Abstract

A distortion mapping and computational image unwarping method based on a network interpolation that uses radial basis functions is presented. The method is applied to correct distortion in an off-axis head-worn display (HWD) presenting up to 23% highly asymmetric distortion over a 27°x21° field of view. A 10−5 mm absolute error of the mapping function over the field of view was achieved. The unwarping efficacy was assessed using the image-rendering feature of optical design software. Correlation coefficients between unwarped images seen through the HWD and the original images, as well as edge superimposition results, are presented. In an experiment, images are prewarped using radial basis functions for a recently built, off-axis HWD with a 20° diagonal field of view in a 4:3 ratio. Real-time video is generated by a custom application with 2 ms added latency and is demonstrated.

© 2012 OSA

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References

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  1. W. Faig, “Calibration of close-range photogrammetric systems: Mathematical formulation,” Photogramm. Eng. Remote Sensing 41, 1479–1486 (1975).
  2. R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
    [CrossRef]
  3. W. Robinett and J.P. Rolland, “A computational model for the stereoscopic optics of a head mounted-display,” Presence (Camb. Mass.) 1, 45–62 (1992).
  4. P. Cerveri, C. Forlani, A. Pedotti, and G. Ferrigno, “Hierarchical radial basis function networks and local polynomial un-warping for X-ray image intensifier distortion correction: a comparison with global techniques,” Med. Biol. Eng. Comput. 41(2), 151–163 (2003).
  5. W. T. Welford, Aberrations of Optical Systems (CRC Press, 1986).
  6. J. M. Sasian, “Image plane tilt in optical systems,” Opt. Eng. 31(3), 527 (1992).
    [CrossRef]
  7. B. A. Watson and L. F. Hodges, “Using texture maps to correct for optical distortion in head-mounted-displays,” in Proceedings of the Virtual Reality Annual Int. Symposium, 172–178 (1995).
  8. J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluations,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
    [CrossRef]
  9. A. Basu, S. Licardie, A. Basu, and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recognit. Lett. 16(4), 433–441 (1995).
  10. F. Devernay, O. Faugeras, F. Devernay, and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13(1), 14–24 (2001).
  11. D. Clause and A. W. Fitzgibbon, “A rational function lens distortion model for general cameras,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 213–219 (2005).
  12. J. P. Rolland, “Wide-angle, off-axis, see-through head-mounted display,” Opt. Eng. 39(7), 1760–1767 (2000).
    [CrossRef]
  13. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011).
    [CrossRef] [PubMed]
  14. C. Slama, Manual of Photogrammetry, (Amer. Soc. of Photogrammetry, 1980).
  15. J. Heikkilä and O. Silven, “A four-step camera calibration procedure with implicit image correction,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 1106–1112 (1997).
  16. J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
    [CrossRef]
  17. A. E. Conrady, “Decentered lens systems,” Monthly Notices of The Royal Astr Society 39, 384–390 (1919).
  18. D. C. Brown, “Decentering distortion of lenses,” Photogramm. Eng. Remote Sensing 32, 444–462 (1966).
  19. R. I. Hartley and T. Saxena, “The cubic rational polynomial camera model,”in Proceedings of Defense Advanced Research Projects Agency Image Understanding Workshop, 649–653 (1997).
  20. G. Q. Wei and S. D. Ma, “A Complete two-plane camera calibration method and experimental comparisons,” in Proceedings of IEEE 4th International Conference on Computer Vision, 439–446 (1993).
  21. D. Ruprecht, H. Muller, D. Ruprecht, and H. Muller, “Image warping with scattered data interpolation,” IEEE Comput. Graph. Appl. 15(2), 37–43 (1995).
  22. C. C. Leung, P. C. K. Kwok, K. Y. Zee, and F. H. Y. Chan, “B-spline interpolation for bend intra-oral radiographs,” Comput. Biol. Med. 37(11), 1565–1571 (2007).
    [CrossRef] [PubMed]
  23. P. Cerveri, S. Ferrari, and N. A. Borghese, “Calibration of TV cameras through RBF networks,” Proc. SPIE 3165, 312–318 (1997).
    [CrossRef]
  24. G. E. Martin, Transformation Geometry: An Introduction to Symmetry (Springer, 1982).
  25. D. N. Fogel, “Image Rectification with Radial Basis Functions: Applications to RS/GIS Data Integration,” in Proceedings of the Third International Conference on Integrating GIS and Environmental Modeling,(Sante Fe, 1996). http://www.ncgia.ucsb.edu/conf/SANTA_FE_CD-ROM/sf_papers/fogel_david/santafe.html
  26. X. Zhu, R. M. Rangayyan, and A. L. Ellis, Digital Image Processing for Ophthalmology: Detection of the Optic Nerve Head (Morgan & Claypool, 2011), Chap 3.
  27. J. P. McGuire., “Next-generation head-mounted display,” Proc. SPIE 7618, 761804, 761804-8 (2010).
    [CrossRef]
  28. B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
    [CrossRef]

2011 (2)

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011).
[CrossRef] [PubMed]

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
[CrossRef]

2010 (1)

J. P. McGuire., “Next-generation head-mounted display,” Proc. SPIE 7618, 761804, 761804-8 (2010).
[CrossRef]

2008 (1)

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
[CrossRef]

2007 (1)

C. C. Leung, P. C. K. Kwok, K. Y. Zee, and F. H. Y. Chan, “B-spline interpolation for bend intra-oral radiographs,” Comput. Biol. Med. 37(11), 1565–1571 (2007).
[CrossRef] [PubMed]

2003 (1)

P. Cerveri, C. Forlani, A. Pedotti, and G. Ferrigno, “Hierarchical radial basis function networks and local polynomial un-warping for X-ray image intensifier distortion correction: a comparison with global techniques,” Med. Biol. Eng. Comput. 41(2), 151–163 (2003).

2001 (1)

F. Devernay, O. Faugeras, F. Devernay, and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13(1), 14–24 (2001).

2000 (1)

J. P. Rolland, “Wide-angle, off-axis, see-through head-mounted display,” Opt. Eng. 39(7), 1760–1767 (2000).
[CrossRef]

1997 (1)

P. Cerveri, S. Ferrari, and N. A. Borghese, “Calibration of TV cameras through RBF networks,” Proc. SPIE 3165, 312–318 (1997).
[CrossRef]

1995 (2)

D. Ruprecht, H. Muller, D. Ruprecht, and H. Muller, “Image warping with scattered data interpolation,” IEEE Comput. Graph. Appl. 15(2), 37–43 (1995).

A. Basu, S. Licardie, A. Basu, and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recognit. Lett. 16(4), 433–441 (1995).

1992 (3)

W. Robinett and J.P. Rolland, “A computational model for the stereoscopic optics of a head mounted-display,” Presence (Camb. Mass.) 1, 45–62 (1992).

J. M. Sasian, “Image plane tilt in optical systems,” Opt. Eng. 31(3), 527 (1992).
[CrossRef]

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluations,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[CrossRef]

1987 (1)

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
[CrossRef]

1975 (1)

W. Faig, “Calibration of close-range photogrammetric systems: Mathematical formulation,” Photogramm. Eng. Remote Sensing 41, 1479–1486 (1975).

1966 (1)

D. C. Brown, “Decentering distortion of lenses,” Photogramm. Eng. Remote Sensing 32, 444–462 (1966).

1919 (1)

A. E. Conrady, “Decentered lens systems,” Monthly Notices of The Royal Astr Society 39, 384–390 (1919).

Basu, A.

A. Basu, S. Licardie, A. Basu, and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recognit. Lett. 16(4), 433–441 (1995).

A. Basu, S. Licardie, A. Basu, and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recognit. Lett. 16(4), 433–441 (1995).

Borghese, N. A.

P. Cerveri, S. Ferrari, and N. A. Borghese, “Calibration of TV cameras through RBF networks,” Proc. SPIE 3165, 312–318 (1997).
[CrossRef]

Brown, D. C.

D. C. Brown, “Decentering distortion of lenses,” Photogramm. Eng. Remote Sensing 32, 444–462 (1966).

Cerveri, P.

P. Cerveri, C. Forlani, A. Pedotti, and G. Ferrigno, “Hierarchical radial basis function networks and local polynomial un-warping for X-ray image intensifier distortion correction: a comparison with global techniques,” Med. Biol. Eng. Comput. 41(2), 151–163 (2003).

P. Cerveri, S. Ferrari, and N. A. Borghese, “Calibration of TV cameras through RBF networks,” Proc. SPIE 3165, 312–318 (1997).
[CrossRef]

Chan, F. H. Y.

C. C. Leung, P. C. K. Kwok, K. Y. Zee, and F. H. Y. Chan, “B-spline interpolation for bend intra-oral radiographs,” Comput. Biol. Med. 37(11), 1565–1571 (2007).
[CrossRef] [PubMed]

Cohen, P.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluations,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[CrossRef]

Conrady, A. E.

A. E. Conrady, “Decentered lens systems,” Monthly Notices of The Royal Astr Society 39, 384–390 (1919).

Devernay, F.

F. Devernay, O. Faugeras, F. Devernay, and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13(1), 14–24 (2001).

F. Devernay, O. Faugeras, F. Devernay, and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13(1), 14–24 (2001).

Faig, W.

W. Faig, “Calibration of close-range photogrammetric systems: Mathematical formulation,” Photogramm. Eng. Remote Sensing 41, 1479–1486 (1975).

Faugeras, O.

F. Devernay, O. Faugeras, F. Devernay, and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13(1), 14–24 (2001).

F. Devernay, O. Faugeras, F. Devernay, and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13(1), 14–24 (2001).

Ferrari, S.

P. Cerveri, S. Ferrari, and N. A. Borghese, “Calibration of TV cameras through RBF networks,” Proc. SPIE 3165, 312–318 (1997).
[CrossRef]

Ferrigno, G.

P. Cerveri, C. Forlani, A. Pedotti, and G. Ferrigno, “Hierarchical radial basis function networks and local polynomial un-warping for X-ray image intensifier distortion correction: a comparison with global techniques,” Med. Biol. Eng. Comput. 41(2), 151–163 (2003).

Flyer, N.

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
[CrossRef]

Forlani, C.

P. Cerveri, C. Forlani, A. Pedotti, and G. Ferrigno, “Hierarchical radial basis function networks and local polynomial un-warping for X-ray image intensifier distortion correction: a comparison with global techniques,” Med. Biol. Eng. Comput. 41(2), 151–163 (2003).

Fornberg, B.

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
[CrossRef]

Fuerschbach, K.

Herniou, M.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluations,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[CrossRef]

Kwok, P. C. K.

C. C. Leung, P. C. K. Kwok, K. Y. Zee, and F. H. Y. Chan, “B-spline interpolation for bend intra-oral radiographs,” Comput. Biol. Med. 37(11), 1565–1571 (2007).
[CrossRef] [PubMed]

Larsson, E.

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
[CrossRef]

Leung, C. C.

C. C. Leung, P. C. K. Kwok, K. Y. Zee, and F. H. Y. Chan, “B-spline interpolation for bend intra-oral radiographs,” Comput. Biol. Med. 37(11), 1565–1571 (2007).
[CrossRef] [PubMed]

Licardie, S.

A. Basu, S. Licardie, A. Basu, and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recognit. Lett. 16(4), 433–441 (1995).

A. Basu, S. Licardie, A. Basu, and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recognit. Lett. 16(4), 433–441 (1995).

Liu, Y.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
[CrossRef]

McGuire, J. P.

J. P. McGuire., “Next-generation head-mounted display,” Proc. SPIE 7618, 761804, 761804-8 (2010).
[CrossRef]

Muller, H.

D. Ruprecht, H. Muller, D. Ruprecht, and H. Muller, “Image warping with scattered data interpolation,” IEEE Comput. Graph. Appl. 15(2), 37–43 (1995).

D. Ruprecht, H. Muller, D. Ruprecht, and H. Muller, “Image warping with scattered data interpolation,” IEEE Comput. Graph. Appl. 15(2), 37–43 (1995).

Pedotti, A.

P. Cerveri, C. Forlani, A. Pedotti, and G. Ferrigno, “Hierarchical radial basis function networks and local polynomial un-warping for X-ray image intensifier distortion correction: a comparison with global techniques,” Med. Biol. Eng. Comput. 41(2), 151–163 (2003).

Rolland, J. P.

Ruprecht, D.

D. Ruprecht, H. Muller, D. Ruprecht, and H. Muller, “Image warping with scattered data interpolation,” IEEE Comput. Graph. Appl. 15(2), 37–43 (1995).

D. Ruprecht, H. Muller, D. Ruprecht, and H. Muller, “Image warping with scattered data interpolation,” IEEE Comput. Graph. Appl. 15(2), 37–43 (1995).

Sasian, J. M.

J. M. Sasian, “Image plane tilt in optical systems,” Opt. Eng. 31(3), 527 (1992).
[CrossRef]

Shi, F.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
[CrossRef]

Thompson, K. P.

Tsai, R. Y.

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
[CrossRef]

Wang, J.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
[CrossRef]

Weng, J.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluations,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[CrossRef]

Zee, K. Y.

C. C. Leung, P. C. K. Kwok, K. Y. Zee, and F. H. Y. Chan, “B-spline interpolation for bend intra-oral radiographs,” Comput. Biol. Med. 37(11), 1565–1571 (2007).
[CrossRef] [PubMed]

Zhang, J.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
[CrossRef]

Comput. Biol. Med. (1)

C. C. Leung, P. C. K. Kwok, K. Y. Zee, and F. H. Y. Chan, “B-spline interpolation for bend intra-oral radiographs,” Comput. Biol. Med. 37(11), 1565–1571 (2007).
[CrossRef] [PubMed]

IEEE Comput. Graph. Appl. (1)

D. Ruprecht, H. Muller, D. Ruprecht, and H. Muller, “Image warping with scattered data interpolation,” IEEE Comput. Graph. Appl. 15(2), 37–43 (1995).

IEEE J. Robot. Autom. (1)

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluations,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[CrossRef]

Mach. Vis. Appl. (1)

F. Devernay, O. Faugeras, F. Devernay, and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13(1), 14–24 (2001).

Med. Biol. Eng. Comput. (1)

P. Cerveri, C. Forlani, A. Pedotti, and G. Ferrigno, “Hierarchical radial basis function networks and local polynomial un-warping for X-ray image intensifier distortion correction: a comparison with global techniques,” Med. Biol. Eng. Comput. 41(2), 151–163 (2003).

Monthly Notices of The Royal Astr Society (1)

A. E. Conrady, “Decentered lens systems,” Monthly Notices of The Royal Astr Society 39, 384–390 (1919).

Opt. Eng. (2)

J. M. Sasian, “Image plane tilt in optical systems,” Opt. Eng. 31(3), 527 (1992).
[CrossRef]

J. P. Rolland, “Wide-angle, off-axis, see-through head-mounted display,” Opt. Eng. 39(7), 1760–1767 (2000).
[CrossRef]

Opt. Express (1)

Pattern Recognit. (1)

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
[CrossRef]

Pattern Recognit. Lett. (1)

A. Basu, S. Licardie, A. Basu, and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recognit. Lett. 16(4), 433–441 (1995).

Photogramm. Eng. Remote Sensing (2)

W. Faig, “Calibration of close-range photogrammetric systems: Mathematical formulation,” Photogramm. Eng. Remote Sensing 41, 1479–1486 (1975).

D. C. Brown, “Decentering distortion of lenses,” Photogramm. Eng. Remote Sensing 32, 444–462 (1966).

Presence (Camb. Mass.) (1)

W. Robinett and J.P. Rolland, “A computational model for the stereoscopic optics of a head mounted-display,” Presence (Camb. Mass.) 1, 45–62 (1992).

Proc. SPIE (2)

P. Cerveri, S. Ferrari, and N. A. Borghese, “Calibration of TV cameras through RBF networks,” Proc. SPIE 3165, 312–318 (1997).
[CrossRef]

J. P. McGuire., “Next-generation head-mounted display,” Proc. SPIE 7618, 761804, 761804-8 (2010).
[CrossRef]

SIAM J. Sci. Comput. (1)

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
[CrossRef]

Other (10)

G. E. Martin, Transformation Geometry: An Introduction to Symmetry (Springer, 1982).

D. N. Fogel, “Image Rectification with Radial Basis Functions: Applications to RS/GIS Data Integration,” in Proceedings of the Third International Conference on Integrating GIS and Environmental Modeling,(Sante Fe, 1996). http://www.ncgia.ucsb.edu/conf/SANTA_FE_CD-ROM/sf_papers/fogel_david/santafe.html

X. Zhu, R. M. Rangayyan, and A. L. Ellis, Digital Image Processing for Ophthalmology: Detection of the Optic Nerve Head (Morgan & Claypool, 2011), Chap 3.

R. I. Hartley and T. Saxena, “The cubic rational polynomial camera model,”in Proceedings of Defense Advanced Research Projects Agency Image Understanding Workshop, 649–653 (1997).

G. Q. Wei and S. D. Ma, “A Complete two-plane camera calibration method and experimental comparisons,” in Proceedings of IEEE 4th International Conference on Computer Vision, 439–446 (1993).

W. T. Welford, Aberrations of Optical Systems (CRC Press, 1986).

B. A. Watson and L. F. Hodges, “Using texture maps to correct for optical distortion in head-mounted-displays,” in Proceedings of the Virtual Reality Annual Int. Symposium, 172–178 (1995).

D. Clause and A. W. Fitzgibbon, “A rational function lens distortion model for general cameras,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 213–219 (2005).

C. Slama, Manual of Photogrammetry, (Amer. Soc. of Photogrammetry, 1980).

J. Heikkilä and O. Silven, “A four-step camera calibration procedure with implicit image correction,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 1106–1112 (1997).

Supplementary Material (1)

» Media 1: MOV (3811 KB)     

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Figures (12)

Fig. 1
Fig. 1

An example of correspondences.

Fig. 2
Fig. 2

(a) 2D layout of the starting design for the off-axis HWD under study [12].(b) CODE V distortion grid computation.

Fig. 3
Fig. 3

A visual representation of the local RBF-based distortion mapping method. An RBF is located at each correspondence, j (bottom). Using the correspondences and Eq. (1), a weighted amplitude is given to each RBF (middle) to create the top interpolated surface. The interpolated surface (top) represents all the possible x-coordinate values in the distorted image for any pixel (x,y) in the original image. A separate, yet similar figure must be made for the distorted y-coordinates using Eq. (2).

Fig. 4
Fig. 4

Comparison of data and RBF modeled distorted points showing normalized x coordinates along the horizontal axis and the normalized y coordinate along the vertical axis: (a) Set of N correspondences only; (b) Set of Ntest = 412 test point including correspondences. The resulting RMS and maximum absolute error between data points and RBF model-computed points is 4 × 10−17 mm and 4 × 10−15 mm, respectively, considering only correspondences; 4 × 10−15 mm and 0.058 mm, respectively, considering Ntest = 1012 test points; 10−5 mm and 0.003 mm considering all Ntest test points, but only over the effective FOV.

Fig. 5
Fig. 5

Images showing RBF-based distortion mapping. (a) Input Image (SVGA resolution), (b) prewarped image (method 1), (c) prewarped image (method 2).

Fig. 6
Fig. 6

Resulting unwarped images (a) Original images. (b) Unwarped images (method 1). (c) Unwarped images (method 2).

Fig. 7
Fig. 7

Side-by-side comparison with original images. (a) Input images after downsampling and cropping to unwarped image size. (b) Unwarped cropped images (method 1). (c) Unwarped cropped images (method 2).

Fig. 8
Fig. 8

Edge comparison between original and unwarped images using method 2 for(a) Landscape 1 and (b) Landscape 2.

Fig. 9
Fig. 9

An optical see-through, eyeglasses-format, head worn display [Courtesy of Optical Research Associates].

Fig. 10
Fig. 10

(a) Original input image, (b) Prewarped image using the exact (gold standard) correspondence table, (c) Prewarped image using an RBF-generated correspondence table,(d) Prewarped image showing the location of the sampled correspondences.

Fig. 11
Fig. 11

(a) Plot showing RMS pixel displacement error for the RBF distortion method for subsets containing different amounts of correspondences. There is no accuracy gain for sampling more than 120 points. (b) Plot showing RMS pixel displacement error for the RBF distortion method for various values of the parameter, χ, in Eq. (3). The optimal value is 24 and the sensitivity to a small change is low.

Fig. 12
Fig. 12

Custom real-time distortion correction application. While the application itself is designed for use with any generic distortion map, the images shown correspond to the ORA HWD detailed above. The input video is split into the separate outputs video feeds for the Left and Right eyes with the correct distortion applied. The right eye output is flipped vertically because it was necessary for the micro display in the HWD to be inverted to comply with constraints of the electronics. A video clip of the application is presented (Media 1).

Tables (1)

Tables Icon

Table 1 Correlation Coefficients between CODE V 2D IMS UnwarpedImages and Original Images

Equations (4)

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

x j= ' i=1 n α x, i R i (d)+ p m ( x j , y j ),
y j= ' i=1 n α y, i R i (d)+ p m ( x j , y j ),
R i (d)= ( d 2 +χ r i 2 ) μ/2 = [ ( x j x basis_center_i ) 2 + ( y j y basis_center_i ) 2 +χ r i 2 ] μ/2
cc= i=1 N ( A i μ A )( B i μ B ) i=1 N ( A i μ A ) 2 i=1 N ( B i μ B ) 2 ,

Metrics