Abstract

We present a new paradigm for performing two-dimensional scanning called dual-beam manually-actuated distortion-corrected imaging (DMDI). DMDI operates by imaging the same object with two spatially-separated beams that are being mechanically scanned rapidly in one dimension with slower manual actuation along a second dimension. Registration of common features between the two imaging channels allows remapping of the images to correct for distortions due to manual actuation. We demonstrate DMDI using a 4.7 mm OD rotationally scanning dual-beam micromotor catheter (DBMC). The DBMC requires a simple, one-time calibration of the beam paths by imaging a patterned phantom. DMDI allows for distortion correction of non-uniform axial speed and rotational motion of the DBMC. We show the utility of this technique by demonstrating en face OCT image distortion correction of a manually-scanned checkerboard phantom and fingerprint scan.

© 2017 Optical Society of America

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References

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  1. D. C. Adams, Y. Wang, L. P. Hariri, and M. J. Suter, “Advances in endoscopic optical coherence tomography catheter designs,” IEEE J. Sel. Top. Quantum Electron. 22(3), 210–221 (2016).
    [Crossref]
  2. J. G. F. Chao Zhou, T. H. Tsai, and H. Mashimo, “Endoscopic optical coherence tomography,” in Optical Coherence Tomography: Technology and Applications, 2nd ed., W. D. a. J. G. Fujimoto, ed. (2015), pp. 2077–2108.
  3. T. Adriaenssens and G. J. Ughi, “Recent advances in the field of optical coherence tomography,” Curr. Cardiovasc. Imaging Rep. 10(7), 23 (2017).
    [Crossref]
  4. T.-H. Tsai, J. G. Fujimoto, and H. Mashimo, “Endoscopic optical coherence tomography for clinical gastroenterology,” Diagnostics (Basel) 4(2), 57–93 (2014).
    [Crossref] [PubMed]
  5. M. J. Gora, M. J. Suter, G. J. Tearney, and X. Li, “Endoscopic optical coherence tomography: technologies and clinical applications [Invited],” Biomed. Opt. Express 8(5), 2405–2444 (2017).
    [Crossref] [PubMed]
  6. A. Ahmad, S. G. Adie, E. J. Chaney, U. Sharma, and S. A. Boppart, “Cross-correlation-based image acquisition technique for manually-scanned optical coherence tomography,” Opt. Express 17(10), 8125–8136 (2009).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  9. Y. Huang, X. Liu, C. Song, and J. U. Kang, “Motion-compensated hand-held common-path Fourier-domain optical coherence tomography probe for image-guided intervention,” Biomed. Opt. Express 3(12), 3105–3118 (2012).
    [Crossref] [PubMed]
  10. N. Uribe-Patarroyo and B. E. Bouma, “Rotational distortion correction in endoscopic optical coherence tomography based on speckle decorrelation,” Opt. Lett. 40(23), 5518–5521 (2015).
    [Crossref] [PubMed]
  11. B. Y. Yeo, R. A. McLaughlin, R. W. Kirk, and D. D. Sampson, “Enabling freehand lateral scanning of optical coherence tomography needle probes with a magnetic tracking system,” Biomed. Opt. Express 3(7), 1565–1578 (2012).
    [Crossref] [PubMed]
  12. N. Iftimia, G. Maguluri, E. W. Chang, S. Chang, J. Magill, and W. Brugge, “Hand scanning optical coherence tomography imaging using encoder feedback,” Opt. Lett. 39(24), 6807–6810 (2014).
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    [Crossref]
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    [Crossref] [PubMed]
  16. D. C. Adler, C. Zhou, T.-H. Tsai, J. Schmitt, Q. Huang, H. Mashimo, and J. G. Fujimoto, “Three-dimensional endomicroscopy of the human colon using optical coherence tomography,” Opt. Express 17(2), 784–796 (2009).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  18. K. Liang, G. Traverso, H.-C. Lee, O. O. Ahsen, Z. Wang, B. Potsaid, M. Giacomelli, V. Jayaraman, R. Barman, A. Cable, H. Mashimo, R. Langer, and J. G. Fujimoto, “Ultrahigh speed en face OCT capsule for endoscopic imaging,” Biomed. Opt. Express 6(4), 1146–1163 (2015).
    [Crossref] [PubMed]
  19. A. M. D. Lee, L. Cahill, K. Liu, C. MacAulay, C. Poh, and P. Lane, “Wide-field in vivo oral OCT imaging,” Biomed. Opt. Express 6(7), 2664–2674 (2015).
    [Crossref] [PubMed]
  20. H. Pahlevaninezhad, A. M. D. Lee, T. Shaipanich, R. Raizada, L. Cahill, G. Hohert, V. X. D. Yang, S. Lam, C. MacAulay, and P. Lane, “A high-efficiency fiber-based imaging system for co-registered autofluorescence and optical coherence tomography,” Biomed. Opt. Express 5(9), 2978–2987 (2014).
    [Crossref] [PubMed]

2017 (2)

T. Adriaenssens and G. J. Ughi, “Recent advances in the field of optical coherence tomography,” Curr. Cardiovasc. Imaging Rep. 10(7), 23 (2017).
[Crossref]

M. J. Gora, M. J. Suter, G. J. Tearney, and X. Li, “Endoscopic optical coherence tomography: technologies and clinical applications [Invited],” Biomed. Opt. Express 8(5), 2405–2444 (2017).
[Crossref] [PubMed]

2016 (3)

D. C. Adams, Y. Wang, L. P. Hariri, and M. J. Suter, “Advances in endoscopic optical coherence tomography catheter designs,” IEEE J. Sel. Top. Quantum Electron. 22(3), 210–221 (2016).
[Crossref]

P. Pande, G. L. Monroy, R. M. Nolan, R. L. Shelton, and S. A. Boppart, “Sensor-based technique for manually scanned hand-held optical coherence tomography imaging,” J. Sens. 2016, 1–7 (2016).
[Crossref]

J. C. Jing, L. Chou, E. Su, B. J. F. Wong, and Z. Chen, “Anatomically correct visualization of the human upper airway using a high-speed long range optical coherence tomography system with an integrated positioning sensor,” Sci. Rep. 6(1), 39443 (2016).
[Crossref] [PubMed]

2015 (4)

2014 (3)

2013 (1)

M. J. Gora, J. S. Sauk, R. W. Carruth, K. A. Gallagher, M. J. Suter, N. S. Nishioka, L. E. Kava, M. Rosenberg, B. E. Bouma, and G. J. Tearney, “Tethered capsule endomicroscopy enables less invasive imaging of gastrointestinal tract microstructure,” Nat. Med. 19(2), 238–240 (2013).
[Crossref] [PubMed]

2012 (3)

2010 (1)

2009 (2)

Adams, D. C.

D. C. Adams, Y. Wang, L. P. Hariri, and M. J. Suter, “Advances in endoscopic optical coherence tomography catheter designs,” IEEE J. Sel. Top. Quantum Electron. 22(3), 210–221 (2016).
[Crossref]

Adie, S. G.

Adler, D. C.

Adriaenssens, T.

T. Adriaenssens and G. J. Ughi, “Recent advances in the field of optical coherence tomography,” Curr. Cardiovasc. Imaging Rep. 10(7), 23 (2017).
[Crossref]

Ahmad, A.

Ahsen, O. O.

Akansu, A.

Barman, R.

Belfield, K. D.

Boppart, S. A.

P. Pande, G. L. Monroy, R. M. Nolan, R. L. Shelton, and S. A. Boppart, “Sensor-based technique for manually scanned hand-held optical coherence tomography imaging,” J. Sens. 2016, 1–7 (2016).
[Crossref]

A. Ahmad, S. G. Adie, E. J. Chaney, U. Sharma, and S. A. Boppart, “Cross-correlation-based image acquisition technique for manually-scanned optical coherence tomography,” Opt. Express 17(10), 8125–8136 (2009).
[Crossref] [PubMed]

Bouma, B. E.

N. Uribe-Patarroyo and B. E. Bouma, “Rotational distortion correction in endoscopic optical coherence tomography based on speckle decorrelation,” Opt. Lett. 40(23), 5518–5521 (2015).
[Crossref] [PubMed]

M. J. Gora, J. S. Sauk, R. W. Carruth, K. A. Gallagher, M. J. Suter, N. S. Nishioka, L. E. Kava, M. Rosenberg, B. E. Bouma, and G. J. Tearney, “Tethered capsule endomicroscopy enables less invasive imaging of gastrointestinal tract microstructure,” Nat. Med. 19(2), 238–240 (2013).
[Crossref] [PubMed]

Brugge, W.

Cable, A.

Cahill, L.

Carruth, R. W.

M. J. Gora, J. S. Sauk, R. W. Carruth, K. A. Gallagher, M. J. Suter, N. S. Nishioka, L. E. Kava, M. Rosenberg, B. E. Bouma, and G. J. Tearney, “Tethered capsule endomicroscopy enables less invasive imaging of gastrointestinal tract microstructure,” Nat. Med. 19(2), 238–240 (2013).
[Crossref] [PubMed]

Chaney, E. J.

Chang, E. W.

Chang, S.

Chen, Z.

J. C. Jing, L. Chou, E. Su, B. J. F. Wong, and Z. Chen, “Anatomically correct visualization of the human upper airway using a high-speed long range optical coherence tomography system with an integrated positioning sensor,” Sci. Rep. 6(1), 39443 (2016).
[Crossref] [PubMed]

Chou, L.

J. C. Jing, L. Chou, E. Su, B. J. F. Wong, and Z. Chen, “Anatomically correct visualization of the human upper airway using a high-speed long range optical coherence tomography system with an integrated positioning sensor,” Sci. Rep. 6(1), 39443 (2016).
[Crossref] [PubMed]

Curatolo, A.

Fujimoto, J. G.

Gallagher, K. A.

M. J. Gora, J. S. Sauk, R. W. Carruth, K. A. Gallagher, M. J. Suter, N. S. Nishioka, L. E. Kava, M. Rosenberg, B. E. Bouma, and G. J. Tearney, “Tethered capsule endomicroscopy enables less invasive imaging of gastrointestinal tract microstructure,” Nat. Med. 19(2), 238–240 (2013).
[Crossref] [PubMed]

Gerstmann, D. K.

Giacomelli, M.

Gora, M. J.

M. J. Gora, M. J. Suter, G. J. Tearney, and X. Li, “Endoscopic optical coherence tomography: technologies and clinical applications [Invited],” Biomed. Opt. Express 8(5), 2405–2444 (2017).
[Crossref] [PubMed]

M. J. Gora, J. S. Sauk, R. W. Carruth, K. A. Gallagher, M. J. Suter, N. S. Nishioka, L. E. Kava, M. Rosenberg, B. E. Bouma, and G. J. Tearney, “Tethered capsule endomicroscopy enables less invasive imaging of gastrointestinal tract microstructure,” Nat. Med. 19(2), 238–240 (2013).
[Crossref] [PubMed]

Hariri, L. P.

D. C. Adams, Y. Wang, L. P. Hariri, and M. J. Suter, “Advances in endoscopic optical coherence tomography catheter designs,” IEEE J. Sel. Top. Quantum Electron. 22(3), 210–221 (2016).
[Crossref]

Hohert, G.

Huang, Q.

Huang, Y.

Hubbi, B.

Iftimia, N.

Jayaraman, V.

Jing, J. C.

J. C. Jing, L. Chou, E. Su, B. J. F. Wong, and Z. Chen, “Anatomically correct visualization of the human upper airway using a high-speed long range optical coherence tomography system with an integrated positioning sensor,” Sci. Rep. 6(1), 39443 (2016).
[Crossref] [PubMed]

Kang, J. U.

Kava, L. E.

M. J. Gora, J. S. Sauk, R. W. Carruth, K. A. Gallagher, M. J. Suter, N. S. Nishioka, L. E. Kava, M. Rosenberg, B. E. Bouma, and G. J. Tearney, “Tethered capsule endomicroscopy enables less invasive imaging of gastrointestinal tract microstructure,” Nat. Med. 19(2), 238–240 (2013).
[Crossref] [PubMed]

Kirk, R. W.

Lam, S.

Lane, P.

Langer, R.

Lau, B.

Lee, A. M. D.

Lee, H.-C.

Li, X.

Liang, K.

Liu, K.

Liu, X.

MacAulay, C.

Magill, J.

Maguluri, G.

Mashimo, H.

McLaughlin, R. A.

Monroy, G. L.

P. Pande, G. L. Monroy, R. M. Nolan, R. L. Shelton, and S. A. Boppart, “Sensor-based technique for manually scanned hand-held optical coherence tomography imaging,” J. Sens. 2016, 1–7 (2016).
[Crossref]

Nishioka, N. S.

M. J. Gora, J. S. Sauk, R. W. Carruth, K. A. Gallagher, M. J. Suter, N. S. Nishioka, L. E. Kava, M. Rosenberg, B. E. Bouma, and G. J. Tearney, “Tethered capsule endomicroscopy enables less invasive imaging of gastrointestinal tract microstructure,” Nat. Med. 19(2), 238–240 (2013).
[Crossref] [PubMed]

Nolan, R. M.

P. Pande, G. L. Monroy, R. M. Nolan, R. L. Shelton, and S. A. Boppart, “Sensor-based technique for manually scanned hand-held optical coherence tomography imaging,” J. Sens. 2016, 1–7 (2016).
[Crossref]

Pahlevaninezhad, H.

Pande, P.

P. Pande, G. L. Monroy, R. M. Nolan, R. L. Shelton, and S. A. Boppart, “Sensor-based technique for manually scanned hand-held optical coherence tomography imaging,” J. Sens. 2016, 1–7 (2016).
[Crossref]

Poh, C.

Potsaid, B.

Raizada, R.

Rosenberg, M.

M. J. Gora, J. S. Sauk, R. W. Carruth, K. A. Gallagher, M. J. Suter, N. S. Nishioka, L. E. Kava, M. Rosenberg, B. E. Bouma, and G. J. Tearney, “Tethered capsule endomicroscopy enables less invasive imaging of gastrointestinal tract microstructure,” Nat. Med. 19(2), 238–240 (2013).
[Crossref] [PubMed]

Sampson, D. D.

Sauk, J. S.

M. J. Gora, J. S. Sauk, R. W. Carruth, K. A. Gallagher, M. J. Suter, N. S. Nishioka, L. E. Kava, M. Rosenberg, B. E. Bouma, and G. J. Tearney, “Tethered capsule endomicroscopy enables less invasive imaging of gastrointestinal tract microstructure,” Nat. Med. 19(2), 238–240 (2013).
[Crossref] [PubMed]

Schmitt, J.

Shaipanich, T.

Sharma, U.

Shelton, R. L.

P. Pande, G. L. Monroy, R. M. Nolan, R. L. Shelton, and S. A. Boppart, “Sensor-based technique for manually scanned hand-held optical coherence tomography imaging,” J. Sens. 2016, 1–7 (2016).
[Crossref]

Song, C.

Su, E.

J. C. Jing, L. Chou, E. Su, B. J. F. Wong, and Z. Chen, “Anatomically correct visualization of the human upper airway using a high-speed long range optical coherence tomography system with an integrated positioning sensor,” Sci. Rep. 6(1), 39443 (2016).
[Crossref] [PubMed]

Suter, M. J.

M. J. Gora, M. J. Suter, G. J. Tearney, and X. Li, “Endoscopic optical coherence tomography: technologies and clinical applications [Invited],” Biomed. Opt. Express 8(5), 2405–2444 (2017).
[Crossref] [PubMed]

D. C. Adams, Y. Wang, L. P. Hariri, and M. J. Suter, “Advances in endoscopic optical coherence tomography catheter designs,” IEEE J. Sel. Top. Quantum Electron. 22(3), 210–221 (2016).
[Crossref]

M. J. Gora, J. S. Sauk, R. W. Carruth, K. A. Gallagher, M. J. Suter, N. S. Nishioka, L. E. Kava, M. Rosenberg, B. E. Bouma, and G. J. Tearney, “Tethered capsule endomicroscopy enables less invasive imaging of gastrointestinal tract microstructure,” Nat. Med. 19(2), 238–240 (2013).
[Crossref] [PubMed]

Tearney, G. J.

M. J. Gora, M. J. Suter, G. J. Tearney, and X. Li, “Endoscopic optical coherence tomography: technologies and clinical applications [Invited],” Biomed. Opt. Express 8(5), 2405–2444 (2017).
[Crossref] [PubMed]

M. J. Gora, J. S. Sauk, R. W. Carruth, K. A. Gallagher, M. J. Suter, N. S. Nishioka, L. E. Kava, M. Rosenberg, B. E. Bouma, and G. J. Tearney, “Tethered capsule endomicroscopy enables less invasive imaging of gastrointestinal tract microstructure,” Nat. Med. 19(2), 238–240 (2013).
[Crossref] [PubMed]

Traverso, G.

Tsai, T.-H.

Ughi, G. J.

T. Adriaenssens and G. J. Ughi, “Recent advances in the field of optical coherence tomography,” Curr. Cardiovasc. Imaging Rep. 10(7), 23 (2017).
[Crossref]

Uribe-Patarroyo, N.

Wang, Y.

Wang, Z.

Wong, B. J. F.

J. C. Jing, L. Chou, E. Su, B. J. F. Wong, and Z. Chen, “Anatomically correct visualization of the human upper airway using a high-speed long range optical coherence tomography system with an integrated positioning sensor,” Sci. Rep. 6(1), 39443 (2016).
[Crossref] [PubMed]

Yang, V. X. D.

Yeo, B. Y.

Zhou, C.

Biomed. Opt. Express (7)

M. J. Gora, M. J. Suter, G. J. Tearney, and X. Li, “Endoscopic optical coherence tomography: technologies and clinical applications [Invited],” Biomed. Opt. Express 8(5), 2405–2444 (2017).
[Crossref] [PubMed]

Y. Wang, Y. Wang, A. Akansu, K. D. Belfield, B. Hubbi, and X. Liu, “Robust motion tracking based on adaptive speckle decorrelation analysis of OCT signal,” Biomed. Opt. Express 6(11), 4302–4316 (2015).
[Crossref] [PubMed]

Y. Huang, X. Liu, C. Song, and J. U. Kang, “Motion-compensated hand-held common-path Fourier-domain optical coherence tomography probe for image-guided intervention,” Biomed. Opt. Express 3(12), 3105–3118 (2012).
[Crossref] [PubMed]

B. Y. Yeo, R. A. McLaughlin, R. W. Kirk, and D. D. Sampson, “Enabling freehand lateral scanning of optical coherence tomography needle probes with a magnetic tracking system,” Biomed. Opt. Express 3(7), 1565–1578 (2012).
[Crossref] [PubMed]

K. Liang, G. Traverso, H.-C. Lee, O. O. Ahsen, Z. Wang, B. Potsaid, M. Giacomelli, V. Jayaraman, R. Barman, A. Cable, H. Mashimo, R. Langer, and J. G. Fujimoto, “Ultrahigh speed en face OCT capsule for endoscopic imaging,” Biomed. Opt. Express 6(4), 1146–1163 (2015).
[Crossref] [PubMed]

A. M. D. Lee, L. Cahill, K. Liu, C. MacAulay, C. Poh, and P. Lane, “Wide-field in vivo oral OCT imaging,” Biomed. Opt. Express 6(7), 2664–2674 (2015).
[Crossref] [PubMed]

H. Pahlevaninezhad, A. M. D. Lee, T. Shaipanich, R. Raizada, L. Cahill, G. Hohert, V. X. D. Yang, S. Lam, C. MacAulay, and P. Lane, “A high-efficiency fiber-based imaging system for co-registered autofluorescence and optical coherence tomography,” Biomed. Opt. Express 5(9), 2978–2987 (2014).
[Crossref] [PubMed]

Curr. Cardiovasc. Imaging Rep. (1)

T. Adriaenssens and G. J. Ughi, “Recent advances in the field of optical coherence tomography,” Curr. Cardiovasc. Imaging Rep. 10(7), 23 (2017).
[Crossref]

Diagnostics (Basel) (1)

T.-H. Tsai, J. G. Fujimoto, and H. Mashimo, “Endoscopic optical coherence tomography for clinical gastroenterology,” Diagnostics (Basel) 4(2), 57–93 (2014).
[Crossref] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

D. C. Adams, Y. Wang, L. P. Hariri, and M. J. Suter, “Advances in endoscopic optical coherence tomography catheter designs,” IEEE J. Sel. Top. Quantum Electron. 22(3), 210–221 (2016).
[Crossref]

J. Sens. (1)

P. Pande, G. L. Monroy, R. M. Nolan, R. L. Shelton, and S. A. Boppart, “Sensor-based technique for manually scanned hand-held optical coherence tomography imaging,” J. Sens. 2016, 1–7 (2016).
[Crossref]

Nat. Med. (1)

M. J. Gora, J. S. Sauk, R. W. Carruth, K. A. Gallagher, M. J. Suter, N. S. Nishioka, L. E. Kava, M. Rosenberg, B. E. Bouma, and G. J. Tearney, “Tethered capsule endomicroscopy enables less invasive imaging of gastrointestinal tract microstructure,” Nat. Med. 19(2), 238–240 (2013).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (2)

Sci. Rep. (1)

J. C. Jing, L. Chou, E. Su, B. J. F. Wong, and Z. Chen, “Anatomically correct visualization of the human upper airway using a high-speed long range optical coherence tomography system with an integrated positioning sensor,” Sci. Rep. 6(1), 39443 (2016).
[Crossref] [PubMed]

Other (1)

J. G. F. Chao Zhou, T. H. Tsai, and H. Mashimo, “Endoscopic optical coherence tomography,” in Optical Coherence Tomography: Technology and Applications, 2nd ed., W. D. a. J. G. Fujimoto, ed. (2015), pp. 2077–2108.

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Figures (5)

Fig. 1
Fig. 1

Dual-channel OCT imaging system and dual beam micromotor catheter (DBMC). i) Schematic diagram of dual-channel OCT imaging system. SS = swept-source laser, PC = polarization controller, C = collimator, RM = reference mirror, CIRC = circulator, BD = balanced detector. ii) Model of DBMC (outermost plastic tubing not shown). DFP = dual fiber pigtail; GRIN = graded-index lens; MM = 4 mm OD micromotor, FPC = flexible printed circuit. Approximate paths of A and B beams at catheter outer diameter are shown in red and blue respectively. iii) Side view of DBMC showing approximate beam paths for A and B beams. iv) Photograph of the DBMC. The rigid length of the DBMC is indicated. Scale bar is 1 cm.

Fig. 2
Fig. 2

Outline of primary DMDI processing steps for (left) calibration and (right) distortion correcting images acquired with DBMCs.

Fig. 3
Fig. 3

Calibration of DBMC using the paper calibration pattern. i) Raw en face OCT images from A and B channels. The range of the n axis is from 1 to 1000 A-lines, with each image column corresponding to one rotation of the prism. ii) Data and fit of positions of horizontal lines in i). iii) Aligned, shear-corrected, and θ calibrated images. The range of the θ axis is from 0° to 360°. iv) Scan pattern data and fit. v) Distortion corrected images. vi) Fixed pattern data and fit.

Fig. 4
Fig. 4

Image correction of 1mm checkerboard pattern with manual axial and rotational actuation. i) Aligned and shear-corrected images of channels A and B. ii) Control points (n = 303) used for A/B image registration. Green points (n = 295) were used for distortion correction, while red points (n = 8) were omitted. iii) Catheter rotation characterization. iv) Axial actuation characterization. v) Rotation distortion-corrected images. vi) Rotation and axial distortion-corrected images.

Fig. 5
Fig. 5

Distortion correction for manually-actuated DBMC imaging across fingers. i) Aligned and shear-corrected images of channels A and B. ii) control points (n = 312) used for image registration. Green points (n = 308) were used for catheter angular and axial speed characterization, while red points (n = 4) were omitted. iii) Catheter angular rotation and axial displacement characterization. iv) Distortion-corrected images.

Tables (2)

Tables Icon

Table 1 Fit parameters for θ calibration

Tables Icon

Table 2 Fit parameters for S(θ) and F(θ) calibrations

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

θ= θ 0 +j(Δθ)
θ (A,B) ( n (A,B) )=g n (A,B) +g k c (A,B),k cos( 2πk N fr ( n (A,B) n (A,B),k,off ) )g C (A,B),0
C ( A,B ),0 = g k c ( A,B ),k cos( 2πk N fr ( n ( A,B ),k, off ) )
S fr ( θ )= 1 2 [ b A ( θ ) b B ( θ ) ]
F fr ( θ )=  1 2 [ b A ( θ )+ b B ( θ ) ]
S( θ )= [ S fr ( θ ) N fr + n A ( θ )+  N ref,A n B ( θ ) N ref,B ] v z f laser
F( θ )= [ F fr ( θ ) N fr + n A ( θ )+  N ref,A n B ( θ ) N ref,B ] v z f laser
S( θ )=  s 0 + k s k sin( 2πk N fr ( θ θ s,k,off ) )
F( θ )=  f 0 + k f k sin( 2πk N fr ( θ θ f,k,off ) )
z A ( θ,t )= z DBMC ( t ) S( θ )F( θ )
z B ( θ,t )= z DBMC ( t ) +S( θ )F( θ )
υ ¯ z,AB,j = S( θ A,j )+ S( θ B,j ) | t A,j t B,j |
ω ¯ AB,j = θ second,j   θ first,j | t A,j t B,j |
υ z,max =  S max f fr