# Error Correction for FSI-Based System without Cooperative Target Using an Adaptive Filtering Method and a Phase-Matching Mosaic Algorithm

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## Abstract

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## Featured Application

**Absolute distance measurement and surface profiling.**

## Abstract

## 1. Introduction

## 2. Theory

#### 2.1. The Principle of FSI Using Frequency Sampling Method

#### 2.2. The Influence of the Erroneous Sampling Points in the Auxiliary Interferometer

#### 2.3. The CRLB When There Are Interfering Signals and Noise in the Measurement Interferometer

^{2}$,{I}_{i}\left(k\right)$ denotes the interfering signals, and $\zeta $ is $\frac{{\pi \tau}_{m}}{{\tau}_{a}}$, $\phi $ is ${2\pi f}_{0}{\tau}_{m}$. The interfering signals ${I}_{i}\left(k\right)$ are caused by the reflected lights from the circulator and the fiber end in Figure 1.

^{2}.

#### 2.4. The Adaptive Filtering Method, the Phase-Matching Mosaic Algorithm, and the Segmentation Mosaic Algorithm

- (1)
- Firstly, we make use of the Hanning window and the wavelet threshold filtering to depress the noises (such as the GWN) and the interfering signals in the auxiliary interferometer and measurement interferometer.
- (2)
- Then, we get the sampling points (all of the maximum and minimum extreme points of the auxiliary interference signal) and set appropriate thresholds to delete most of the erroneous sampling points caused by mode-hopping signals.
- (3)
- Lastly, according to the rule that the maximum and minimum values occur alternately, we remove the rest of the erroneous sampling points.

_{k}) is the amplitude of the extreme point, ${A}_{ma}$ is the maximum amplitude of the auxiliary interference signal, ${A}_{mi}$ is the minimum amplitude of the auxiliary interference signal, k is the index of the extreme points, and S is the length of $Ep$. Then, we set the time interval thresholds to delete part of the erroneous sampling points, and the rest of the sampling points can be written as:

_{k}) is the time interval of the extreme point, ${\Delta T}_{me}$ denotes the mean time interval of all of the points in ${Ep}_{1}$, and K is the length of ${Ep}_{1}$.

## 3. Simulation

## 4. Experiment and Analysis

^{2}/km, and the tuning bandwidth was 10 nm, so the distance offset was 2.3 μm.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Barwood, G.P.; Gill, P.; Rowley, W.R.C. High-accuracy length metrology using multiple-stage swept-frequency interferometry with laser diodes. Meas. Sci. Technol.
**1999**, 9, 1036–1041. [Google Scholar] [CrossRef] - Stejskal, A.; Stone, J.A.; Howard, L. Absolute interferometry with a 670-nm external cavity diode laser. Appl. Opt.
**1999**, 38, 5981–5994. [Google Scholar] - Le, F.S.; Salvadé, Y.; Mitouassiwou, R.; Favre, P. Radio frequency controlled synthetic wavelength sweep for absolute distance measurement by optical interferometry. Appl. Opt.
**2008**, 47, 3027–3031. [Google Scholar] - Gao, R.; Wang, L.; Teti, R.; Dornfeld, D.; Kumara, S.; Mori, M.; Helu, M. Cloud-enabled prognosis for manufacturing. CIRP Ann. Manuf. Technol.
**2015**, 64, 749–772. [Google Scholar] [CrossRef][Green Version] - Abou-Zeid, A.; Pollinger, F.; Meiners-Hagen, K.; Wedde, M. Diode-laser-based high-precision absolute distance interferometer of 20 m range. Appl. Opt.
**2009**, 48, 6188–6194. [Google Scholar] - Kinder, T.; Salewski, K.D. Absolute distance interferometer with grating-stabilized tunable diode laser at 633 nm. J. Opt. Apure Appl. Opt.
**2002**, 4, S364–S368. [Google Scholar] [CrossRef] - Dale, J.; Hughes, B.; Lancaster, A.J.; Lewis, A.J.; Reichold, A.J.H.; Warden, M.S. Multi-channel absolute distancemeasurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorptioncells. Opt. Express
**2014**, 22, 24869–24893. [Google Scholar] [CrossRef] [PubMed] - Prellinger, G.; Meinershagen, K.; Pollinger, F. Spectroscopically in situ traceable heterodyne frequency-scanning interferometry for distances up to 50 m. Meas. Sci. Technol.
**2015**, 26, 084003. [Google Scholar] [CrossRef] - Wang, L.T.; Iiyama, K.; Tsukada, F.; Yoshida, N.; Hayashi, K.I. Loss measurement in optical waveguide devices by coherentfrequency-modulated continuous-wave reflectometry. Opt. Lett.
**1993**, 18, 1095–1097. [Google Scholar] [CrossRef] [PubMed] - Zheng, K.; Liu, B.; Huang, C.; Brezinski, M.E. Experimental confirmation of potential swept source optical coherence tomography performance limitations. Appl. Opt.
**2008**, 47, 6151–6158. [Google Scholar] [CrossRef] [PubMed][Green Version] - Baumann, E.; Giorgetta, F.R.; Deschênes, J.D.; Swann, W.C.; Coddington, I.; Newbury, N.R. Comb-calibrated laser ranging for three-dimensional surface profiling with micrometer-level precision at a distance. Opt. Express
**2014**, 22, 24914–24928. [Google Scholar] [CrossRef] [PubMed] - Ahn, T.J.; Kim, D.Y. Analysis of nonlinear frequency sweep in high-speed tunable laser sources using a self-homodyne measurement and Hilbert transformation. Appl. Opt.
**2007**, 46, 2394–2400. [Google Scholar] [CrossRef] [PubMed] - Coddington, I.; Swann, W.C.; Nenadovic, L. Rapid and precise absolute distance measurements at long range. Nat. Photonics
**2009**, 3, 351–356. [Google Scholar] [CrossRef] - Barber, Z.W.; Babbitt, W.R.; Kaylor, B.; Reibel, R.R.; Roos, P.A. Accuracy of active chirp linearization for broadband frequency modulated continuous wave ladar. Appl. Opt.
**2010**, 49, 213–219. [Google Scholar] [CrossRef] [PubMed] - Yuksel, K.; Wuilpart, M.; Mégret, P. Analysis and suppression of nonlinear frequency modulation in an optical frequency-domain reflectometer. Opt. Express
**2009**, 17, 5845–5851. [Google Scholar] [CrossRef] [PubMed] - Lu, C.; Liu, G.D.; Liu, B.G. Absolute distance measurement system with micron-grade measurement uncertainty and 24 m range using frequency scanning interferometry with compensation of environmental vibration. Opt. Express
**2016**, 24, 30215–30224. [Google Scholar] [CrossRef] [PubMed] - Deng, Z.; Liu, Z.; Li, B.; Liu, Z. Precision improvement in frequency scanning interferometry based on suppressing nonlinear optical frequency sweeping. Opt. Rev.
**2015**, 22, 724–730. [Google Scholar] [CrossRef] - Ahn, T.J.; Lee, J.Y.; Kim, D.Y. Suppression of nonlinear frequency sweep in an optical frequency-domain reflectometer by use of Hilbert transformation. Appl. Opt.
**2005**, 44, 7630–7634. [Google Scholar] [CrossRef] [PubMed] - Satyan, N.; Vasilyev, A. Phase-locking and coherent power combining of broadband linearly chirped optical waves. Opt. Express
**2012**, 20, 25213–25227. [Google Scholar] [CrossRef] [PubMed] - Lippok, N.; Coen, S.; Nielsen, P. Dispersion compensation in Fourier domain optical coherence tomography using the fractional Fourier transform. Opt. Express
**2012**, 20, 23398–23413. [Google Scholar] [CrossRef] [PubMed] - Liu, G.; Xu, X.; Liu, B.F. Dispersion compensation method based on focus definition evaluation functions for high-resolution laser frequency scanning interference measurement. Opt. Commun.
**2017**, 386, 57–64. [Google Scholar] - Gifford, D.K.; Soller, B.J.; Wolfe, M.S. Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion. Appl. Opt.
**2005**, 44, 7282–7286. [Google Scholar] [CrossRef] [PubMed] - Gart, J. An extension of the Cramér–Rao inequality. Ann. Math. Stat.
**1958**, 29, 367–380. [Google Scholar] [CrossRef] - Hussain, M. Mammogram Enhancement Using Lifting Dyadic Wavelet Transform and Normalized Tsallis Entropy. J. Comput. Sci. Technol.
**2014**, 29, 1048–1057. [Google Scholar] [CrossRef] - Guang, S.; Wen, W. Single laser complex method to improve the resolution of FMCW laser ranging. J. Infrared Millim. Waves
**2016**, 35, 363–367. [Google Scholar]

**Figure 1.**Schematic diagram of our frequency scanning interferometry (FSI)-based system and the experimental setup.

**Figure 2.**(

**a**) Extreme points in the experiment when there are noises in the auxiliary interferometer. (

**b**) Extreme points in the experiment when there is the mode-hopping signal in the auxiliary interferometer.

**Figure 3.**(

**a**) The simulation result of 10 erroneous sampling points on accuracy when the ideal value of P is 9000 and N is 60,000. (

**b**) The simulation result of 100 erroneous sampling points on accuracy when the ideal value of P is 9000 and N is 60,000.

**Figure 7.**Part (

**a**,

**b**) respectively represent the experiment results of not using andusing the adaptive filtering method and the mosaic algorithm with the same data.

**Figure 9.**Distance residual between the measured results of our FSI-based system and the laser interferometer.

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**MDPI and ACS Style**

Xiong, X.-T.; Qu, X.-H.; Zhang, F.-M. Error Correction for FSI-Based System without Cooperative Target Using an Adaptive Filtering Method and a Phase-Matching Mosaic Algorithm. *Appl. Sci.* **2018**, *8*, 1954.
https://doi.org/10.3390/app8101954

**AMA Style**

Xiong X-T, Qu X-H, Zhang F-M. Error Correction for FSI-Based System without Cooperative Target Using an Adaptive Filtering Method and a Phase-Matching Mosaic Algorithm. *Applied Sciences*. 2018; 8(10):1954.
https://doi.org/10.3390/app8101954

**Chicago/Turabian Style**

Xiong, Xing-Ting, Xing-Hua Qu, and Fu-Min Zhang. 2018. "Error Correction for FSI-Based System without Cooperative Target Using an Adaptive Filtering Method and a Phase-Matching Mosaic Algorithm" *Applied Sciences* 8, no. 10: 1954.
https://doi.org/10.3390/app8101954