Abstract

Different techniques have been proposed to assess the modulation transfer function (MTF) of sampled imaging systems. Some of these are based on the use of periodic targets made up of thin lines or points. The main potential problem of implementation concerning these methods is the fact that, in specific experiments, it may be difficult to ensure a good balance in intensity between the individual lines or points. An analytical model permitting a first estimation of the actual importance of this problem is presented. The error in the MTF assessment for two generic cameras is then estimated, taking into account the experimental process.

© 1997 Optical Society of America

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References

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  1. J. M. Lloyd, “Fundamentals of electro-optical imaging systems analysis,” in Infrared and Electro-Optical Systems Handbook, M. C. Dudzik, ed. (ERIM, Ann Arbor, Mich., 1993), Vol. 4, pp. 44–48.
  2. W. Wittenstein, J. C. Fontanella, A. R. Newbury, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Acta 29, 41–52 (1982).
    [CrossRef]
  3. R. F. Rauchmiller, R. A. Schowengerdt, “Measurement of the Landsat thematic mapper modulation transfer function using an array of point sources,” Opt. Eng. 27, 334–343 (1988).
    [CrossRef]
  4. G. C. Holst, “Infrared imaging system testing,” in Infrared and Electro-Optical Systems Handbook, M. C. Dudzik, ed. (ERIM, Ann Arbor, Mich., 1993), Vol. 4, pp. 223–232.
  5. J. Primot, M. Girard, M. Chambon, “Modulation transfer function assessment for sampled imaging systems: a generalization of the line spread function,” J. Mod. Opt. 41, 1301–1306 (1994).
    [CrossRef]
  6. M. Chambon, J. Primot, M. Girard, “Modulation transfer function assessment for sampled imaging systems: application of the generalized line spread function to a standard infrared camera,” Infrared Phys. Technol. 37, 619–626 (1996).
    [CrossRef]
  7. N. L. Johnson, S. Kotz, Distributions in Statistics: Continuous Univariate Distributions (Wiley Interscience, New York, 1970).

1996

M. Chambon, J. Primot, M. Girard, “Modulation transfer function assessment for sampled imaging systems: application of the generalized line spread function to a standard infrared camera,” Infrared Phys. Technol. 37, 619–626 (1996).
[CrossRef]

1994

J. Primot, M. Girard, M. Chambon, “Modulation transfer function assessment for sampled imaging systems: a generalization of the line spread function,” J. Mod. Opt. 41, 1301–1306 (1994).
[CrossRef]

1988

R. F. Rauchmiller, R. A. Schowengerdt, “Measurement of the Landsat thematic mapper modulation transfer function using an array of point sources,” Opt. Eng. 27, 334–343 (1988).
[CrossRef]

1982

W. Wittenstein, J. C. Fontanella, A. R. Newbury, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Acta 29, 41–52 (1982).
[CrossRef]

Baars, J.

W. Wittenstein, J. C. Fontanella, A. R. Newbury, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Acta 29, 41–52 (1982).
[CrossRef]

Chambon, M.

M. Chambon, J. Primot, M. Girard, “Modulation transfer function assessment for sampled imaging systems: application of the generalized line spread function to a standard infrared camera,” Infrared Phys. Technol. 37, 619–626 (1996).
[CrossRef]

J. Primot, M. Girard, M. Chambon, “Modulation transfer function assessment for sampled imaging systems: a generalization of the line spread function,” J. Mod. Opt. 41, 1301–1306 (1994).
[CrossRef]

Fontanella, J. C.

W. Wittenstein, J. C. Fontanella, A. R. Newbury, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Acta 29, 41–52 (1982).
[CrossRef]

Girard, M.

M. Chambon, J. Primot, M. Girard, “Modulation transfer function assessment for sampled imaging systems: application of the generalized line spread function to a standard infrared camera,” Infrared Phys. Technol. 37, 619–626 (1996).
[CrossRef]

J. Primot, M. Girard, M. Chambon, “Modulation transfer function assessment for sampled imaging systems: a generalization of the line spread function,” J. Mod. Opt. 41, 1301–1306 (1994).
[CrossRef]

Holst, G. C.

G. C. Holst, “Infrared imaging system testing,” in Infrared and Electro-Optical Systems Handbook, M. C. Dudzik, ed. (ERIM, Ann Arbor, Mich., 1993), Vol. 4, pp. 223–232.

Johnson, N. L.

N. L. Johnson, S. Kotz, Distributions in Statistics: Continuous Univariate Distributions (Wiley Interscience, New York, 1970).

Kotz, S.

N. L. Johnson, S. Kotz, Distributions in Statistics: Continuous Univariate Distributions (Wiley Interscience, New York, 1970).

Lloyd, J. M.

J. M. Lloyd, “Fundamentals of electro-optical imaging systems analysis,” in Infrared and Electro-Optical Systems Handbook, M. C. Dudzik, ed. (ERIM, Ann Arbor, Mich., 1993), Vol. 4, pp. 44–48.

Newbury, A. R.

W. Wittenstein, J. C. Fontanella, A. R. Newbury, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Acta 29, 41–52 (1982).
[CrossRef]

Primot, J.

M. Chambon, J. Primot, M. Girard, “Modulation transfer function assessment for sampled imaging systems: application of the generalized line spread function to a standard infrared camera,” Infrared Phys. Technol. 37, 619–626 (1996).
[CrossRef]

J. Primot, M. Girard, M. Chambon, “Modulation transfer function assessment for sampled imaging systems: a generalization of the line spread function,” J. Mod. Opt. 41, 1301–1306 (1994).
[CrossRef]

Rauchmiller, R. F.

R. F. Rauchmiller, R. A. Schowengerdt, “Measurement of the Landsat thematic mapper modulation transfer function using an array of point sources,” Opt. Eng. 27, 334–343 (1988).
[CrossRef]

Schowengerdt, R. A.

R. F. Rauchmiller, R. A. Schowengerdt, “Measurement of the Landsat thematic mapper modulation transfer function using an array of point sources,” Opt. Eng. 27, 334–343 (1988).
[CrossRef]

Wittenstein, W.

W. Wittenstein, J. C. Fontanella, A. R. Newbury, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Acta 29, 41–52 (1982).
[CrossRef]

Infrared Phys. Technol.

M. Chambon, J. Primot, M. Girard, “Modulation transfer function assessment for sampled imaging systems: application of the generalized line spread function to a standard infrared camera,” Infrared Phys. Technol. 37, 619–626 (1996).
[CrossRef]

J. Mod. Opt.

J. Primot, M. Girard, M. Chambon, “Modulation transfer function assessment for sampled imaging systems: a generalization of the line spread function,” J. Mod. Opt. 41, 1301–1306 (1994).
[CrossRef]

Opt. Acta

W. Wittenstein, J. C. Fontanella, A. R. Newbury, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Acta 29, 41–52 (1982).
[CrossRef]

Opt. Eng.

R. F. Rauchmiller, R. A. Schowengerdt, “Measurement of the Landsat thematic mapper modulation transfer function using an array of point sources,” Opt. Eng. 27, 334–343 (1988).
[CrossRef]

Other

G. C. Holst, “Infrared imaging system testing,” in Infrared and Electro-Optical Systems Handbook, M. C. Dudzik, ed. (ERIM, Ann Arbor, Mich., 1993), Vol. 4, pp. 223–232.

J. M. Lloyd, “Fundamentals of electro-optical imaging systems analysis,” in Infrared and Electro-Optical Systems Handbook, M. C. Dudzik, ed. (ERIM, Ann Arbor, Mich., 1993), Vol. 4, pp. 44–48.

N. L. Johnson, S. Kotz, Distributions in Statistics: Continuous Univariate Distributions (Wiley Interscience, New York, 1970).

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Figures (8)

Fig. 1
Fig. 1

Spectrum of the image of a Dirac comb with a spacing of (4 + 1/2) pixels. Bold lines correspond to the analyzed MTF; thin lines correspond to the aliased components.

Fig. 2
Fig. 2

Spectrum of the image of a Dirac comb with a spacing of (4 + 1/4) pixel. Bold lines correspond to the right wing of the analyzed MTF; thin lines correspond to the aliased components.

Fig. 3
Fig. 3

Transfer function of the low-resolution camera.

Fig. 4
Fig. 4

Theoretical evaluation of the measured transfer function for the low-resolution camera, with ±1σ error bars, considering a 10% intensity variation and a target made up of four lines.

Fig. 5
Fig. 5

Theoretical evaluation of the MTF for the low-resolution camera. The solid curve corresponds to the actual MTF; the dotted lines correspond to the measured MTF.

Fig. 6
Fig. 6

Transfer function of the high-resolution camera.

Fig. 7
Fig. 7

Global behavior of the variances of R(u) and I (u) for the high-resolution camera.

Fig. 8
Fig. 8

Theoretical evaluation of the MTF of the high-resolution camera. The solid curve corresponds to the actual MTF; the dotted lines correspond to the measured MTF. The ±1σ error bars are plotted, considering a 10% intensity variation and a target made up of four lines.

Equations (46)

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tx=k=-KK-1δx-kxp,
xp=N+1/2s,
itx=tx*hx1/s combx/s,
Itu=TuHu*combus,
Tu=k=-KK-1exp2iπxpku.
Itu=TuHu+T1/s-uH1/s-u.
un=nxp, 0nN.
Tun=k=-KK-1exp2iπxpkun=k=-KK-1exp2iπkn=2K,
T1/s-un=k=-KK-1exp2iπxpk1s-un=k=-KK-1exp2iπN+12sk1s-un=k=-KK-1exp2iπkN+12-n=k=-KK-1-1k=0,
Itun=2K Hun.
MTFun=Hun=12KItun.
PTFun=argHun=argItun.
tax=tx+bx.
bx=k=-KK-1bkδx-kxp,
k=-KK-1bk=0.
itax=tx+bx*hxIIIsx.
Itaun=TunHun+T1/s-unH1/s-un+BunHun+B1/s-unH1/s-us.
Bun=k=-KK-1 bk exp2iπxpkun=k=-KK-1bk=0,
B1s-un=k=-KK-1exp2iπxpk1s-un=k=-KK-1 bk-1k.
Hmun=Hun+12Kk=-KK-1bk-1kH1s-un.
t2x=k=-2K2K-1 δx-kxp2,
xp2=N+1/4s.
It2aun=T2unHun+T21/s-unTF1/s-un+T21/s+unH1/s+un+T22/s-un×H2/s-un+BunHun+B1/s-un×H1/s-un+B1/s+unH1/s+un+B2/s-unH2/s-un.
Hmun=Hun+14KsKH1s-un+14KsK×H1s+un+14Ks2KH2s-un,
sK=k=-2K2K-1bkik=k=-KK-1b2ki2k+k=-KK-1b2k+1i2k+1=k=-KK-1b2k-1k+ik=-KK-1b2k+1-1k+1, sK
s2K=k=-2K2K-1bk-1k, s2K.
Hlowu=oudu,
ou=1π2 acosus-sin2 acosus.
du=sinπusπus.
Hmu=Hu+12KcKH1s-u,
CK=k=-KK-1 bk-1k.
cK=k=-2K2K-1bk-1k=0,
cK2=k=-KK-1l=-KK-1bkbl-1k+l=k=-KK-1bk2=2Kσb2, σcK2=2Kσb2.
σfold2u=σb22KH1s-u2.
MTFmeanu=2πσfolduexp-Hu22σfoldu2+MTFu1-Φ-Huσfoldu,
Φy=12π-yexp-z22dz,
σMTF2u=σfold2u+MTFu2-MTFmeanu2.
H2u=o2udu,
o2u=1π2 acosus2-sin2 acosus2.
H2mu=H2u+14KsKH21s-u+14KsKH21s+u+14Ks2KH22s-u,
H2mu=H2u+ResKH21/s-u+H21/s+u+s2KH22/s-u+i ImsKH21/s-u+H21/s+u.
s2KResK=k=-2K2K-1l=-KK-1bkb2l-1k+l=k=-KK-1b2k2-1k=σb2k=-KK-1-1k=0.
H2mu=H2u+Ru+iIu,
σRu2=σb28KH21s-u+H21s+u2+2H22s-u2,σIu2=σb28KH21s-u+H21s+u2,
MTF2meanu=π2σRu exp-H2u24σRu2×1+H2u22σRu2I0H2u24σRu2+H2u22σRu2I1H2u24σRu2,
σMTF2u2=σRu22+H2u2σRu2.

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