Abstract

Present determination of optical imaging systems specifications are based on performance values and modulation transfer function results carried with a 1D resolution template (such as the USAF resolution target or spoke templates). Such a template allows determining image quality, resolution limit, and contrast. Nevertheless, the conventional 1D template does not provide satisfactory results, since most optical imaging systems handle 2D objects for which imaging system response may be different by virtue of some not readily observable spatial frequencies. In this paper we derive and analyze contrast transfer function results obtained with 1D as well as 2D templates.

© 2012 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
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    [CrossRef]
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    [CrossRef]
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  7. V. Ronchi, “Le frange di combinazioni nello studio delle superficie e dei sistemi ottici [Combination fringes in the study of surfaces and optical systems],” Riv. Ottica Mecc. Precis. [J. Opt. Prec. Mech.] 2, 9–35 (1923).
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    [CrossRef]
  9. E. Marom, B. Milgrom, and N. Konforti, “Two-dimensional modulation transfer function: a new perspective,” Appl. Opt. 49, 6749–6755 (2010).
    [CrossRef]

2010

1998

1995

1976

1954

1923

V. Ronchi, “Le frange di combinazioni nello studio delle superficie e dei sistemi ottici [Combination fringes in the study of surfaces and optical systems],” Riv. Ottica Mecc. Precis. [J. Opt. Prec. Mech.] 2, 9–35 (1923).

Bhogra, R. K.

Boreman, G. D.

Coltman, J. W.

Ferrell, R. K.

Goddard, J. S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Jaiswal, A. K.

Konforti, N.

Marom, E.

Milgrom, B.

Osterberg, H.

H. Osterberg, “Evaluation phase optical tests,” in Military Standardization Handbook: Optical Design (Defense Supply Agency, 1962), pp. 1–8.

Rogers, G. L.

Ronchi, V.

V. Ronchi, “Le frange di combinazioni nello studio delle superficie e dei sistemi ottici [Combination fringes in the study of surfaces and optical systems],” Riv. Ottica Mecc. Precis. [J. Opt. Prec. Mech.] 2, 9–35 (1923).

Sitter, D. N.

Yang, S.

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Figures (9)

Fig. 1.
Fig. 1.

USAF—1D resolution target.

Fig. 2.
Fig. 2.

Input function and output function for (a) ffc, (b) ffc/3, and (c) f0.99fc. Note abscissa scale changes.

Fig. 3.
Fig. 3.

MTF versus 1D CTF (infinite target as well as three bars target). The normalized frequency refers to f/fc.

Fig. 4.
Fig. 4.

2D target.

Fig. 5.
Fig. 5.

Fourier transform of an infinite checkerboard.

Fig. 6.
Fig. 6.

Fourier transform of a 5×5 checkerboard pattern.

Fig. 7.
Fig. 7.

MTF versus 2D CTF (5×5 cells target) and 1D CTF (three bars targets).

Fig. 8.
Fig. 8.

Asymmetric 2D template.

Fig. 9.
Fig. 9.

Aspect ratio effect on CTF for 2D infinite asymmetric patterns.

Equations (25)

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OTF(fx,fy)=P(x+λzifx2,y+λzify2)·P*(xλzifx2,yλzify2)dxdy|P(x,y)|2dxdy,
OTF(f)={2π[arccos(ffc)ffc·1(ffc)2]ffc0otherwise,
Contrast=ImaxIminImax+Imin,
Iin=12(1+sin(2πfx)),
Iout=12(1+MTF(f)·sin(2πfx)).
Ssqw(x)=12(1+4πn=1,3,5sin(nπxa)n)=12+2π(sin(πxa)+13sin(3πxa)+15sin(5πxa)+),
Ssqwout(x)=12(1+4πn=1,3,5sin(nπxa)n·MTF(n2a))=12+2π(MTF(12a)·sin(πxa)+13MTF(32a)sin(3πxa)+15MTF(52a)sin(5πxa)+).
g(x,y)=h(x,y)*r(x,y),
h(x,y)=comb(xa2a)·comb(y2a)+comb(x2a)·comb(ya2a)r(x,y)=rect(xa)·rect(ya),
G(fx,fy)=H(fx,fy)·R(fx,fy).
t(x,y)=rect(x5a)·rect(y5a).
Schecker(x,y)=12(1+(4π)2n=1,3,5sin(nπxa)nm=1,3,5sin(mπya)m).
u2D(x,y)=12(1+sin(2πfxx)·sin(2πfyy)).
U2D(ξ,η)=12(1+14[δ(ξfx)δ(ξ+fx)]*[δ(η+fy)δ(ηfy)])=12(1+14[δ(ξfx,η+fy)+δ(ξ+fx,ηfy)δ(ξfx,ηfy)δ(ξ+fx,η+fy)]),
MTF(fx2+fy2)=MTF(n2+m22a).
u2Dout(x,y)=12(1+sin(2πfxx)·sin(2πfyy)·MTF(fx2+fy2))=12(1+sin(nπxa)·sin(mπya)·MTF(n2+m22a)).
u2Dout(x,y=a/2)=12(1+sin(πxa)·MTF(22a)).
u1Dout(x)=12(1+sin(πxa)·MTF(12a)).
Scheckerout(x,y)=12(1+(4π)2n=1,3,5m=1,3,5sin(nπxa)n·sin(mπya)m·MTF(n2+m22a)).
Scheckerout(x,y=a/2)=12(1+(4π)2n=1,3,5m=1,3,5sin(nπxa)n·(1)m12m·MTF(n2+m22a)).
Scheckerout(x,y=a/2)=12(1+(4π)2sin(2πfxx)·MTF(2fx)).
CTFchecker(fx=1/3fc)=(4π)2·MTF(fc23)0.6855>MTF(fc3)0.5836.
(4π)2·MTF(22a)=MTF(12a)12a/fc=fxfc0.41.
Schessout(x,y)=12(1+(4π)2n=1,3,5m=1,3,5sin(nπxa)n·sin(mπyb)m·MTF(b2n2+a2m22ab)).
fc=fc/(ab)2+1,

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