Abstract

The Coltman series for obtaining the optical transfer function from measurements of the bar transfer function is mathematically derived. The bar transfer function rather than the contrast transfer function is defined so that the relation remains valid when an image exhibits phase reversal.

© 1998 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), p. 126.
  2. G. D. Boreman, S. Yang, “Modulation transfer function measurement using three- and four-bar targets,” Appl. Opt. 34, 8050–8052 (1995).
    [CrossRef] [PubMed]
  3. J. W. Coltman, “The specification of imaging properties by response to a sine wave input,” J. Opt. Soc. Am. 44, 468–471 (1954).
    [CrossRef]
  4. W. Altar, Westinghouse Research Memorandum 60-94410-14-19, as cited in Ref. 3, p. 470.

1995 (1)

1954 (1)

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Other (2)

W. Altar, Westinghouse Research Memorandum 60-94410-14-19, as cited in Ref. 3, p. 470.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), p. 126.

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Figures (1)

Fig. 1
Fig. 1

Spatial waveforms (a) 0.5[1 + cos(2πfx)], and (b) 0.5[1 + sqc(2πfx)] used to measure the optical transfer function and the bar transfer function, respectively. Sqc(x) = sign[cos(2πfx)], sqc(x) = +1 for cos(2πfx) ≥ 0, sqc(x) = -1 for cos(2πfx) < 0.

Equations (20)

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I i x = 1 2 1 + cos 2 π fx sensor 1 2 × 1 + OTF f cos 2 π fx = I o x ,
I i x = 1 2 1 + sqc 2 π fx sensor 1 2 × 1 + BTF f sqc 2 π fx = I o x ,
I o 0 - I o 1 / 2 f I o 0 + I o 1 / 2 f = OTF f     or     BTF f .
I max - I min I max + I min MTF f     or     CTF f ,
sqc 2 π fx = 4 π cos 2 π fx - 1 3 cos 2 π 3 f x + 1 5 cos 2 π 5 f x - = 4 π l = 1 - 1 l - 1 2 l - 1 cos 2 π 2 l - 1 fx .
BTF f = 4 π l = 1 - 1 l - 1 2 l - 1 OTF 2 l - 1 f .
BTF mf = 4 π l = 1 - 1 l - 1 2 l - 1 OTF 2 l - 1 mf ,
BTF f BTF 3 f BTF 5 f BTF 7 f BTF 9 f = 4 π 1 - 1 / 3 + 1 / 5 - 1 / 7 + 1 / 9 0 1 0 0 - 1 / 3 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 × OTF f OTF 3 f OTF 5 f OTF 7 f OTF 9 f .
OTF f = π 4 k = 1 A 2 k - 1 2 k - 1 BTF 2 k - 1 f ,
OTF f = k = 1 A 2 k - 1 2 k - 1 l = 1 - 1 l - 1 2 l - 1 OTF 2 l - 1 × 2 k - 1 f ,
OTF f = A 1 T f 1 - T 3 f 3 + T 5 f 5 - T 7 f 7 + T 9 f 9 - T 11 f 11 + T 13 f 13 - T 15 f 15 + + A 3 T 3 f 3 - T 9 f 9 + T 15 f 15 - T 21 f 21 + + A 5 T 5 f 5 - T 15 f 15 + T 25 f 25 - + A 7 T 7 f 7 - T 21 f 21 + + A 9 T 9 f 9 - T 27 f 27 + + A 11 T 11 f 11 - + A 13 T 13 f 13 - + A 15 T 15 f 15 - +
OTF f = A 1 OTF f     A 1 = 1 , OTF 3 f 3 A 3 - A 1 = 0     A 3 = 1 , OTF 5 f 5 A 5 + A 1 = 0     A 5 = - 1 , OTF 7 f 7 A 7 - A 1 = 0     A 7 = 1 , OTF 9 f 9 A 9 - A 3 + A 1 = 0     A 9 = 0 , OTF 11 f 11 A 11 - A 1 = 0     A 11 = 1 , OTF 13 f 13 A 13 + A 1 = 0     A 13 = - 1 , OTF 15 f 15 A 15 - A 5 + A 3 - A 1 = 0     A 15 = - 1 , etc .
- 1 n = - 1 - n = + 1 n   even - 1 n   odd , - 1 n + 1 = - 1 n - 1 ,   - 1 n - 2 = - 1 n .
- 1 n 1,2 q - 1 = - 1 q +   i = 1 q n i .
0 = 1 - 1 p = i = 0 p - 1 i C p i ,
i = 0 p - 1 - 1 i C p i = - - 1 p C p p = - 1 p - 1 .
A 1 = 1 ; for   n 2   A 2 n - 1 = - 1 n + p - 1 if the   p   prime factors of   2 n - 1 are all distinct 0 otherwise .
0 = A 2 n - 1 + C p 0 - 1 p +   i = 1 p n i + C p 1 - 1 p - 1 +   i = 1 p n i + C p 2 - 1 p - 2 +   i = 1 p n i + + C p p - 1 - 1 1 +   i = 1 p n i = A 2 n - 1 + - 1 p +   i = 1 p n i C p 0 - C p 1 + C p 2 - + - 1 p - 1 C p p - 1 = A 2 n - 1 + - 1 p +   i = 1 p n i i = 0 p - 1 - 1 i C p i = A 2 n - 1 + - 1 - 1 + i = 1 p n i ,
A 2 n - 1 = - 1 i = 1 p   n i = - 1 n + p - 1 .
OTF f = π 4 n = 1 A 2 n - 1 2 n - 1 BTF 2 n - 1 f ,

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