Abstract

We previously developed a technique to acquire a SLO (scanning laser ophthalmoscope) like fundus intensity image from the raw spectra measured with spectral-domain optical coherence tomography (OCT), the same spectra used to generate a 3D OCT data set. This technique offers simultaneous fundus and OCT images and, therefore, solves the problem of registering a cross sectional OCT image to fundus features. However, the registration of high density OCT images is still an unsolved problem because no useful fundus image can be generated from the high density scans. High density OCT images can significantly improve the image quality and enhance the visualization of retinal structure, especially the structure of small lesions. We have developed a feature-based algorithm, which can register a high density OCT image on the fundus image generated from normal density scans. The algorithm was successfully tested for both normal and diseased eyes.

©2006 Optical Society of America

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References

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  1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregori, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [Crossref] [PubMed]
  2. A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Optics Commun. 117, 43–48 (1995).
    [Crossref]
  3. M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “in vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomedical Opt. 7, 457–463 (2002).
    [Crossref]
  4. N. A. Nassif, B. Cense, B. H. Park, M. C. Pierce, S. H. Yun, B. E. Bouma, G. J. Tearney, T. C. Chen, and J. F. de Boer, “In vivo high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve,” Opt. Express 12, 367–376 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-367
    [Crossref] [PubMed]
  5. B. Cense, N. A. Nassif, T. C. Chen, M. C. Pierce, S. Yun, B. H. Park, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “Ultrahigh-resolution high-speed retinal imaging using spectral-domain optical coherence tomography,” Opt. Express 12, 2435–2447 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2435
    [Crossref] [PubMed]
  6. R. A. Leitgeb, W. Drexler, A. Unterhuber, B. Hermann, T. Bajraszewski, T. Le, A. Stingl, and A. F. Fercher, “Ultrahigh resolution Fourier domain optical coherence tomography,” Opt. Express 12, 2156–2165 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2156
    [Crossref] [PubMed]
  7. M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12, 2404–2422 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2404.
    [Crossref] [PubMed]
  8. S. Jiao, R. Knighton, X. Huang, G. Gregori, and C. A. Puliafito, “Simultaneous acquisition of sectional and fundus ophthalmic images with spectral-domain optical coherence tomography,” Opt. Express 13, 444–452 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-444
    [Crossref] [PubMed]
  9. Maciej Wojtkowski, Vivek Srinivasan, James G. Fujimoto, Tony Ko, Joel S. Schuman, Andrzej Kowalczyk, and Jay S. Duker, “Three-dimensional retinal imaging with high-speed ultrahigh-resolution optical coherence tomography,” Ophthalmology 112, 1734–1746 (2005).
    [Crossref] [PubMed]
  10. C. K. Hitzenberger, P. Trost, P. Lo, and Q. Zhou, “Three-dimensional imaging of the human retina by high-speed optical coherence tomography,” Opt. Express 11, 2753–2761 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-21-2753
    [Crossref] [PubMed]
  11. B. Zitová and J. Flusser, “Image registration methods: a survey,” Image and Vision Computing 21, 977–1000 (2003).
    [Crossref]
  12. D. L. G. Hill, P. G. Batchelor, M. Holden, and D. J. Hawkes, “Medical image registration,” Phys. Med. Biol. 46, R1–R45 (2001).
    [Crossref] [PubMed]
  13. G. Häusler and M. W. Lindner, “Coherence radar and spectral radar—new tools for dermatological diagnosis,” J. Biomedical Opt. 3, 21–31 (1998).
    [Crossref]

2005 (2)

S. Jiao, R. Knighton, X. Huang, G. Gregori, and C. A. Puliafito, “Simultaneous acquisition of sectional and fundus ophthalmic images with spectral-domain optical coherence tomography,” Opt. Express 13, 444–452 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-444
[Crossref] [PubMed]

Maciej Wojtkowski, Vivek Srinivasan, James G. Fujimoto, Tony Ko, Joel S. Schuman, Andrzej Kowalczyk, and Jay S. Duker, “Three-dimensional retinal imaging with high-speed ultrahigh-resolution optical coherence tomography,” Ophthalmology 112, 1734–1746 (2005).
[Crossref] [PubMed]

2004 (4)

2003 (2)

2002 (1)

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “in vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomedical Opt. 7, 457–463 (2002).
[Crossref]

2001 (1)

D. L. G. Hill, P. G. Batchelor, M. Holden, and D. J. Hawkes, “Medical image registration,” Phys. Med. Biol. 46, R1–R45 (2001).
[Crossref] [PubMed]

1998 (1)

G. Häusler and M. W. Lindner, “Coherence radar and spectral radar—new tools for dermatological diagnosis,” J. Biomedical Opt. 3, 21–31 (1998).
[Crossref]

1995 (1)

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Optics Commun. 117, 43–48 (1995).
[Crossref]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregori, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Bajraszewski, T.

Batchelor, P. G.

D. L. G. Hill, P. G. Batchelor, M. Holden, and D. J. Hawkes, “Medical image registration,” Phys. Med. Biol. 46, R1–R45 (2001).
[Crossref] [PubMed]

Bouma, B. E.

Cense, B.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregori, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Chen, T. C.

de Boer, J. F.

Drexler, W.

Duker, J. S.

Duker, Jay S.

Maciej Wojtkowski, Vivek Srinivasan, James G. Fujimoto, Tony Ko, Joel S. Schuman, Andrzej Kowalczyk, and Jay S. Duker, “Three-dimensional retinal imaging with high-speed ultrahigh-resolution optical coherence tomography,” Ophthalmology 112, 1734–1746 (2005).
[Crossref] [PubMed]

El-Zaiat, S. Y.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Optics Commun. 117, 43–48 (1995).
[Crossref]

Fercher, A. F.

R. A. Leitgeb, W. Drexler, A. Unterhuber, B. Hermann, T. Bajraszewski, T. Le, A. Stingl, and A. F. Fercher, “Ultrahigh resolution Fourier domain optical coherence tomography,” Opt. Express 12, 2156–2165 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2156
[Crossref] [PubMed]

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “in vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomedical Opt. 7, 457–463 (2002).
[Crossref]

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Optics Commun. 117, 43–48 (1995).
[Crossref]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregori, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Flusser, J.

B. Zitová and J. Flusser, “Image registration methods: a survey,” Image and Vision Computing 21, 977–1000 (2003).
[Crossref]

Fujimoto, J. G.

Fujimoto, James G.

Maciej Wojtkowski, Vivek Srinivasan, James G. Fujimoto, Tony Ko, Joel S. Schuman, Andrzej Kowalczyk, and Jay S. Duker, “Three-dimensional retinal imaging with high-speed ultrahigh-resolution optical coherence tomography,” Ophthalmology 112, 1734–1746 (2005).
[Crossref] [PubMed]

Gregori, G.

Gregori, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregori, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Häusler, G.

G. Häusler and M. W. Lindner, “Coherence radar and spectral radar—new tools for dermatological diagnosis,” J. Biomedical Opt. 3, 21–31 (1998).
[Crossref]

Hawkes, D. J.

D. L. G. Hill, P. G. Batchelor, M. Holden, and D. J. Hawkes, “Medical image registration,” Phys. Med. Biol. 46, R1–R45 (2001).
[Crossref] [PubMed]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregori, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Hermann, B.

Hill, D. L. G.

D. L. G. Hill, P. G. Batchelor, M. Holden, and D. J. Hawkes, “Medical image registration,” Phys. Med. Biol. 46, R1–R45 (2001).
[Crossref] [PubMed]

Hitzenberger, C. K.

C. K. Hitzenberger, P. Trost, P. Lo, and Q. Zhou, “Three-dimensional imaging of the human retina by high-speed optical coherence tomography,” Opt. Express 11, 2753–2761 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-21-2753
[Crossref] [PubMed]

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Optics Commun. 117, 43–48 (1995).
[Crossref]

Holden, M.

D. L. G. Hill, P. G. Batchelor, M. Holden, and D. J. Hawkes, “Medical image registration,” Phys. Med. Biol. 46, R1–R45 (2001).
[Crossref] [PubMed]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregori, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Huang, X.

Jiao, S.

Kamp, G.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Optics Commun. 117, 43–48 (1995).
[Crossref]

Knighton, R.

Ko, T. H.

Ko, Tony

Maciej Wojtkowski, Vivek Srinivasan, James G. Fujimoto, Tony Ko, Joel S. Schuman, Andrzej Kowalczyk, and Jay S. Duker, “Three-dimensional retinal imaging with high-speed ultrahigh-resolution optical coherence tomography,” Ophthalmology 112, 1734–1746 (2005).
[Crossref] [PubMed]

Kowalczyk, A.

Kowalczyk, Andrzej

Maciej Wojtkowski, Vivek Srinivasan, James G. Fujimoto, Tony Ko, Joel S. Schuman, Andrzej Kowalczyk, and Jay S. Duker, “Three-dimensional retinal imaging with high-speed ultrahigh-resolution optical coherence tomography,” Ophthalmology 112, 1734–1746 (2005).
[Crossref] [PubMed]

Le, T.

Leitgeb, R.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “in vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomedical Opt. 7, 457–463 (2002).
[Crossref]

Leitgeb, R. A.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregori, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Lindner, M. W.

G. Häusler and M. W. Lindner, “Coherence radar and spectral radar—new tools for dermatological diagnosis,” J. Biomedical Opt. 3, 21–31 (1998).
[Crossref]

Lo, P.

Nassif, N. A.

Park, B. H.

Pierce, M. C.

Puliafito, C. A.

S. Jiao, R. Knighton, X. Huang, G. Gregori, and C. A. Puliafito, “Simultaneous acquisition of sectional and fundus ophthalmic images with spectral-domain optical coherence tomography,” Opt. Express 13, 444–452 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-444
[Crossref] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregori, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregori, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Schuman, Joel S.

Maciej Wojtkowski, Vivek Srinivasan, James G. Fujimoto, Tony Ko, Joel S. Schuman, Andrzej Kowalczyk, and Jay S. Duker, “Three-dimensional retinal imaging with high-speed ultrahigh-resolution optical coherence tomography,” Ophthalmology 112, 1734–1746 (2005).
[Crossref] [PubMed]

Srinivasan, V. J.

Srinivasan, Vivek

Maciej Wojtkowski, Vivek Srinivasan, James G. Fujimoto, Tony Ko, Joel S. Schuman, Andrzej Kowalczyk, and Jay S. Duker, “Three-dimensional retinal imaging with high-speed ultrahigh-resolution optical coherence tomography,” Ophthalmology 112, 1734–1746 (2005).
[Crossref] [PubMed]

Stingl, A.

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregori, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregori, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Tearney, G. J.

Trost, P.

Unterhuber, A.

Wojtkowski, M.

Wojtkowski, Maciej

Maciej Wojtkowski, Vivek Srinivasan, James G. Fujimoto, Tony Ko, Joel S. Schuman, Andrzej Kowalczyk, and Jay S. Duker, “Three-dimensional retinal imaging with high-speed ultrahigh-resolution optical coherence tomography,” Ophthalmology 112, 1734–1746 (2005).
[Crossref] [PubMed]

Yun, S.

Yun, S. H.

Zhou, Q.

Zitová, B.

B. Zitová and J. Flusser, “Image registration methods: a survey,” Image and Vision Computing 21, 977–1000 (2003).
[Crossref]

Image and Vision Computing (1)

B. Zitová and J. Flusser, “Image registration methods: a survey,” Image and Vision Computing 21, 977–1000 (2003).
[Crossref]

J. Biomedical Opt. (2)

G. Häusler and M. W. Lindner, “Coherence radar and spectral radar—new tools for dermatological diagnosis,” J. Biomedical Opt. 3, 21–31 (1998).
[Crossref]

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “in vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomedical Opt. 7, 457–463 (2002).
[Crossref]

Ophthalmology (1)

Maciej Wojtkowski, Vivek Srinivasan, James G. Fujimoto, Tony Ko, Joel S. Schuman, Andrzej Kowalczyk, and Jay S. Duker, “Three-dimensional retinal imaging with high-speed ultrahigh-resolution optical coherence tomography,” Ophthalmology 112, 1734–1746 (2005).
[Crossref] [PubMed]

Opt. Express (6)

C. K. Hitzenberger, P. Trost, P. Lo, and Q. Zhou, “Three-dimensional imaging of the human retina by high-speed optical coherence tomography,” Opt. Express 11, 2753–2761 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-21-2753
[Crossref] [PubMed]

N. A. Nassif, B. Cense, B. H. Park, M. C. Pierce, S. H. Yun, B. E. Bouma, G. J. Tearney, T. C. Chen, and J. F. de Boer, “In vivo high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve,” Opt. Express 12, 367–376 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-367
[Crossref] [PubMed]

B. Cense, N. A. Nassif, T. C. Chen, M. C. Pierce, S. Yun, B. H. Park, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “Ultrahigh-resolution high-speed retinal imaging using spectral-domain optical coherence tomography,” Opt. Express 12, 2435–2447 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2435
[Crossref] [PubMed]

R. A. Leitgeb, W. Drexler, A. Unterhuber, B. Hermann, T. Bajraszewski, T. Le, A. Stingl, and A. F. Fercher, “Ultrahigh resolution Fourier domain optical coherence tomography,” Opt. Express 12, 2156–2165 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2156
[Crossref] [PubMed]

M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12, 2404–2422 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2404.
[Crossref] [PubMed]

S. Jiao, R. Knighton, X. Huang, G. Gregori, and C. A. Puliafito, “Simultaneous acquisition of sectional and fundus ophthalmic images with spectral-domain optical coherence tomography,” Opt. Express 13, 444–452 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-444
[Crossref] [PubMed]

Optics Commun. (1)

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Optics Commun. 117, 43–48 (1995).
[Crossref]

Phys. Med. Biol. (1)

D. L. G. Hill, P. G. Batchelor, M. Holden, and D. J. Hawkes, “Medical image registration,” Phys. Med. Biol. 46, R1–R45 (2001).
[Crossref] [PubMed]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregori, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

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Figures (7)

Fig. 1.
Fig. 1.

OCT fundus image for the fovea of a normal human eye with different scan densities. The images covered an area of 6mm × 6mm on the retina. (a) 2048×32 pixels (horizontal × vertical); (b) 512 × 128 pixels.

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Fig. 2.
Fig. 2.

Schematic of the experimental system. SLD: superluminescent diode; PC: polarization controller. The spectrometer consists of a f=50 mm collimating lens, a 1200 line/mm transmission grating and a f=200 mm imaging lens. The CCD camera is a 2048 element linear array.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3.

Fundus images for the optic disc of a normal human eye with an elliptical high-pass filter of order 1 with different normalized cut off frequencies. (a) 0.1; (b) 0.2; (c) 0.3; (d) 0.5.

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4.

(a) high density OCT image of the fovea of a normal human eye consisting of 2048 A-scans; (b) contour of the retinal surface in the high density image; (c) contours of the retinal surface in the normal density OCT images (31 frames from No. 50 to No. 80); (d) the calculated maximal correlation coefficients vs frame number; (e) the enhanced correlation coefficients vs frame number.

Download Full Size | PPT Slide | PDF

Fig. 5.
Fig. 5.

(a) The picked high density OCT image; (b) the fundus image generated from the normal density scans; (c) the corresponding frame of the normal density scans (frame No. 64 in the 128 OCT images). The white line on the fundus image represents the determined location of the high density image.

Download Full Size | PPT Slide | PDF

Fig. 6.
Fig. 6.

(a) The picked high density OCT image; (b) the fundus image generated from the normal density scans; (c) the corresponding frame of the normal density OCT images. The white line on the fundus image represents the determined location of the high density OCT image.

Download Full Size | PPT Slide | PDF

Fig. 7.
Fig. 7.

(a) The picked high density OCT image; (b) the fundus image generated from the normal density scans; (c) the corresponding frame of the normal density OCT images. The white line on the fundus image represents the determined location of the high density OCT image.

Download Full Size | PPT Slide | PDF

Equations (7)

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

G d ( ν ) = G s ( ν ) { 1 + ∑ n R n + 2 ∑ n ≠ m R n R m cos ⁡ [ 2 πν ( τ n − τ m ) ] + 2 ∑ n R n cos ⁡ [ 2 πν ( τ n − τ r ) ] } ,
I f ( τ ) = I ( τ ) H ( τ ) ,
A = { x A : x A ∈ Ω A }
B = { x B : x B ∈ Ω B } ,
x z = t x t z + s cos ⁡ θ − sin ⁡ θ sin ⁡ θ cos ⁡ θ x ' z ' ,
C = ∑ x A [ A ( x A ) − A ¯ ] { B [ T ( x B ) ] − B ¯ } ∑ x A [ A ( x A ) − A ¯ ] 2 ∑ x B { B [ T ( x B ) ] − B ¯ } 2 ,
C e = max [ C ( θ , t x ) ] ∑ m { z A ( m ) − T [ z B ( m ) ] } 2 ,

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