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Abstract
The accuracy of two-dimensional (2D) shape reconstruction is highly susceptible to fake peaks in the strain distribution measured by optical frequency domain reflectometry (OFDR). In this paper, a post-processing method using first-order differential local filtering is proposed to suppress fake peaks and further improve the accuracy of shape reconstruction. By analyzing the principles of 2D shape reconstruction, an explanation of how fake peaks lead to shape reconstruction errors is provided, along with the introduction of an error evaluation standard. The principle of first-order differential local filtering is presented, and its feasibility is verified by simulation. An OFDR 2D shape reconstruction system is built, with three groups of 2D shape reconstruction experiments carried out, including up bending, down bending and arch bending. The experimental results show that the end errors of the three groups of shape reconstruction are respectively reduced from 2.33%, 2.97%, and 1.07% to 0.25%, 0.78%, and 0.20%, at the shape reconstruction length of 0.5 m. The research demonstrates that the accuracy of OFDR 2D shape reconstruction can be improved by using first-order differential local filtering.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical fiber sensing technology has broad application prospects in the field of shape sensing due to its advantages of anti-electromagnetic interference, high sensitivity and easy embedding in materials [1]. By measuring the strain distribution of the sensing fiber and processed with shape reconstruction algorithm, the shape sensing can be realized [2,3]. Several shape sensing methods were proposed, which are based on phase-sensitive optical time domain reflectometry (phase-OTDR) [4], Brillouin optical time-domain analysis (BOTDA) [5], Brillouin optical time domain reflectometry (BOTDR) [6], and fiber Bragg grating (FBG) [7–10].
However, the aforementioned shape sensing and reconstruction approaches are short at their cm-level sensing spatial resolution, which limits the shape reconstruction performance [7]. Optical frequency domain reflectometry (OFDR) is an optical fiber sensing technology with the advantages of high spatial resolution, which makes it suitable for shape sensing applications [11,12]. In order to further improve the shape reconstruction spatial resolution, several shape reconstruction schemes based on OFDR have been proposed [13,14]. However, the conventional OFDR shape reconstruction methods are suspicious to several noises in signals which stem from nonlinear sweep and white noise of the laser source, and may further deteriorate the sensing accuracy. A set of solutions have been proposed for resolving the noise problem and enhancing the sensing accuracy. In 2019, Yin et al. proposed the local spectral method to suppress the fake peaks in the cross-correlation process, which improves the signal quality and shape reconstruction accuracy [15]. In 2021, Liu et al. proposed a shape reconstruction model for the accuracy improvement of shape reconstruction under large curvature [16]. In addition, the modification of the sensing fiber can also improve the signal quality and the shape reconstruction accuracy. In 2017, Parent et al. enhanced the Rayleigh scattering intensity of ordinary single-mode fiber (SMF) through ultraviolet exposure, improved the signal-to-noise ratio of OFDR measurements, and increased the average measurement accuracy of the three-dimensional (3D) needle end by 47% [17]. In 2019, Beisenova et al. carried out multi-channel MgO doping on single-mode fiber to enhance Rayleigh scattering intensity and improve the accuracy of OFDR shape reconstruction [18]. In 2022, Sun et al. improved the spatial resolution of strain measurement and the accuracy of shape reconstruction by using scattering enhanced helical multi-fiber cables [19]. With the advancement of hardware-based shape sensing approaches and denoising algorithms, the shape sensing accuracy has been improved to a new level. However, the complexity of shape sensing system in both hardware and software rise with the development of aforementioned approaches, which limits the application of shape sensing.
This paper proposes a method to improve the accuracy of shape reconstruction by using first-order differential local filtering. The proposed method is a post-processing algorithm for the signal, which has low computational complexity and no modification in hardware. The theory of OFDR two-dimensional (2D) shape reconstruction and first-order differential local filtering are expounded. The feasibility of first-order differential local filtering algorithm to improve the accuracy of OFDR 2D shape reconstruction is verified by simulation. The OFDR shape reconstruction experimental system is built for 3 sets of shape reconstruction experiments, including up-bend, down-bend and arch. The experimental results show that with the proposed method, the end errors of the 3 shape reconstruction results are reduced from 2.33%, 2.97%, 1.07% to 0.25%, 0.78%, 0.20%, respectively. Notably, a complete 2D shape reconstruction process costs about 38.43s, while the proposed filtering algorithm only takes ∼0.01s with extremely low computational cost.
2. Principle of measurement
2.1 Principle of two-dimensional shape reconstruction
The principle of the proposed OFDR 2D shape reconstruction method is shown in Fig. 1. The system uses the strain data from the upper and lower surfaces of the object under test. Real-time Rayleigh wavelength shift (RWS) signals are obtained through the sensing fibers on the two sides of object under test, which are then analyzed for strain measurement and 2D shape reconstruction.
Fig. 1. 2D shape reconstruction principle based on OFDR. (a) The object under test, (b) strain sensor result under ideal conditions, (c) strain sensing result with noise, (d) the reconstruction model, (e) shape reconstruction results.
The shape reconstruction using Frenet-Serret formulas is based on the micro-element theory [11]. As shown in Fig. 1(a), the fiber under test (FUT) is arranged on the upper and lower surfaces of the object under test. When the object under test deforms in the manner shown in Fig. 1(a), the L1 segment on the upper surface of FUT is stretched, and the L2 segment on the lower surface is compressed. The relationship between the strain ε and the Rayleigh wavelength shift $\Delta \lambda $ is as follows:
where a represents the strain-wavelength shift coefficient. Under ideal conditions, the strain sensing results are shown in Fig. 1(b). At this time, the linear relationship between the strain and curvature of any element ${r_i}$ on the object under test is as follows: where i represents the sequence number of each micro element, ${k_i}$ is the curvature of ${r_i}$, $\Delta {\varepsilon _i}$ denotes the difference between strain on the upper and lower surfaces of ${r_i}$, and d represents the distance between the fiber cores on the upper and lower surfaces.Through the curvature information of each element ${r_i}$ in the reconstruction model in Fig. 1(d), the shape reconstruction results can be obtained. The micro-element ${r_i}$ with radius of curvature R in Frenet coordinate system can be obtained by calculating its tangent vector Ni and normal vector Ti, which are obtained by curvature information. The relationship between the unit normal vector Ni, unit tangent vector Ti and curvature ${k_i}$ of the element ${r_i}$ is as follows:
By substituting conditions ${k_0}$= 0, ${T_0}$ = (1,0), ${N_0}$ = (0,1) into Eq. (2), Eq. (3), and Eq. (4), the representation of the object in Frenet coordinate system can be obtained.
The representation of ${r_i}$ in Cartesian coordinate system is then obtained through coordinate transformation, which are shown in Eq. (5).
where ${x_i}$ and ${y_i}$ are the projection of ${r_i}$ on the X-axis and Y-axis, respectively, and L is the length of the element, which is related to the spatial resolution of the system.After the representation of each element in Cartesian coordinate system is obtained, the shape reconstruction result of the object under test is then obtained by splicing each element. Ideally, the shape reconstruction result is shown in S1 of Fig. 1(e), which is not disturbed by noise. Nevertheless, in practice scenarios with noise, the wavelength shift signal is disturbed, and the strain sensing result is shown in Fig. 1(c). Fake peaks that appear in Fig. 1(c) are the result of signal disturbance, which will cause curvature mutation in the shape reconstruction result, and further restrict the shape reconstruction accuracy. At this time, the 2D shape reconstruction result which stem from the strain sensing result with noise disturbance is shown in S2 of Fig. 1(e), which demonstrates the deterioration of shape reconstruction accuracy.
In order to estimate the accuracy of shape reconstruction, the end errors of the aforementioned shape reconstruction results are calculated. The end error refers to the deviation between the measured and actual end positions of the object under test, which is a standard to measure the accuracy of shape reconstruction. The end error E can be obtained by Eq. (6).
where D is the length of the object under test. $({{x_r},{y_r}} )$ and $({{x_d},{y_d}} )$ are the end coordinates of the actual shape and the measured shape, respectively. ${e_L}$ represents the distance between the end points of the measured shape and the actual shape shown in Fig. 1(e). In the process of 2D shape reconstruction, errors accumulate constantly, which can be estimated in the end error.2.2 Principle of first-order differential local filtering
The signal disturbed by noise is shown in Fig. 2(a). The most part of noise that disturbs the signal is salt-and-pepper noise, which is presented as fake peaks and mutations in the wavelength shift signal. In the strain-based shape reconstruction system, the wavelength shift signal should be continuous and smooth, and its first-order differential $d(i )$ in Eq. (7) ought to be relatively smooth.
where s(i) is the original wavelength offset signal, and i is the sequence number of the point that being processed.Fig. 2. The processing of first-order differential local filtering. (a) The signal disturbed by noise, (b) first-order differential signal, (c) first-order differential signal after filtering, (d) the signal after filtering.
Therefore, the abrupt mutations at first-order differential signal of wavelength shift are caused by noise, as shown in Fig. 2(b). Eliminating the fake peaks in the first-order differential signal can also eliminate the fake peaks in the distance domain, which is the purpose of the method proposed in this paper.
The method proposed in this paper can process the noise in the wavelength shift signal, while avoiding influence on the non-noise information in the signal. Firstly, a threshold discrimination is performed on the first-order differential signal. The values above the threshold are identified as noise, and the noise is processed by a statistical sorting filter, as shown in Eq. (8).
Statistical sorting filtering is a non-linear filtering algorithm with capability in salt-and-pepper noise reduction, while maintaining the original signal unaffected, which has a wide range of applications in the field of image processing. In this paper, a median-based statistical sorting filter is used to process the first-order differential signal. The method derives the sequence with N data points from each outlier point neighborhood, and calculates the median of the sequence. The median of the sequence takes place of the original outlier data for the signal noise mitigation, which is shown in Eq. (9).
where N is the window width of the filtering, which affects the overall smoothness. If the outlier data point is located at the beginning or the end of signal, where the remaining signal length is less than the window width N, the signal will be symmetrically extended to complete the data, as follows: where L is the signal length. After the processing of the proposed first-order differential local filtering, the fake peaks in the differential signal have been effectively suppressed, as shown in Fig. 2(c).The differential signal after the aforementioned processing is then transformed back to distance domain by integral, with the processing result $s^{\prime}$ as follows:
The wavelength shift signal after the first-order differential local filtering is shown in Fig. 2(d), with the fake peaks and mutations suppressed.
To sum up, the first-order differential local filtering can remove the salt-and-pepper noise in the signal, while maintaining the non-noise information unaffected. This method can eliminate the influence of coherent fading and spectrum leakage on the system performance, and can further improve the accuracy of 2D shape reconstruction without affecting other performance.
2.3 Simulation of first-order differential local filtering
In order to verify the feasibility of first-order differential local filtering in mitigation of the noise, a shape sensing simulation is carried out on an object under test with the length of 4.25 m. The fiber arrangement on the object under test is shown in Fig. 3, where the L1, L2, and L3 respectively represent the fibers arranged on the lower surface, junction area, and upper surface. The lengths of the L1, L2, and L3 are 4.25 m, 1.6 m and 4.25 m, respectively.
The simulated strain distribution on both surfaces is compiled according to the Frenet formulas, with several fake peaks and a mutation added as the outliers. The simulated original wavelength shift signal is shown in Fig. 4(a). Analog fake peaks with amplitudes of 2500 pm, −1700 pm, 1800 pm, 2000 pm and −400 pm is added at positions 1 m, 1.1 m, 2.1 m, 3.45 m, and 8.8 m, respectively. Additionally, a mutation with an amplitude of 2500 pm is added between the 6.2 m and 6.28 m positions.
Fig. 4. The simulation process and results. (a) The simulated original wavelength shift signal, (b) first-order differential signal, (c) first-order differential signal after filtering, (d) the simulated wavelength shift signal after filtering.
The fake peaks may lead to the reconstruction distortion of the 2D shape. The first-order differential signal of wavelength shift signal is shown in Fig. 4(b), with numerous fake peaks at the positions which are similar to the fake peaks in Fig. 4(a), and is processed with first-order differential filtering algorithm to get rid of the fake peaks. The first-order differential filtering algorithm result is shown in Fig. 4(c), where fake peaks are effectively suppressed. The processed first-order differential result is integrated to obtain corrected wavelength shift signal shown in Fig. 4(d), where fake peaks are also effectively suppressed. The simulation results show that first-order differential local filtering can effectively suppress the influence of fake peaks on the strain sensing results.
In order to demonstrate the advantages of first-order differential local filtering over distance-domain filtering, this study employed identical algorithms to process both distance-domain and first-order differential signals. The simulated original wavelength shift signal is shown in Fig. 5(a) which is identical to Fig. 4(a). The original signal is processed using the algorithm as shown in Eq. (9), with a window width of 4. The result is shown in Fig. 5(b), where mutation is not completely eliminated between the 6.2 m and 6.28 m positions. And there are distortions in the previous fake peak positions at 3.45 m and 8.8 m. If the window width is increased, because filtering is applied to the entire signal, the unaffected parts of the signal may experience over-smoothing and distortion. Nevertheless, using first-order differential local filtering can effectively avoid this situation. The result after first-order differential filtering with a window width of 4 is shown in Fig. 5(c). It can be observed that all noise is well suppressed, with no significant distortion. The simulation results show that compared to distance-domain filtering, first-order differential local filtering can more effectively remove noise.
Fig. 5. Comparison of filtering results between distance domain and differential domain. (a) The simulated original wavelength shift signal, (b) the first-order differential local filtering result with a window width of 4.
3. Experimental results
Aiming at verifying the feasibility of first-order differential local filtering in improvement of shape sensing accuracy, an OFDR-based shape sensing experimental system is established with configuration shown in Fig. 6.
The tunable laser source (TLS) used is the SANTEC TSL-710 with a sweep range of 1480-1640 nm, a linewidth of 100 MHz, and a maximum output power of 13 dBm. The bandwidth of the photodetector (PD) is 120 MHz. The specific parameters of the main devices are shown in the Table 1.
Table 1. Parameter setting of 2D shape reconstruction system
When the system is activated, the TLS emits a wavelength swept laser with a scanning speed of 100 nm/s and a scanning range from 1545 nm to 1555 nm. The swept laser passes through a coupler (Coupler1) with a splitting ratio of 99:1 into two Mach-Zehnder interferometers separately. 1% of the laser enters the auxiliary interferometer, which consists of 2 delay fibers with different optical paths. The effect of the sweeping nonlinearity can be compensated by the signal from auxiliary interferometer. 99% of the laser enters the main interferometer, which includes a coupler (Coupler4) with splitting ratio of 90:10, a coupler (Coupler5) with splitting ratio of 50:50, a polarization controller (PC), a polarization beam splitter (PBS), an optical circulator (OC), and the object under test. 90% of the laser is injected into the measurement arm and 10% is injected into the reference arm. The laser in the measurement arm enters the FUT through the OC, the generated Rayleigh backscattering light beats with the laser from the reference arm. The beat signal is converted into an electrical signal by the PD, which is then sampled and processed by the upper computer.
The object under test is a homogeneous flexible steel sheet with length of 50 cm and thickness of 0.13 mm. The FUT is arranged on the object under test with configuration shown in Fig. 7. The FUT is divided into 3 segments, which are the 3 segments in shown Fig. 7(b) with the color of red, yellow, and blue. The segment between 10.00 m and 10.50 m of FUT (the blue segment) is tightly attached and glued to the lower surface of the object under test, and the segment between 10.83 m and 11.33 m (the red segment) is tightly attached and glued to the upper surface of the object under test. The second segment with yellow color is the transition segment between the first and third segments, which is 0.33 m long, and is relaxed without strain.
Fig. 7. The original shape of the object under test and its fiber arrangement. (a) The object under test, (b) the fiber arrangement of object under test.
3.1 Calibration of strain-wavelength shift coefficient
In order to calibrate the strain-wavelength shift coefficient, a calibration with 10 m long FUT was conducted. The segment between 4.5 m and 4.9 m of FUT was set to be the strain area. Strains ranging from 0 to 1000µε were applied to this area, with the step of 100µε. In order to make an accurate calibration, the RWS measurement was conducted 10 times independently, with the average value as the sensing results, which are shown in Fig. 8(a).
Fig. 8. The calibration results of strain-wavelength shift coefficient. (a) The strain sensing results, (b) the result of linear regression fitting.
The RWS in the strain area shows a linear relationship with the applied strain. A linear fitting was performed between the strain and RWS, with fitting result shown in Fig. 8(b). The strain-wavelength shift coefficient is 1.07 pm/µε, and the R-squared is 0.99969, indicating an excellent linear fit.
3.2 Experiment of 2D shape reconstruction accuracy improvement
After the strain-wavelength shift coefficient calibration, the shape reconstruction experiment was conducted with the object under test bent in 3 different shapes. On the optical test bench, the starting point of the object under test was secured by screws, a coordinate paper was used to record each position coordinates of object under test before and after bending. The preset shapes are shown in Fig. 9.
Fig. 9. The 3 different shapes of the object under test. (a) Physical diagram, (b) schematic diagram.
The wavelength shift signal of the up-bend shape is shown in Fig. 10(a), with the demodulated strain sensing results shown in Fig. 10(b). The fiber arranged on the lower surface is stretched, leading to a positive strain value in the first half. Conversely, the fiber arranged on the upper surface is compressed, leading to a negative strain value in the second half. The unprocessed strain sensing result shown in Fig. 10(b) has numerous fake peaks, which are the source of shape reconstruction errors.
Fig. 10. The wavelength shift signal and original strain sensing result. (a) The wavelength shift signal, (b) the original strain sensing result.
The original strain signal in Fig. 10(b) was performed with first-order differential. The first-order differential signal is shown in Fig. 11(a). The first-order differential signal is processed using the proposed local filtering, with the filtered result shown in Fig. 11(b). Numerous fake peaks appear in the signal before local filtering in Fig. 11(a), while no fake peaks can be found in the signal after local filtering in Fig. 11(b).
Fig. 11. The first-order differential signals before and after processing. (a) The first-order differential signal before processing, (b) the first-order differential signal after processing.
After the local filtering, the processed first-order differential signal is integrated to obtain the corrected strain sensing result, as shown in Fig. 12(a). Compared to the strain signal before processing shown in Fig. 10(b), the fake peaks on strain signal after processing have been effectively suppressed.
Fig. 12. The corrected strain sensing result and shape reconstruction results. (a) The corrected strain sensing result, (b) the shape reconstruction results.
In order to verify the shape reconstruction accuracy enhancement of the proposed first-order differential local filtering, 2D shape reconstruction based on the aforementioned Frenet formulas were performed to both of the strain sensing results before and after processing. The reconstruction results are compared in Fig. 12(b) with the reference of the actual shape. The curvature mutation in the process of shape reconstruction is compensated, which demonstrates the feasibility of the proposed method in improvement of shape reconstruction accuracy.
In addition to the up-bend shape, the wavelength shift signals of the down-bend shape and the arch shape are also demodulated as shown in Fig. 13(a) and (b), respectively.
Fig. 13. The wavelength shift signals of the down-bend shape and the arch shape. (a) The signal of down-bend shape, (b) the signal of arch shape.
The original strain sensing results and the corrected strain sensing results are shown in Fig. 14 and Fig. 15, respectively. The reconstructed down-bend shape before and after processing are show in Fig. 16(a). While the reconstructed arch shape before and after processing are show in Fig. 16(b).
Fig. 14. The original strain sensing results of the down-bend shape and the arch shape. (a) The result of down-bend shape, (b) the result of arch shape.
Fig. 15. The corrected strain sensing results of the down-bend shape and the arch shape. (a) The result of down-bend shape, (b) the result of arch shape.
Fig. 16. The shape reconstruction results of the down-bend shape and the arch shape. (a) The results of down-bend shape, (b) the results of arch shape.
The error of each point on the three sets of original shapes and the corrected shapes are compared with the actual shape, with the results shown in Fig. 17. The end errors of the three 2D shapes reconstructed based on the original strain sensing data are 2.33%, 2.97%, and 1.06%, respectively. While the errors of the three 2D shapes reconstructed based on the corrected strain sensing data by the first-order differential local filtering are 0.25%, 0.78%, and 0.20%, respectively. The experimental results show that after the first-order differential local filtering algorithm, the end error of shape reconstruction is reduced, and the reconstructed shape resembles the actual shape more closely, which demonstrates the capability in enhancement of 2D shape reconstruction accuracy.
Fig. 17. The errors of each point before and after correction. (a) Error of up-bend shape, (b) error of down-bend shape, (c) error of arch shape.
4. Conclusion
This paper proposed the first-order differential local filtering for the enhancement of OFDR 2D shape reconstruction accuracy. The process of OFDR 2D shape reconstruction was analyzed with the principle of first-order differential local filtering. A simulation was performed to demonstrate feasibility of the proposed method. A set of OFDR-based strain and shape sensing experiments were conducted on an OFDR 2D shape reconstruction system for 3 different shape reconstruction. The experimental results show that the end errors of the aforementioned 3 shape reconstruction results are reduced from 2.33%, 2.97%, 1.07% to 0.25%, 0.78%, 0.20%, respectively, which demonstrates the improvement of 2D shape reconstruction accuracy.
The proposed approach can improve the accuracy of 2D shape reconstruction without introducing additional hardware devices or increasing time complexity. This approach can be applied to scenarios requiring high real-time performance and accuracy, such as in the medical field, equipment manufacturing, high-precision measurement and so on.
Funding
Key Research and Development (R&D) Projects of Shanxi Province (202102150101005, 202202130501004, 202302150101014); Fundamental Research Program of Shanxi Province (202303021221032).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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