Abstract

The phase of the optical transfer function is advocated as an important tool in the characterization of modern incoherent imaging systems. It is shown that knowledge of the phase transfer function (PTF) can benefit a diverse array of applications involving both traditional and computational imaging systems. Areas of potential benefits are discussed, and three applications are presented, demonstrating the utility of the phase of the complex frequency response in practical scenarios. In traditional imaging systems, the PTF is shown via simulation results to be strongly coupled with odd-order aberrations and hence useful in misalignment detection and correction. In computational imaging systems, experimental results confirm that the PTF can be successfully applied to subpixel shift estimation and wavefront coding characterization tasks.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. H. Hopkins, “Image shift, phase distortion and the optical transfer function,” Opt. Acta 31, 345–368 (1984).
    [CrossRef]
  2. K.-J. Rosenbruch and R. Gerschler, “The meaning of the phase transfer function and the modular transfer function in using OTF as a criterion for image quality,” Optik 55, 173–182 (1980) (in German).
  3. C. S. Williams and O. A. Becklund, in Introduction to the Optical Transfer Function (Wiley, 1988), pp. 207–208.
  4. V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Effects of sampling on the phase transfer function of incoherent imaging systems,” Opt. Express 19, 24609–24626 (2011).
    [CrossRef]
  5. E. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [CrossRef]
  6. W. Singer, M. Totzeck, and H. Gross, in Handbook of Optical Systems, Vol. 2, Physical Image Formation, 1st ed. (Wiley-VCH, 2005), p. 446.
  7. M. Somayaji and M. P. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using a cubic phase mask,” Appl. Opt. 45, 2911–2923 (2006).
    [CrossRef]
  8. S. Barwick, “Defocus sensitivity optimization using the defocus Taylor expansion of the optical transfer function,” Appl. Opt. 47, 5893–5902 (2008).
    [CrossRef]
  9. M. Demenikov and A. R. Harvey, “Image artifacts in hybrid imaging systems with a cubic phase mask,” Opt. Express 18, 8207–8212 (2010).
    [CrossRef]
  10. J. W. Goodman, in Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 146–151.
  11. R. Barakat and A. Houston, “Transfer function of an optical system in the presence of off-axis aberrations,” J. Opt. Soc. Am. 55, 1142–1148 (1965).
    [CrossRef]
  12. A. Utkin, R. Vilar, and A. J. Smirnov, “On the relation between the wave aberration function and the phase transfer function for an incoherent imaging system with circular pupil,” Eur. Phys. J. D 17, 145–148 (2001).
    [CrossRef]
  13. J. Gaskill, in Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), pp. 60–62.
  14. C. D. Claxton and R. C. Staunton, “Measurement of the point-spread function of a noisy imaging system,” J. Opt. Soc. Am. A 25, 159–170 (2008).
    [CrossRef]
  15. N. Joshi, R. Szeliski, and D. Kriegman, “PSF estimation using sharp edge prediction,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2008 (IEEE, 2008), pp. 1–8.
  16. MITRE Corporation, “Image quality evaluation,” http://www.mitre.org/tech/mtf/ , 2001.
  17. B. Tatian, “Method for obtaining the transfer function from the edge response function,” J. Opt. Soc. Am. 55, 1014–1019 (1965).
    [CrossRef]
  18. S. Reichenbach, S. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30, 170–177 (1991).
    [CrossRef]
  19. D. Williams and P. Burns, “Low-frequency MTF estimation for digital imaging devices using slanted-edge analysis,” Proc. SPIE 5294, 93–101 (2003).
    [CrossRef]
  20. V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Image-based measurement of phase transfer function,” in Digital Image Processing and Analysis, OSA Technical Digest (CD) (Optical Society of America, 2010), paper DMD1.
  21. V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Phase transfer function of sampled imaging systems,” in Computational Optical Sensing and Imaging, OSA Technical Digest (Optical Society of America, 2011), paper CTuB1.
  22. D. Keren, S. Peleg, and R. Brada, “Image sequence enhancement using sub-pixel displacement,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1988), pp. 742–746.
  23. J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in Proceedings of the 2nd European Conference on Computer Vision (ECCV ’92), Lecture Notes in Computer Science (Springer-Verlag, 1992), pp. 237–252.
  24. B. Zitova and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21, 977–1000 (2003).
    [CrossRef]
  25. P. Vandewalle, S. Süsstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. (special issue on super-resolution) 2006, 71459 (2006).
    [CrossRef]
  26. H. Foroosh, J. B. Zerubia, and M. Berthod, “Extension of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–200 (2002).
    [CrossRef]
  27. L. Lucchese and G. M. Cortelazzo, “A noise-robust frequency domain technique for estimating planar roto-translations,” IEEE Trans. Signal Process. 48, 1769–1786 (2000).
    [CrossRef]
  28. H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54(1966).
    [CrossRef]
  29. Gonzalo Muyo and Andy R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. 30, 2715–2717 (2005).
    [CrossRef]

2011 (1)

2010 (1)

2008 (2)

2006 (2)

P. Vandewalle, S. Süsstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. (special issue on super-resolution) 2006, 71459 (2006).
[CrossRef]

M. Somayaji and M. P. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using a cubic phase mask,” Appl. Opt. 45, 2911–2923 (2006).
[CrossRef]

2005 (1)

2003 (2)

D. Williams and P. Burns, “Low-frequency MTF estimation for digital imaging devices using slanted-edge analysis,” Proc. SPIE 5294, 93–101 (2003).
[CrossRef]

B. Zitova and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21, 977–1000 (2003).
[CrossRef]

2002 (1)

H. Foroosh, J. B. Zerubia, and M. Berthod, “Extension of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–200 (2002).
[CrossRef]

2001 (1)

A. Utkin, R. Vilar, and A. J. Smirnov, “On the relation between the wave aberration function and the phase transfer function for an incoherent imaging system with circular pupil,” Eur. Phys. J. D 17, 145–148 (2001).
[CrossRef]

2000 (1)

L. Lucchese and G. M. Cortelazzo, “A noise-robust frequency domain technique for estimating planar roto-translations,” IEEE Trans. Signal Process. 48, 1769–1786 (2000).
[CrossRef]

1995 (1)

1991 (1)

S. Reichenbach, S. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30, 170–177 (1991).
[CrossRef]

1984 (1)

H. H. Hopkins, “Image shift, phase distortion and the optical transfer function,” Opt. Acta 31, 345–368 (1984).
[CrossRef]

1980 (1)

K.-J. Rosenbruch and R. Gerschler, “The meaning of the phase transfer function and the modular transfer function in using OTF as a criterion for image quality,” Optik 55, 173–182 (1980) (in German).

1966 (1)

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54(1966).
[CrossRef]

1965 (2)

Anandan, P.

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in Proceedings of the 2nd European Conference on Computer Vision (ECCV ’92), Lecture Notes in Computer Science (Springer-Verlag, 1992), pp. 237–252.

Barakat, R.

Barwick, S.

Becklund, O. A.

C. S. Williams and O. A. Becklund, in Introduction to the Optical Transfer Function (Wiley, 1988), pp. 207–208.

Bergen, J. R.

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in Proceedings of the 2nd European Conference on Computer Vision (ECCV ’92), Lecture Notes in Computer Science (Springer-Verlag, 1992), pp. 237–252.

Berthod, M.

H. Foroosh, J. B. Zerubia, and M. Berthod, “Extension of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–200 (2002).
[CrossRef]

Bhakta, V. R.

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Effects of sampling on the phase transfer function of incoherent imaging systems,” Opt. Express 19, 24609–24626 (2011).
[CrossRef]

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Image-based measurement of phase transfer function,” in Digital Image Processing and Analysis, OSA Technical Digest (CD) (Optical Society of America, 2010), paper DMD1.

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Phase transfer function of sampled imaging systems,” in Computational Optical Sensing and Imaging, OSA Technical Digest (Optical Society of America, 2011), paper CTuB1.

Brada, R.

D. Keren, S. Peleg, and R. Brada, “Image sequence enhancement using sub-pixel displacement,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1988), pp. 742–746.

Burns, P.

D. Williams and P. Burns, “Low-frequency MTF estimation for digital imaging devices using slanted-edge analysis,” Proc. SPIE 5294, 93–101 (2003).
[CrossRef]

Cathey, W. T.

Christensen, M. P.

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Effects of sampling on the phase transfer function of incoherent imaging systems,” Opt. Express 19, 24609–24626 (2011).
[CrossRef]

M. Somayaji and M. P. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using a cubic phase mask,” Appl. Opt. 45, 2911–2923 (2006).
[CrossRef]

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Image-based measurement of phase transfer function,” in Digital Image Processing and Analysis, OSA Technical Digest (CD) (Optical Society of America, 2010), paper DMD1.

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Phase transfer function of sampled imaging systems,” in Computational Optical Sensing and Imaging, OSA Technical Digest (Optical Society of America, 2011), paper CTuB1.

Claxton, C. D.

Cortelazzo, G. M.

L. Lucchese and G. M. Cortelazzo, “A noise-robust frequency domain technique for estimating planar roto-translations,” IEEE Trans. Signal Process. 48, 1769–1786 (2000).
[CrossRef]

Demenikov, M.

Dowski, E.

Flusser, J.

B. Zitova and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21, 977–1000 (2003).
[CrossRef]

Foroosh, H.

H. Foroosh, J. B. Zerubia, and M. Berthod, “Extension of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–200 (2002).
[CrossRef]

Gaskill, J.

J. Gaskill, in Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), pp. 60–62.

Gerschler, R.

K.-J. Rosenbruch and R. Gerschler, “The meaning of the phase transfer function and the modular transfer function in using OTF as a criterion for image quality,” Optik 55, 173–182 (1980) (in German).

Goodman, J. W.

J. W. Goodman, in Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 146–151.

Gross, H.

W. Singer, M. Totzeck, and H. Gross, in Handbook of Optical Systems, Vol. 2, Physical Image Formation, 1st ed. (Wiley-VCH, 2005), p. 446.

Hanna, K. J.

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in Proceedings of the 2nd European Conference on Computer Vision (ECCV ’92), Lecture Notes in Computer Science (Springer-Verlag, 1992), pp. 237–252.

Harvey, A. R.

Harvey, Andy R.

Hingorani, R.

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in Proceedings of the 2nd European Conference on Computer Vision (ECCV ’92), Lecture Notes in Computer Science (Springer-Verlag, 1992), pp. 237–252.

Hopkins, H. H.

H. H. Hopkins, “Image shift, phase distortion and the optical transfer function,” Opt. Acta 31, 345–368 (1984).
[CrossRef]

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54(1966).
[CrossRef]

Houston, A.

Joshi, N.

N. Joshi, R. Szeliski, and D. Kriegman, “PSF estimation using sharp edge prediction,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2008 (IEEE, 2008), pp. 1–8.

Keren, D.

D. Keren, S. Peleg, and R. Brada, “Image sequence enhancement using sub-pixel displacement,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1988), pp. 742–746.

Kriegman, D.

N. Joshi, R. Szeliski, and D. Kriegman, “PSF estimation using sharp edge prediction,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2008 (IEEE, 2008), pp. 1–8.

Lucchese, L.

L. Lucchese and G. M. Cortelazzo, “A noise-robust frequency domain technique for estimating planar roto-translations,” IEEE Trans. Signal Process. 48, 1769–1786 (2000).
[CrossRef]

Muyo, Gonzalo

Narayanswamy, R.

S. Reichenbach, S. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30, 170–177 (1991).
[CrossRef]

Park, S.

S. Reichenbach, S. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30, 170–177 (1991).
[CrossRef]

Peleg, S.

D. Keren, S. Peleg, and R. Brada, “Image sequence enhancement using sub-pixel displacement,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1988), pp. 742–746.

Reichenbach, S.

S. Reichenbach, S. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30, 170–177 (1991).
[CrossRef]

Rosenbruch, K.-J.

K.-J. Rosenbruch and R. Gerschler, “The meaning of the phase transfer function and the modular transfer function in using OTF as a criterion for image quality,” Optik 55, 173–182 (1980) (in German).

Singer, W.

W. Singer, M. Totzeck, and H. Gross, in Handbook of Optical Systems, Vol. 2, Physical Image Formation, 1st ed. (Wiley-VCH, 2005), p. 446.

Smirnov, A. J.

A. Utkin, R. Vilar, and A. J. Smirnov, “On the relation between the wave aberration function and the phase transfer function for an incoherent imaging system with circular pupil,” Eur. Phys. J. D 17, 145–148 (2001).
[CrossRef]

Somayaji, M.

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Effects of sampling on the phase transfer function of incoherent imaging systems,” Opt. Express 19, 24609–24626 (2011).
[CrossRef]

M. Somayaji and M. P. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using a cubic phase mask,” Appl. Opt. 45, 2911–2923 (2006).
[CrossRef]

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Phase transfer function of sampled imaging systems,” in Computational Optical Sensing and Imaging, OSA Technical Digest (Optical Society of America, 2011), paper CTuB1.

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Image-based measurement of phase transfer function,” in Digital Image Processing and Analysis, OSA Technical Digest (CD) (Optical Society of America, 2010), paper DMD1.

Staunton, R. C.

Süsstrunk, S.

P. Vandewalle, S. Süsstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. (special issue on super-resolution) 2006, 71459 (2006).
[CrossRef]

Szeliski, R.

N. Joshi, R. Szeliski, and D. Kriegman, “PSF estimation using sharp edge prediction,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2008 (IEEE, 2008), pp. 1–8.

Tatian, B.

Tiziani, H. J.

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54(1966).
[CrossRef]

Totzeck, M.

W. Singer, M. Totzeck, and H. Gross, in Handbook of Optical Systems, Vol. 2, Physical Image Formation, 1st ed. (Wiley-VCH, 2005), p. 446.

Utkin, A.

A. Utkin, R. Vilar, and A. J. Smirnov, “On the relation between the wave aberration function and the phase transfer function for an incoherent imaging system with circular pupil,” Eur. Phys. J. D 17, 145–148 (2001).
[CrossRef]

Vandewalle, P.

P. Vandewalle, S. Süsstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. (special issue on super-resolution) 2006, 71459 (2006).
[CrossRef]

Vetterli, M.

P. Vandewalle, S. Süsstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. (special issue on super-resolution) 2006, 71459 (2006).
[CrossRef]

Vilar, R.

A. Utkin, R. Vilar, and A. J. Smirnov, “On the relation between the wave aberration function and the phase transfer function for an incoherent imaging system with circular pupil,” Eur. Phys. J. D 17, 145–148 (2001).
[CrossRef]

Williams, C. S.

C. S. Williams and O. A. Becklund, in Introduction to the Optical Transfer Function (Wiley, 1988), pp. 207–208.

Williams, D.

D. Williams and P. Burns, “Low-frequency MTF estimation for digital imaging devices using slanted-edge analysis,” Proc. SPIE 5294, 93–101 (2003).
[CrossRef]

Zerubia, J. B.

H. Foroosh, J. B. Zerubia, and M. Berthod, “Extension of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–200 (2002).
[CrossRef]

Zitova, B.

B. Zitova and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21, 977–1000 (2003).
[CrossRef]

Appl. Opt. (3)

Br. J. Appl. Phys. (1)

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54(1966).
[CrossRef]

Eur. Phys. J. D (1)

A. Utkin, R. Vilar, and A. J. Smirnov, “On the relation between the wave aberration function and the phase transfer function for an incoherent imaging system with circular pupil,” Eur. Phys. J. D 17, 145–148 (2001).
[CrossRef]

EURASIP J. Appl. Signal Process. (1)

P. Vandewalle, S. Süsstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. (special issue on super-resolution) 2006, 71459 (2006).
[CrossRef]

IEEE Trans. Image Process. (1)

H. Foroosh, J. B. Zerubia, and M. Berthod, “Extension of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–200 (2002).
[CrossRef]

IEEE Trans. Signal Process. (1)

L. Lucchese and G. M. Cortelazzo, “A noise-robust frequency domain technique for estimating planar roto-translations,” IEEE Trans. Signal Process. 48, 1769–1786 (2000).
[CrossRef]

Image Vis. Comput. (1)

B. Zitova and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21, 977–1000 (2003).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

H. H. Hopkins, “Image shift, phase distortion and the optical transfer function,” Opt. Acta 31, 345–368 (1984).
[CrossRef]

Opt. Eng. (1)

S. Reichenbach, S. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30, 170–177 (1991).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Optik (1)

K.-J. Rosenbruch and R. Gerschler, “The meaning of the phase transfer function and the modular transfer function in using OTF as a criterion for image quality,” Optik 55, 173–182 (1980) (in German).

Proc. SPIE (1)

D. Williams and P. Burns, “Low-frequency MTF estimation for digital imaging devices using slanted-edge analysis,” Proc. SPIE 5294, 93–101 (2003).
[CrossRef]

Other (10)

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Image-based measurement of phase transfer function,” in Digital Image Processing and Analysis, OSA Technical Digest (CD) (Optical Society of America, 2010), paper DMD1.

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Phase transfer function of sampled imaging systems,” in Computational Optical Sensing and Imaging, OSA Technical Digest (Optical Society of America, 2011), paper CTuB1.

D. Keren, S. Peleg, and R. Brada, “Image sequence enhancement using sub-pixel displacement,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1988), pp. 742–746.

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in Proceedings of the 2nd European Conference on Computer Vision (ECCV ’92), Lecture Notes in Computer Science (Springer-Verlag, 1992), pp. 237–252.

J. Gaskill, in Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), pp. 60–62.

C. S. Williams and O. A. Becklund, in Introduction to the Optical Transfer Function (Wiley, 1988), pp. 207–208.

W. Singer, M. Totzeck, and H. Gross, in Handbook of Optical Systems, Vol. 2, Physical Image Formation, 1st ed. (Wiley-VCH, 2005), p. 446.

J. W. Goodman, in Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 146–151.

N. Joshi, R. Szeliski, and D. Kriegman, “PSF estimation using sharp edge prediction,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2008 (IEEE, 2008), pp. 1–8.

MITRE Corporation, “Image quality evaluation,” http://www.mitre.org/tech/mtf/ , 2001.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1.

Relationship between PSF, MTF, and PTF in an incoherent imaging system. The leftmost column shows PSFs for three different imaging conditions, namely, a diffraction-limited system (top row), a defocus-only system (center row), and a system with coma (bottom row), respectively. The center column shows the corresponding MTFs, and the rightmost column shows the PTFs.

Fig. 2.
Fig. 2.

Effect of the PTF on a sinusoidal target. The leftmost column shows three different types of PSFs. The center column shows the effects of these PSFs on a sinusoidal target, while intensity values of a slice from each image (along the horizontal colored lines) are shown on the right.

Fig. 3.
Fig. 3.

One of five subpixel-shifted Air Force resolution target images as captured by an aliased imaging system. The PTFs for each value of subpixel shift Δx were measured using the straightedge-based measurement method discussed in Subsection 2.B, in the region highlighted by the yellow box.

Fig. 4.
Fig. 4.

Measured PTFs for different subpixel shifts based on the set of images, one of which is shown in Fig. 3.

Fig. 5.
Fig. 5.

Estimated values of subpixel shifts obtained from the set of Air Force resolution target images as shown in Fig. 3. The subpixel shifts estimated from the proposed PTF-based method are shown, along with estimates obtained by using the Keren and the Vandewalle methods.

Fig. 6.
Fig. 6.

MTF and PTF data for a Cooke triplet designed in ZEMAX, whose middle element was decentered by various amounts of shifts denoted by Δs. The variation in MTF is negligible even for Δs=±20μm, whereas the PTFs noticeably diverge from zero even for Δs=±5μm. Furthermore, the PTFs also indicates the direction of decenter, an attribute not evident upon inspecting the MTFs.

Fig. 7.
Fig. 7.

Through-focus MTF and PTF plots for a cubic phase mask wavefront coding imager for various normalized spatial frequencies u. It is seen in (a) that the MTFs do not readily indicate the location of zero defocus, whereas the plots in (b) clearly identify the in-focus plane.

Fig. 8.
Fig. 8.

Experimentally measured through-focus PTF (red curve) at 50cyc/mm for the CPM127-R40 cubic phase mask imager with α=25.88. The blue curve indicates the corresponding theoretical through-focus PTF, while the dashed green curve marks the curve-fitted experimental data.

Equations (9)

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

Φx=2πΔxp,
hs(xs)=[ho(xsΔx)hd(x)]×comb(xp),
comb(xp)=pk=δ(xkp),
Hs(u)=k=H(u˜2ku˜n)ej2π(u˜2ku˜n)Δx,
Θs(0)=0.
Θs(u˜n)={2mπ;Re{Hs(u˜n)}0(2m1)π;Re{Hs(u˜n)}<0,integerm.
Θs(0)=Θ(0)2πΔx;u˜o:0<u˜o2u˜n.
Θs(u˜n)=Θ(u˜n)2πΔx+M(u˜n)M(u˜n)tan{Θ(u˜n)2πu˜nΔx};u˜o:u˜n<u˜o2u˜n.
H(u,ψ)={(π24α|u|)1/2exp[2j(αu3ψ2u3α)]×12{C(b(u))C(a(u))+jS(b(u))jS(a(u))}0<|u|11u=0,

Metrics