Abstract

We present a method for measuring the optical transfer function (OTF) of a camera lens using a tartan test pattern containing sinusoidal functions with multiple frequencies and orientations. The method is designed to optimize measurement accuracy for an adjustable set of sparse spatial frequencies and be reliable and fast in a wide range of measurement conditions. We describe the pattern design and the algorithm for estimating the OTF accurately from a captured image. Simulations show the tartan method is significantly more accurate than the International Organization for Standardization 12233 standard slanted-edge method. Experimental results from the tartan method were reproducible to 0.01 root mean square and in reasonable agreement with the slanted-edge method.

© 2011 Optical Society of America

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  1. F. J. P. Consitt and W. Mandler, “OTF techniques in the routine testing of production lenses,” J. Mod. Opt. 18, 123–131 (1971).
    [CrossRef]
  2. H. Kondo, T. Watanabe, and H. Yamaoka, “Criteria for the evaluation of photographic lenses,” J. Mod. Opt. 22, 353–363(1975).
    [CrossRef]
  3. N. Joshi, R. Szeliski, and D. Kriegman, “PSF estimation using sharp edge prediction,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition 2008 (Institute of Electrical and Electronics Engineers, 2008), pp. 1–8.
    [CrossRef]
  4. A. Levin, S. Hasinoff, P. Green, F. Durand, and W. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” in Proceedings of ACM SIGGRAPH 2009 (ACM, 2009), article 97.
  5. P. Lindberg, “Measurement of contrast transmission characteristics in optical image formation,” J. Mod. Opt. 1, 80–89(1954).
    [CrossRef]
  6. S. E. Reichenbach, S. K. Park, and R. Narayanswarmy, “Characterizing digital image acquisition devices,” Opt. Eng. 30, 170–177 (1991).
    [CrossRef]
  7. International Organization for Standardization, ISO 12233:2000 standard: photography—electronic still-picture cameras—resolution measurements (2000).
  8. H. Frieser, “Le Pouvier résolvant des Couches photographiques (The resolving power of photographic coatings),” in IX Congrès International de Photographic Scientifique et Appliquée (Masson SA, 1936), pp. 207–218.
  9. Sine Patterns, “Composite Design,” http://web.archive.org/web/20080327091116/www.sinepatterns.com/S_Comp2.htm (2006, accessed June 2010).
  10. K. G. Larkin and P. A. Fletcher, “Frequency estimation under affine distortion,” Australian patent application 2008211991(4 September 2008).
  11. M. R. Arnison, S. J. Hardy, and K. G. Larkin, “Optical transfer function measurement system,” Australian patent application 2005203031 (12 July 2005).
  12. A. D. Edgar and A. S. Bopardikar, “System and method of testing imaging equipment using transformed patterns,” U.S. patent application 20070230776 (13 June 2006).
  13. H. J. Levin, “Discrete Fourier transform (DFT) leakage removal,” U.S. patent 6,882,947 (19 April 2005).
  14. D. J. Fleet, D. J. Heeger, T. A. Cass, and D. L. Hecht, “Automatic geometric image transformations using embedded signals,” U.S. patent 5,949,055 (7 September 1999).
  15. D. Williams and P. D. Burns, “Applying and extending ISO/TC42 digital camera resolution standards to mobile imaging products,” Proc. SPIE 6494, 64940H (2007).
    [CrossRef]
  16. N. Koren, “Sharpness: what is it and how is it measured?” http://www.imatest.com/docs/sharpness.html (Imatest, 2010, accessed June 2010).
  17. P. A. Stokseth, “Properties of a defocused optical system,” J. Opt. Soc. Am. 59, 1314–1321 (1969).
    [CrossRef]
  18. G. C. Brock, A. C. Marchant, and T. L. Williams, “OTF standards for aerial mapping lenses,” J. Mod. Opt. 19, 953–972(1972).
    [CrossRef]
  19. D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, P. A. Fletcher, and K. G. Larkin, “Two-dimensional measurement of the lens optical transfer function from a digital image,” Proc. SPIE 7876, 78760H (2011).
    [CrossRef]
  20. D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, S. J. Hardy, and K. G. Larkin, “Geometric parameter measurement of an imaging device,” U.S. patent application 20090161945(21 November 2008).
  21. A. C. Marchant, E. A. Ironside, J. F. Attryde, and T. L. Williams, “The reproducibility of MTF measurements,” J. Mod Opt. 22, 249–264 (1975).
    [CrossRef]

2011

D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, P. A. Fletcher, and K. G. Larkin, “Two-dimensional measurement of the lens optical transfer function from a digital image,” Proc. SPIE 7876, 78760H (2011).
[CrossRef]

2007

D. Williams and P. D. Burns, “Applying and extending ISO/TC42 digital camera resolution standards to mobile imaging products,” Proc. SPIE 6494, 64940H (2007).
[CrossRef]

1991

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

1975

H. Kondo, T. Watanabe, and H. Yamaoka, “Criteria for the evaluation of photographic lenses,” J. Mod. Opt. 22, 353–363(1975).
[CrossRef]

A. C. Marchant, E. A. Ironside, J. F. Attryde, and T. L. Williams, “The reproducibility of MTF measurements,” J. Mod Opt. 22, 249–264 (1975).
[CrossRef]

1972

G. C. Brock, A. C. Marchant, and T. L. Williams, “OTF standards for aerial mapping lenses,” J. Mod. Opt. 19, 953–972(1972).
[CrossRef]

1971

F. J. P. Consitt and W. Mandler, “OTF techniques in the routine testing of production lenses,” J. Mod. Opt. 18, 123–131 (1971).
[CrossRef]

1969

1954

P. Lindberg, “Measurement of contrast transmission characteristics in optical image formation,” J. Mod. Opt. 1, 80–89(1954).
[CrossRef]

Arnison, M. R.

D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, P. A. Fletcher, and K. G. Larkin, “Two-dimensional measurement of the lens optical transfer function from a digital image,” Proc. SPIE 7876, 78760H (2011).
[CrossRef]

D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, S. J. Hardy, and K. G. Larkin, “Geometric parameter measurement of an imaging device,” U.S. patent application 20090161945(21 November 2008).

M. R. Arnison, S. J. Hardy, and K. G. Larkin, “Optical transfer function measurement system,” Australian patent application 2005203031 (12 July 2005).

Attryde, J. F.

A. C. Marchant, E. A. Ironside, J. F. Attryde, and T. L. Williams, “The reproducibility of MTF measurements,” J. Mod Opt. 22, 249–264 (1975).
[CrossRef]

Bopardikar, A. S.

A. D. Edgar and A. S. Bopardikar, “System and method of testing imaging equipment using transformed patterns,” U.S. patent application 20070230776 (13 June 2006).

Brock, G. C.

G. C. Brock, A. C. Marchant, and T. L. Williams, “OTF standards for aerial mapping lenses,” J. Mod. Opt. 19, 953–972(1972).
[CrossRef]

Burns, P. D.

D. Williams and P. D. Burns, “Applying and extending ISO/TC42 digital camera resolution standards to mobile imaging products,” Proc. SPIE 6494, 64940H (2007).
[CrossRef]

Cass, T. A.

D. J. Fleet, D. J. Heeger, T. A. Cass, and D. L. Hecht, “Automatic geometric image transformations using embedded signals,” U.S. patent 5,949,055 (7 September 1999).

Consitt, F. J. P.

F. J. P. Consitt and W. Mandler, “OTF techniques in the routine testing of production lenses,” J. Mod. Opt. 18, 123–131 (1971).
[CrossRef]

Deller, C. A.

D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, P. A. Fletcher, and K. G. Larkin, “Two-dimensional measurement of the lens optical transfer function from a digital image,” Proc. SPIE 7876, 78760H (2011).
[CrossRef]

D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, S. J. Hardy, and K. G. Larkin, “Geometric parameter measurement of an imaging device,” U.S. patent application 20090161945(21 November 2008).

Durand, F.

A. Levin, S. Hasinoff, P. Green, F. Durand, and W. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” in Proceedings of ACM SIGGRAPH 2009 (ACM, 2009), article 97.

Edgar, A. D.

A. D. Edgar and A. S. Bopardikar, “System and method of testing imaging equipment using transformed patterns,” U.S. patent application 20070230776 (13 June 2006).

Fleet, D. J.

D. J. Fleet, D. J. Heeger, T. A. Cass, and D. L. Hecht, “Automatic geometric image transformations using embedded signals,” U.S. patent 5,949,055 (7 September 1999).

Fletcher, P. A.

D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, P. A. Fletcher, and K. G. Larkin, “Two-dimensional measurement of the lens optical transfer function from a digital image,” Proc. SPIE 7876, 78760H (2011).
[CrossRef]

K. G. Larkin and P. A. Fletcher, “Frequency estimation under affine distortion,” Australian patent application 2008211991(4 September 2008).

Freeman, W.

A. Levin, S. Hasinoff, P. Green, F. Durand, and W. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” in Proceedings of ACM SIGGRAPH 2009 (ACM, 2009), article 97.

Frieser, H.

H. Frieser, “Le Pouvier résolvant des Couches photographiques (The resolving power of photographic coatings),” in IX Congrès International de Photographic Scientifique et Appliquée (Masson SA, 1936), pp. 207–218.

Green, P.

A. Levin, S. Hasinoff, P. Green, F. Durand, and W. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” in Proceedings of ACM SIGGRAPH 2009 (ACM, 2009), article 97.

Hardy, S. J.

M. R. Arnison, S. J. Hardy, and K. G. Larkin, “Optical transfer function measurement system,” Australian patent application 2005203031 (12 July 2005).

D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, S. J. Hardy, and K. G. Larkin, “Geometric parameter measurement of an imaging device,” U.S. patent application 20090161945(21 November 2008).

Hasinoff, S.

A. Levin, S. Hasinoff, P. Green, F. Durand, and W. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” in Proceedings of ACM SIGGRAPH 2009 (ACM, 2009), article 97.

Hecht, D. L.

D. J. Fleet, D. J. Heeger, T. A. Cass, and D. L. Hecht, “Automatic geometric image transformations using embedded signals,” U.S. patent 5,949,055 (7 September 1999).

Heeger, D. J.

D. J. Fleet, D. J. Heeger, T. A. Cass, and D. L. Hecht, “Automatic geometric image transformations using embedded signals,” U.S. patent 5,949,055 (7 September 1999).

Ironside, E. A.

A. C. Marchant, E. A. Ironside, J. F. Attryde, and T. L. Williams, “The reproducibility of MTF measurements,” J. Mod Opt. 22, 249–264 (1975).
[CrossRef]

Joshi, N.

N. Joshi, R. Szeliski, and D. Kriegman, “PSF estimation using sharp edge prediction,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition 2008 (Institute of Electrical and Electronics Engineers, 2008), pp. 1–8.
[CrossRef]

Kondo, H.

H. Kondo, T. Watanabe, and H. Yamaoka, “Criteria for the evaluation of photographic lenses,” J. Mod. Opt. 22, 353–363(1975).
[CrossRef]

Koren, N.

N. Koren, “Sharpness: what is it and how is it measured?” http://www.imatest.com/docs/sharpness.html (Imatest, 2010, accessed June 2010).

Kriegman, D.

N. Joshi, R. Szeliski, and D. Kriegman, “PSF estimation using sharp edge prediction,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition 2008 (Institute of Electrical and Electronics Engineers, 2008), pp. 1–8.
[CrossRef]

Larkin, K. G.

D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, P. A. Fletcher, and K. G. Larkin, “Two-dimensional measurement of the lens optical transfer function from a digital image,” Proc. SPIE 7876, 78760H (2011).
[CrossRef]

D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, S. J. Hardy, and K. G. Larkin, “Geometric parameter measurement of an imaging device,” U.S. patent application 20090161945(21 November 2008).

M. R. Arnison, S. J. Hardy, and K. G. Larkin, “Optical transfer function measurement system,” Australian patent application 2005203031 (12 July 2005).

K. G. Larkin and P. A. Fletcher, “Frequency estimation under affine distortion,” Australian patent application 2008211991(4 September 2008).

Levin, A.

A. Levin, S. Hasinoff, P. Green, F. Durand, and W. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” in Proceedings of ACM SIGGRAPH 2009 (ACM, 2009), article 97.

Levin, H. J.

H. J. Levin, “Discrete Fourier transform (DFT) leakage removal,” U.S. patent 6,882,947 (19 April 2005).

Lindberg, P.

P. Lindberg, “Measurement of contrast transmission characteristics in optical image formation,” J. Mod. Opt. 1, 80–89(1954).
[CrossRef]

Mandler, W.

F. J. P. Consitt and W. Mandler, “OTF techniques in the routine testing of production lenses,” J. Mod. Opt. 18, 123–131 (1971).
[CrossRef]

Marchant, A. C.

A. C. Marchant, E. A. Ironside, J. F. Attryde, and T. L. Williams, “The reproducibility of MTF measurements,” J. Mod Opt. 22, 249–264 (1975).
[CrossRef]

G. C. Brock, A. C. Marchant, and T. L. Williams, “OTF standards for aerial mapping lenses,” J. Mod. Opt. 19, 953–972(1972).
[CrossRef]

Morgan-Mar, D. P.

D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, P. A. Fletcher, and K. G. Larkin, “Two-dimensional measurement of the lens optical transfer function from a digital image,” Proc. SPIE 7876, 78760H (2011).
[CrossRef]

D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, S. J. Hardy, and K. G. Larkin, “Geometric parameter measurement of an imaging device,” U.S. patent application 20090161945(21 November 2008).

Narayanswarmy, R.

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

Park, S. K.

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

Reichenbach, S. E.

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

Stokseth, P. A.

Szeliski, R.

N. Joshi, R. Szeliski, and D. Kriegman, “PSF estimation using sharp edge prediction,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition 2008 (Institute of Electrical and Electronics Engineers, 2008), pp. 1–8.
[CrossRef]

Watanabe, T.

H. Kondo, T. Watanabe, and H. Yamaoka, “Criteria for the evaluation of photographic lenses,” J. Mod. Opt. 22, 353–363(1975).
[CrossRef]

Williams, D.

D. Williams and P. D. Burns, “Applying and extending ISO/TC42 digital camera resolution standards to mobile imaging products,” Proc. SPIE 6494, 64940H (2007).
[CrossRef]

Williams, T. L.

A. C. Marchant, E. A. Ironside, J. F. Attryde, and T. L. Williams, “The reproducibility of MTF measurements,” J. Mod Opt. 22, 249–264 (1975).
[CrossRef]

G. C. Brock, A. C. Marchant, and T. L. Williams, “OTF standards for aerial mapping lenses,” J. Mod. Opt. 19, 953–972(1972).
[CrossRef]

Yamaoka, H.

H. Kondo, T. Watanabe, and H. Yamaoka, “Criteria for the evaluation of photographic lenses,” J. Mod. Opt. 22, 353–363(1975).
[CrossRef]

J. Mod Opt.

A. C. Marchant, E. A. Ironside, J. F. Attryde, and T. L. Williams, “The reproducibility of MTF measurements,” J. Mod Opt. 22, 249–264 (1975).
[CrossRef]

J. Mod. Opt.

P. Lindberg, “Measurement of contrast transmission characteristics in optical image formation,” J. Mod. Opt. 1, 80–89(1954).
[CrossRef]

F. J. P. Consitt and W. Mandler, “OTF techniques in the routine testing of production lenses,” J. Mod. Opt. 18, 123–131 (1971).
[CrossRef]

H. Kondo, T. Watanabe, and H. Yamaoka, “Criteria for the evaluation of photographic lenses,” J. Mod. Opt. 22, 353–363(1975).
[CrossRef]

G. C. Brock, A. C. Marchant, and T. L. Williams, “OTF standards for aerial mapping lenses,” J. Mod. Opt. 19, 953–972(1972).
[CrossRef]

J. Opt. Soc. Am.

Opt. Eng.

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

Proc. SPIE

D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, P. A. Fletcher, and K. G. Larkin, “Two-dimensional measurement of the lens optical transfer function from a digital image,” Proc. SPIE 7876, 78760H (2011).
[CrossRef]

D. Williams and P. D. Burns, “Applying and extending ISO/TC42 digital camera resolution standards to mobile imaging products,” Proc. SPIE 6494, 64940H (2007).
[CrossRef]

Other

N. Koren, “Sharpness: what is it and how is it measured?” http://www.imatest.com/docs/sharpness.html (Imatest, 2010, accessed June 2010).

D. P. Morgan-Mar, M. R. Arnison, C. A. Deller, S. J. Hardy, and K. G. Larkin, “Geometric parameter measurement of an imaging device,” U.S. patent application 20090161945(21 November 2008).

N. Joshi, R. Szeliski, and D. Kriegman, “PSF estimation using sharp edge prediction,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition 2008 (Institute of Electrical and Electronics Engineers, 2008), pp. 1–8.
[CrossRef]

A. Levin, S. Hasinoff, P. Green, F. Durand, and W. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” in Proceedings of ACM SIGGRAPH 2009 (ACM, 2009), article 97.

International Organization for Standardization, ISO 12233:2000 standard: photography—electronic still-picture cameras—resolution measurements (2000).

H. Frieser, “Le Pouvier résolvant des Couches photographiques (The resolving power of photographic coatings),” in IX Congrès International de Photographic Scientifique et Appliquée (Masson SA, 1936), pp. 207–218.

Sine Patterns, “Composite Design,” http://web.archive.org/web/20080327091116/www.sinepatterns.com/S_Comp2.htm (2006, accessed June 2010).

K. G. Larkin and P. A. Fletcher, “Frequency estimation under affine distortion,” Australian patent application 2008211991(4 September 2008).

M. R. Arnison, S. J. Hardy, and K. G. Larkin, “Optical transfer function measurement system,” Australian patent application 2005203031 (12 July 2005).

A. D. Edgar and A. S. Bopardikar, “System and method of testing imaging equipment using transformed patterns,” U.S. patent application 20070230776 (13 June 2006).

H. J. Levin, “Discrete Fourier transform (DFT) leakage removal,” U.S. patent 6,882,947 (19 April 2005).

D. J. Fleet, D. J. Heeger, T. A. Cass, and D. L. Hecht, “Automatic geometric image transformations using embedded signals,” U.S. patent 5,949,055 (7 September 1999).

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Figures (11)

Fig. 1
Fig. 1

Geometry for camera lens OTF measurement. The image position ( x , y ) and focus z are defined relative to the optical axis at best focus in the image plane. The transverse magnification M T = z i / z o is the ratio between the image size and the object size. For a lens designed for a DSLR camera with an APS-C size sensor, imaging at a transverse magnification of 1 / 50 , the image width is about 20 mm and the object width is about 1 m . With the lens aperture open, the focus depth at the image plane is of the order of micrometers. The spatial frequency ( u , v ) is defined in the image plane. The meridional (M) and sagittal (S) spatial frequency orientations are shown in the top-right object corner. Not shown are the lens focal length f L , the relative lens aperture A V , and the illumination wavelength λ.

Fig. 2
Fig. 2

Slanted-edge test pattern. A typical slanted-edge pattern has only a small angle relative to the sensor axes. However, in this article we use a slanted edge with a meridional orientation, suitable for OTF measurement in the top-right or bottom-left image corner.

Fig. 3
Fig. 3

The tiled tartan pattern t T (3) is constructed by combining sinusoidal waves with multiple orientations (1) and frequencies (2). Each sinusoidal wave corresponds to a Hermitian pair of peaks in the Fourier domain, rounded to the nearest integer pixel position. The ( 0 , 0 ) frequency is in the center of the Fourier domain.

Fig. 4
Fig. 4

Signal strength of test patterns along the meridional axis in the spatial frequency domain: (a) edge signal strength G and tartan signal strength T C , (b) the ratio T C / G at the tartan peak locations.

Fig. 5
Fig. 5

Effects of defocus and windowing on the tartan pattern: (a) tiled tartan design t T with (b) Fourier transform T T , (c) imaged tartan p with z = 45 μm of defocus and alignment perturbation with (d) Fourier transform P, and (e) a surface plot of three tartan peaks near the center of the Fourier transform P. The side lobes caused by the spatial domain windowing are clearly visible in (d) and (e). The reduction in peak strength with increasing spatial frequency due to defocus can also be seen. The contrast has been inverted and adjusted in (b) and (d) to clarify the image structure.

Fig. 6
Fig. 6

Defocus OTF H G used for the simulations, calculated at a range of defocus distances z from 5 to 45 μm .

Fig. 7
Fig. 7

Ideal MTF (thick, dark line) at (a)  z = 15 μm and (b)  z = 25 μm compared with the simulated meridional MTF measurements from the slanted-edge (multiple thin, light lines) and simulated tartan measurements over 90 random cases with 5% imaging noise. The vertical bars indicate the ± 1 σ ranges of the slanted-edge method (hollow bars) and the tartan method (solid bars) at each tartan spatial frequency. The tartan frequencies are between each adjoining pair of hollow and solid bars. The slanted-edge results were measured using Imatest Master 3.6.

Fig. 8
Fig. 8

RMS error of simulated MTF measurements with defocus z = 25 μm over 90 cases of random alignment with (a) 1% imaging noise and (b) 5% imaging noise. The tartan results are shown for the meridional (diamonds) and sagittal (squares) orientations, while the slanted-edge results are meridional only (circles).

Fig. 9
Fig. 9

Effect of the side-lobe reduction algorithm on the accuracy of the simulated tartan MTF measurements. The maximum error in measurements at the meridional orientation over 90 cases at z = 25 μm with 1% imaging noise is shown for direct peak finding (squares) and side-lobe reduction (circles).

Fig. 10
Fig. 10

Plots of the MTF of an EF-S 18 55 mm IS lens at focal length f L = 18 mm measured across the entire image plane at 55 lppmm with our tartan pattern. We measured the MTF on a set of evenly spaced tiles, 50 tiles horizontally by 33 tiles vertically. The sampling grid is three times as dense as the grid lines shown on the plots. The plot on the left shows frequencies oriented at angle ψ M , meridionally with respect to the lower-left to top-right diagonal of the sensor. The plot on the right shows frequencies oriented at ψ S , 90 ° to ψ M . The frequency orientations are fixed across the chart for these measurements, which is in contrast with the standard polar variation of meridional and sagittal orientations.

Fig. 11
Fig. 11

Plots of the MTF versus spatial frequency of an EF-S 18 55 mm IS lens at focal length f L = 18 mm as measured by our tartan method (points) and Imatest’s implementation of the ISO standard slanted-edge method (continuous lines). The spatial frequencies are meridional and measured in two different image locations partway between the image center and the bottom left corner, as indicated by the captions on each graph. The maximum difference between the tartan and Imatest MTF results on the plot at 26% bottom left is 0.11, which is the largest difference that we found for this set of measurements. The RMS difference over six image locations was 0.06.

Tables (1)

Tables Icon

Table 1 Measured Effective Printer MTFs of a Printed Tartan Chart

Equations (22)

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g ( x , y ) = { 0 , if     x < 0 , 1 , if     x 0 ,
x = x cos ψ + y sin ψ , y = x sin ψ + y cos ψ .
ψ M = arctan ( 2 / 3 ) , ψ S = arctan ( 2 / 3 ) + π 2 .
G ( u , v ) = g ( x , y ) e 2 π i ( u x + v y ) = { 1 2 , if     u = 0 , i 2 π u , if     u 0 ,
t ( x , y ) = 2 j = 1 M T j [ 1 + cos ( 2 π u j x + 2 π v j y + φ j ( T ) ) ] ,
T ( u , v ) = t ( x , y ) exp [ 2 π i ( u x + v y ) ] d x d y = 2 δ ( u , v ) j = 1 M T j + j = 1 M T j exp ( i φ j ( T ) ) δ ( u u j , v v j ) + j = 1 M T j exp ( i φ j ( T ) ) δ ( u + u j , v + v j ) ,
F ( u , v ) = c H ( u , v ) T ( u , v ) ,
F ( u , v ) = c T 0 δ ( u , v ) + c j = 1 M H j T j exp [ i ( φ j ( T ) + φ j ( H ) ) ] δ ( u u j , v v j ) + c j = 1 M H j T j exp [ i ( φ j ( T ) + φ j ( H ) ) ] δ ( u + u j , v + v j ) .
H j = T 0 F j T j F 0 , ϕ j ( H ) = ϕ j ( F ) ϕ j ( T ) ,
w ( x ) = { 1 , if     | x | a 2 0 , if     | x | > a 2 ,
W ( u ) = | a | sin ( a π u ) a π u = | a | sinc ( a π u ) .
P ( u , v ) = W ( u , v ) F ( u , v ) = c T 0 W ( u , v ) + c j = 1 M H j T j exp [ i ( φ j ( T ) + φ j ( H ) ) ] W ( u u j , v v j ) + c j = 1 M H j T j exp [ i ( φ j ( T ) + φ j ( H ) ) ] W ( u + u j , v + v j ) .
P = W · B ,
P = ( P 0 P 1 P k P K ) T ,
W = ( W 00 W 01 .. W 0 k .. W 0 K W 10 W 11 .. W 1 k .. W 1 K .. .. .. W .. k .. W .. K W j 0 W j 1 W j .. W j k .. W j K .. .. .. .. .. W .. K W K 0 W K 1 W K .. W K k W K .. W K K ) ,
B = ( B 0 B 1 B k B K ) T .
( u H k , v H k ) = { ( 0 , 0 ) k = 0 ( u A k , v A k ) k Z even + ( u A k , v A k ) k Z odd +
B k = { c T 0 k = 0 F k / 2 exp ( i φ k ( F ) ) k Z even + F ( k 1 ) / 2 exp ( i φ ( k 1 ) / 2 ( F ) ) k Z odd +
P k = P ( u H k , v H k )
W j k = W ( u H j u H k , v H j v H k ) .
z = 4 π A V z n arcsin ( 1 2 A V ) 2 ( 1 z f L + z ) ,
H G ( q ) = 2 J 1 ( z q ) z q ,

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