# An Exact Formula for Calculating Inverse Radial Lens Distortions

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Previous Work

#### 2.1. Calibration Approach

#### 2.2. Inverse Radial Distortion

- Approximation. Mallon [35], Heikkilä [30,36] and then Wei and Ma [37] proposed inverse approximations of a Taylor expansion including first order derivatives. According to Mallon and Welhan, “This is sometimes assumed to be the actual model and indeed suffices for small distortion levels.” [35]. A global approach, inverse distortion plus image interpolation, is presented in a patent held by Adobe Systems Incorporated [38].
- Look-up table. All the pixels are previously computed and a look-up table is generated to store the mapping (as for example in OpenCV).

Algorithm 1 Iterative algorithm to compute the inverse distortion |

Require: point ${P}_{n}$ |

${P}_{c}={P}_{n}$ |

repeat |

$r=\left|\right|{P}_{c}\left|\right|$ |

$dr=1+{k}_{2}{r}^{2}+{k}_{4}{r}^{4}+...$ |

${P}_{c}={P}_{n}/dr$ |

until Convergence of ${P}_{c}$ |

return${P}_{c}$ |

## 3. Exact Formula for Inverse Radial Distortion: An Original Approach

#### 3.1. Lens Distortion Models

#### 3.2. General Framework

#### Radial distortion

**Proposition 1.**

#### Remark 1

#### Remark 2

## 4. Results and Experimental Section

- First, we begin by testing the accuracy of the inverse formula by applying the forward/inverse formula recursively within a loop. Hence, the inverse of the inverse radial distortion is computed 10,000 times and compared to the original distortion coefficients.
- Then, for a given calibrated camera, we compute the residual after applying and compensating the distortion along the camera frame. A residual curve shows the results of the inverse camera in all the frame.
- The last experiment is the use of inverse distortion model on a image made with a metric camera built with a large eccentricity (a film-based Wild P32 camera), without distortion. We apply a strong distortion and then compensate it and finally compare it to the original image.

#### 4.1. Inverse Distortion Loop

#### 4.2. Inverse Distortion Computation onto a Frame

#### 4.3. Inverse Distortion Computation on an Image Done with a Metric Camera

## 5. Conclusions and Discussion

#### 5.1. A Bridge between PhotoModeler and Agisoft for Radial Distortion

- Given ${k}_{1}...{k}_{3}$ as the coefficients of the polynom modeling the radial distortion in PhotoModeler. Note that PhotoModeler uses only ${k}_{1}...{k}_{3}$ coefficient.
- ${k}_{1}={k}_{1}*focalm{m}^{2}$${k}_{2}={k}_{2}*focalm{m}^{4}$${k}_{3}={k}_{3}*focalm{m}^{6}$

- Given ${k}_{1}...{k}_{4}$ as the coefficient of the polynom modeling the radial distortion in PhotoScan.
- ${k}_{1}={k}_{1}/focalm{m}^{2}$${k}_{2}={k}_{2}/focalm{m}^{4}$${k}_{3}={k}_{3}/focalm{m}^{6}$${k}_{4}={k}_{4}/focalm{m}^{8}$
- compute ${k}_{1}^{\prime}...{k}_{3}^{\prime}$ (PhotoModeler uses ${k}_{3}$) according to Appendix C.

#### 5.2. Possible Limitations

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## Appendix B

Algorithm B1 Computation of $p(j,k)$ |

Require: coefficients ${a}_{1},{a}_{2},{a}_{3},{a}_{4}$, integer N |

Define an array p of size $N\times 2N-1$ |

for $k=0:2(N-1)$ do |

$p(0,k)=1$ |

end for |

for $k=1:2(N-1)$ do |

for $j=1:2(N-1)$ do |

$p(j,k)=p(j,k-1)+{a}_{1}p(j-1,k-1)+{a}_{2}p(j-2,k-1)+{a}_{3}p(j-3,k-1)+{a}_{4}p(j-4,k-1)$ |

▹ With $p(j,k)=0$ as soon as $j<0$ or $k\le 0$ |

end for |

end for |

return p |

Algorithm B2 Computation of coefficients ${b}_{n}$ |

Require: coefficients ${a}_{1},{a}_{2},{a}_{3},{a}_{4}$, integer N |

Define an array $q=[0,0,0,0]$ |

Define an array b of size N |

$b\left(0\right)=1$ |

Compute p with Algorithm B1 |

for $n=1:N$ do |

$tmp=0$ |

for $k=1:4$ do |

$tmp=tmp-{a}_{k}q(k-1)$ |

end for |

$b\left(n\right)=b\left(n\right)+tmp$ |

for $k=n/9:(n-1)$ do |

$b\left(n\right)=b\left(n\right)-b\left(k\right)p(n-k,2k)$ |

end for |

$q(1:3)=q(0:2)$ and $q\left(0\right)=tmp$ |

end for |

return b |

## Appendix C

$\begin{array}{cc}{b}_{1}\hfill & =-{k}_{1}\hfill \end{array}$ |

$\begin{array}{cc}{b}_{2}\hfill & =3{k}_{1}^{2}-{k}_{2}\hfill \end{array}$ |

$\begin{array}{cc}{b}_{3}\hfill & =-12{k}_{1}^{3}+8{k}_{1}{k}_{2}-{k}_{3}\hfill \end{array}$ |

$\begin{array}{cc}{b}_{4}\hfill & =55{k}_{1}^{4}-55{k}_{1}^{2}{k}_{2}+5{k}_{2}^{2}+10{k}_{1}{k}_{3}-{k}_{4}\hfill \end{array}$ |

$\begin{array}{cc}{b}_{5}\hfill & =-273{k}_{1}^{5}+364{k}_{1}^{3}{k}_{2}-78{k}_{1}{k}_{2}^{2}-78{k}_{1}^{2}{k}_{3}+12{k}_{2}{k}_{3}+12{k}_{1}{k}_{4}\hfill \end{array}$ |

$\begin{array}{cc}{b}_{6}\hfill & =1428{k}_{1}^{6}-2380{k}_{1}^{4}{k}_{2}+840{k}_{1}^{2}{k}_{2}^{2}-35{k}_{2}^{3}+560{k}_{1}^{3}{k}_{3}-210{k}_{1}{k}_{2}{k}_{3}+7{k}_{3}^{2}-105{k}_{1}^{2}{k}_{4}+14{k}_{2}{k}_{4}\hfill \end{array}$ |

$\begin{array}{cc}{b}_{7}\hfill & =-7752{k}_{1}^{7}+15504{k}_{1}^{5}{k}_{2}-7752{k}_{1}^{3}{k}_{2}^{2}+816{k}_{1}{k}_{2}^{3}-3876{k}_{1}^{4}{k}_{3}+2448{k}_{1}^{2}{k}_{2}{k}_{3}-136{k}_{2}^{2}{k}_{3}-136{k}_{1}{k}_{3}^{2}\hfill \\ & +816{k}_{1}^{3}{k}_{4}-272{k}_{1}{k}_{2}{k}_{4}+16{k}_{3}{k}_{4}\hfill \end{array}$ |

$\begin{array}{cc}{b}_{8}=\hfill & 43263{k}_{1}^{8}-100947{k}_{1}^{6}{k}_{2}+65835{k}_{1}^{4}{k}_{2}^{2}-11970{k}_{1}^{2}{k}_{2}^{3}+285{k}_{2}^{4}+26334{k}_{1}^{5}{k}_{3}-23940{k}_{1}^{3}{k}_{2}{k}_{3}\hfill \\ & +3420{k}_{1}{k}_{2}^{2}{k}_{3}+1710{k}_{1}^{2}{k}_{3}^{2}-171{k}_{2}{k}_{3}^{2}-5985{k}_{1}^{4}{k}_{4}+3420{k}_{1}^{2}{k}_{2}{k}_{4}-171{k}_{2}^{2}{k}_{4}-342{k}_{1}{k}_{3}{k}_{4}+9{k}_{4}^{2}\hfill \end{array}$ |

$\begin{array}{cc}{b}_{9}\hfill & =-246675{k}_{1}^{9}+657800{k}_{1}^{7}{k}_{2}-531300{k}_{1}^{5}{k}_{2}^{2}+141680{k}_{1}^{3}{k}_{2}^{3}-8855{k}_{1}{k}_{2}^{4}-177100{k}_{1}^{6}{k}_{3}\hfill \\ & +212520{k}_{1}^{4}{k}_{2}{k}_{3}-53130{k}_{1}^{2}{k}_{2}^{2}{k}_{3}+1540{k}_{2}^{3}{k}_{3}-17710{k}_{1}^{3}{k}_{3}^{2}+4620{k}_{1}{k}_{2}{k}_{3}^{2}-70{k}_{3}^{3}+42504{k}_{1}^{5}{k}_{4}\hfill \\ & -35420{k}_{1}^{3}{k}_{2}{k}_{4}+4620{k}_{1}{k}_{2}^{2}{k}_{4}+4620{k}_{1}^{2}{k}_{3}{k}_{4}-420{k}_{2}{k}_{3}{k}_{4}-210{k}_{1}{k}_{4}^{2}\hfill \end{array}$ |

## References

- Arfaoui, A. Geometric image rectification: A review of most commonly used calibration patterns. Int. J. Signal Image Process. Issues
**2015**, 1, 1–8. [Google Scholar] - Rahul, S.; Nayar, S.K. Nonmetric calibration of Wide-Angle Lenses and Polycameras. IEEE Trans. Pattern Anal. Mach. Intell.
**2000**, 8, 1172–1178. [Google Scholar] - Michael, C.; Wolfgang, K.; Jan, S.; Norbert, H.; Silvia, N.; Wallgrun, J. Data Capture. In Handbook of Geographic Information; Wolfgang, K., Danko, D.M., Eds.; Publishing House: Berlin, Germany, 2012; pp. 212–297. [Google Scholar]
- Ben, T.; Murray, D.W. The impact of radial distortion on the self-calibration of rotating cameras. Comput. Vis. Image Underst.
**2004**, 96, 17–34. [Google Scholar] - Conrady, A.E. Lens-systems, Decentered. Mon. Not. R. Astron. Soc.
**1919**, 79, 384–390. [Google Scholar] [CrossRef] - Brown, D.C. Close-range camera calibration. Photom. Eng.
**1971**, 37, 855–866. [Google Scholar] - Brown, D.C. Decentering Distortion of Lenses. Photom. Eng.
**1966**, 32, 444–462. [Google Scholar] - Santana-Cedrés, D.; Gómez, L.; Alemán-Flores, M.; Salgado, A.; Esclarín, J.; Mazorra, L.; Álvarez, L. An Iterative Optimization Algorithm for Lens Distortion Correction Using Two-Parameter Models. IPOL J. Image Process. Line
**2015**, 1. [Google Scholar] [CrossRef] - Tordoff, B.; Murray, D.W. Violating rotating camera geometry: The effect of radial distortion on self-calibration. In Proceedings of the 15th International Conference on Pattern Recognition, Barcelona, Spain, 3 September 2000; pp. 423–427.
- Dougherty, G. Digital Image Processing for Medical Applications; Editor, F., Meditor, A., Eds.; Cambridge University Press: New York, NY, USA, 2009. [Google Scholar]
- De Villiers, J.P.; Wilhelm, L.F.; Geldenhuys, R. Centi-pixel accurate real-time inverse distortion correction. In Proceedings of the Optomechatronic Technologies, 7266 726611-1, San Diego, CA, USA, 17 November 2008.
- Mondrian, P. Public Domain, 2015. Available online: http://www.aventdudomainepublic.org/mondrian (accessed on 25 May 2016).
- Papadaki, A.I.; Georgopoulos, A. Development, comparison, and evaluation of software for radial distortion elimination. In Proceedings of the SPIE 9528, Videometrics, Range Imaging, and Applications XIII, Munich, Germany, 21 June 2015. [CrossRef]
- Kasser, M. Photogrammetrie et vision par ordinateur. XYZ
**2008**, 12, 49–54. [Google Scholar] - Eos Systems Inc. PhotoModeler Software. 2016. Available online: http://www.photomodeler.com/index.html (accessed on 29 May 2016).
- Agisoft. PhotoScan Software. 2016. Available online: http://www.agisoft.com/ (accessed on 29 May 2016).
- OpenCV. Toolbox. 2016. Available online: http://opencv.org// (accessed on 29 May 2016).
- Fraser, C.S. Photogrammetric Camera Component Calibration: A Review of Analytical Techniques. In Calibration and Orientation of Cameras in Computer Vision; Gruen, A., Huang, T., Eds.; Springer Verlag: Berlin, Germany, 2001; pp. 95–119. [Google Scholar]
- Zhang, Z. Camera Calibration. In Emerging Topics in Computer Vision; Medioni, G., Kang, S.B., Eds.; Prentice Hall Technical References: Upper Saddle River, NJ, USA, 2004; pp. 1–37. [Google Scholar]
- Shortis, M. Calibration Techniques for Accurate Measurements by Underwater Camera Systems. Sensors
**2015**, 15, 30810–30826. [Google Scholar] [CrossRef] [PubMed] - Hartley, R.I. Camera Calibration Using Line Correspondences. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Seattle, WA, USA, 21–23 June 1994; pp. 361–366.
- Hartley, R.I. Projective reconstruction from line correspondences. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR ’94), Seattle, WA, USA, 21–23 June 1994; pp. 903–907.
- Devernay, F.; Faugeras, O.D. Straight lines have to be straight. Mach. Vis. Appl.
**2001**, 13, 14–24. [Google Scholar] [CrossRef] - Nomura, Y.; Sagara, M.; Naruse, H.; Ide, A. Self-calibration of a general radially symmetric distortion model. IEEE Trans. Pattern Anal. Mach. Intell.
**2002**, 14, 1095–1099. [Google Scholar] [CrossRef] - Claus, D.; Fitzgibbon, A.W. A Plumbline Constraint for the Rational Function Lens Distortion Model. In Proceedings of the British Machine Vision Conference, Oxford, UK, 15 September 2005; pp. 99–108.
- Tardif, J.-P.; Sturm, P.; Roy, S. Self-calibration of a general radially symmetric distortion model. In Proceedings of the 9th European Conference on Computer Vision, Graz, Austria, 7–13 May 2006.
- Rosten, E.; Loveland, R. Camera distortion self-calibration using the plumb-line constraint and minimal Hough entropy. Mach. Vis. Appl.
**2009**, 22, 77–85. [Google Scholar] [CrossRef] - Slama, C.C. Manual of Photogrammetry 4 Edition; Slama, C.C., Theurer, C., Hendrikson, S.W., Eds.; American Society of Photogrammetry: Falls Church, VA, USA, 1980; pp. 32–58. [Google Scholar]
- Atkinson, K. Close Range Photogrammetry and Machine Vision; Atkinson, F.K., Fryer, J.G., Eds.; Whittles Publishing: Dunbeath, UK, 2001. [Google Scholar]
- Heikkilä, J.; Silven, O. A Four-Step Camera Calibration Procedure with Implicit Image Correction. CVPR97
**1997**, 22, 1106–1112. [Google Scholar] - Basu, A.; Licardie, S. Alternative models for fish-eye lenses. Pattern Recognit. Lett.
**1995**, 16, 433–441. [Google Scholar] [CrossRef] - Hughes, C.; Glavin, M.; Jones, E.; Denny, P. Review of geometric distortion compensation in fish-eye cameras. In Proceedings of the Signals and Systems Conference, Galway, Irish, 18–19 June 2008; pp. 162–167.
- Hughes, C.; Glavin, M.; Jones, E.; Denny, P. Accuracy of fish-eye lens models. Appl. Opt.
**2010**, 49, 3338–3347. [Google Scholar] [CrossRef] [PubMed] - De Villiers, J.P.; Nicolls, F. Application of neural networks to inverse lens distortion modelling. In Proceedings of the 21st Annual Symposium of the Pattern Recognition Society of South Africa (PRASA), Stellenbosch, South Africa, 22–23 November 2010; pp. 63–68.
- Mallon, J.; Whelan, P.F. Precise Radial Un-distortion of Images. In Proceedings of the 17th International Conference on Pattern Recognition (ICPR 04), Cambridge, UK, 23–26 August 2004.
- Heikkilä, J. Geometric Camera Calibration Using Circular Control Points. IEEE Trans. Pattern Anal. Mach. Intell.
**2000**, 22, 1066–1077. [Google Scholar] [CrossRef] - Wei, G.-Q.; Ma, S. Implicit and explicit camera calibration: Theory and experiments. IEEE Trans. Pattern Anal. Mach. Intell.
**2002**, 16, 469–480. [Google Scholar] - Jin, H. Method and Apparatus for Removing General Lens Distortion From Images. US Patent 8,265,422, 23 September 2014. [Google Scholar]
- Han, D. Real-Time Digital Image Warping for Display Distortion Correction. In Proceedings of the Second International Conference on Image Analysis and Recognition ICIAR’05, Toronto, ON, Canada, 2005; pp. 1258–1265.
- Abeles, P. Inverse Radial Distortion Formula. Less Than Optimal. Available online: http://peterabeles.com/blog/?p=73 (accessed on 29 May 2016).
- Alvarez, L.; Gómez, L.; Sendra, J.R. An Algebraic Approach to Lens Distortion by Line Rectification. J. Math. Imaging Vis.
**2009**, 35. [Google Scholar] [CrossRef] - Drap, P.; Grussenmeter, P.; Hartmann-Virnich, A. Photogrammetric stone-bystone survey and archeological knowledge: An application on the Romanesque Priory Church Notre-Dame d’ Aleyrac (Provence, France). In Proceedings of the VAST2000 Euroconference, Prato, Italy, 24–25 November 2000; pp. 139–145.
- Ramis, J.P. Les séries k-sommables et leurs applications, Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory. In Proceedings of the Colloquium Held at Les Houches, Centre de Physique, Berlin, Germany, 23 September 1979; pp. 178–199.
- Ramis, J.P. Séries divergentes et procédés de resommation. Available online: http://www.math.polytechnique.fr/xups/xups91.pdf (accessed on 29 May 2016).

**Figure 2.**In the center, a painting from Piet Mondrian [12] (which is now in the public domain since 1 January 2016); on the left, the painting with a barrel effect; and on the right, the same image with pincushion distortion.

**Figure 3.**Iterative method applied to a Nikon D700 camera with a 14mm lens. Along the frame diagonal the points are first compensated by what is a normal process of the radial distortion using Equation (1) and then the distortion is applied with the iterative process [40] and the result compared to the original point. On Y-axis, the distance between the original point and the computed reverse point.

**Figure 4.**Inverse residual distortion with ${k}_{1}..{k}_{9}$ Coefficient Computed for the entire frame. (

**a**) Inverse residual distortion with ${k}_{1}...{k}_{4}$ Coefficient Computed from 0 to $maxX$ frame; (

**b**) Inverse residual distortion with ${k}_{1}..{k}_{9}$ Coefficient Computed from 0 to $maxX$ frame.

**Figure 5.**Inverse residual distortion with ${k}_{1}...{k}_{9}$ coef. Computed on the entire frame. (

**a**) Axonometric view; (

**b**) Top view.

**Figure 6.**Distortion-less image taken using a small-format Wild P32 metric camera and application of an artificial distortion. (

**a**) On the left, original image taken with P32 Wild metric camera; (

**b**) On the right, pincushion distortion applied on this original image whithout interpolation. As images have not the same pixel size some vacant pixel are visible as black lines (see Hughues [32]). These lines surround the distortion center, here located on a fiducial mark, strongly shifted from the image center.

**Figure 7.**Distortion compensation applied on the pincushion image obtained in Figure 6b. In green the image corrected by inverse distortion; in black and white the original image.

**Table 1.**Radial distortion calibration in Column two. Column three, inverse of radial distortion with coefficient from ${k}_{1}...{k}_{9}.$

Radial Distortion Coefficients | Original Value | Computed Inverse Value |
---|---|---|

${k}_{1}$ | $1.532\times {10}^{-4}$ | $1.532\times {10}^{-4}$ |

${k}_{2}$ | $-9.656\times {10}^{-8}$ | $1.6697072\times {10}^{-7}$ |

${k}_{3}$ | $7.245\times {10}^{-11}$ | $-2.33941625216\times {10}^{-10}$ |

${k}_{4}$ | $0.0$ | $3.1255518770316804\times {10}^{-13}$ |

${k}_{5}$ | $0.0$ | $-4.774156462972984\times {10}^{-16}$ |

${k}_{6}$ | $0.0$ | $7.680785197322419\times {10}^{-19}$ |

${k}_{7}$ | $0.0$ | $-1.1582853960835112\times {10}^{-21}$ |

${k}_{8}$ | $0.0$ | $2.1694555835054252\times {10}^{-24}$ |

${k}_{9}$ | $0.0$ | $-3.779164309884112\times {10}^{-27}$ |

**Table 2.**Radial distortion inverse loop and residual between coefficients of the orignal distortion and after n inversions (n = 2 and n = 10,000).

Coefficient | Original | Inverse 1 | Delta Loop 1 | Delta Loop 10,000 |
---|---|---|---|---|

${k}_{1}$ | $1.532\times {10}^{-4}$ | $-1.532\times {10}^{-4}$ | 0.0 | 0.0 |

${k}_{2}$ | $-9.656\times {10}^{-8}$ | $1.6697072\times {10}^{-7}$ | 0.0 | 0.0 |

${k}_{3}$ | $7.245\times {10}^{-11}$ | $-2.33941625216\times {10}^{-10}$ | $1.292469707114\times {10}^{-26}$ | $1.292469707\times {10}^{-26}$ |

${k}_{4}$ | 0.0 | $3.1255518770316804\times {10}^{-13}$ | $1.009741958682\times {10}^{-28}$ | $1.009842932\times {10}^{-24}$ |

Pixel by Residual | Nb Concerned Pixel |
---|---|

Pixels | 10,000.0 |

Pixels with residual < 0.2 | 9344.0 (93.44 %) |

Pixels with residual < 1.0 | 9732.0 (97.32 %) |

Pixels with residual > 1.0 | 268.0 (2.68 %) |

**Table 4.**Radial distortion comensation and then application of the inverse used with the image taken with the P32 camera.

Coef. | Original | Inverse |
---|---|---|

${k}_{1}$ | 0.09532 | −0.09532 |

${k}_{2}$ | $-9.656\times {10}^{-8}$ | 0.02725780376 |

${k}_{3}$ | $7.245\times {10}^{-11}$ | −0.010392892306459602 |

${k}_{4}$ | 0.0 | 0.004540497555744342 |

${k}_{5}$ | 0.0 | −0.0021482705738196948 |

${k}_{6}$ | 0.0 | 0.0010711249019932042 |

${k}_{7}$ | 0.0 | $-5.542464764540273\times {10}^{-4}$ |

${k}_{8}$ | 0.0 | $2.948490225469636\times {10}^{-4}$ |

${k}_{9}$ | 0.0 | $-1.6024842649677896\times {10}^{-4}$ |

**Table 5.**Photograph taken with the Wild P32 camera: data and some results of the Internal Orientation.

Param. | X Value | Y Value |
---|---|---|

mm 2 Pixel | 58.885 | 58.885 |

Frame size in pixel | 4860 | 3575 |

Frame size in mm | 82.54 | 60.71 |

CoD, ppx ppy (pixel) | 2423.212 | 2377.528 |

Fiducial mark-up- (mm) | 0.0 | 37.5 |

Fiducial mark-right- (mm) | 37.5 | 0.0 |

Fiducial mark-down- (mm) | 0.0 | −17.5 |

Fiducial mark-left- (mm) | −37.5 | 37.5 |

Fiducial mark-center- (mm) | 0.0 | 0.0 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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Drap, P.; Lefèvre, J.
An Exact Formula for Calculating Inverse Radial Lens Distortions. *Sensors* **2016**, *16*, 807.
https://doi.org/10.3390/s16060807

**AMA Style**

Drap P, Lefèvre J.
An Exact Formula for Calculating Inverse Radial Lens Distortions. *Sensors*. 2016; 16(6):807.
https://doi.org/10.3390/s16060807

**Chicago/Turabian Style**

Drap, Pierre, and Julien Lefèvre.
2016. "An Exact Formula for Calculating Inverse Radial Lens Distortions" *Sensors* 16, no. 6: 807.
https://doi.org/10.3390/s16060807