)={\int }_{-\infty }^{\infty }{g}_i(u^{\prime })h(u-u^{\prime })\mathrm{d}{u}^{\prime }.$$ (23)

In practice, this convolution is efficiently implemented in the frequency domain via the Fast Fourier Transform.

Next, we extend this process to the full 3D cone‐beam projection. A key simplification arises from our earlier detector pose correction: since the corrected column vector is now parallel to the effective rotation axis $n_{\text{rot}}$, ramp filtering can be applied independently to each detector row. For each row, the filtering origin is shifted according to the row‐wise virtual‐detector construction described above, and the projection data are pre‐weighted using a vector‐based FDK factor:

$$W_i(u,v)=\cos {\alpha }_{u,v}\frac{R_i}{D_i},$$ (24)

where, ${\alpha }_{u,v}$ is the angle between the ray from the source ${\mathbf{s}}_i$ to the detector coordinate $(u,v)$ and the central ray. The weighted projection data are then filtered along the corrected row direction .

The final step is to backproject the filtered projections ${\stackrel{\sim }{P}}_i^{\prime \prime }$ into the 3D volume grid $f(x)$. For each voxel $x$, a distance‐dependent FDK weight is applied using the corrected detector plane normal . Specifically, we define,

(25)

To compensate for nonuniform angular sampling introduced by the robotic arm motion, an additional angular weight $w_{\text{ang},i}$ is applied. We adopt a weighting scheme analogous to that proposed for fan‐beam reconstruction by, ⁵⁰ which is based on the principles of numerical integration over a nonuniform grid. The weight for projection $i$ is thus set proportionally to its local integration interval, which we approximate as the average of its angular separation from its immediate neighbors:

$$w_{\text{ang},i}=\frac{1}{2}\left(\mathrm{\Delta }{\beta }_{i-1,i}+\mathrm{\Delta }{\beta }_{i,i+1}\right),$$ (26)

where, $\mathrm{\Delta }{\beta }_{j,k}$ is the angle between the vectors from the conceptual isocenter $o_i$ to the source positions $s_j$ and $s_k$, computed as:

$$\mathrm{\Delta }{\beta }_{j,k}=arccos\left(\frac{(s_j-o_i)·(s_k-o_i)}{\parallel s_j-o_i\parallel \parallel s_k-o_i\parallel }\right).$$ (27)

The final reconstructed volume is obtained by summing the weighted backprojection contributions from all $N$ projections:

$$f(x)=\sum _{i=1}^N{w}_{\text{ang},i}·\frac{1}{U_i{(x)}^2}·{\stackrel{\sim }{P}}_i^{\prime \prime }\left(u(x),v(x)\right).$$ (28)

This formulation completes our geometry‐adapted FDK framework for reconstruction from projection‐wise robotic scan geometries.

2.3. Experimental setup

This subsection describes the validation design for evaluating the proposed SyCaR framework using complementary numerical and physical studies. The physical robotic acquisitions covered approximately ${190}^{\circ }$, corresponding to a short‐scan setting imposed by the current hardware and workspace constraints. Because the current PD‐FDK implementation is an FDK type analytical reconstruction method without dedicated short‐scan compensation, the numerical study used a ${360}^{\circ }$ trajectory completed from a robotic scan measured by the motion capture system and a 3D Shepp‐Logan phantom to reduce the confounding effect of angular incompleteness while preserving measured robotic trajectory deviations. The same numerical setting was also used to evaluate the optional online refinement by applying known residual perturbations and measuring their recovery behavior.

The physical study retained the original short‐scan acquisitions and used two phantoms, shown in Figure 5, to compare reconstructions obtained from different geometry inputs and reconstruction algorithms under real acquisition conditions. Reference scans were used to quantify the mitigation of geometry artifacts, rather than to compare acquisition speed, dose efficiency, or short‐scan reconstruction performance across protocols. Accordingly, the present validation should be interpreted as a feasibility evaluation of projection‐wise geometric correction and FDK type reconstruction for near‐circular robotic trajectories with measured pose deviations, rather than as a validation for arbitrary noncircular trajectories.

FIGURE 5.

Physical phantoms used for real‐data validation. The upper shows Phantom I, a transparent acrylic cylinder containing high‐contrast aluminum filaments and low‐contrast acrylic rods. The lower shows Phantom II, a resin relief phantom containing multiple raised line‐shaped and curved structures, which provides a denser set of high‐gradient edges and spatial patterns for reconstruction assessment.

2.3.1. Reconstruction protocol and evaluation metrics

The compared reconstruction methods included FDK, SIRT, Parametric FDK, ⁴⁷ and the proposed PD‐FDK. FDK and SIRT were implemented using ASTRA Toolbox v2.4.1: ⁴⁹ , ⁵¹ FDK used FDK_CUDA with cone_vec geometry, and SIRT used SIRT3D_CUDA with the same vector geometry for 200 iterations unless otherwise specified. Parametric FDK denotes our implementation of the independent source–detector rotation FBP formulation of Rit et al., ⁴⁷ for which the projection‐wise vector geometry was converted into the required scan parameters. No additional detector‐pose correction or projection resampling was applied before Parametric FDK. In contrast, the proposed PD‐FDK used the same preprocessed projection stack and the original vector geometry directly, with detector‐pose correction followed by the extended FDK reconstruction described in Section 2.2.

For both numerical and physical reconstructions, the reconstruction field of view (FOV) was determined from the common imaging region covered by the source–detector imaging zone over the projection views, following the FOV definition used by Rit et al. ⁴⁷ For the trajectories evaluated in this study, the estimated in‐plane FOV was approximately 180 mm, which was sufficient to cover both physical phantoms. Therefore, all numerical and physical datasets were reconstructed using a common $180\times 180\times 180{\mathrm{mm}}^3$ reconstruction volume, with an in‐plane voxel size of 0.5 mm and a slice thickness of 2 mm. Unless otherwise stated, displayed slices correspond to the effective central slice of the corresponding trajectory.

For numerical experiments, PSNR, SSIM, and RMSE were computed against the known numerical phantom after intensity normalization. The numerical metrics were averaged over ten neighboring 2‐mm‐thick slices around the effective central slice, so that slices slightly away from the central plane were also included. For physical experiments, image quality metrics were computed against the registered reference scan when applicable. Line profiles were extracted at fixed representative locations through high‐gradient structures and displayed with the corresponding reconstructed slices.

2.3.2. Numerical validation

To reduce the confounding effect of angular incompleteness in the controlled reconstruction comparison, a complementary numerical study was performed using a ${360}^{\circ }$ trajectory completed from the robotic trajectory measured by the motion capture system. This design retained the non‐ideal motion characteristics of the robotic acquisition while providing full angular coverage for evaluating the reconstruction algorithms. The measured projection‐wise vector geometry was first ordered by scan angle. Small angular gaps were bridged by interpolation, and the trajectory was then completed by applying a ${180}^{\circ }$ rigid rotation about the fitted source trajectory axis. The completed trajectory was used only for numerical validation and should not be interpreted as a general trajectory completion method.

Two numerical analyses were performed. First, a 3D Shepp–Logan phantom was generated and centered within the estimated reconstruction FOV. Simulated projections were generated using the completed trajectory with the same detector sampling and reconstruction settings as those used for the physical datasets. The projection matrix was set to $1024\times 768$ with an effective detector pixel pitch of 388 $\mu \mathrm{m}$. The simulated data were then reconstructed with the same known geometry. The data flow was:

$${\mathit{\omega }}_{\mathrm{meas}}\to {\mathit{\omega }}_{360}\to p_{\mathrm{sim}}=\mathcal{P}(f_{\mathrm{true}},{\mathit{\omega }}_{360})\to \widehat{f}=\mathcal{R}(p_{\mathrm{sim}},{\mathit{\omega }}_{360}).$$ (29)

where, ${\mathit{\omega }}_{\mathrm{meas}}$ denotes the measured geometry, ${\mathit{\omega }}_{360}$ denotes the completed geometry, $f_{\mathrm{true}}$ is the Shepp–Logan phantom, $\mathcal{P}$ is the forward projector, $p_{\mathrm{sim}}$ is the simulated projection data, and $\mathcal{R}$ is the reconstruction operator.

Second, a limited diagnostic experiment was performed to examine the optional ECC refinement. Known bounded residual perturbations were added to ${\mathit{\omega }}_{360}$, yielding ${\mathit{\omega }}_{\mathrm{pert}}$, and simulated projections were generated from this perturbed geometry. ECC was initialized with the unperturbed completed geometry:

$$\begin{array}{ccc}\hfill {\mathit{\omega }}_{360}\to {\mathit{\omega }}_{\mathrm{pert}}\to p_{\mathrm{sim}}& =& \mathcal{P}(f_{\mathrm{true}},{\mathit{\omega }}_{\mathrm{pert}})\to {\mathit{\omega }}_{\mathrm{ECC}}\hfill \\ & =& {\mathcal{C}}_{\mathrm{ECC}}(p_{\mathrm{sim}},{\mathit{\omega }}_{360}).\hfill \end{array}$$ (30)

The refined geometry ${\mathit{\omega }}_{\mathrm{ECC}}$ was compared with the known perturbed geometry ${\mathit{\omega }}_{\mathrm{pert}}$. This experiment tested the local capture behavior of ECC around an already available tracking‐based geometry, rather than its ability to recover gross robotic pose errors. Consistent with its intended role as an optional residual correction module, ECC was evaluated diagnostically and was not used as the main evidence for the proposed reconstruction method.

2.3.3. Physical phantom validation

Two physical phantoms were used for real‐data validation. Both phantoms were scanned using the dual‐arm robotic CBCT system. For reference‐based image‐quality evaluation, a reference volume was acquired using a commercial CBCT system (EVISION‐720T), which used the same nominal X‐ray source and flat‐panel detector models as the dual‐arm robotic CBCT system. The commercial reference scan used a pulsed‐exposure protocol at 110 kV and 10 mA, acquired 720 projections over a full ${360}^{\circ }$ rotation, and included standard CBCT hardware such as beam collimation. The resulting volume was used as a reference for quantifying reconstruction consistency and geometry‐related artifact reduction. In contrast, because the source‐control synchronization of the current robotic prototype is still under development, the robotic scans were performed using continuous X‐ray exposure. The tube voltage was set to 110 kV, and the tube current was reduced to 2 mA to allow continuous operation throughout the robotic scan. The small focal spot was used, corresponding to a nominal focal spot size of 0.3 mm. The detector was operated in a $2\times 2$ binned mode, yielding a $1024\times 768$ projection matrix with an effective detector pixel pitch of 388 $\mu \mathrm{m}$. Because the commercial reference and robotic acquisitions used different exposure modes, angular coverage, and acquisition protocols, the reference scan was not treated as an exact ground truth for system‐level dose or speed comparison, but only as a practical registered reference for image‐domain evaluation. Phantom I was a custom‐designed transparent acrylic cylinder containing five high‐contrast aluminum cylindrical filaments with diameters of 1, 2, 3, 5, and 7 mm and three larger low‐contrast acrylic rods, all positioned approximately parallel to the cylinder axis. Phantom II was a resin relief phantom containing multiple raised line‐shaped and curved structures. Providing a denser set of high‐gradient edges and spatial patterns. Representative photographs of both phantoms are shown in Figure 5.

For each phantom and each planned robotic trajectory, the scan was repeated three times. To isolate the effect of geometry input from the effect of reconstruction algorithm, the geometry‐input comparison was performed using a fixed SIRT reconstruction protocol. Specifically, reconstructions obtained from robot‐encoder geometry, motion‐capture geometry, and ECC‐refined geometry were compared under the same reconstruction setting. The geometry setting selected from this comparison was then used for the subsequent different reconstruction algorithms comparison, unless otherwise specified.

Encoder and motion‐capture geometries were compared after transforming both into the same reconstruction coordinate system. Because the robot encoder records and projection images were paired view by view, the source position and detector center were compared projection‐wise without applying an additional rigid alignment between trajectories. Source and detector position discrepancies were summarized as Euclidean distances. Detector orientation discrepancies were computed from the relative detector rotations and decomposed into yaw, pitch, and roll components using the same detector‐frame convention as in Section 2.2.

For repeated trajectory analysis, the first scan of each trajectory was used as the reference trajectory. For each projection in the remaining repeated scans, the corresponding point was projected onto the closest local segment of the reference source trajectory. The reference source and detector positions were obtained by linear interpolation along this segment, and the reference detector orientation was obtained by spherical linear interpolation of the endpoint quaternions. The Euclidean source‐position and detector‐center errors and the relative detector‐orientation errors were then used to characterize scan‐to‐scan repeatability.

For the physical ECC diagnostic, ECC was applied after the motion‐capture geometry had been obtained. The correction magnitude was computed by comparing the motion‐capture geometry before and after ECC refinement. Source and detector translational corrections were summarized as the Euclidean 3D shifts, and detector angular corrections were summarized from the relative detector rotation.

3. RESULTS

3.1. Numerical reconstruction results

Figure 6 compares cone‐beam CT reconstructions of the 3D Shepp–Logan phantom from simulated projections generated using the completed robotic trajectory. Compared with standard FDK, both Parametric FDK and PD‐FDK better accommodated the non‐ideal trajectory geometry. The alternating bright and dark line‐like intensity nonuniformities visible in the standard FDK reconstruction were substantially reduced in the two geometry‐adapted analytical reconstructions. This artifact pattern likely reflects the application of standard FDK to a trajectory with non‐uniform angular sampling, where explicit view‐dependent angular integration weighting is not incorporated in the same way as in the geometry‐adapted formulations.

FIGURE 6.

Numerical reconstruction of the 3D Shepp–Logan phantom using the completed ${360}^{\circ }$ robotic trajectory. The top row shows the ground truth (GT) and reconstructions obtained with FDK, Parametric FDK, PD‐FDK, and SIRT. The bottom row contains voxel‐value profiles extracted along the red dashed lines drawn on the corresponding slices

All FDK type reconstructions exhibited small local fluctuations, whereas SIRT produced a smoother appearance. The line profiles in Figure 6 show that SIRT was closer to the ground truth in this displayed slice, both in terms of reduced background fluctuation and smoother recovery of high‐contrast structures. Standard FDK showed the largest profile discrepancy. In contrast, Parametric FDK and PD‐FDK produced very similar profiles in this effective central slice. Parametric FDK was even slightly more stable within some object regions in the displayed profile. This may be because the displayed slice is close to the effective central plane, where detector pitch perturbations have limited influence, whereas the detector‐pose resampling step in PD‐FDK can introduce additional interpolation error.

The quantitative results in Table 1 were computed by averaging over ten slices around the effective central slice, with a 2‐mm slice thickness. Under this evaluation setting, PD‐FDK was slightly better than Parametric FDK. This suggests that, when slices slightly away from the central plane are included, the detector‐pose correction in PD‐FDK can partially compensate for pitch‐ and roll‐related deviations and improve the average reconstruction accuracy. SIRT achieved the highest SSIM, which is consistent with its smoother image appearance, but its PSNR and RMSE were lower than those of the two geometry‐adapted FDK methods.

TABLE 1.

Numerical reconstruction using the completed trajectory. Projection data were generated from the 3D Shepp–Logan phantom using ${\mathit{\omega }}_{360}$, and the same known geometry was supplied during reconstruction. Quantitative metrics were computed against the numerical ground truth after intensity normalization and averaged over ten neighboring 2‐mm‐thick slices around the effective central slice.

Method	PSNR (dB)	SSIM	RMSE
FDK	28.99	0.674	0.036
Parametric FDK	32.46	0.690	0.024
PD‐FDK(ours)	32.72	0.695	0.023
SIRT	30.14	0.837	0.031

As an optional diagnostic check, ECC was further tested on the same completed trajectory with known residual perturbations. A component‐wise mean absolute error reduction ratio was used to summarize whether the refined geometry moved closer to the known perturbed geometry. In isolated perturbation tests, ECC reduced selected residual errors, for example detector‐center translation along the detector column direction $d_y$ ($\rho =0.45$), detector roll rotation about the detector normal $r_z$ ($\rho =0.98$), and detector pitch rotation about the detector row direction $r_x$ ($\rho =0.43$). Here, the detector column direction is approximately aligned with the effective rotation‐axis direction in the completed trajectory. However, the recovery was not uniformly stable when multiple geometric parameters were coupled: improvements in some components were accompanied by error amplification in others. These results support the intended role of ECC as a local residual‐correction module initialized by an already available tracking‐based geometry. In the SyCaR framework, motion‐capture geometry provides the primary compensation for gross robotic pose errors, whereas ECC is used only to optionally compensate smaller residual errors associated with marker‐to‐component calibration and pose‐measurement uncertainty. Thus, the simulation results show that ECC can recover selected residual components, while the coupled‐parameter tests indicate that it should not be treated as a required calibration step.

3.2. Physical reconstruction results

For the physical phantom experiments, each planned robotic trajectory was repeated three times. We first examined the influence of geometry input while keeping the reconstruction method fixed. To minimize confounding from algorithm effects, SIRT was used for the geometry‐input comparison shown in Figures 7 and 8(b)–(d). Reconstructions based only on the robot‐encoder geometry exhibited severe streaking artifacts and structural inconsistency, making the internal structures difficult to identify. Replacing the encoder geometry with motion‐capture geometry substantially reduced these artifacts and restored the visibility of the phantom structures.

FIGURE 7.

Physical reconstruction comparison for Phantom I. The first row compares geometry inputs using the encoder geometry, motion‐capture geometry, and ECC‐refined geometry. The second row compares FDK, Parametric FDK, PD‐FDK, and SIRT using the selected geometry. The third row shows line profiles extracted from the reference image and the four reconstruction results along the indicated red dashed line.

FIGURE 8.

Physical reconstruction comparison for Phantom II. The first row compares geometry inputs using the encoder geometry, motion‐capture geometry, and ECC‐refined geometry. The second row compares FDK, Parametric FDK, PD‐FDK, and SIRT using the selected geometry. The third row shows line profiles extracted from the reference image and the four reconstruction results along the indicated red dashed line.

The effect of the optional ECC refinement was more case‐dependent. For Phantom I, ECC‐refined geometry produced a slight visual improvement compared with unrefined geometry, with a modest reduction in residual streaking. For Phantom II, the reconstructions obtained with motion‐capture geometry and ECC‐refined geometry were visually similar. Therefore, for the subsequent reconstruction‐method comparison, a single geometry setting was fixed for each phantom: ECC‐refined geometry for Phantom I and unrefined geometry for Phantom II.

Table 2 summarizes the quantitative reconstruction results computed against the registered reference scans, and Figures 7 and 8 show the corresponding representative slices and line profiles. In the current physical system, SIRT generally produced smoother reconstructions and higher metrics. This behavior likely reflects the smoothing effect of the iterative reconstruction process, which can suppress noise and residual streaking in real data. This should not be interpreted as a purely geometric advantage, because the physical acquisitions also contain noise, scatter, beam‐hardening effects, short‐scan incompleteness, and other differences that are not present in the controlled numerical experiment.

TABLE 2.

Physical reconstruction results for two custom phantoms using four reconstruction methods. Values were computed from aligned reference‐ROI images and are reported as mean $\pm$ standard deviation over repeated scans.

Dataset	Method	PSNR (dB)	SSIM	RMSE
Phantom I	FDK	30.21 $\pm$ 0.28	0.892 $\pm$ 0.016	0.030 $\pm$ 0.001
Phantom I	Parametric FDK	28.70 $\pm$ 0.42	0.912 $\pm$ 0.002	0.037 $\pm$ 0.002
Phantom I	PD‐FDK	30.78 $\pm$ 0.30	0.890 $\pm$ 0.005	0.029 $\pm$ 0.001
Phantom I	SIRT	33.17 $\pm$ 0.32	0.940 $\pm$ 0.001	0.022 $\pm$ 0.001
Phantom II	FDK	15.46 $\pm$ 0.15	0.677 $\pm$ 0.009	0.169 $\pm$ 0.003
Phantom II	Parametric FDK	14.97 $\pm$ 0.84	0.649 $\pm$ 0.049	0.179 $\pm$ 0.017
Phantom II	PD‐FDK	15.42 $\pm$ 0.34	0.678 $\pm$ 0.017	0.170 $\pm$ 0.007
Phantom II	SIRT	15.39 $\pm$ 0.42	0.699 $\pm$ 0.036	0.170 $\pm$ 0.008

For Phantom I, FDK and PD‐FDK produced comparable reconstructions, with PD‐FDK showing a slightly higher PSNR. Parametric FDK showed lower reconstructed values along the selected line profile, especially within the high‐contrast structures, which contributed to its lower quantitative metrics. This behavior may be related to the larger detector‐orientation deviations in the Phantom I trajectory and the strong contrast differences between the aluminum filaments and acrylic background, although this interpretation remains qualitative. In Phantom II, where the object material was more homogeneous, the differences among the FDK type methods were less pronounced, and their line profiles were largely similar.

The physical results differed from the numerical Shepp–Logan experiment. This is expected because the numerical experiment isolated the effect of nonideal trajectory geometry under a completed ${360}^{\circ }$ trajectory, whereas the physical scans were short‐scan acquisitions and included additional real‐data factors such as noise, scatter, beam hardening, exposure differences, and imperfect registration to the reference volume. These factors can particularly affect analytical FDK type reconstructions and reduce the extent to which the physical results reflect geometry errors alone.

To quantify the scale of geometric deviations in the physical scans, encoder geometry and motion‐capture geometry were compared projection by projection after transformation into the common reconstruction coordinate system. As summarized in Table 3, the encoder geometry differed from the motion‐capture geometry by several millimeters in both source position and detector‐center position. These values explain why encoder‐only reconstructions showed severe artifacts and support the use of external tracking as the primary geometry source.

TABLE 3.

Geometry‐source discrepancy and repeated‐trajectory variation in physical scans. Enc.–mocap rows report the mean absolute discrepancy between encoder geometry and motion‐capture geometry, summarized as mean $\pm$ standard deviation across repeated scans. Mocap rep. rows report the scan‐to‐scan variation of repeated motion‐capture trajectories, summarized as the mean $\pm$ standard deviation over projection views.

Analysis	Source (mm)	Detector (mm)	Yaw (deg)	Pitch (deg)	Roll (deg)
P‐I Enc.–mocap	$8.51\pm 0.46$	$6.04\pm 0.44$	$1.83\pm 0.16$	$0.87\pm 0.12$	$0.72\pm 0.10$
P‐II Enc.–mocap	$7.69\pm 0.38$	$7.59\pm 0.05$	$2.39\pm 0.95$	$0.59\pm 0.09$	$0.48\pm 0.11$
P‐I Mocap rep.	$1.44\pm 0.54$	$0.96\pm 0.29$	$0.12\pm 0.01$	$0.16\pm 0.05$	$0.11\pm 0.01$
P‐II Mocap rep.	$0.66\pm 0.27$	$0.49\pm 0.17$	$0.08\pm 0.01$	$0.15\pm 0.03$	$0.17\pm 0.01$

The repeated motion‐capture trajectories showed smaller scan‐to‐scan variation in most cases. This is consistent with the expected behavior of robotic arms: repeated execution of the same commanded trajectory can be relatively stable, with submillimeter to millimeter‐level positional variation and sub‐degree orientation variation in most tested scans. However, the proposed framework is primarily concerned with absolute projection‐wise CT geometry rather than repeatability alone. Therefore, the discrepancy of encoder and motion capture is more informative for this study, because it reflects the deviation between the encoder geometry and the externally measured geometry.

Finally, ECC was evaluated as an optional residual‐refinement module after motion‐capture geometry had been obtained. The ECC correction magnitude was on the order of a few millimeters for detector‐center translation in the selected scans, with detector‐position corrections ranging from approximately 1.2 – 3.3 mm, and the corresponding ray‐angle changes were approximately 0.18–0.54. The effect of ECC on physical reconstructions was not uniformly beneficial. In some cases, ECC slightly reduced residual artifacts, as observed for Phantom I, whereas in other cases the reconstruction metrics were unchanged or worsened. This behavior is consistent with the controlled ECC simulation, where isolated residual perturbations could be partially recovered but coupled parameter schedules produced parameter‐dependent error amplification. It is also consistent with prior observations that projection‐domain consistency optimization can be affected by object content, noise, truncation, scatter, occlusion, and residual modeling mismatch. ²⁰ Therefore, in the SyCaR framework, ECC is retained as an optional residual‐refinement module initialized by motion‐capture geometry, rather than as the primary calibration source or a required reconstruction component.

4. DISCUSSION AND CONCLUSIONS

Our results suggest that the proposed SyCaR framework can improve the geometric consistency of dual‐arm robotic CBCT under the evaluated conditions. The geometry‐source and reconstruction comparisons (Tables 3 and 2) indicate that external metrology is important for estimating projection‐wise geometry and reducing the effect of gross robotic pose errors, for which internal encoders alone may be insufficient. The optional online refinement can further compensate for selected residual deviations in controlled cases, but its effect was not uniformly reliable across coupled parameters. Thus, in the current prototype, external tracking provides the primary geometric estimate, whereas online refinement is better interpreted as a residual‐correction and diagnostic step.

With projection‐wise geometry estimates established, our PD‐FDK algorithm achieves favorable reconstruction performance compared with the evaluated FDK type methods under the tested trajectories (Tables 1 and 2). Conventional reconstruction algorithms, including advanced parametric methods, ⁴⁷ are well suited for geometries that can be described by their assumed parametric models, such as mechanically coupled C‐arm systems. The challenge arises when applying these reduced models to a dual‐arm robot, as it may force the system's complex, flexible trajectory into an overly restrictive parametric description, creating a potential mismatch. The strength of our PD‐FDK lies in its direct vector‐based formulation, which uses the per‐projection pose data provided by robotic and motion‐capture systems and reduces the need for model‐fitting approximations. This makes it a computationally efficient and extensible reconstruction approach for non‐ideal robotic scan paths.

Several limitations should be noted. The short‐scan trajectories used in the physical experiments introduce limited‐angle artifacts, which were outside the primary scope of this work on geometric correction. The current PD‐FDK implementation and validation were also limited to non‐truncated projection conditions, where the main object support was sufficiently covered by the detector; detector truncation and corresponding truncation‐compensation strategies were not addressed in this study. The physical validation was limited to two phantoms and repeated scans on the current prototype, and the motion‐capture geometry should be regarded as an external pose estimate rather than ground truth. Residual errors may still arise from marker‐to‐component calibration uncertainty, tracking uncertainty, focal‐spot drift, scatter, beam hardening, and object‐dependent behavior of projection‐consistency optimization. In addition, the numerical 360 trajectory was completed from a measured robotic trajectory using a specific extension, which reduces angular incompleteness but should not be interpreted as a general trajectory‐completion strategy. Future work will address short‐scan data insufficiency, further streamline the calibration workflow, and evaluate the framework using broader object types, independent geometric validation, and more general robotic scanning protocols.

In summary, this paper introduced SyCaR, a computational framework that combines external metrology, optional residual refinement, and pose‐driven analytical reconstruction for dual‐arm robotic CBCT. Numerical and physical experiments showed that the proposed framework can mitigate geometry‐related artifacts and improve reconstruction consistency relative to the evaluated baselines under the tested conditions. For repeatedly executed fixed scan protocols, trajectory‐specific pre‐calibration or other simpler calibration strategies may provide sufficiently stable geometry by exploiting the high repeatability of industrial robots. The projection‐wise design of SyCaR is expected to be more useful when scan trajectories are modified for different objects or imaging tasks, where relying on separately pre‐calibrated trajectories becomes less practical. Overall, this work supports further development of flexible dual‐arm robotic CBCT for industrial inspection and medical imaging, while broader validation remains necessary before general deployment claims can be made.

CONFLICT OF INTEREST STATEMENT

The authors declare no conflict of interest.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.