High-Precision Endoscopic Shape Sensing Using Two Calibrated Outer Cores of MC-FBG Array

1. Introduction

Three-dimensional (3D) shape sensing is reshaping endoscopy for minimally invasive surgery. Conventional systems provide only two-dimensional (2D) views with limited depth cues, complicating navigation through tortuous anatomy [1,2,3]. By reconstructing the 3D configuration of endoscope in real time-position, curvature and tip pose-shape sensing restores spatial awareness, enables precise lesion localization and instrument guidance, and reduces iatrogenic injury. Continuous shape monitoring also flags excessive bending or looping that can increase patient discomfort and procedural risk, while data-driven feedback supports operator training and helps standardize performance [4,5,6]. When integrated with image guidance or robotic platforms, it enables closed-loop control and more reliable, efficient workflows [7,8,9,10]. Together, these capabilities mark a decisive step toward intelligent, data-driven endoscopy.

Established methods (e.g., electromagnetic tracking [11,12,13], image-based techniques [14,15,16]) provide partial solutions but have limitations including interference, tissue deformation, motion blur, and complex calibration; by contrast, fiber-optic shape sensing (FOSS) enables continuous, high-precision, real-time endoscope shape reconstruction, making it a promising alternative [17,18]. The principle of the FOSS is to convert multi-channel strain measurements from fully/quasi distributed sensors into 3D curvature, and then recover the 3D spatial coordinates of the sensing fiber via a shape-reconstruction algorithm. For short-range applications (from tens of centimeters to tens of meters), the distributed strain sensing approaches are optical frequency-domain reflectometry (OFDR) [25,27] and wavelength-division multiplexing (WDM) [4,7,17]. OFDR systems achieve distributed, high-spatial-resolution strain measurements along the fiber—up to tens of micrometers—by beat-frequency detection; however, the need for an expensive tunable laser source (TLS) limits their practical deployment. In contrast, WDM systems estimate strain from shifts in the Bragg wavelength of a fiber Bragg grating (FBG) array. By precisely controlling the FBG spacing during fabrication, spatial resolutions of a few millimeters to a few centimeters can be obtained. As a more mature and cost-effective demodulation scheme, WDM is well suited for use as an auxiliary modality for endoscopic navigation [4]. For fiber-optic shape sensors, configurations based on fiber bundles [21,22] or on bonding multiple fibers to an elastic substrate (e.g., nickel-titanium, NiTi, wire) [23,24] offer high sensitivity; however, they involve complex packaging and exhibit pronounced geometric mismatch, which can compromise reconstruction accuracy. In contrast, MCF has compact core layout and well-defined geometry, enables accurate 3D shape sensing. Current MCF-based shape-sensing fibers typically employ three or more outer cores surrounding a central core [25,26], and estimate curvature magnitude and bending direction using the apparent-curvature method (ACM). Nevertheless, each outer-core channel is subject to strain-measurement error; therefore, using an excessive number of outer cores can degrade sensing accuracy and increase computational complexity, which is undesirable for real-time, high-precision endoscopic shape sensing. Meng et al. [27] proposed a shape-reconstruction strategy based on projecting the curvature vector onto the directions of two outer cores, which substantially reduces the required number of cores and simplifies the system. However, the method does not calibrate the geometric parameters of the outer cores (i.e., core spacing and core angles); consequently, its shape-reconstruction error is comparable to that of multi-outer-core configurations that use the ACM.

To overcome limitations of conventional MCF-based shape sensing, we propose a method that employs only two outer cores arranged non-collinearly with the central core in an MC-FBG array, together with precise geometric calibration of these two cores. Leveraging the underlying geometry, we derive analytical expressions for the curvature magnitude and bending angle specific to this two-outer-core configuration. The calibration compensates for manufacturing tolerances and packaging-induced deviations in core spacings and angles, thereby reducing systematic bias in the strain-to-curvature mapping. Building on these components, our reconstruction pipeline accurately recovers both 2D and 3D standard shapes and provides a uniform basis for comparing alternative outer-core layouts. Overall, the proposed architecture delivers high accuracy with lower hardware complexity and computational burden, making it well suited for real-time, high-precision, and resource-efficient endoscopic shape sensing.

2. Sensing Principle

The main principle of MC-FBG array-based shape sensing is to derive key 3D curve parameters (i.e., curvature and bending angle) from the discrete strain of multiple outer cores, then reconstruct 3D spatial coordinates iteratively via the moving frame method. According to the Euler-Bernoulli beam theory [30], the strain in the outer core i of a MCF (as shown in Figure 1a) can be expressed as:

$${\epsilon }_i=\kappa r_i\cos \left({\theta }_B-{\theta }_i+{\theta }_{{\tau }_0}\right)+{\epsilon }_1,i\in \left\{2,\mathrm{3,4},\mathrm{5,6},7\right\}$$

(1)

where $\kappa$ is the curvature, $r_i$ is the core spacing of the ith outer core, ${\theta }_B$ denotes the bending angle, ${\theta }_i$ represents the angular of the ith outer core relative to the local x-axis, and ${\theta }_{{\tau }_0}$ is the small torsion angle introduced during fabrication, which can be compensated through calibration. ${\epsilon }_1$ denotes the strain experienced by the central core, accounting for both temperature and axial strain.

In conventional seven-core fiber, the outer cores are typically arranged in geometries such as regular hexagon, equilateral triangle (the most common), rectangle, or right triangle, as illustrated in Figure 1b. When the geometric parameters are assumed to take their theoretical values, the apparent curvature method [25] can be used to compute the curvature and bending angle from the measured strains of multiple outer cores. However, since only $\kappa$ and ${\theta }_B$ are unknowns in Equation (1), two independent equations are sufficient to determine them. As reported in [28], the accuracy of shape reconstruction is largely influenced by strain measurement errors; hence, using a greater number of outer cores introduces more sources of error. In addition, the internal geometric parameters of the MCF after packaging can be calibrated using our previously reported method [29] to reduce measurement errors in curvature and bending angle. Therefore, a shape sensing and reconstruction method based on two calibrated outer cores can effectively reduce computational complexity while improving measurement accuracy.

Taking two outer cores i and j arranged non-collinearly with the central core in MCF as an example (as illustrated in Figure 1c), the projection of the curvature on the line connecting each outer core to the central core can be expressed as:

$${\kappa }_{i/j}=\kappa \cos \left({\theta }_B-{\theta }_{i/j}+{\theta }_{{\tau }_0}\right)={\displaystyle \frac{{\epsilon }_{i/j}-{\epsilon }_1}{r_{i/j}}}$$

(2)

As shown in Figure 1d, a geometric relationship can be established in the right triangle:

$${\kappa }_{m/n}={\displaystyle \frac{{\kappa }_{i/j}}{\mathrm{c}\mathrm{o}\mathrm{s}({\theta }_j-{\theta }_i-\pi /2)}}={\displaystyle \frac{{\kappa }_{i/j}}{\mathrm{s}\mathrm{i}\mathrm{n}({\theta }_j-{\theta }_i)}}$$

(3)

Accordingly, based on the cosine law, the curvature magnitude can be derived as:

$$\kappa =\sqrt{{\kappa }_m^2+{\kappa }_n^2-2\bullet {\kappa }_m\bullet {\kappa }_n\mathrm{c}\mathrm{o}\mathrm{s}({\theta }_j-{\theta }_i)}$$

(4)

and the bending angle is given by:

$${\theta }_B={\theta }_i+\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left({\displaystyle \frac{{\kappa }_i}{\kappa }}\right)$$

(5)

Thus, the projections of the curvature in the local Cartesian coordinate system can be expressed as:

$${\kappa }_1=\kappa \bullet \mathrm{c}\mathrm{o}\mathrm{s}{\theta }_B,{\kappa }_2=\kappa \bullet \mathrm{s}\mathrm{i}\mathrm{n}{\theta }_B$$

(6)

To enhance the effective spatial resolution, ${\kappa }_1$ and ${\kappa }_2$ are interpolated using cubic splines to yield ${\kappa }_1\left(s\right)$and ${\kappa }_2\left(s\right)$, with $s$ representing the arc length of the curve.Subsequently, the Bishop frame [25] is adopted as the moving frame for reconstruction, in which the unit tangent vector T and two mutually orthogonal normal vectors N₁ and N₂ satisfy the following relationship:

$$\left[\begin{array}{c}\mathit{T}\mathit{\text{'}}\\ {\mathit{N}}_1^{\mathit{\text{'}}}\\ {\mathit{N}}_2^{\mathit{\text{'}}}\end{array}\right]=\left[\begin{array}{ccc}0& {\kappa }_1\left(s\right)& {\kappa }_2\left(s\right)\\ -{\kappa }_1\left(s\right)& 0& 0\\ -{\kappa }_2\left(s\right)& 0& 0\end{array}\right]\left[\begin{array}{c}\mathit{T}\\ {\mathit{N}}_1\\ {\mathit{N}}_2\end{array}\right]$$

(7)

By integrating the iteratively computed unit tangent vectors, the 3D spatial coordinates along the shape sensor can be reconstructed as:

$$r\left(s\right)=r\left(0\right)+{\int }_0^s\mathit{T}ds$$

(8)

3. Experiments and Results

3.1. Experimental Setup

As shown in Figure 2, the 3D shape-sensing system based on an MC-FBG array comprises an endoscope (CONCEMED Co., Ltd.) integrated with a fiber-optic shape sensor, a fan-in/fan-out (FIFO; OPTOWEAVE Co., Ltd.) device for the MCF, a seven-core MCF (YOFC Co., Ltd.) with a core spacing of 41.5 µm and a regular hexagonal outer-core layout, an eight-channel FBG interrogator (Wuhan Smart Fiber Co., Ltd.; wavelength range 1522–1572 nm; sampling rate 100 Hz; wavelength resolution 1 pm), and a host PC. The 3D shape sensor was fabricated by packaging the MC-FBG array as described in our previous work [29]; characterization results for the MC-FBG array are provided in Section 3.2. The FIFO connects the shape sensor to the interrogator, which demodulates the Bragg wavelengths, whereas the PC executes the 3D shape-reconstruction algorithm.

3.2. Testing and Calibration

Prior to curvature interpolation, the locations of all sensing points and the two outer-core geometric parameters (i.e., core spacing and core angles) are identified through experimental characterization and geometric calibration of the MC-FBG array. The reflection spectrum of core 1 measured by the FBG interrogator is shown in Figure 3a. Each core in the MC-FBG array contains 27 FBGs. The Bragg-wavelength spacing between the first five FBGs is approximately 3 nm (e.g., 3.23 nm between #1 and #2), whereas that of the remaining FBGs is about 1.5 nm (e.g., 1.41 nm between #5 and #6). Concurrently, the axial spatial distribution of core 1 was characterized using OFDR (IF-LAB laboratory), as shown in Figure 3b. Each FBG length was set to 5 mm to mitigate the influence of non-uniform strain during sensing [31]. The array has a total length of 1235.91 mm; the spacing between adjacent FBGs is ~25 mm for the first five sensors (e.g., #3–#4: 24.99 mm) and ~50 mm for the subsequent ones (e.g., #15–#16: 49.61 mm). This combined wavelength and spatial layout prevents spectral overlap between neighboring FBGs in the highly bendable distal segment of the endoscope. A notably larger gap (~100 mm) is observed between #7 and #8 in Figure 3b due to a missing FBG during fabrication; nonetheless, as evidenced by the shape-sensing and reconstruction results in Section 3.3, this irregularity had negligible impact on overall performance.

According to Equation (1), the core spacing and relative core angle of the two outer cores can be calibrated based on the cosine dependence of Bragg wavelength shifts on bending angles. Figure 3c shows the raw data and cosine fits for cores 2 and 4 at the 1st sensing point, yielding $r_2=40.48\mu \mathrm{m}$, $r_4=39.74\mu \mathrm{m}$, and ${\theta }_{24}={\theta }_4-{\theta }_2={130.29}^{\circ }$.Similarly, for the 27th sensing point, the calibrated parameters are $r_2=41.09\mu \mathrm{m}$, $r_4=41.34\mu \mathrm{m}$, and ${\theta }_{24}={126.39}^{\circ }$. These results indicate that the MCF outer-core geometric parameters deviate slightly from their nominal specifications; therefore, calibration is necessary to obtain accurate estimates of curvature and bending angle.

3.3. Standard Shape Sensing

To evaluate the accuracy of the proposed two-core-based shape sensing method, both 2D and 3D standard shapes were reconstructed using three different outer-core configurations: the equilateral triangle, the regular hexagon, and the two-core configuration with a 120° core separation. For the 2D case, a circular arc with a radius of 300 mm was adopted. As shown in Figure 4a1–4a2, without geometric calibration the deviations of the estimated curvature and bending angle from the theoretical values (3.3 × 10⁻³ mm⁻¹ and 0 rad) are comparable across the three configurations considered, confirming that the proposed non-collinear two-outer-core scheme already yields valid estimates. After calibration, the deviations for the two-outer-core case decrease markedly, demonstrating the effectiveness of the calibration procedure. This improvement is also evident in the reconstruction results: as shown in Figure 4a3, the calibrated two-outer-core scheme produces a 2D curve visibly closer to the ground truth than the other three configurations; the quantitative error analysis in Figure 4a4 and Table 1 indicates a maximum relative error (${\delta }^{rel}=\Vert {\mathit{r}}^{rec}-{\mathit{r}}^{tru}\Vert /s$, where ${\mathit{r}}^{rec}$denotes the reconstructed position and ${\mathit{r}}^{tru}$the ground-truth position; $s$ denotes the arc length)of only 1.62%. A similar conclusion holds for a 3D shape—specifically, a left-handed cylindrical helix with a base radius of 100 mm and a pitch of 50 mm—see Figure 4b1–4b4; the maximum relative error is only 2.81%. These results show that shape sensing using two calibrated outer cores in an MC-FBG array effectively improves reconstruction accuracy while maintaining low hardware complexity.

3.4. Arbitrary Shape Sensing

During conventional endoscopic interventions, only the distal tip is visualized, making shaft looping difficult to anticipate; consequently, reconstructing common loop types is necessary. To demonstrate the applicability of the proposed method to realistic endoscopic shape sensing, we reconstructed three commonly occurring loops formed within the body—the α-loop, the reversed α-loop, and the N-loop. The reconstructed shapes closely matched the corresponding endoscopic images (Figure 5a–5c). In addition, arbitrary 3D shapes were reconstructed with similarly high fidelity (Figure 5d). These results confirm that the proposed method can accurately capture endoscope configurations under realistic clinical conditions.

4. Conclusions

In this work, we present a lightweight approach to endoscopic 3-D shape sensing that uses only two non-collinear outer cores, geometrically referenced to the center core of a MC-FBG array. By exploiting inter-core geometric constraints, we derive closed-form expressions for curvature and bending angle and introduce parameter calibration to suppress systematic bias and error propagation, thereby achieving high-accuracy reconstruction with minimal hardware and algorithmic complexity. On canonical shapes, the method attains maximum relative position errors of 1.62% for a 2-D circular arc and 2.81% for a 3-D helix, outperforming uncalibrated multi-outer-core baselines; on clinically relevant configurations (α-, reverse-α-, and N-loops), reconstructions closely match the actual endoscope geometry, demonstrating robustness and practical applicability. These results highlight the method’s effectiveness in providing a low-resource, high-precision solution for practical endoscopic shape sensing applications.