Accurate real-time depth control for CP-SSOCT distal sensor based handheld microsurgery tools

Gyeong Woo Cheon; Yong Huang; Jaepyeng Cha; Peter L. Gehlbach; Jin U. Kang

doi:10.1364/BOE.6.001942

Accurate real-time depth control for CP-SSOCT distal sensor based handheld microsurgery tools

Gyeong Woo Cheon, Yong Huang, Jaepyeng Cha, Peter L. Gehlbach, and Jin U. Kang

Biomedical Optics Express
Vol. 6,
Issue 5,
pp. 1942-1953
(2015)
•https://doi.org/10.1364/BOE.6.001942

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Gyeong Woo Cheon, Yong Huang, Jaepyeng Cha, Peter L. Gehlbach, and Jin U. Kang, "Accurate real-time depth control for CP-SSOCT distal sensor based handheld microsurgery tools," Biomed. Opt. Express 6, 1942-1953 (2015)

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Related Topics
Table of Contents Category
- Ophthalmology Applications
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Fiber optics sensors (060.2370)
- Robotic and machine control (150.5758)
- Ophthalmic optics and devices (170.4460)
- Optical coherence tomography (170.4500)

About this Article
History
- Original Manuscript: March 20, 2015
- Revised Manuscript: April 21, 2015
- Manuscript Accepted: April 23, 2015
- Published: April 30, 2015

Accessible

Open Access

Abstract

This paper presents a novel intuitive targeting and tracking scheme that utilizes a common-path swept source optical coherence tomography (CP-SSOCT) distal sensor integrated handheld microsurgical tool. To achieve micron-order precision control, a reliable and accurate OCT distal sensing method is required; simultaneously, a prediction algorithm is necessary to compensate for the system delay associated with the computational, mechanical and electronic latencies. Due to the multi-layered structure of retina, it is necessary to develop effective surface detection methods rather than simple peak detection. To achieve this, a shifted cross-correlation method is applied for surface detection in order to increase robustness and accuracy in distal sensing. A predictor based on Kalman filter was implemented for more precise motion compensation. The performance was first evaluated using an established dry phantom consisting of stacked cellophane tape. This was followed by evaluation in an ex-vivo bovine retina model to assess system accuracy and precision. The results demonstrate highly accurate depth targeting with less than 5 μm RMSE depth locking.

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References

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Publication

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T. Ueta, Y. Yamaguchi, Y. Shirakawa, T. Nakano, R. Ideta, Y. Noda, A. Morita, R. Mochizuki, N. Sugita, M. Mitsuishi, and Y. Tamaki, “Robot-Assisted Vitreoretinal Surgery: Development of a Prototype and Feasibility Studies in an Animal Model,” Ophthalmology 116(8), 1543 (2009).
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[Crossref]
B. Gonenc, J. T. Handa, P. L. Gehlbach, R. H. Taylor, and I. I. Iordachita, “Design of 3-DOF force sensing micro-forceps for robot assisted vitreoretinal surgery,” in Proceedings of the 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBS (2013), pp. 5686–5689.
[Crossref]
H. Yu, J.-H. Shen, R. J. Shah, N. Simaan, and K. M. Joos, “Evaluation of microsurgical tasks with OCT-guided and/or robot-assisted ophthalmic forceps,” Biomed. Opt. Express 6(2), 457–472 (2015).
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[Crossref] [PubMed]
K. M. Joos and J.-H. Shen, “Miniature real-time intraoperative forward-imaging optical coherence tomography probe,” Biomed. Opt. Express 4(8), 1342–1350 (2013).
[Crossref] [PubMed]
K. Zhang and J. U. Kang, “Graphics processing unit accelerated non-uniform fast Fourier transform for ultrahigh-speed, real-time Fourier-domain OCT,” Opt. Express 18(22), 23472–23487 (2010).
[Crossref] [PubMed]
K. Zhang and J. U. Kang, “Real-time intraoperative 4D full-range FD-OCT based on the dual graphics processing units architecture for microsurgery guidance,” Biomed. Opt. Express 2(4), 764–770 (2011).
[Crossref] [PubMed]
X. Liu, I. I. Iordachita, X. He, R. H. Taylor, and J. U. Kang, “Miniature fiber-optic force sensor based on low-coherence Fabry-Pérot interferometry for vitreoretinal microsurgery,” Biomed. Opt. Express 3(5), 1062–1076 (2012).
[Crossref] [PubMed]
X. Liu, M. Balicki, R. H. Taylor, and J. U. Kang, “Towards automatic calibration of Fourier-Domain OCT for robot-assisted vitreoretinal surgery,” Opt. Express 18(23), 24331–24343 (2010).
[Crossref] [PubMed]
J. U. Kang, Y. Huang, K. Zhang, Z. Ibrahim, J. Cha, W. P. Lee, G. Brandacher, and P. L. Gehlbach, “Real-time three-dimensional Fourier-domain optical coherence tomography video image guided microsurgeries,” J. Biomed. Opt. 17(8), 081403 (2012).
[Crossref] [PubMed]
K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. 56(9), 2318–2321 (2009).
[Crossref] [PubMed]
J. U. Kang, J. H. Han, X. Liu, and K. Zhang, “Common-path optical coherence tomography for biomedical imaging and sensing,” J. Opt. Soc. Korea 14(1), 1–13 (2010).
[Crossref] [PubMed]
K. Zhang and J. U. Kang, “Common-path low-coherence interferometry fiber-optic sensor guided microincision,” J. Biomed. Opt. 16(9), 095003 (2011).
[Crossref] [PubMed]
Y. Huang, K. Zhang, C. Lin, and J. U. Kang, “Motion compensated fiber-optic confocal microscope based on a common-path optical coherence tomography distance sensor,” Opt. Eng. 50(8), 083201 (2011).
[Crossref]
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]
C. Song, P. L. Gehlbach, and J. U. Kang, “Active tremor cancellation by a “smart” handheld vitreoretinal microsurgical tool using swept source optical coherence tomography,” Opt. Express 20(21), 23414–23421 (2012).
[Crossref] [PubMed]
C. Song, D. Y. Park, P. L. Gehlbach, S. J. Park, and J. U. Kang, “Fiber-optic OCT sensor guided “SMART” micro-forceps for microsurgery,” Biomed. Opt. Express 4(7), 1045–1050 (2013).
[Crossref] [PubMed]
K. Briechle and U. D. Hanebeck, “Template matching using fast normalized cross correlation,” Proc. SPIE 4387, 95–102 (2001).
[Crossref]
R. E. Kalman, “A new approach to linear filtering and prediction problems,” Trans. ASME-J. Basic Eng. 82(1), 35–45 (1960).
[Crossref]
S.-K. Weng, C.-M. Kuo, and S.-K. Tu, “Video object tracking using adaptive Kalman filter,” J. Vis. Commun. Image Represent. 17(6), 1190–1208 (2006).
[Crossref]
Y. Zheng, S. Chen, W. Tan, R. Kinnick, and J. F. Greenleaf, “Detection of tissue harmonic motion induced by ultrasonic radiation force using pulse-echo ultrasound and Kalman filter,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(2), 290–300 (2007).
[Crossref] [PubMed]
S. H. P. Won, F. Golnaraghi, and W. W. Melek, “A fastening tool tracking system using an IMU and a position sensor with Kalman filters and a fuzzy expert system,” IEEE Trans. Ind. Electron. 56(5), 1782–1792 (2009).
[Crossref]
R. K. Mudi and N. R. Pal, “A robust self-tuning scheme for PI- and PD-type fuzzy controllers,” IEEE Trans. Fuzzy Syst. 7(1), 2–16 (1999).
[Crossref]
A. M. Bagci, M. Shahidi, R. Ansari, M. Blair, N. P. Blair, and R. Zelkha, “Thickness Profiles of Retinal Layers by Optical Coherence Tomography Image Segmentation,” Am. J. Ophthalmol. 146(5), 679–687 (2008).
[Crossref] [PubMed]
D. Cabrera Fernández, H. M. Salinas, and C. A. Puliafito, “Automated detection of retinal layer structures on optical coherence tomography images,” Opt. Express 13(25), 10200–10216 (2005).
[Crossref] [PubMed]

2015 (1)

H. Yu, J.-H. Shen, R. J. Shah, N. Simaan, and K. M. Joos, “Evaluation of microsurgical tasks with OCT-guided and/or robot-assisted ophthalmic forceps,” Biomed. Opt. Express 6(2), 457–472 (2015).
[Crossref] [PubMed]

2014 (3)

Y. K. Tao, S. K. Srivastava, and J. P. Ehlers, “Microscope-integrated intraoperative OCT with electrically tunable focus and heads-up display for imaging of ophthalmic surgical maneuvers,” Biomed. Opt. Express 5(6), 1877–1885 (2014).
[Crossref] [PubMed]

J. P. Ehlers, S. K. Srivastava, D. Feiler, A. I. Noonan, A. M. Rollins, and Y. K. Tao, “Integrative advances for OCT-guided ophthalmic surgery and intraoperative OCT: microscope integration, surgical instrumentation, and heads-up display surgeon feedback,” PLoS ONE 9(8), e105224 (2014).
[Crossref] [PubMed]

J. P. Ehlers, T. Tam, P. K. Kaiser, D. F. Martin, G. M. Smith, and S. K. Srivastava, “Utility of intraoperative optical coherence tomography during vitrectomy surgery for vitreomacular traction syndrome,” Retina 34(7), 1341–1346 (2014).
[Crossref] [PubMed]

2013 (4)

P. Hahn, J. Migacz, R. O’Connell, J. A. Izatt, and C. A. Toth, “Unprocessed real-time imaging of vitreoretinal surgical maneuvers using a microscope-integrated spectral-domain optical coherence tomography system,” Graefes Arch. Clin. Exp. Ophthalmol. 251(1), 213–220 (2013).
[Crossref] [PubMed]

P. Hahn, J. Migacz, R. O’Donnell, S. Day, A. Lee, P. Lin, R. Vann, A. Kuo, S. Fekrat, P. Mruthyunjaya, E. A. Postel, J. A. Izatt, and C. A. Toth, “Preclinical evaluation and intraoperative human retinal imaging with a high-resolution microscope-integrated spectral domain optical coherence tomography device,” Retina 33(7), 1328–1337 (2013).
[Crossref] [PubMed]

C. Song, D. Y. Park, P. L. Gehlbach, S. J. Park, and J. U. Kang, “Fiber-optic OCT sensor guided “SMART” micro-forceps for microsurgery,” Biomed. Opt. Express 4(7), 1045–1050 (2013).
[Crossref] [PubMed]

K. M. Joos and J.-H. Shen, “Miniature real-time intraoperative forward-imaging optical coherence tomography probe,” Biomed. Opt. Express 4(8), 1342–1350 (2013).
[Crossref] [PubMed]

2012 (5)

X. Liu, I. I. Iordachita, X. He, R. H. Taylor, and J. U. Kang, “Miniature fiber-optic force sensor based on low-coherence Fabry-Pérot interferometry for vitreoretinal microsurgery,” Biomed. Opt. Express 3(5), 1062–1076 (2012).
[Crossref] [PubMed]

C. Song, P. L. Gehlbach, and J. U. Kang, “Active tremor cancellation by a “smart” handheld vitreoretinal microsurgical tool using swept source optical coherence tomography,” Opt. Express 20(21), 23414–23421 (2012).
[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]

J. U. Kang, Y. Huang, K. Zhang, Z. Ibrahim, J. Cha, W. P. Lee, G. Brandacher, and P. L. Gehlbach, “Real-time three-dimensional Fourier-domain optical coherence tomography video image guided microsurgeries,” J. Biomed. Opt. 17(8), 081403 (2012).
[Crossref] [PubMed]

R. A. Maclachlan, B. C. Becker, J. C. Tabarés, G. W. Podnar, L. A. Lobes, and C. N. Riviere, “Micron: An actively stabilized handheld tool for microsurgery,” IEEE Trans. Robot. 28(1), 195–212 (2012).
[Crossref] [PubMed]

2011 (4)

J. P. Ehlers, Y. K. Tao, S. Farsiu, R. Maldonado, J. A. Izatt, and C. A. Toth, “Integration of a spectral domain optical coherence tomography system into a surgical microscope for intraoperative imaging,” Invest. Ophthalmol. Vis. Sci. 52(6), 3153–3159 (2011).
[Crossref] [PubMed]

K. Zhang and J. U. Kang, “Common-path low-coherence interferometry fiber-optic sensor guided microincision,” J. Biomed. Opt. 16(9), 095003 (2011).
[Crossref] [PubMed]

Y. Huang, K. Zhang, C. Lin, and J. U. Kang, “Motion compensated fiber-optic confocal microscope based on a common-path optical coherence tomography distance sensor,” Opt. Eng. 50(8), 083201 (2011).
[Crossref]

K. Zhang and J. U. Kang, “Real-time intraoperative 4D full-range FD-OCT based on the dual graphics processing units architecture for microsurgery guidance,” Biomed. Opt. Express 2(4), 764–770 (2011).
[Crossref] [PubMed]

2010 (3)

J. U. Kang, J. H. Han, X. Liu, and K. Zhang, “Common-path optical coherence tomography for biomedical imaging and sensing,” J. Opt. Soc. Korea 14(1), 1–13 (2010).
[Crossref] [PubMed]

K. Zhang and J. U. Kang, “Graphics processing unit accelerated non-uniform fast Fourier transform for ultrahigh-speed, real-time Fourier-domain OCT,” Opt. Express 18(22), 23472–23487 (2010).
[Crossref] [PubMed]

X. Liu, M. Balicki, R. H. Taylor, and J. U. Kang, “Towards automatic calibration of Fourier-Domain OCT for robot-assisted vitreoretinal surgery,” Opt. Express 18(23), 24331–24343 (2010).
[Crossref] [PubMed]

2009 (3)

S. H. P. Won, F. Golnaraghi, and W. W. Melek, “A fastening tool tracking system using an IMU and a position sensor with Kalman filters and a fuzzy expert system,” IEEE Trans. Ind. Electron. 56(5), 1782–1792 (2009).
[Crossref]

K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. 56(9), 2318–2321 (2009).
[Crossref] [PubMed]

T. Ueta, Y. Yamaguchi, Y. Shirakawa, T. Nakano, R. Ideta, Y. Noda, A. Morita, R. Mochizuki, N. Sugita, M. Mitsuishi, and Y. Tamaki, “Robot-Assisted Vitreoretinal Surgery: Development of a Prototype and Feasibility Studies in an Animal Model,” Ophthalmology 116(8), 1543 (2009).
[Crossref] [PubMed]

2008 (1)

A. M. Bagci, M. Shahidi, R. Ansari, M. Blair, N. P. Blair, and R. Zelkha, “Thickness Profiles of Retinal Layers by Optical Coherence Tomography Image Segmentation,” Am. J. Ophthalmol. 146(5), 679–687 (2008).
[Crossref] [PubMed]

2007 (1)

Y. Zheng, S. Chen, W. Tan, R. Kinnick, and J. F. Greenleaf, “Detection of tissue harmonic motion induced by ultrasonic radiation force using pulse-echo ultrasound and Kalman filter,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(2), 290–300 (2007).
[Crossref] [PubMed]

2006 (2)

S.-K. Weng, C.-M. Kuo, and S.-K. Tu, “Video object tracking using adaptive Kalman filter,” J. Vis. Commun. Image Represent. 17(6), 1190–1208 (2006).
[Crossref]

C. N. Riviere, J. Gangloff, and M. De Mathelin, “Robotic compensation of biological motion to enhance surgical accuracy,” Proc. IEEE 94(9), 1705–1716 (2006).
[Crossref]

2005 (1)

D. Cabrera Fernández, H. M. Salinas, and C. A. Puliafito, “Automated detection of retinal layer structures on optical coherence tomography images,” Opt. Express 13(25), 10200–10216 (2005).
[Crossref] [PubMed]

2003 (2)

C. N. Riviere, W. T. Ang, and P. K. Khosla, “Toward Active Tremor Canceling in Handheld Microsurgical Instruments,” IEEE Trans. Robot. Autom. 19(5), 793–800 (2003).
[Crossref]

R. H. Taylor and D. Stoianovici, “Medical Robotics in Computer-Integrated Surgery,” IEEE Trans. Robot. Autom. 19(5), 765–781 (2003).
[Crossref]

2001 (1)

K. Briechle and U. D. Hanebeck, “Template matching using fast normalized cross correlation,” Proc. SPIE 4387, 95–102 (2001).
[Crossref]

1999 (2)

R. H. Taylor, P. Jensen, L. Whitcomb, A. Barnes, R. Kumar, D. Stoianovici, P. Gupta, Z. Wang, E. DeJuan, and L. Kavoussi, “A Steady-Hand Robotic System for Microsurgical Augmentation,” Int. J. Robot. Res. 18(12), 1201–1210 (1999).
[Crossref]

R. K. Mudi and N. R. Pal, “A robust self-tuning scheme for PI- and PD-type fuzzy controllers,” IEEE Trans. Fuzzy Syst. 7(1), 2–16 (1999).
[Crossref]

1996 (1)

C. N. Riviere and N. V. Thakor, “Modeling and canceling tremor in human-machine interfaces,” IEEE Eng. Med. Biol. Mag. 15(3), 29–36 (1996).
[Crossref]

1977 (1)

M. Patkin, “Ergonomics applied to the practice of microsurgery,” Aust. N. Z. J. Surg. 47(3), 320–329 (1977).
[Crossref] [PubMed]

1960 (1)

R. E. Kalman, “A new approach to linear filtering and prediction problems,” Trans. ASME-J. Basic Eng. 82(1), 35–45 (1960).
[Crossref]

Ang, W. T.

C. N. Riviere, W. T. Ang, and P. K. Khosla, “Toward Active Tremor Canceling in Handheld Microsurgical Instruments,” IEEE Trans. Robot. Autom. 19(5), 793–800 (2003).
[Crossref]

W. T. Ang, P. K. Pradeep, and C. N. Riviere, “Active tremor compensation in microsurgery,” in Proceedings of the 26th Annual International Conference of IEEE Engineering in Medicine and Biology Society, EMBS (2004), 1, pp. 2738–2741.
[Crossref]

Ansari, R.

Bagci, A. M.

Balicki, M.

S. Yang, M. Balicki, R. a. MacLachlan, X. Liu, J. U. Kang, R. H. Taylor, and C. N. Riviere, “Optical coherence tomography scanning with a handheld vitreoretinal micromanipulator,” in Proceedings of the 34th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBS (2012), pp. 948–951.

S. Yang, M. Balicki, T. S. Wells, R. A. MacLachlan, X. Liu, J. U. Kang, J. T. Handa, R. H. Taylor, and C. N. Riviere, “Improvement of Optical Coherence Tomography using Active Handheld Micromanipulator in Vitreoretinal Surgery,” in Proceedings of the 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBS (2013), pp. 5674–5677.

I. Kuru, B. Gonenc, M. Balicki, J. T. Handa, P. L. Gehlbach, R. H. Taylor, and I. I. Iordachita, “Force sensing micro-forceps for robot assisted retinal surgery,” in Proceedings of the 34th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBS (2012), pp. 1401–1404.
[Crossref]

Barnes, A.

Becker, B. C.

Blair, M.

Blair, N. P.

Brandacher, G.

Briechle, K.

K. Briechle and U. D. Hanebeck, “Template matching using fast normalized cross correlation,” Proc. SPIE 4387, 95–102 (2001).
[Crossref]

Cabrera Fernández, D.

Cha, J.

Chen, S.

Day, S.

De Mathelin, M.

C. N. Riviere, J. Gangloff, and M. De Mathelin, “Robotic compensation of biological motion to enhance surgical accuracy,” Proc. IEEE 94(9), 1705–1716 (2006).
[Crossref]

DeJuan, E.

Ehlers, J. P.

Farsiu, S.

Feiler, D.

Fekrat, S.

Gangloff, J.

C. N. Riviere, J. Gangloff, and M. De Mathelin, “Robotic compensation of biological motion to enhance surgical accuracy,” Proc. IEEE 94(9), 1705–1716 (2006).
[Crossref]

Gehlbach, P. L.

B. Gonenc, J. T. Handa, P. L. Gehlbach, R. H. Taylor, and I. I. Iordachita, “Design of 3-DOF force sensing micro-forceps for robot assisted vitreoretinal surgery,” in Proceedings of the 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBS (2013), pp. 5686–5689.
[Crossref]

Golnaraghi, F.

Gonenc, B.

Greenleaf, J. F.

Gupta, P.

Hahn, P.

Han, J.

Han, J. H.

J. U. Kang, J. H. Han, X. Liu, and K. Zhang, “Common-path optical coherence tomography for biomedical imaging and sensing,” J. Opt. Soc. Korea 14(1), 1–13 (2010).
[Crossref] [PubMed]

Handa, J. T.

Hanebeck, U. D.

K. Briechle and U. D. Hanebeck, “Template matching using fast normalized cross correlation,” Proc. SPIE 4387, 95–102 (2001).
[Crossref]

He, X.

Huang, Y.

Ibrahim, Z.

Ideta, R.

Iordachita, I. I.

Izatt, J. A.

Jensen, P.

Joos, K. M.

K. M. Joos and J.-H. Shen, “Miniature real-time intraoperative forward-imaging optical coherence tomography probe,” Biomed. Opt. Express 4(8), 1342–1350 (2013).
[Crossref] [PubMed]

Kaiser, P. K.

Kalman, R. E.

R. E. Kalman, “A new approach to linear filtering and prediction problems,” Trans. ASME-J. Basic Eng. 82(1), 35–45 (1960).
[Crossref]

Kang, J. U.

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Fig. 1 Three consecutive A-scan data of 20-stacked layers of adherent cellophane tape near the DC line (multi-layered phantom sample); the upper right inserted graph is a partial magnified graph of area in dotted red square in lower entire graph.

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Fig. 2 a) Active depth targeting and locking microsurgery system, b) CAD cross-sectional image of handheld tool (NTC: needle-to-tube connector, PM: piezo-motor, OF: optical fiber).

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Fig. 3 a) A-scan data of ex-vivo open bovine retina, b) three consecutive A-scan data of stacked layers of adherent cellophane tape, c) data processing flowchart consisting of 1) OCT data process, 2) Surface detection, and 3) Motor control.

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Fig. 4 Dry phantom experiment: a) evaluation test for the surface detection algorithm with the auxiliary shifted cross-correlation in non-depth-locking freehand, b) evaluation test for the depth-locking with the predictor based on Kalman filter: the upper inserted graph is a magnified graph of area in dotted yellow trapezoid in the entire graph.

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Fig. 5 Bovine retina experiment: a) bovine retina after removing cornea, lens, and vitreous humor, b) snapshot of the experiment, c) depth targeting and locking experiment result with six different jumping distance from 10 µm to 60 µm.

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Fig. 6 Averages of RMSE with standard deviation error bar of the bovine retina experiment. The x-axis labels mean jumping distance and the y-axis indicates the means and standard deviations of the measurement specified in Table 1.

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Tables (1)

Table 1 Optical Constants of Thin Films of Materials [µm]

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Equations (14)

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

X_{i} * Y_{i + n} = \frac{\sum_{i} (X_{i} - m_{X}) (Y_{i + n} - m_{Y})}{\sqrt{\sum_{i} {(X_{i} - m_{X})}^{2} \sum_{i} {(Y_{i + n} - m_{Y})}^{2}}} = ρ_{- n}

ρ_{n}^{n u m} = \sum_{i} X_{i}^{1} Y_{n - i}^{2}

ρ_{n}^{n u m} = ℱ^{- 1} (ℱ (X^{1}) \cdot ℱ (Y^{2})) = ℱ^{- 1} (ℱ (X^{1}) \cdot ℱ^{*} (Y^{1}))

{\hat{x}}_{t | t - 1} = F_{t} {\hat{x}}_{t - 1 | t - 1} + B_{t} u_{t}

P_{t | t - 1} = F_{t} P_{t - 1 | t - 1} F_{t}^{T} + Q_{t}

{\hat{x}}_{t | t} = {\hat{x}}_{t | t - 1} + K_{t} (y_{t} - H_{t} {\hat{x}}_{t | t - 1})

K_{t} = P_{t | t - 1} H_{t}^{T} {(H_{t} P_{t | t - 1} H_{t}^{T} + R_{t})}^{- 1}

P_{t | t} = (I - K_{t} H_{t}) P_{t | t - 1}

x_{t} = [\begin{matrix} x_{t} \\ v_{t} \end{matrix}] = [\begin{matrix} 1 & △ t \\ 0 & 1 \end{matrix}] [\begin{matrix} x_{t - 1} \\ v_{t - 1} \end{matrix}] + a_{t} [\begin{matrix} △ t^{2} / 2 \\ △ t \end{matrix}] = F_{t} x_{t - 1} + w_{t}

y_{t} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] x_{t} + [\begin{matrix} r_{t} \\ 0 \end{matrix}] = H_{t} x_{t} + b_{t}

Q_{t} = v a r (w_{t}) = E [w_{t} w_{t}^{T}] = σ^{2} [\begin{matrix} △ t^{4} / 4 & △ t^{3} / 2 \\ △ t^{3} / 2 & △ t^{2} \end{matrix}]

R_{t} = v a r (b_{t}) = E [b_{t} b_{t}^{T}] = [\begin{matrix} r_{t}^{2} & 0 \\ 0 & 0 \end{matrix}] \to [\begin{matrix} r_{t}^{2} & 0 \\ 0 & c_{1} \end{matrix}]

P_{0} = [\begin{matrix} c_{2} & 0 \\ 0 & c_{2} \end{matrix}]

B_{t} u_{t} = [\begin{matrix} - 1 & 1 \\ 0 & 0 \end{matrix}] [\begin{matrix} y \\ y_{t g t} \end{matrix}]

	∆10	∆20	∆30	∆40	∆50	∆60
1	5.264	4.562	6.750	2.944	8.129	5.939
2	5.080	3.774	6.222	3.108	7.404	3.950
3	6.473	4.297	7.161	3.038	5.447	5.488
4	4.683	3.416	5.494	3.376	5.630	3.141
5	5.233	5.017	6.114	3.891	5.061	4.823
6	6.286	4.007	5.838	3.170	5.761	4.981
7	6.367	4.078	6.427	2.600	5.513	4.344
8	6.830	3.938	6.377	2.887	5.642	3.937
9	5.148	4.187	6.685	3.181	5.275	5.118
10	5.259	3.200	7.342	2.104	6.826	2.941
11	6.063	4.255	6.603	2.982	7.277	5.388