Abstract

Corneal topography is an essential tool in ophthalmology, in particular for surgical planning and diagnostics. Optical coherence tomography (OCT) enables cross-sectional or volumetric imaging with high resolution. It is, however, not widely used for corneal topography. A major reason for this is that conventional beam-scanning OCT is susceptible to eye motion compared to established modalities, which measure corneal shape in a single shot. To overcome this limitation, we propose a novel pipeline for motion-compensated OCT-based corneal topography. The pipeline includes three main features: (1) continuous, two-dimensional scanning; (2) the three-dimensional continuous motion compensation in postprocessing; and (3) regularised Zernike reconstruction. First, we evaluated our method on an eye phantom that is moved to mimic typical eye motion. The proposed motion compensation was able to determine and correct the movements of the phantom. Second, we performed an in vivo study on 48 eyes, measuring each eye twice with our OCT-based topography, Placido disc topography (Atlas 9000, Carl Zeiss Meditec), and Scheimpflug (Pentacam, Oculus) topography. We then compared the performance of the OCT-based topography to the reference topographies in terms of repeatability and equivalence. The results confirm the necessity and efficiency of the presented motion compensation and validate the proposed methods for scanning and reconstruction.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  21. E. S. Bennett and B. A. Weissman, Clinical Contact Lens Practice (Lippincott Williams & Wilkins, 2005).
  22. R. P. McNabb, S. Farsiu, S. S. Stinnett, J. A. Izatt, and A. N. Kuo, “Optical coherence tomography accurately measures corneal power change from laser refractive surgery,” Ophthalmology 122(4), 677–686 (2015).
    [Crossref]

2019 (1)

2017 (2)

2015 (1)

R. P. McNabb, S. Farsiu, S. S. Stinnett, J. A. Izatt, and A. N. Kuo, “Optical coherence tomography accurately measures corneal power change from laser refractive surgery,” Ophthalmology 122(4), 677–686 (2015).
[Crossref]

2013 (1)

2012 (3)

2011 (2)

S. Ortiz, D. Siedlecki, P. Pérez-Merino, N. Chia, A. de Castro, M. Szkulmowski, M. Wojtkowski, and S. Marcos, “Corneal topography from spectral optical coherence tomography (soct),” Biomed. Opt. Express 2(12), 3232–3247 (2011).
[Crossref]

D. Gatinel, J. Malet, T. Hoang-Xuan, and D. T. Azar, “Corneal elevation topography: best fit sphere, elevation distance, asphericity, toricity and clinical implications,” Cornea 30(5), 508–515 (2011).
[Crossref]

2010 (3)

2009 (2)

B. Braaf, T. C. van de Watering, K. Spruijt, R. G. van der Heijde, and V. A. D. Sicam, “Calculating angle lambda (λ) using zernike tilt measurements in specular reflection corneal topography,” J. Optom. 2(4), 207–214 (2009).
[Crossref]

E. L. Lamoureux, S.-M. Saw, J. Thumboo, H. L. Wee, T. Aung, P. Mitchell, and T. Y. Wong, “The impact of corrected and uncorrected refractive error on visual functioning: the singapore malay eye study,” Invest. Ophthalmol. Visual Sci. 50(6), 2614–2620 (2009).
[Crossref]

Alejandre, N.

Anderson, T.

T. Anderson, A. Segref, G. Frisken, and S. Frisken, “3d spectral imaging system for anterior chamber metrology,” in Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XIX, vol. 9312 (International Society for Optics and Photonics, 2015), p. 93120N.

Auksorius, E.

Aung, T.

E. L. Lamoureux, S.-M. Saw, J. Thumboo, H. L. Wee, T. Aung, P. Mitchell, and T. Y. Wong, “The impact of corrected and uncorrected refractive error on visual functioning: the singapore malay eye study,” Invest. Ophthalmol. Visual Sci. 50(6), 2614–2620 (2009).
[Crossref]

Azar, D. T.

D. Gatinel, J. Malet, T. Hoang-Xuan, and D. T. Azar, “Corneal elevation topography: best fit sphere, elevation distance, asphericity, toricity and clinical implications,” Cornea 30(5), 508–515 (2011).
[Crossref]

Baumann, B.

Bennett, E. S.

E. S. Bennett and B. A. Weissman, Clinical Contact Lens Practice (Lippincott Williams & Wilkins, 2005).

Bock, R.

Braaf, B.

B. Braaf, T. C. van de Watering, K. Spruijt, R. G. van der Heijde, and V. A. D. Sicam, “Calculating angle lambda (λ) using zernike tilt measurements in specular reflection corneal topography,” J. Optom. 2(4), 207–214 (2009).
[Crossref]

Cattin, P. C.

J. Wagner, D. Goldblum, and P. C. Cattin, “Golden angle based scanning for robust corneal topography with OCT,” Biomed. Opt. Express 8(2), 475–483 (2017).
[Crossref]

J. Wagner, S. Pezold, and P. C. Cattin, “Model-driven 3-D regularisation for robust segmentation of the refractive corneal surfaces in spiral OCT scans,” 4th MICCAI Workshop on Ophthalmic Medical Image Analysis (2017).

Chen, A.

M. Tang, A. Chen, Y. Li, and D. Huang, “Corneal power measurement with fourier-domain optical coherence tomography,” J. Cataract Refractive Surg. 36(12), 2115–2122 (2010).
[Crossref]

Chen, Y.

Chia, N.

de Castro, A.

Farsiu, S.

R. P. McNabb, S. Farsiu, S. S. Stinnett, J. A. Izatt, and A. N. Kuo, “Optical coherence tomography accurately measures corneal power change from laser refractive surgery,” Ophthalmology 122(4), 677–686 (2015).
[Crossref]

R. P. McNabb, F. LaRocca, S. Farsiu, A. N. Kuo, and J. A. Izatt, “Distributed scanning volumetric SDOCT for motion corrected corneal biometry,” Biomed. Opt. Express 3(9), 2050–2065 (2012).
[Crossref]

Ferchera, A. F.

R. J. Zawadzki, C. Leissera, R. Leitgeba, M. Pirchera, and A. F. Ferchera, “Three-dimensional ophthalmic optical coherence tomography with a refraction correction algorithm,” in European Conference on Biomedical Optics, (Optical Society of America, 2003), p. 5140_20.

Frisken, G.

T. Anderson, A. Segref, G. Frisken, and S. Frisken, “3d spectral imaging system for anterior chamber metrology,” in Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XIX, vol. 9312 (International Society for Optics and Photonics, 2015), p. 93120N.

Frisken, S.

T. Anderson, A. Segref, G. Frisken, and S. Frisken, “3d spectral imaging system for anterior chamber metrology,” in Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XIX, vol. 9312 (International Society for Optics and Photonics, 2015), p. 93120N.

Fujimoto, J. G.

Gambra, E.

Garstecki, P.

Gatinel, D.

D. Gatinel, J. Malet, T. Hoang-Xuan, and D. T. Azar, “Corneal elevation topography: best fit sphere, elevation distance, asphericity, toricity and clinical implications,” Cornea 30(5), 508–515 (2011).
[Crossref]

Goldblum, D.

Grulkowski, I.

Hoang-Xuan, T.

D. Gatinel, J. Malet, T. Hoang-Xuan, and D. T. Azar, “Corneal elevation topography: best fit sphere, elevation distance, asphericity, toricity and clinical implications,” Cornea 30(5), 508–515 (2011).
[Crossref]

Hong, Y.-J.

Hornegger, J.

Huang, D.

M. Tang, A. Chen, Y. Li, and D. Huang, “Corneal power measurement with fourier-domain optical coherence tomography,” J. Cataract Refractive Surg. 36(12), 2115–2122 (2010).
[Crossref]

Izatt, J. A.

Jimenez-Alfaro, I.

Kozon, L.

Kraus, M. F.

Kuo, A. N.

Lamoureux, E. L.

E. L. Lamoureux, S.-M. Saw, J. Thumboo, H. L. Wee, T. Aung, P. Mitchell, and T. Y. Wong, “The impact of corrected and uncorrected refractive error on visual functioning: the singapore malay eye study,” Invest. Ophthalmol. Visual Sci. 50(6), 2614–2620 (2009).
[Crossref]

LaRocca, F.

Leissera, C.

R. J. Zawadzki, C. Leissera, R. Leitgeba, M. Pirchera, and A. F. Ferchera, “Three-dimensional ophthalmic optical coherence tomography with a refraction correction algorithm,” in European Conference on Biomedical Optics, (Optical Society of America, 2003), p. 5140_20.

Leitgeba, R.

R. J. Zawadzki, C. Leissera, R. Leitgeba, M. Pirchera, and A. F. Ferchera, “Three-dimensional ophthalmic optical coherence tomography with a refraction correction algorithm,” in European Conference on Biomedical Optics, (Optical Society of America, 2003), p. 5140_20.

Li, Y.

M. Tang, A. Chen, Y. Li, and D. Huang, “Corneal power measurement with fourier-domain optical coherence tomography,” J. Cataract Refractive Surg. 36(12), 2115–2122 (2010).
[Crossref]

Liu, J. J.

Makita, S.

Malet, J.

D. Gatinel, J. Malet, T. Hoang-Xuan, and D. T. Azar, “Corneal elevation topography: best fit sphere, elevation distance, asphericity, toricity and clinical implications,” Cornea 30(5), 508–515 (2011).
[Crossref]

Marcos, S.

Mayer, M. A.

McNabb, R. P.

Mitchell, P.

E. L. Lamoureux, S.-M. Saw, J. Thumboo, H. L. Wee, T. Aung, P. Mitchell, and T. Y. Wong, “The impact of corrected and uncorrected refractive error on visual functioning: the singapore malay eye study,” Invest. Ophthalmol. Visual Sci. 50(6), 2614–2620 (2009).
[Crossref]

Ortiz, S.

Pascual, D.

Pérez-Merino, P.

Pezold, S.

J. Wagner, S. Pezold, and P. C. Cattin, “Model-driven 3-D regularisation for robust segmentation of the refractive corneal surfaces in spiral OCT scans,” 4th MICCAI Workshop on Ophthalmic Medical Image Analysis (2017).

Pirchera, M.

R. J. Zawadzki, C. Leissera, R. Leitgeba, M. Pirchera, and A. F. Ferchera, “Three-dimensional ophthalmic optical coherence tomography with a refraction correction algorithm,” in European Conference on Biomedical Optics, (Optical Society of America, 2003), p. 5140_20.

Potsaid, B.

Remon, L.

Saw, S.-M.

E. L. Lamoureux, S.-M. Saw, J. Thumboo, H. L. Wee, T. Aung, P. Mitchell, and T. Y. Wong, “The impact of corrected and uncorrected refractive error on visual functioning: the singapore malay eye study,” Invest. Ophthalmol. Visual Sci. 50(6), 2614–2620 (2009).
[Crossref]

Segref, A.

T. Anderson, A. Segref, G. Frisken, and S. Frisken, “3d spectral imaging system for anterior chamber metrology,” in Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XIX, vol. 9312 (International Society for Optics and Photonics, 2015), p. 93120N.

Sicam, V. A. D.

B. Braaf, T. C. van de Watering, K. Spruijt, R. G. van der Heijde, and V. A. D. Sicam, “Calculating angle lambda (λ) using zernike tilt measurements in specular reflection corneal topography,” J. Optom. 2(4), 207–214 (2009).
[Crossref]

Siedlecki, D.

Spruijt, K.

B. Braaf, T. C. van de Watering, K. Spruijt, R. G. van der Heijde, and V. A. D. Sicam, “Calculating angle lambda (λ) using zernike tilt measurements in specular reflection corneal topography,” J. Optom. 2(4), 207–214 (2009).
[Crossref]

Stinnett, S. S.

R. P. McNabb, S. Farsiu, S. S. Stinnett, J. A. Izatt, and A. N. Kuo, “Optical coherence tomography accurately measures corneal power change from laser refractive surgery,” Ophthalmology 122(4), 677–686 (2015).
[Crossref]

Stremplewski, P.

Szkulmowski, M.

Tang, M.

M. Tang, A. Chen, Y. Li, and D. Huang, “Corneal power measurement with fourier-domain optical coherence tomography,” J. Cataract Refractive Surg. 36(12), 2115–2122 (2010).
[Crossref]

Thumboo, J.

E. L. Lamoureux, S.-M. Saw, J. Thumboo, H. L. Wee, T. Aung, P. Mitchell, and T. Y. Wong, “The impact of corrected and uncorrected refractive error on visual functioning: the singapore malay eye study,” Invest. Ophthalmol. Visual Sci. 50(6), 2614–2620 (2009).
[Crossref]

van de Watering, T. C.

B. Braaf, T. C. van de Watering, K. Spruijt, R. G. van der Heijde, and V. A. D. Sicam, “Calculating angle lambda (λ) using zernike tilt measurements in specular reflection corneal topography,” J. Optom. 2(4), 207–214 (2009).
[Crossref]

van der Heijde, R. G.

B. Braaf, T. C. van de Watering, K. Spruijt, R. G. van der Heijde, and V. A. D. Sicam, “Calculating angle lambda (λ) using zernike tilt measurements in specular reflection corneal topography,” J. Optom. 2(4), 207–214 (2009).
[Crossref]

Wagner, J.

J. Wagner, D. Goldblum, and P. C. Cattin, “Golden angle based scanning for robust corneal topography with OCT,” Biomed. Opt. Express 8(2), 475–483 (2017).
[Crossref]

J. Wagner, S. Pezold, and P. C. Cattin, “Model-driven 3-D regularisation for robust segmentation of the refractive corneal surfaces in spiral OCT scans,” 4th MICCAI Workshop on Ophthalmic Medical Image Analysis (2017).

Wee, H. L.

E. L. Lamoureux, S.-M. Saw, J. Thumboo, H. L. Wee, T. Aung, P. Mitchell, and T. Y. Wong, “The impact of corrected and uncorrected refractive error on visual functioning: the singapore malay eye study,” Invest. Ophthalmol. Visual Sci. 50(6), 2614–2620 (2009).
[Crossref]

Weissman, B. A.

E. S. Bennett and B. A. Weissman, Clinical Contact Lens Practice (Lippincott Williams & Wilkins, 2005).

Wnuk, P.

Wojtkowski, M.

Wong, T. Y.

E. L. Lamoureux, S.-M. Saw, J. Thumboo, H. L. Wee, T. Aung, P. Mitchell, and T. Y. Wong, “The impact of corrected and uncorrected refractive error on visual functioning: the singapore malay eye study,” Invest. Ophthalmol. Visual Sci. 50(6), 2614–2620 (2009).
[Crossref]

Yasuno, Y.

Zawadzki, R. J.

R. J. Zawadzki, C. Leissera, R. Leitgeba, M. Pirchera, and A. F. Ferchera, “Three-dimensional ophthalmic optical coherence tomography with a refraction correction algorithm,” in European Conference on Biomedical Optics, (Optical Society of America, 2003), p. 5140_20.

Zhao, M.

Biomed. Opt. Express (6)

Cornea (1)

D. Gatinel, J. Malet, T. Hoang-Xuan, and D. T. Azar, “Corneal elevation topography: best fit sphere, elevation distance, asphericity, toricity and clinical implications,” Cornea 30(5), 508–515 (2011).
[Crossref]

Invest. Ophthalmol. Visual Sci. (1)

E. L. Lamoureux, S.-M. Saw, J. Thumboo, H. L. Wee, T. Aung, P. Mitchell, and T. Y. Wong, “The impact of corrected and uncorrected refractive error on visual functioning: the singapore malay eye study,” Invest. Ophthalmol. Visual Sci. 50(6), 2614–2620 (2009).
[Crossref]

J. Cataract Refractive Surg. (1)

M. Tang, A. Chen, Y. Li, and D. Huang, “Corneal power measurement with fourier-domain optical coherence tomography,” J. Cataract Refractive Surg. 36(12), 2115–2122 (2010).
[Crossref]

J. Optom. (1)

B. Braaf, T. C. van de Watering, K. Spruijt, R. G. van der Heijde, and V. A. D. Sicam, “Calculating angle lambda (λ) using zernike tilt measurements in specular reflection corneal topography,” J. Optom. 2(4), 207–214 (2009).
[Crossref]

Ophthalmology (1)

R. P. McNabb, S. Farsiu, S. S. Stinnett, J. A. Izatt, and A. N. Kuo, “Optical coherence tomography accurately measures corneal power change from laser refractive surgery,” Ophthalmology 122(4), 677–686 (2015).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Optica (1)

Other (7)

ISO 15004-2:2007, “Ophthalmic instruments - fundamental requirements and test methods - part 2: light hazard protection,” International Organization for Standardization (2007).

E. S. Bennett and B. A. Weissman, Clinical Contact Lens Practice (Lippincott Williams & Wilkins, 2005).

T. Anderson, A. Segref, G. Frisken, and S. Frisken, “3d spectral imaging system for anterior chamber metrology,” in Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XIX, vol. 9312 (International Society for Optics and Photonics, 2015), p. 93120N.

R. J. Zawadzki, C. Leissera, R. Leitgeba, M. Pirchera, and A. F. Ferchera, “Three-dimensional ophthalmic optical coherence tomography with a refraction correction algorithm,” in European Conference on Biomedical Optics, (Optical Society of America, 2003), p. 5140_20.

J. Wagner, S. Pezold, and P. C. Cattin, “Model-driven 3-D regularisation for robust segmentation of the refractive corneal surfaces in spiral OCT scans,” 4th MICCAI Workshop on Ophthalmic Medical Image Analysis (2017).

ANSI Z80.28-2017, “Methods for reporting optical aberrations of eyes,” American National Standards Institute (2017).

ISO 19980:2012, “Ophthalmic instruments - corneal topographers,” International Organization for Standardization (2012).

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Figures (13)

Fig. 1.
Fig. 1. The different steps in our pipeline.
Fig. 2.
Fig. 2. Schematic of the swept-source OCT system with its components. LS: Laser Source; PC1, PC2: Polarisation controllers; FC: Fibre coupler; CL: Collimating lens; MS: 2-D MEMS beam scanner; MO: Measurement objective; D: Detector; FM: Fibre mirror.
Fig. 3.
Fig. 3. (a) One frame of our scan pattern and (b) its projections onto the $x$ and $y$ axis. The color coding of the scan points indicate the point in time and is consistent between both subfigures.
Fig. 4.
Fig. 4. Section of the oversampled gradient image with masked out eye lid (reddish overlay), the ROI (blue) and the fine segmentation of the anterior corneal surface (green).
Fig. 5.
Fig. 5. The movement of the eye and its axes.
Fig. 6.
Fig. 6. Section of a corneal scan as acquired (no motion correction applied). The reconstruction of the anterior corneal surface is delineated by sampling the reconstructed corneal model $Z$ at the scan positions $x_i, y_i$. Yellow: Sampling of the static model, representing the axial distances to the virtual motion-free anterior corneal surface ($Z_i$). Cyan: Sampling of the model incorporating the motion profile $f$ determined during motion compensation, recreating the axial distances, as acquired ($Z_i^*$).
Fig. 7.
Fig. 7. (b) Axial curvature map of our OCT-based topography. (c) Axial curvature map of the Pentacam. (d) The mapped differences between the two topographies. The axial curvature maps are color-coded with increments of 0.5 D between the shades of color (a). The difference map is color-coded with increments of 0.2 D between the shades of color (e). The pentacam map is cropped to a square area with 8 mm lateral extent.
Fig. 8.
Fig. 8. Influence of the regularisation on (a) the repeatability of the aberrations and (b) the equivalence of the aberrations with Atlas (solid) and Pentacam (dashed), for the central zone (red), middle zone (green) and outer zone (blue).
Fig. 9.
Fig. 9. Repeatability of the topographies from phantom measurements: (a) The repeatability of the mean and (b) the repeatability of the aberrations. The repeatability for the moved phantom is given without motion compensation and without outlier removal (blue), without motion correction but with outlier removal (orange) and with motion compensation and outlier removal (green). The repeatability for the static phantom is shown in red. The 95 % confidence interval for the repeatability of the mean is indicated by the gray bars.
Fig. 10.
Fig. 10. (a) The compensated motion (red) and the motion applied to the phantom (blue) for a measurement of the moved phantom. The axial motion obtained by prior in vivo measurements (solid blue) was complemented by an unnatural linear part to obtain periodic motion (dotted blue). (b) The difference between the compensated motion and the applied motion.
Fig. 11.
Fig. 11. Effect of the motion compensation on the topography for a measurement of the moved phantom. (a) Theoretical axial curvature map. (b) Axial curvature map without motion compensation. (c) Axial curvature map with motion compensation. (d) The mapped differences between the curvature map with motion compensation and the theoretical axial curvature map. The axial curvature maps are color-coded with increments of 0.5 D (Fig. 7(a)). The difference map is color-coded with increments of 0.1 D (e). The corresponding difference measures can be found in Table 1.
Fig. 12.
Fig. 12. Repeatability of the topographies from in vivo measurements: (a) repeatability of the mean and (b) repeatability of the aberrations without motion compensation (blue), with motion compensation (orange), for the Atlas (green) and for the Pentacam (red).
Fig. 13.
Fig. 13. Difference mean (a) and standard deviation of the differences (b) between the devices for the different zones. The mean and standard deviation of the difference mean as well as the mean of the standard deviation of the differences are listed in Table 2.

Tables (2)

Tables Icon

Table 1. Effect of the motion compensation on the difference to the theoretical topography for the measurement of the moved phantom shown in Fig. 11.

Tables Icon

Table 2. Mean and the standard deviation (SD) of the zone differences between our method, Atlas and Pentacam. The p -value for the mean of the difference mean is calculated using the one sample t-test for the null hypothesis of zero mean difference.

Equations (34)

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

P SP ( t ) = ( x SP ( t ) y SP ( t ) ) = R SP sin ( a ω t ) ( sin ( b ω t ) cos ( b ω t ) ) ,
f ( t ) = ( f x ( t ) f y ( t ) f z ( t ) ) = i = 0 Z 1 boxcar i ( t ) ( α i x 0 1 + α i x 1 ( t t i m ) 1 + + α i x O x ( t t i m ) O x α i y 0 1 + α i y 1 ( t t i m ) 1 + + α i y O y ( t t i m ) O y α i z 0 1 + α i z 1 ( t t i m ) 1 + + α i z O z ( t t i m ) O z ) ,
min β , α ( Z M 0 C ) ( β α ) ( z 0 ) 2 2 ,
M = ( M 0 0 0 0 M 1 0 0 0 M Z 1 ) ,
α = ( α 0 α 1 α Z 1 ) T .
M i = ( M i x M i y M i z ) ,
α i = ( α i x α i y α i z ) T = ( α i x 0 α i x O x α i y 0 α i y O y α i z 0 α i z O z ) T .
M i x n , m = d d x Z ( x n , y n ) × ( t n T i ) m ,
M i y n , m = d d y Z ( x n , y n ) × ( t n T i ) m   .
M i z n , m = ( t n T i ) m .
C = s reg ( R ( t 0 c t 0 m ) R ( t 0 c t 1 m ) 0 0 0 0 R ( t 1 c t 1 m ) R ( t 1 c t 2 m ) 0 0 0 0 0 R ( t Z 3 c t Z 2 m ) 0 0 0 0 R ( t Z 2 c t Z 2 m ) R ( t Z 2 c t Z 1 m ) S 0 0 0 0 ) .
R ( t ) = ( R x ( t ) 0 0 0 R y ( t ) 0 0 0 R z ( t ) ) ,
R x ( t ) = ( 1 t t 2 t x O 0 1 2 t O x t O x 1 ) ,
R y ( t ) = ( 1 t t 2 t y O 0 1 2 t O y t O y 1 ) ,
R z ( t ) = ( 1 t t 2 t z O 0 1 2 t O z t O z 1 ) .
S = ( 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 ) .
P n = P n f ( t n ) for  n = 0 , 1 , , N t 1 .
Z n m ( ρ , φ ) = R n m ( ρ ) { cos ( m φ ) if  m  is positive sin ( m φ ) if  m  is negative ,
R n m ( ρ ) = k = 0 n m 2 ( 1 ) k ( n k ) ! k ! ( n + m 2 k ) ! ( n m 2 k ) ! ρ n 2 k .
K a = 0 r p K m ( r ) d r r p ,
K m = 2 M ( r ) r 2 ( 1 + ( M ( r ) r ) 2 ) 3 2 ,
R n m ( r ) r = 1 R R n m ( ρ ) ρ = 1 R k = 0 n m 2 ( 1 ) k ( n k ) ! k ! ( n + m 2 k ) ! ( n m 2 k ) ! ( n 2 k ) ρ n 2 k 1 ,
β ^ = a r g m i n β z Z β 2 + λ N w ( β β ref ) 2 ,
w j = k = 0 n m 2 ( 1 ) k ( n k ) ! k ! ( n + m 2 k ) ! ( n m 2 k ) ! ( n 2 k ) ,
β ^ = a r g m i n β y X β 2
X = ( Z λ N I w ) , y = ( z λ N β ref ) ,
( X T X ) β ^ = X T y .
Δ D i j k = w k ( D i k D j k ) ,
w k = n r k k = 1 n r k ,
M i j = 1 n k = 1 n Δ D i j k ,
s i j = k = 1 n ( Δ D i j k M i j ) 2 n 1
= k = 1 n ( w k ( D i k D j k ) M i j ) 2 n 1 .
s i j = k = 1 n ( w k m i j m i j ) 2 n 1 = m i j 2 k = 1 n ( w k 1 ) 2 n 1 .
s i j = k = 1 n ( w k ( D i k D j k M i j ) ) 2 n 1 .

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