Abstract

To accurately model lens-based functions such as accommodation and image formation on the retina, it is essential to know anterior chamber depth, anterior segment length, lens thickness, and, most importantly, lens curvature both on the surfaces and internally. With the exception of lens curvatures, all these data can be obtained with a high degree of precision by one or more techniques (i.e., A-scan ultrasonography and pachymetry). Lens curvatures can be collected by Scheimpflug slit lamp photography, but the curvature data must be extracted from these images, a problem complicated by both linear and nonlinear image distortion. Previous approaches have involved significant magnification of the image combined with major subjective input and judgment. We present here a computer-based application of the Hough technique for measurement of curvature of lens surfaces observed in Scheimpflug slit lamp photography and related evaluation of (and solutions for) the associated image distortion. Minimal user input is required for successful application of this method, but the time required to obtain a fit for each surface is >1 min. Results obtained by this technique on test images compare favorably with those obtained by independent methods.

© 1991 Optical Society of America

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References

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  1. N. P. Brown, “An Advanced Slit-Lamp Camera,” Br. J. Opthalmol. 56, 624–631 (1972).
    [CrossRef]
  2. J. F. Koretz, P. L. Kaufman, M. W. Neider, P. A. Goeckner, “Accommodation and Presbyopia in the Human Eye—Aging of the Anterior Segment,” Vision Res. 29, 1685–1692 (1989).
    [CrossRef] [PubMed]
  3. J. F. Koretz, P. L. Kaufman, M. W. Neider, P. A. Goeckner, “Accommodation and Presbyopia in the Human Eye. 1: Evaluation of In Vivo Measurement Techniques,” Appl. Opt. 28, 1097–1102 (1989).
    [CrossRef] [PubMed]
  4. J. F. Koretz, G. H. Handelman, N. P. Brown, “Analysis of Human Crystalline Lens Curvature as a Function of Accommodative State and Age,” Vision Res. 24, 1141–1151 (1984).
    [CrossRef] [PubMed]
  5. A. Hachicha et al., “The Use of Gray-Level Information and Fitting Techniques for Precise Measurement of Corneal Curvature and Thickness,” Comput. Vision Graphics Image Process. 47, 131–164 (1989).
    [CrossRef]
  6. C. Kimme, D. H. Ballard, J. Slansky, “Finding Circles by an Array of Accumulators,” Commun. ACM 18, 120–122 (1975).
    [CrossRef]
  7. H. Wechsler, J. Slansky, “Automatic Detection of Ribs In Chest Radiograms,” Pattern Recognition 9, 21–30 (1977).
    [CrossRef]
  8. J. Illingworth, J. Kittler, “A Survey of the Hough Transform,” Comput. Vision Graphics Image Process. 44, 87–116 (1988).
    [CrossRef]
  9. C. A. Cook, J. F. Koretz, “An Efficient Implementation of the Hough Transform for the Detection of the Curvatures of the Human Crystalline Lens,” in preparation.
  10. Documentation for Automatix Image Analyst.
  11. J. F. Koretz, G. H. Handelman, “The ‘Lens Paradox’ and Image Formation in Accommodating Human Eyes,” in The Lens: Transparency and Cataract, G. Duncan, Ed., Topics in Aging Research in Europe6, 57–64 (1986).
  12. R. O. Duda, P. E. Hart, “Use of the Hough Transformation to Detect Lines and Curves in Pictures,” Commun. ACM 15, 11–16 (1972).
    [CrossRef]
  13. R. O. Duda, P. E. Hart, Pattern Recognition and Scene Analysis (Wiley, New York, 1973).
  14. S. D. Shapiro, “Transformations for the Computer Detection of Curves in Noisy Pictures,” Comput. Graphics Image Process. 4, 328–338 (1975).
    [CrossRef]
  15. S. D. Shapiro, “Generalization of the Hough Transform for Curve Detection in Noisy Digital Images,” in Fourth IJCPR, Kyoto (1978), pp. 710–714.
  16. D. H. Ballard, C. M. Brown, Computer Vision (Prentice-Hall, Englewood Cliffs, NJ, 1982).
  17. R. J. Schalkoff, Digital Image Processing and Computer Vision (Wiley, New York, 1989).
  18. R. S. Conkner, “A Dual Plane Variation of the Hough Transform for Detecting Non-Concentric Circles of Different Radii,” Comput. Vision Graphics Image Process. 43, 115–132 (1988).
    [CrossRef]
  19. S. Tsuji, F. Matsumoto, “Detection of Elliptic and Linear Edges by Searching Two Parameter Spaces,” IEEE Trans. Comput. C-27, 777–781 (1979).
    [CrossRef]
  20. W. H. Press, Numerical Recipes (Cambridge U. Press, London, 1986).
  21. J. Illingworth, J. Kittler, “The Adaptive Hough Transform,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-9, 690–697 (1987).
    [CrossRef]
  22. T. M. Van Veen, F. C. A. Groen, “Discretization Errors in the Hough Transform,” Pattern Recognition 14, 137–145 (1981).
    [CrossRef]
  23. C. M. Brown, “Inherent Bias and Noise in the Hough Transform,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-5, 439–505 (1983).
    [CrossRef]

1989 (3)

A. Hachicha et al., “The Use of Gray-Level Information and Fitting Techniques for Precise Measurement of Corneal Curvature and Thickness,” Comput. Vision Graphics Image Process. 47, 131–164 (1989).
[CrossRef]

J. F. Koretz, P. L. Kaufman, M. W. Neider, P. A. Goeckner, “Accommodation and Presbyopia in the Human Eye—Aging of the Anterior Segment,” Vision Res. 29, 1685–1692 (1989).
[CrossRef] [PubMed]

J. F. Koretz, P. L. Kaufman, M. W. Neider, P. A. Goeckner, “Accommodation and Presbyopia in the Human Eye. 1: Evaluation of In Vivo Measurement Techniques,” Appl. Opt. 28, 1097–1102 (1989).
[CrossRef] [PubMed]

1988 (2)

J. Illingworth, J. Kittler, “A Survey of the Hough Transform,” Comput. Vision Graphics Image Process. 44, 87–116 (1988).
[CrossRef]

R. S. Conkner, “A Dual Plane Variation of the Hough Transform for Detecting Non-Concentric Circles of Different Radii,” Comput. Vision Graphics Image Process. 43, 115–132 (1988).
[CrossRef]

1987 (1)

J. Illingworth, J. Kittler, “The Adaptive Hough Transform,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-9, 690–697 (1987).
[CrossRef]

1984 (1)

J. F. Koretz, G. H. Handelman, N. P. Brown, “Analysis of Human Crystalline Lens Curvature as a Function of Accommodative State and Age,” Vision Res. 24, 1141–1151 (1984).
[CrossRef] [PubMed]

1983 (1)

C. M. Brown, “Inherent Bias and Noise in the Hough Transform,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-5, 439–505 (1983).
[CrossRef]

1981 (1)

T. M. Van Veen, F. C. A. Groen, “Discretization Errors in the Hough Transform,” Pattern Recognition 14, 137–145 (1981).
[CrossRef]

1979 (1)

S. Tsuji, F. Matsumoto, “Detection of Elliptic and Linear Edges by Searching Two Parameter Spaces,” IEEE Trans. Comput. C-27, 777–781 (1979).
[CrossRef]

1977 (1)

H. Wechsler, J. Slansky, “Automatic Detection of Ribs In Chest Radiograms,” Pattern Recognition 9, 21–30 (1977).
[CrossRef]

1975 (2)

C. Kimme, D. H. Ballard, J. Slansky, “Finding Circles by an Array of Accumulators,” Commun. ACM 18, 120–122 (1975).
[CrossRef]

S. D. Shapiro, “Transformations for the Computer Detection of Curves in Noisy Pictures,” Comput. Graphics Image Process. 4, 328–338 (1975).
[CrossRef]

1972 (2)

N. P. Brown, “An Advanced Slit-Lamp Camera,” Br. J. Opthalmol. 56, 624–631 (1972).
[CrossRef]

R. O. Duda, P. E. Hart, “Use of the Hough Transformation to Detect Lines and Curves in Pictures,” Commun. ACM 15, 11–16 (1972).
[CrossRef]

Ballard, D. H.

C. Kimme, D. H. Ballard, J. Slansky, “Finding Circles by an Array of Accumulators,” Commun. ACM 18, 120–122 (1975).
[CrossRef]

D. H. Ballard, C. M. Brown, Computer Vision (Prentice-Hall, Englewood Cliffs, NJ, 1982).

Brown, C. M.

C. M. Brown, “Inherent Bias and Noise in the Hough Transform,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-5, 439–505 (1983).
[CrossRef]

D. H. Ballard, C. M. Brown, Computer Vision (Prentice-Hall, Englewood Cliffs, NJ, 1982).

Brown, N. P.

J. F. Koretz, G. H. Handelman, N. P. Brown, “Analysis of Human Crystalline Lens Curvature as a Function of Accommodative State and Age,” Vision Res. 24, 1141–1151 (1984).
[CrossRef] [PubMed]

N. P. Brown, “An Advanced Slit-Lamp Camera,” Br. J. Opthalmol. 56, 624–631 (1972).
[CrossRef]

Conkner, R. S.

R. S. Conkner, “A Dual Plane Variation of the Hough Transform for Detecting Non-Concentric Circles of Different Radii,” Comput. Vision Graphics Image Process. 43, 115–132 (1988).
[CrossRef]

Cook, C. A.

C. A. Cook, J. F. Koretz, “An Efficient Implementation of the Hough Transform for the Detection of the Curvatures of the Human Crystalline Lens,” in preparation.

Duda, R. O.

R. O. Duda, P. E. Hart, “Use of the Hough Transformation to Detect Lines and Curves in Pictures,” Commun. ACM 15, 11–16 (1972).
[CrossRef]

R. O. Duda, P. E. Hart, Pattern Recognition and Scene Analysis (Wiley, New York, 1973).

Goeckner, P. A.

J. F. Koretz, P. L. Kaufman, M. W. Neider, P. A. Goeckner, “Accommodation and Presbyopia in the Human Eye—Aging of the Anterior Segment,” Vision Res. 29, 1685–1692 (1989).
[CrossRef] [PubMed]

J. F. Koretz, P. L. Kaufman, M. W. Neider, P. A. Goeckner, “Accommodation and Presbyopia in the Human Eye. 1: Evaluation of In Vivo Measurement Techniques,” Appl. Opt. 28, 1097–1102 (1989).
[CrossRef] [PubMed]

Groen, F. C. A.

T. M. Van Veen, F. C. A. Groen, “Discretization Errors in the Hough Transform,” Pattern Recognition 14, 137–145 (1981).
[CrossRef]

Hachicha, A.

A. Hachicha et al., “The Use of Gray-Level Information and Fitting Techniques for Precise Measurement of Corneal Curvature and Thickness,” Comput. Vision Graphics Image Process. 47, 131–164 (1989).
[CrossRef]

Handelman, G. H.

J. F. Koretz, G. H. Handelman, N. P. Brown, “Analysis of Human Crystalline Lens Curvature as a Function of Accommodative State and Age,” Vision Res. 24, 1141–1151 (1984).
[CrossRef] [PubMed]

J. F. Koretz, G. H. Handelman, “The ‘Lens Paradox’ and Image Formation in Accommodating Human Eyes,” in The Lens: Transparency and Cataract, G. Duncan, Ed., Topics in Aging Research in Europe6, 57–64 (1986).

Hart, P. E.

R. O. Duda, P. E. Hart, “Use of the Hough Transformation to Detect Lines and Curves in Pictures,” Commun. ACM 15, 11–16 (1972).
[CrossRef]

R. O. Duda, P. E. Hart, Pattern Recognition and Scene Analysis (Wiley, New York, 1973).

Illingworth, J.

J. Illingworth, J. Kittler, “A Survey of the Hough Transform,” Comput. Vision Graphics Image Process. 44, 87–116 (1988).
[CrossRef]

J. Illingworth, J. Kittler, “The Adaptive Hough Transform,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-9, 690–697 (1987).
[CrossRef]

Kaufman, P. L.

J. F. Koretz, P. L. Kaufman, M. W. Neider, P. A. Goeckner, “Accommodation and Presbyopia in the Human Eye—Aging of the Anterior Segment,” Vision Res. 29, 1685–1692 (1989).
[CrossRef] [PubMed]

J. F. Koretz, P. L. Kaufman, M. W. Neider, P. A. Goeckner, “Accommodation and Presbyopia in the Human Eye. 1: Evaluation of In Vivo Measurement Techniques,” Appl. Opt. 28, 1097–1102 (1989).
[CrossRef] [PubMed]

Kimme, C.

C. Kimme, D. H. Ballard, J. Slansky, “Finding Circles by an Array of Accumulators,” Commun. ACM 18, 120–122 (1975).
[CrossRef]

Kittler, J.

J. Illingworth, J. Kittler, “A Survey of the Hough Transform,” Comput. Vision Graphics Image Process. 44, 87–116 (1988).
[CrossRef]

J. Illingworth, J. Kittler, “The Adaptive Hough Transform,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-9, 690–697 (1987).
[CrossRef]

Koretz, J. F.

J. F. Koretz, P. L. Kaufman, M. W. Neider, P. A. Goeckner, “Accommodation and Presbyopia in the Human Eye—Aging of the Anterior Segment,” Vision Res. 29, 1685–1692 (1989).
[CrossRef] [PubMed]

J. F. Koretz, P. L. Kaufman, M. W. Neider, P. A. Goeckner, “Accommodation and Presbyopia in the Human Eye. 1: Evaluation of In Vivo Measurement Techniques,” Appl. Opt. 28, 1097–1102 (1989).
[CrossRef] [PubMed]

J. F. Koretz, G. H. Handelman, N. P. Brown, “Analysis of Human Crystalline Lens Curvature as a Function of Accommodative State and Age,” Vision Res. 24, 1141–1151 (1984).
[CrossRef] [PubMed]

C. A. Cook, J. F. Koretz, “An Efficient Implementation of the Hough Transform for the Detection of the Curvatures of the Human Crystalline Lens,” in preparation.

J. F. Koretz, G. H. Handelman, “The ‘Lens Paradox’ and Image Formation in Accommodating Human Eyes,” in The Lens: Transparency and Cataract, G. Duncan, Ed., Topics in Aging Research in Europe6, 57–64 (1986).

Matsumoto, F.

S. Tsuji, F. Matsumoto, “Detection of Elliptic and Linear Edges by Searching Two Parameter Spaces,” IEEE Trans. Comput. C-27, 777–781 (1979).
[CrossRef]

Neider, M. W.

J. F. Koretz, P. L. Kaufman, M. W. Neider, P. A. Goeckner, “Accommodation and Presbyopia in the Human Eye—Aging of the Anterior Segment,” Vision Res. 29, 1685–1692 (1989).
[CrossRef] [PubMed]

J. F. Koretz, P. L. Kaufman, M. W. Neider, P. A. Goeckner, “Accommodation and Presbyopia in the Human Eye. 1: Evaluation of In Vivo Measurement Techniques,” Appl. Opt. 28, 1097–1102 (1989).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, Numerical Recipes (Cambridge U. Press, London, 1986).

Schalkoff, R. J.

R. J. Schalkoff, Digital Image Processing and Computer Vision (Wiley, New York, 1989).

Shapiro, S. D.

S. D. Shapiro, “Transformations for the Computer Detection of Curves in Noisy Pictures,” Comput. Graphics Image Process. 4, 328–338 (1975).
[CrossRef]

S. D. Shapiro, “Generalization of the Hough Transform for Curve Detection in Noisy Digital Images,” in Fourth IJCPR, Kyoto (1978), pp. 710–714.

Slansky, J.

H. Wechsler, J. Slansky, “Automatic Detection of Ribs In Chest Radiograms,” Pattern Recognition 9, 21–30 (1977).
[CrossRef]

C. Kimme, D. H. Ballard, J. Slansky, “Finding Circles by an Array of Accumulators,” Commun. ACM 18, 120–122 (1975).
[CrossRef]

Tsuji, S.

S. Tsuji, F. Matsumoto, “Detection of Elliptic and Linear Edges by Searching Two Parameter Spaces,” IEEE Trans. Comput. C-27, 777–781 (1979).
[CrossRef]

Van Veen, T. M.

T. M. Van Veen, F. C. A. Groen, “Discretization Errors in the Hough Transform,” Pattern Recognition 14, 137–145 (1981).
[CrossRef]

Wechsler, H.

H. Wechsler, J. Slansky, “Automatic Detection of Ribs In Chest Radiograms,” Pattern Recognition 9, 21–30 (1977).
[CrossRef]

Appl. Opt. (1)

Br. J. Opthalmol. (1)

N. P. Brown, “An Advanced Slit-Lamp Camera,” Br. J. Opthalmol. 56, 624–631 (1972).
[CrossRef]

Commun. ACM (2)

C. Kimme, D. H. Ballard, J. Slansky, “Finding Circles by an Array of Accumulators,” Commun. ACM 18, 120–122 (1975).
[CrossRef]

R. O. Duda, P. E. Hart, “Use of the Hough Transformation to Detect Lines and Curves in Pictures,” Commun. ACM 15, 11–16 (1972).
[CrossRef]

Comput. Graphics Image Process. (1)

S. D. Shapiro, “Transformations for the Computer Detection of Curves in Noisy Pictures,” Comput. Graphics Image Process. 4, 328–338 (1975).
[CrossRef]

Comput. Vision Graphics Image Process. (3)

J. Illingworth, J. Kittler, “A Survey of the Hough Transform,” Comput. Vision Graphics Image Process. 44, 87–116 (1988).
[CrossRef]

R. S. Conkner, “A Dual Plane Variation of the Hough Transform for Detecting Non-Concentric Circles of Different Radii,” Comput. Vision Graphics Image Process. 43, 115–132 (1988).
[CrossRef]

A. Hachicha et al., “The Use of Gray-Level Information and Fitting Techniques for Precise Measurement of Corneal Curvature and Thickness,” Comput. Vision Graphics Image Process. 47, 131–164 (1989).
[CrossRef]

IEEE Trans. Comput. (1)

S. Tsuji, F. Matsumoto, “Detection of Elliptic and Linear Edges by Searching Two Parameter Spaces,” IEEE Trans. Comput. C-27, 777–781 (1979).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell. (2)

J. Illingworth, J. Kittler, “The Adaptive Hough Transform,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-9, 690–697 (1987).
[CrossRef]

C. M. Brown, “Inherent Bias and Noise in the Hough Transform,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-5, 439–505 (1983).
[CrossRef]

Pattern Recognition (2)

T. M. Van Veen, F. C. A. Groen, “Discretization Errors in the Hough Transform,” Pattern Recognition 14, 137–145 (1981).
[CrossRef]

H. Wechsler, J. Slansky, “Automatic Detection of Ribs In Chest Radiograms,” Pattern Recognition 9, 21–30 (1977).
[CrossRef]

Vision Res. (2)

J. F. Koretz, P. L. Kaufman, M. W. Neider, P. A. Goeckner, “Accommodation and Presbyopia in the Human Eye—Aging of the Anterior Segment,” Vision Res. 29, 1685–1692 (1989).
[CrossRef] [PubMed]

J. F. Koretz, G. H. Handelman, N. P. Brown, “Analysis of Human Crystalline Lens Curvature as a Function of Accommodative State and Age,” Vision Res. 24, 1141–1151 (1984).
[CrossRef] [PubMed]

Other (8)

C. A. Cook, J. F. Koretz, “An Efficient Implementation of the Hough Transform for the Detection of the Curvatures of the Human Crystalline Lens,” in preparation.

Documentation for Automatix Image Analyst.

J. F. Koretz, G. H. Handelman, “The ‘Lens Paradox’ and Image Formation in Accommodating Human Eyes,” in The Lens: Transparency and Cataract, G. Duncan, Ed., Topics in Aging Research in Europe6, 57–64 (1986).

R. O. Duda, P. E. Hart, Pattern Recognition and Scene Analysis (Wiley, New York, 1973).

S. D. Shapiro, “Generalization of the Hough Transform for Curve Detection in Noisy Digital Images,” in Fourth IJCPR, Kyoto (1978), pp. 710–714.

D. H. Ballard, C. M. Brown, Computer Vision (Prentice-Hall, Englewood Cliffs, NJ, 1982).

R. J. Schalkoff, Digital Image Processing and Computer Vision (Wiley, New York, 1989).

W. H. Press, Numerical Recipes (Cambridge U. Press, London, 1986).

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Figures (10)

Fig. 1
Fig. 1

Reproductions of digitized, Scheimpflug slit lamp photographs of an 18-yr old, unaccommodated human lens. (a) The slit lamp image is positioned, in relation to the digitizing camera, to fill the field of view. (b) The curves of the cornea are obtained separately from those of the lens; the anterior lens surface curve is used as a reference to correlate lens and cornea curves, obtaining the overall geometry of the optical system.

Fig. 2
Fig. 2

Screen reproductions illustrating processing carried out on the anterior lens surface of the image in Fig. 1(a) after initial angular orientation of the slide (±2°). (a) Points obtained by edge enhancement and grey scale gradient scanning are shown in their vector form: a 5 × 5 Sobel kernel was used in edge enhancement; gradient scanning was performed with a sparseness of 5 and a threshold of 120. The anterior surface is well defined (long projections) in spite of noise (short, randomly oriented segments) and the surface lying just behind it. (b) Points obtained by restricting those in (a) to a surface normal angle of 90° ± 30° are shown as crosses. (c) The curve and optical center line obtained by applying the first Hough technique to the points obtained in (b). The first grid mesh used was 10:1, with a 9 × 7 × 50 accumulator array for the second search; coefficient c is restricted to the range of 0.004 ± 0.0035.

Fig. 3
Fig. 3

(a) Points obtained through the process as in Fig. 2(a) except using a gradient threshold of 80 for the posterior lens surface. The reduction in the gradient necessary to obtain approximately the same density of points for both regions indicates the reduction in contrast with progression through the lens; this effect is much more dramatic in older lenses, requiring thresholds down to 10 to resolve the back surface. (b) A significant reduction in noise at the same gradient threshold is achieved through averaging although obscuring internal curves. (c) The curve shown is obtained from the points shown in (b). After having reoriented the image, the center line found in Fig. 4(b) is used in conjunction with the y value of a user input point to center the xy location of a 5 × 7 × 50 accumulator array, with mesh of 1; the value c was restricted to −0.0035 ± 0.003. Note that the final curve is accurate despite gaps resulting from localized irregularities along the surface.

Fig. 4
Fig. 4

(a) Points obtained using processing as in Fig. 2(a) and spatial averaging of grey scale values; averaging has been extended somewhat beyond the processing window for clarity. After applying the angular restriction (90° ± 30°) for this surface, the remaining points define the anterior surface exclusively. (b) The curve and center line shown are obtained using the processing as in Fig. 2(b) and including the overall angular parameter ϕ. The resulting curve, generated using the value of ϕ found, represents the optimal curve as it would appear if the image was rotated through this angle about the origin. The image has been realigned using this angle (or to the optimal curve) for subsequent illustrations.

Fig. 5
Fig. 5

(a) Boundary points are obtained using the processing as in Fig. 3(b) and a gradient threshold of 20. Although points defining internal curves are now clearly discerned, they are not easily separated from the points defining the posterior surface. (b) and (c) The internal curves were obtained, after sorting by normal angles 90° ± 40° and 270° ± 40°, respectively, from the points in (a) using processing as in Fig. 3(c); the value of c, however, was restricted to a much narrower range (−0.001 ± 0.0005) to minimize the peaks in the accumulator array caused by points from the posterior surface.

Fig. 6
Fig. 6

Photographs of a vertically and horizontally oriented, respectively, 200-line pairs/in, grating taken in place of the eye with the Scheimpflug slit lamp camera are used to determine the distortion introduced into the image of the lens by the camera. The lines in the horizontal grating appear to be stretched linearly, whereas vertical lines show an additional radial distortion.

Fig. 7
Fig. 7

(a) The distortion along the vertical center field of view of the camera, obtained from the vertical grating, is shown as a polynomial. Since all the images fall beyond the highly distorted light band seen in Fig. 6(a), it is not necessary that the polynomial fit exactly in this region; a fifth-order description for the overall form is adequate. (b) The family of curves shown represent the difference between the curve in (a) and similar curves fit along vertical lines at 1-mm intervals from the center line. (c) The horizontal scaling factor is obtained as a function of the uncorrected vertical distance; vertical points are subsequently corrected along the center using the form in (a).

Fig. 8
Fig. 8

Curves obtained from the 18-yr old lens used in Figs. 15 and 9 are shown in this composite plot in screen units (pixels). The corresponding coefficients for the curves are given in Table II.

Fig. 9
Fig. 9

Points (a) and final curve (b) for the anterior surface of the cornea, shown in Fig. 1(b), were obtained using processing identical to that used in Figs. 2(a)–(c). The resulting curve illustrates clearly the adequacy of a parabolic model for these optical surfaces across their incorporated width.

Fig. 10
Fig. 10

Primary curves for the unaccommodated lens from Fig. 3 are shown in their final, corrected form alongside curves for the same lens in a fully accommodated state obtained using similar processing (not shown). Curve coefficients, given in Table III, give geometric dimensions for the lens and anterior segment that are in agreement with those obtained by ultrasonography and optical pachymetry. Curves are plotted in units of millimeters.

Tables (3)

Tables Icon

Table I Coefficient Matrix for the Vertical, Slit Lamp Camera Distortion Correction

Tables Icon

Table II Coefficients for the Curves Plotted In Fig. 3 Given In the Form (A, B, C) Obtained Directly from Image Processing as In Figs. 15 and 9 and Aligned (x0,y0,c) as Plotted(Units are Pixels)

Tables Icon

Table III Corrected Coefficients for (a) the Cornea, (b) the Unaccommodated Lens, and (c) the Fully Accommodated Lens Shown In Fig. 10

Equations (6)

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

c = 1 2 ( A [ n : x ] - x 0 ) tan - 1 [ A [ n : G x ] A [ n : G y ] ] ,             y 0 = c ( A [ n : x ] - x 0 ) 2 - A [ n : y ] .
A [ n : x ] = A [ n : x ] cos ϕ + A [ n : y ] sin ϕ , A [ n : y ] = A [ n : y ] cos ϕ - A [ n : x ] sin ϕ ,
y * = α y ,             x * = β x ,
y = c { x - x 0 } 2 + y 0 ,             y * = c * { x * - x 0 * } 2 + y 0 * .
x 0 * = β x 0 ,             y 0 * = α y 0 ,             c * = α c β 2 .
y = n = 1 6 [ m = 1 5 ζ [ m , n ] ( x * ) m - 1 ] ( y * ) n - 1 .

Metrics