Abstract

Stereoscopic imagers are widely used in machine vision for three-dimensional (3D) visualization as well as in non-destructive testing for quantitative characterization of cracks, delamination and other defects. Measurement capability in these systems is provided by a proper combination of the optical parameters and data processing techniques. Conventional approach to their design consists of two sequential stages: optical system design and optimization of calibration and image processing algorithms. Such two-stage procedure often complicates both the hardware and the software, and results in a time-ineffective design procedure and cost-ineffective solution. We demonstrate a more effective approach and propose to estimate errors of 3D measurements at the early optical design stage. We show that computer simulation using optical design software allows not only optimizing optical parameters of the imager but also choosing the most effective mathematical model of the system and the equipment necessary for calibration. We tested the proposed approach on the design of miniature prism-based stereoscopic system and analyzed the impact of various factors (aberrations, tolerances, etc.) as on the image quality, so on the quality of calibration and 3D measurements accuracy. The proposed joint design approach may be highly effective for various measurement systems and applications when both optical parameters and image processing algorithms are not defined in advance and are necessary to be optimized.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (2)

A. S. Machikhin, A. V. Gorevoy, D. D. Khokhlov, and A. O. Kuznetsov, “Modification of calibration and image processing procedures for precise 3D measurements in arbitrary spectral bands by means of a stereoscopic prism-based imager,” Opt. Eng. 58(3), 033104 (2019).
[Crossref]

A. S. Machikhin, V. I. Batshev, A. V. Gorevoy, D. D. Khokhlov, A. A. Naumov, and A. O. Kuznetsov, “Compact stereoscopic prism-based optical system with an improved accuracy of 3-D geometrical measurements,” Optik (Stuttg.) 185, 1172–1181 (2019).
[Crossref]

2018 (3)

A. V. Gorevoy, V. Y. Kolyuchkin, and A. S. Machikhin, “Estimation of the geometrical measurement error at the stage of stereoscopic system design,” Comput. Opt. 42(6), 985–997 (2018).
[Crossref]

J. N. Mait, G. W. Euliss, and R. A. Athale, “Computational imaging,” Adv. Opt. Photonics 10(2), 409–483 (2018).
[Crossref]

S. Barone, P. Neri, A. Paoli, and A. V. Razionale, “Catadioptric stereo-vision system using a spherical mirror,” Procedia Structural Integrity 8, 83–91 (2018).
[Crossref]

2017 (4)

A. V. Gorevoy and A. S. Machikhin, “Optimal calibration of a prism-based videoendoscopic system for precise 3D measurements,” Comput. Opt. 41(4), 535–545 (2017).
[Crossref]

D. G. Stork, “Toward a signal-processing foundation for computational sensing and imaging: electrooptical basis and merit functions,” APSIPA Trans. Signal. Inf. Process. 6, E8 (2017).
[Crossref]

V. Batshev, A. Machikhin, and Y. Kachurin, “Stereoscopic tip for a video endoscope: problems in design,” Proc. SPIE 10466, 104664D (2017).

T. Matsuzawa, “Camera calibration based on the principal rays model of imaging optical systems,” J. Opt. Soc. Am. A 34(4), 624–639 (2017).
[Crossref] [PubMed]

2016 (4)

A.-S. Poulin-Girard, S. Thibault, and D. Laurendeau, “Influence of camera calibration conditions on the accuracy of 3D reconstruction,” Opt. Express 24(3), 2678–2686 (2016).
[Crossref] [PubMed]

J. Li, F. Xing, D. Chu, and Z. Liu, “High-accuracy self-calibration for smart, optical orbiting payloads integrated with attitude and position determination,” Sensors (Basel) 16(8), 1176 (2016).
[Crossref] [PubMed]

L. Yu and B. Pan, “Structure parameter analysis and uncertainty evaluation for single-camera stereo-digital image correlation with a four-mirror adapter,” Appl. Opt. 55(25), 6936–6946 (2016).
[Crossref] [PubMed]

S.-H. Baek and M. H. Kim, “Stereo fusion: combining refractive and binocular disparity,” Comput. Vis. Image Underst. 146, 52–66 (2016).
[Crossref]

2015 (1)

2014 (2)

2013 (2)

K. Genovese, L. Casaletto, J. Rayas, V. Flores, and A. Martinez, “Stereo-digital image correlation (DIC) measurements with a single camera using a biprism,” Opt. Lasers Eng. 51(3), 278–285 (2013).
[Crossref]

Z. Chen, K.-Y. K. Wong, Y. Matsushita, and X. Zhu, “Depth from refraction using a transparent medium with unknown pose and refractive index,” Int. J. Comput. Vis. 102(1–3), 3–17 (2013).
[Crossref]

2012 (2)

2011 (2)

O. S. Cossairt, D. Miau, and S. K. Nayar, “Scaling law for computational imaging using spherical optics,” J. Opt. Soc. Am. A 28(12), 2540–2553 (2011).
[Crossref] [PubMed]

C. Zhou and S. K. Nayar, “Computational cameras: convergence of optics and processing,” IEEE Trans. Image Process. 20(12), 3322–3340 (2011).
[Crossref] [PubMed]

2010 (1)

P. Sturm, S. Ramalingam, T.-P. Tardif, S. Gasparini, and J. Barreto, “Camera models and fundamental concepts used in geometric computer vision,” Found. Trends Comput. Graph. Vis. 6(1-2), 1–183 (2010).
[Crossref]

2008 (2)

2007 (1)

J. Mallon and P. F. Whelan, “Which pattern? Biasing aspects of planar calibration patterns and detection methods,” Pattern Recognit. Lett. 28(8), 921–930 (2007).
[Crossref]

2006 (1)

J. Kannala and S. S. Brandt, “A generic camera model and calibration method for conventional, wide-angle, and fish-eye lenses,” IEEE Trans. Pattern Anal. Mach. Intell. 28(8), 1335–1340 (2006).
[Crossref] [PubMed]

2003 (1)

Agrawal, A.

A. Agrawal, Y. Taguchi, and S. Ramalingam, “Beyond Alhazen’s problem: analytical projection model for non-central catadioptric cameras with quadric mirrors,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2011), pp. 2993–3000.
[Crossref]

Athale, R. A.

J. N. Mait, G. W. Euliss, and R. A. Athale, “Computational imaging,” Adv. Opt. Photonics 10(2), 409–483 (2018).
[Crossref]

Baek, S.-H.

S.-H. Baek and M. H. Kim, “Stereo fusion: combining refractive and binocular disparity,” Comput. Vis. Image Underst. 146, 52–66 (2016).
[Crossref]

Bai, J.

Barone, S.

S. Barone, P. Neri, A. Paoli, and A. V. Razionale, “Catadioptric stereo-vision system using a spherical mirror,” Procedia Structural Integrity 8, 83–91 (2018).
[Crossref]

Barreto, J.

P. Sturm, S. Ramalingam, T.-P. Tardif, S. Gasparini, and J. Barreto, “Camera models and fundamental concepts used in geometric computer vision,” Found. Trends Comput. Graph. Vis. 6(1-2), 1–183 (2010).
[Crossref]

Batshev, V.

V. Batshev, A. Machikhin, and Y. Kachurin, “Stereoscopic tip for a video endoscope: problems in design,” Proc. SPIE 10466, 104664D (2017).

Batshev, V. I.

A. S. Machikhin, V. I. Batshev, A. V. Gorevoy, D. D. Khokhlov, A. A. Naumov, and A. O. Kuznetsov, “Compact stereoscopic prism-based optical system with an improved accuracy of 3-D geometrical measurements,” Optik (Stuttg.) 185, 1172–1181 (2019).
[Crossref]

Brandt, S. S.

J. Kannala and S. S. Brandt, “A generic camera model and calibration method for conventional, wide-angle, and fish-eye lenses,” IEEE Trans. Pattern Anal. Mach. Intell. 28(8), 1335–1340 (2006).
[Crossref] [PubMed]

Casaletto, L.

K. Genovese, L. Casaletto, J. Rayas, V. Flores, and A. Martinez, “Stereo-digital image correlation (DIC) measurements with a single camera using a biprism,” Opt. Lasers Eng. 51(3), 278–285 (2013).
[Crossref]

Cathey, W.

Chen, Z.

Z. Chen, K.-Y. K. Wong, Y. Matsushita, and X. Zhu, “Depth from refraction using a transparent medium with unknown pose and refractive index,” Int. J. Comput. Vis. 102(1–3), 3–17 (2013).
[Crossref]

Chu, D.

J. Li, F. Xing, D. Chu, and Z. Liu, “High-accuracy self-calibration for smart, optical orbiting payloads integrated with attitude and position determination,” Sensors (Basel) 16(8), 1176 (2016).
[Crossref] [PubMed]

Cossairt, O. S.

Cui, X.

Dallaire, X.

Dowski, E.

Euliss, G. W.

J. N. Mait, G. W. Euliss, and R. A. Athale, “Computational imaging,” Adv. Opt. Photonics 10(2), 409–483 (2018).
[Crossref]

Flores, V.

K. Genovese, L. Casaletto, J. Rayas, V. Flores, and A. Martinez, “Stereo-digital image correlation (DIC) measurements with a single camera using a biprism,” Opt. Lasers Eng. 51(3), 278–285 (2013).
[Crossref]

Gasparini, S.

P. Sturm, S. Ramalingam, T.-P. Tardif, S. Gasparini, and J. Barreto, “Camera models and fundamental concepts used in geometric computer vision,” Found. Trends Comput. Graph. Vis. 6(1-2), 1–183 (2010).
[Crossref]

Geng, J.

J. Geng and J. Xie, “Review of 3-D endoscopic surface imaging techniques,” IEEE Sens. J. 14(4), 945–960 (2014).
[Crossref]

Genovese, K.

K. Genovese, L. Casaletto, J. Rayas, V. Flores, and A. Martinez, “Stereo-digital image correlation (DIC) measurements with a single camera using a biprism,” Opt. Lasers Eng. 51(3), 278–285 (2013).
[Crossref]

Gorevoy, A. V.

A. S. Machikhin, A. V. Gorevoy, D. D. Khokhlov, and A. O. Kuznetsov, “Modification of calibration and image processing procedures for precise 3D measurements in arbitrary spectral bands by means of a stereoscopic prism-based imager,” Opt. Eng. 58(3), 033104 (2019).
[Crossref]

A. S. Machikhin, V. I. Batshev, A. V. Gorevoy, D. D. Khokhlov, A. A. Naumov, and A. O. Kuznetsov, “Compact stereoscopic prism-based optical system with an improved accuracy of 3-D geometrical measurements,” Optik (Stuttg.) 185, 1172–1181 (2019).
[Crossref]

A. V. Gorevoy, V. Y. Kolyuchkin, and A. S. Machikhin, “Estimation of the geometrical measurement error at the stage of stereoscopic system design,” Comput. Opt. 42(6), 985–997 (2018).
[Crossref]

A. V. Gorevoy and A. S. Machikhin, “Optimal calibration of a prism-based videoendoscopic system for precise 3D measurements,” Comput. Opt. 41(4), 535–545 (2017).
[Crossref]

Guo, Q.

Hou, X. Y.

Huang, Z.

Kachurin, Y.

V. Batshev, A. Machikhin, and Y. Kachurin, “Stereoscopic tip for a video endoscope: problems in design,” Proc. SPIE 10466, 104664D (2017).

Kannala, J.

J. Kannala and S. S. Brandt, “A generic camera model and calibration method for conventional, wide-angle, and fish-eye lenses,” IEEE Trans. Pattern Anal. Mach. Intell. 28(8), 1335–1340 (2006).
[Crossref] [PubMed]

Khokhlov, D. D.

A. S. Machikhin, V. I. Batshev, A. V. Gorevoy, D. D. Khokhlov, A. A. Naumov, and A. O. Kuznetsov, “Compact stereoscopic prism-based optical system with an improved accuracy of 3-D geometrical measurements,” Optik (Stuttg.) 185, 1172–1181 (2019).
[Crossref]

A. S. Machikhin, A. V. Gorevoy, D. D. Khokhlov, and A. O. Kuznetsov, “Modification of calibration and image processing procedures for precise 3D measurements in arbitrary spectral bands by means of a stereoscopic prism-based imager,” Opt. Eng. 58(3), 033104 (2019).
[Crossref]

Kim, M. H.

S.-H. Baek and M. H. Kim, “Stereo fusion: combining refractive and binocular disparity,” Comput. Vis. Image Underst. 146, 52–66 (2016).
[Crossref]

Kolyuchkin, V. Y.

A. V. Gorevoy, V. Y. Kolyuchkin, and A. S. Machikhin, “Estimation of the geometrical measurement error at the stage of stereoscopic system design,” Comput. Opt. 42(6), 985–997 (2018).
[Crossref]

Kubala, K.

Kuznetsov, A. O.

A. S. Machikhin, V. I. Batshev, A. V. Gorevoy, D. D. Khokhlov, A. A. Naumov, and A. O. Kuznetsov, “Compact stereoscopic prism-based optical system with an improved accuracy of 3-D geometrical measurements,” Optik (Stuttg.) 185, 1172–1181 (2019).
[Crossref]

A. S. Machikhin, A. V. Gorevoy, D. D. Khokhlov, and A. O. Kuznetsov, “Modification of calibration and image processing procedures for precise 3D measurements in arbitrary spectral bands by means of a stereoscopic prism-based imager,” Opt. Eng. 58(3), 033104 (2019).
[Crossref]

Laurendeau, D.

Li, J.

J. Li, F. Xing, D. Chu, and Z. Liu, “High-accuracy self-calibration for smart, optical orbiting payloads integrated with attitude and position determination,” Sensors (Basel) 16(8), 1176 (2016).
[Crossref] [PubMed]

Lim, K. B.

Liu, Z.

J. Li, F. Xing, D. Chu, and Z. Liu, “High-accuracy self-calibration for smart, optical orbiting payloads integrated with attitude and position determination,” Sensors (Basel) 16(8), 1176 (2016).
[Crossref] [PubMed]

Machikhin, A.

V. Batshev, A. Machikhin, and Y. Kachurin, “Stereoscopic tip for a video endoscope: problems in design,” Proc. SPIE 10466, 104664D (2017).

Machikhin, A. S.

A. S. Machikhin, A. V. Gorevoy, D. D. Khokhlov, and A. O. Kuznetsov, “Modification of calibration and image processing procedures for precise 3D measurements in arbitrary spectral bands by means of a stereoscopic prism-based imager,” Opt. Eng. 58(3), 033104 (2019).
[Crossref]

A. S. Machikhin, V. I. Batshev, A. V. Gorevoy, D. D. Khokhlov, A. A. Naumov, and A. O. Kuznetsov, “Compact stereoscopic prism-based optical system with an improved accuracy of 3-D geometrical measurements,” Optik (Stuttg.) 185, 1172–1181 (2019).
[Crossref]

A. V. Gorevoy, V. Y. Kolyuchkin, and A. S. Machikhin, “Estimation of the geometrical measurement error at the stage of stereoscopic system design,” Comput. Opt. 42(6), 985–997 (2018).
[Crossref]

A. V. Gorevoy and A. S. Machikhin, “Optimal calibration of a prism-based videoendoscopic system for precise 3D measurements,” Comput. Opt. 41(4), 535–545 (2017).
[Crossref]

Mait, J. N.

J. N. Mait, G. W. Euliss, and R. A. Athale, “Computational imaging,” Adv. Opt. Photonics 10(2), 409–483 (2018).
[Crossref]

Mallon, J.

J. Mallon and P. F. Whelan, “Which pattern? Biasing aspects of planar calibration patterns and detection methods,” Pattern Recognit. Lett. 28(8), 921–930 (2007).
[Crossref]

Martinez, A.

K. Genovese, L. Casaletto, J. Rayas, V. Flores, and A. Martinez, “Stereo-digital image correlation (DIC) measurements with a single camera using a biprism,” Opt. Lasers Eng. 51(3), 278–285 (2013).
[Crossref]

Matsushita, Y.

Z. Chen, K.-Y. K. Wong, Y. Matsushita, and X. Zhu, “Depth from refraction using a transparent medium with unknown pose and refractive index,” Int. J. Comput. Vis. 102(1–3), 3–17 (2013).
[Crossref]

Matsuzawa, T.

Miau, D.

Naumov, A. A.

A. S. Machikhin, V. I. Batshev, A. V. Gorevoy, D. D. Khokhlov, A. A. Naumov, and A. O. Kuznetsov, “Compact stereoscopic prism-based optical system with an improved accuracy of 3-D geometrical measurements,” Optik (Stuttg.) 185, 1172–1181 (2019).
[Crossref]

Nayar, S. K.

O. S. Cossairt, D. Miau, and S. K. Nayar, “Scaling law for computational imaging using spherical optics,” J. Opt. Soc. Am. A 28(12), 2540–2553 (2011).
[Crossref] [PubMed]

C. Zhou and S. K. Nayar, “Computational cameras: convergence of optics and processing,” IEEE Trans. Image Process. 20(12), 3322–3340 (2011).
[Crossref] [PubMed]

Neri, P.

S. Barone, P. Neri, A. Paoli, and A. V. Razionale, “Catadioptric stereo-vision system using a spherical mirror,” Procedia Structural Integrity 8, 83–91 (2018).
[Crossref]

Pan, B.

Paoli, A.

S. Barone, P. Neri, A. Paoli, and A. V. Razionale, “Catadioptric stereo-vision system using a spherical mirror,” Procedia Structural Integrity 8, 83–91 (2018).
[Crossref]

Poulin-Girard, A.-S.

Ramalingam, S.

P. Sturm, S. Ramalingam, T.-P. Tardif, S. Gasparini, and J. Barreto, “Camera models and fundamental concepts used in geometric computer vision,” Found. Trends Comput. Graph. Vis. 6(1-2), 1–183 (2010).
[Crossref]

A. Agrawal, Y. Taguchi, and S. Ramalingam, “Beyond Alhazen’s problem: analytical projection model for non-central catadioptric cameras with quadric mirrors,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2011), pp. 2993–3000.
[Crossref]

Rayas, J.

K. Genovese, L. Casaletto, J. Rayas, V. Flores, and A. Martinez, “Stereo-digital image correlation (DIC) measurements with a single camera using a biprism,” Opt. Lasers Eng. 51(3), 278–285 (2013).
[Crossref]

Razionale, A. V.

S. Barone, P. Neri, A. Paoli, and A. V. Razionale, “Catadioptric stereo-vision system using a spherical mirror,” Procedia Structural Integrity 8, 83–91 (2018).
[Crossref]

Robinson, M. D.

Stork, D. G.

Sturm, P.

P. Sturm, S. Ramalingam, T.-P. Tardif, S. Gasparini, and J. Barreto, “Camera models and fundamental concepts used in geometric computer vision,” Found. Trends Comput. Graph. Vis. 6(1-2), 1–183 (2010).
[Crossref]

Taguchi, Y.

A. Agrawal, Y. Taguchi, and S. Ramalingam, “Beyond Alhazen’s problem: analytical projection model for non-central catadioptric cameras with quadric mirrors,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2011), pp. 2993–3000.
[Crossref]

Tardif, T.-P.

P. Sturm, S. Ramalingam, T.-P. Tardif, S. Gasparini, and J. Barreto, “Camera models and fundamental concepts used in geometric computer vision,” Found. Trends Comput. Graph. Vis. 6(1-2), 1–183 (2010).
[Crossref]

Thibault, S.

Wang, D.

Whelan, P. F.

J. Mallon and P. F. Whelan, “Which pattern? Biasing aspects of planar calibration patterns and detection methods,” Pattern Recognit. Lett. 28(8), 921–930 (2007).
[Crossref]

Wong, K.-Y. K.

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[Crossref]

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J. Li, F. Xing, D. Chu, and Z. Liu, “High-accuracy self-calibration for smart, optical orbiting payloads integrated with attitude and position determination,” Sensors (Basel) 16(8), 1176 (2016).
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Yu, L.

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Z. Zhang, “Flexible camera calibration by viewing a plane from unknown orientations,” in Proc. International Conference on Computer Vision (IEEE, 1999), pp. 666–673.
[Crossref]

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C. Zhou and S. K. Nayar, “Computational cameras: convergence of optics and processing,” IEEE Trans. Image Process. 20(12), 3322–3340 (2011).
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Appl. Opt. (5)

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A. V. Gorevoy and A. S. Machikhin, “Optimal calibration of a prism-based videoendoscopic system for precise 3D measurements,” Comput. Opt. 41(4), 535–545 (2017).
[Crossref]

A. V. Gorevoy, V. Y. Kolyuchkin, and A. S. Machikhin, “Estimation of the geometrical measurement error at the stage of stereoscopic system design,” Comput. Opt. 42(6), 985–997 (2018).
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S.-H. Baek and M. H. Kim, “Stereo fusion: combining refractive and binocular disparity,” Comput. Vis. Image Underst. 146, 52–66 (2016).
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P. Sturm, S. Ramalingam, T.-P. Tardif, S. Gasparini, and J. Barreto, “Camera models and fundamental concepts used in geometric computer vision,” Found. Trends Comput. Graph. Vis. 6(1-2), 1–183 (2010).
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A. S. Machikhin, A. V. Gorevoy, D. D. Khokhlov, and A. O. Kuznetsov, “Modification of calibration and image processing procedures for precise 3D measurements in arbitrary spectral bands by means of a stereoscopic prism-based imager,” Opt. Eng. 58(3), 033104 (2019).
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A. S. Machikhin, V. I. Batshev, A. V. Gorevoy, D. D. Khokhlov, A. A. Naumov, and A. O. Kuznetsov, “Compact stereoscopic prism-based optical system with an improved accuracy of 3-D geometrical measurements,” Optik (Stuttg.) 185, 1172–1181 (2019).
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V. Batshev, A. Machikhin, and Y. Kachurin, “Stereoscopic tip for a video endoscope: problems in design,” Proc. SPIE 10466, 104664D (2017).

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S. Barone, P. Neri, A. Paoli, and A. V. Razionale, “Catadioptric stereo-vision system using a spherical mirror,” Procedia Structural Integrity 8, 83–91 (2018).
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J. Li, F. Xing, D. Chu, and Z. Liu, “High-accuracy self-calibration for smart, optical orbiting payloads integrated with attitude and position determination,” Sensors (Basel) 16(8), 1176 (2016).
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Figures (15)

Fig. 1
Fig. 1 The framework of conventional two-stage and joint design concepts for optical system, calibration equipment and software.
Fig. 2
Fig. 2 Basic layout of prism-based stereoscopic system.
Fig. 3
Fig. 3 The pinhole camera model with the inverse distortion transformation (a). The modification of the model accounting for pupil aberrations (b). The ray tracing model for prism-based stereoscopic system (c).
Fig. 4
Fig. 4 Design layout of analyzed prism-based stereoscopic system (top), ray diagrams and entrance pupil positions (middle) and grid distortion in image plane (bottom) for two variants of the system.
Fig. 5
Fig. 5 Computer simulation of calibration and measurements.
Fig. 6
Fig. 6 Positions of the points used for calibration (a) and as test sequences (b). Overlay of image points for calibration (c).
Fig. 7
Fig. 7 Reconstructed 3D points for ‘Usual_IR’(a) and ‘Prism_IR’ (b).
Fig. 8
Fig. 8 Comparison of camera models by systematic errors for Variant 1. Dependence of mean value (solid) and STD (dotted) of length measurement error for the 1 mm segment along x (left), y (center) and z (right) axes on the distance to the target.
Fig. 9
Fig. 9 Comparison of camera models by systematic errors for Variant 2. Dependence of mean value (solid) and STD (dotted) of length measurement error for the 1 mm segment along x (left), y (center) and z (right) axes on the distance to the target.
Fig. 10
Fig. 10 Dependence of mean value (solid red) and STD (dashed blue) of length measurement error for 50 calibration trials with added image noise using ‘Poly_IP’ (top) and ‘Prism_IRP’ (bottom). Black curves indicate mean (solid) and STD (dashed) for noise-free calibration.
Fig. 11
Fig. 11 Dependence of mean value (solid red) and STD (dashed blue) of length measurement error for 50 calibration trials with random deviations of design parameters using ‘Poly_IP’ (top) and ‘Prism_IRP’ (bottom). Black curves indicate mean (solid) and STD (dashed) for nominal design.
Fig. 12
Fig. 12 Mean value (dash) and 95% confidence intervals (bar) of length measurement error for the 1 mm segment along all axes over the whole measurement volume caused by different factors.
Fig. 13
Fig. 13 Calibration setup (a) and images of the test chart used for calibration (b).
Fig. 14
Fig. 14 Experimental results. Dependence of mean value (solid) and STD (dotted) of length measurement error for the 1 mm segment along x (left), y (center) and z (right) axes on the distance to the target.
Fig. 15
Fig. 15 Mean value (dash) and 95% confidence intervals (bar) of length measurement error for the 1 mm segment along all axes over the whole measurement volume for the computer simulation (blue) and the experiments with the prototype (red).

Tables (2)

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Table 1 Considered camera models

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Table 2 RMS error of 3D coordinates for considered camera models

Equations (16)

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p i = P i E i ( x w )= A i D i F E i ( x w ),
x i = D i ( x i ): ( x i y i )=( 1+ ρ 1 r 2 + ρ 2 r 4 + ρ 3 r 6 )( x i y i )+( ρ 4 ( r 2 +2 x i 2 )+2 ρ 5 x i y i 2 ρ 4 x i y i + ρ 5 ( r 2 +2 y i 2 ) ),
l w = E i 1 F 1 D i 1 A i 1 ( p i ).
x i = D i 1 ( x i ): ( x i y i )=( 1+ ρ 1 r 2 + ρ 2 r 4 + ρ 3 r 6 )( x i y i )+( ρ 4 ( r 2 +2 x i 2 )+2 ρ 5 x i y i 2 ρ 4 x i y i + ρ 5 ( r 2 +2 y i 2 ) ),
x i = D i 1 ( x i ): ( x i y i )= 1 1+ ρ 1 r 2 + ρ 2 r 4 + ρ 3 r 6 ( x i y i )+( ρ 4 ( r 2 +2 x i 2 )+2 ρ 5 x i y i 2 ρ 4 x i y i + ρ 5 ( r 2 +2 y i 2 ) ).
l i = F 1 ( x i )= ( ( 0,0,0 ),( x i , y i ,1 )/| ( x i , y i ,1 ) | ) T .
( x p i y p i )=( 1+ ρ 6 r 2 + ρ 7 r 4 + ρ 8 r 6 )( x i y i )+( ρ 9 ( r 2 +2 x i 2 )+2 ρ 10 x i y i 2 ρ 9 x i y i + ρ 10 ( r 2 +2 y i 2 ) ).
l i = F 1 ( x i x p i ): l i = ( ( x p i , y p i ,0 ),( x i , y i ,1 )/| ( x i , y i ,1 ) | ) T , ( x i x p i )= D i 1 ( x i ),
( x i ' y i ' )= m=0 m max n=0 n max ( α m,n β m,n ) x i ''m y i ''n ( x p i y p i )= m=0 m pmax n=0 n p max ( ξ m,n η m,n ) x i ''m y i ''n ,
l w i = E 1 ( l 2,i )= E 1 S 2,i S 1,i F 1 D 1 A 1 ( p i ),
x ^ w = argmin x w (C( x w ,p,k)),
C( x w ,p,k)= i=1 i max p i = P i E i ( x w ) 2 .
C( x w ,p,k )= i=1 i max d 2 ( x w , E i 1 P i 1 ( p i ) ) = i=1 i max ( I d 3×3 v w i v w i T )( c w i x w ) 2 ,
k ^ , k ^ t = argmin k, k t (C( x t ,p,k, k t )).
C( x t ,p,k, k t )= i,j,k p i,j,k P i E i E k ' ( x t j ) 2 ,
C( x t ,p,k, k t )= i,j,k x tj x ti,j,k 2 .

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