Abstract

In this paper we derive a formula for calculating the point-spread function (PSF) of a rotationally symmetric imaging system from measurements along a line through the image of an arbitrary separable input object. An important special case of this formula is when the input object is a finite-length slit. The set of measurements in this case is called the finite-length line-spread function (FLSF). The FLSF differs from the infinite-length line-spread function (LSF) only in the assumed finite length of the line that is input into the system. This difference between the FLSF and the LSF becomes important for imaging systems for which the PSF is large in extent and in which the isoplanatic patch is relatively small. The usual LSF-to-PSF conversion formulas cannot be applied accurately to such systems.

© 1987 Optical Society of America

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References

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  1. E. W. Marchand, “Derivation of the point spread function from the line spread function,”J. Opt. Soc. Am. 54, 915–919 (1964).
    [Crossref]
  2. E. W. Marchand, “From line to point spread function: the general case,”J. Opt. Soc. Am. 55, 352–354 (1965).
    [Crossref]
  3. J. C. Dainty, R. Shaw, Image Science (Academic, London, 1974) pp. 209–211.
  4. H. Slevogt, “Ein Vorschlag zur Darstellung des Lichtgebirges (Punktbildes), der zweidimensionalen Uebertragungsfunktion und des Linienbildes I,” Optik 22, 391–398 (1965).
  5. A. Papoulis, “Optical systems, singularity functions, complex Hankel transforms,”J. Opt. Soc. Am. 57, 207–213 (1967).
    [Crossref]
  6. A. Papoulis, “Approximation of point spreads for deconvolution,”J. Opt. Soc. Am. 62, 77–80 (1972).
    [Crossref]
  7. J. Ojeda-Castañeda, “Focus-error operator and related special functions,”J. Opt. Soc. Am. 73, 1042–1047 (1983).
    [Crossref]
  8. B. R. Frieden, “Use of a scanning slit in determination of a radial irradiance distribution,”J. Opt. Soc. Am. 55, 1696–1697 (1965).
    [Crossref]
  9. B. D. Cook, “Proposed mapping of ultrasonic fields with conventional light diffraction,”J. Opt. Soc. Am. 65, 682–684 (1975).
    [Crossref]
  10. N. A. Nill, “A visual model weighted cosine transform for image compression and quality assessment,”IEEE Trans. Commun. COM-33, 551–557 (1985).
    [Crossref]
  11. H.-P. Chan, K. Doi, “Physical characteristics of scattered radiation in diagnostic radiology: Monte Carlo simulation studies,” Med. Phys. 12, 152–165 (1985).
    [Crossref] [PubMed]
  12. C. H. Hessler, C. D. Depeursinge, M. Grecescu, Y. Pochon, S. Raimondi, J. F. Valley, “Objective asessment of mammography systems,” Radiology 156, 215–219 (1985).
    [PubMed]
  13. K. Rossmann, “Point spread function, line spread function, and modulation transfer function,” Radiology 93, 257–272 (1969).
    [PubMed]
  14. R. N. Bracewell, “Strip integration in astronomy,” Aust. J. Phys. 9, 198–217 (1956).
    [Crossref]

1985 (3)

N. A. Nill, “A visual model weighted cosine transform for image compression and quality assessment,”IEEE Trans. Commun. COM-33, 551–557 (1985).
[Crossref]

H.-P. Chan, K. Doi, “Physical characteristics of scattered radiation in diagnostic radiology: Monte Carlo simulation studies,” Med. Phys. 12, 152–165 (1985).
[Crossref] [PubMed]

C. H. Hessler, C. D. Depeursinge, M. Grecescu, Y. Pochon, S. Raimondi, J. F. Valley, “Objective asessment of mammography systems,” Radiology 156, 215–219 (1985).
[PubMed]

1983 (1)

1975 (1)

1972 (1)

1969 (1)

K. Rossmann, “Point spread function, line spread function, and modulation transfer function,” Radiology 93, 257–272 (1969).
[PubMed]

1967 (1)

1965 (3)

1964 (1)

1956 (1)

R. N. Bracewell, “Strip integration in astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[Crossref]

Bracewell, R. N.

R. N. Bracewell, “Strip integration in astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[Crossref]

Chan, H.-P.

H.-P. Chan, K. Doi, “Physical characteristics of scattered radiation in diagnostic radiology: Monte Carlo simulation studies,” Med. Phys. 12, 152–165 (1985).
[Crossref] [PubMed]

Cook, B. D.

Dainty, J. C.

J. C. Dainty, R. Shaw, Image Science (Academic, London, 1974) pp. 209–211.

Depeursinge, C. D.

C. H. Hessler, C. D. Depeursinge, M. Grecescu, Y. Pochon, S. Raimondi, J. F. Valley, “Objective asessment of mammography systems,” Radiology 156, 215–219 (1985).
[PubMed]

Doi, K.

H.-P. Chan, K. Doi, “Physical characteristics of scattered radiation in diagnostic radiology: Monte Carlo simulation studies,” Med. Phys. 12, 152–165 (1985).
[Crossref] [PubMed]

Frieden, B. R.

Grecescu, M.

C. H. Hessler, C. D. Depeursinge, M. Grecescu, Y. Pochon, S. Raimondi, J. F. Valley, “Objective asessment of mammography systems,” Radiology 156, 215–219 (1985).
[PubMed]

Hessler, C. H.

C. H. Hessler, C. D. Depeursinge, M. Grecescu, Y. Pochon, S. Raimondi, J. F. Valley, “Objective asessment of mammography systems,” Radiology 156, 215–219 (1985).
[PubMed]

Marchand, E. W.

Nill, N. A.

N. A. Nill, “A visual model weighted cosine transform for image compression and quality assessment,”IEEE Trans. Commun. COM-33, 551–557 (1985).
[Crossref]

Ojeda-Castañeda, J.

Papoulis, A.

Pochon, Y.

C. H. Hessler, C. D. Depeursinge, M. Grecescu, Y. Pochon, S. Raimondi, J. F. Valley, “Objective asessment of mammography systems,” Radiology 156, 215–219 (1985).
[PubMed]

Raimondi, S.

C. H. Hessler, C. D. Depeursinge, M. Grecescu, Y. Pochon, S. Raimondi, J. F. Valley, “Objective asessment of mammography systems,” Radiology 156, 215–219 (1985).
[PubMed]

Rossmann, K.

K. Rossmann, “Point spread function, line spread function, and modulation transfer function,” Radiology 93, 257–272 (1969).
[PubMed]

Shaw, R.

J. C. Dainty, R. Shaw, Image Science (Academic, London, 1974) pp. 209–211.

Slevogt, H.

H. Slevogt, “Ein Vorschlag zur Darstellung des Lichtgebirges (Punktbildes), der zweidimensionalen Uebertragungsfunktion und des Linienbildes I,” Optik 22, 391–398 (1965).

Valley, J. F.

C. H. Hessler, C. D. Depeursinge, M. Grecescu, Y. Pochon, S. Raimondi, J. F. Valley, “Objective asessment of mammography systems,” Radiology 156, 215–219 (1985).
[PubMed]

Aust. J. Phys. (1)

R. N. Bracewell, “Strip integration in astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[Crossref]

IEEE Trans. Commun. (1)

N. A. Nill, “A visual model weighted cosine transform for image compression and quality assessment,”IEEE Trans. Commun. COM-33, 551–557 (1985).
[Crossref]

J. Opt. Soc. Am. (7)

Med. Phys. (1)

H.-P. Chan, K. Doi, “Physical characteristics of scattered radiation in diagnostic radiology: Monte Carlo simulation studies,” Med. Phys. 12, 152–165 (1985).
[Crossref] [PubMed]

Optik (1)

H. Slevogt, “Ein Vorschlag zur Darstellung des Lichtgebirges (Punktbildes), der zweidimensionalen Uebertragungsfunktion und des Linienbildes I,” Optik 22, 391–398 (1965).

Radiology (2)

C. H. Hessler, C. D. Depeursinge, M. Grecescu, Y. Pochon, S. Raimondi, J. F. Valley, “Objective asessment of mammography systems,” Radiology 156, 215–219 (1985).
[PubMed]

K. Rossmann, “Point spread function, line spread function, and modulation transfer function,” Radiology 93, 257–272 (1969).
[PubMed]

Other (1)

J. C. Dainty, R. Shaw, Image Science (Academic, London, 1974) pp. 209–211.

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Equations (52)

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i ( x , y ) = - - o ( x - u , y - v ) p ( u , v ) d u d v .
i ( x ) = - - o x ( x - u ) o y ( - v ) p [ ( u 2 + v 2 ) 1 / 2 ] d u d v .
I ( ξ ) = - i ( x ) exp ( - 2 π i ξ x ) d x .
P ( ρ ) = 2 π 0 J 0 ( 2 π ρ r ) p ( r ) r d r ,
f ^ ( α ) = 2 0 f ( x ) exp ( - 2 π i α x 2 ) x d x , f ( x ) = - f ^ ( α ) exp ( 2 π i α x 2 ) d α .
i ( x ) = 2 0 o y ( y ) p [ ( x 2 + y 2 ) 1 / 2 ] d y .
t ( y ) = o y ( y ) / y .
i ( x ) = 2 0 t ( y ) p [ ( x 2 + y 2 ) 1 / 2 ] y d y .
i ^ ( γ ) = 8 0 0 - 0 t ^ ( α ) p ^ ( β ) exp ( 2 π i α y ) 2 × exp [ 2 π i β ( x 2 + y 2 ) ] exp ( - 2 π i γ x 2 ) x y d α d β d x d y .
2 0 exp ( - 2 π i α x 2 ) x d x = Δ ( α ) ,
i ^ ( γ ) = t ^ ( - γ ) p ^ ( γ ) .
p ^ ( γ ) = i ^ ( γ ) t ^ ( - γ ) .
I ( ξ , η ) = O ( ξ , η ) P ( ξ , η ) .
I ( ξ ) = - O x ( ξ ) O y ( η ) P ( ξ , η ) d η ,
I ( ξ ) = O x ( ξ ) - O y ( η ) P [ ( ξ 2 + η 2 ) 1 / 2 ] d η .
I f ( ξ ) = I ( ξ ) / O x ( ξ ) .
T ( η ) = O y ( η ) / η .
I f ( ξ ) = - T ( η ) P [ ( ξ 2 + η 2 ) 1 / 2 ] η d η .
P ^ ( γ ) = I ^ f ( γ ) / T ^ ( - γ ) .
o ( x , y ) = o x ( x ) o y ( y ) = δ ( x ) rect ( y / L ) .
t ( y ) = o y ( y ) / y = rect ( y / L ) / y .
t ^ ( - α ) = 2 0 [ rect ( y / L ) y ] exp ( 2 π i α y 2 ) y d y = 2 0 L / 2 exp ( 2 π i α y 2 ) d y = 2 0 L / 2 [ cos ( 2 π α y 2 ) + i sin ( 2 π α y 2 ) ] d y = 1 α 1 / 2 { C [ ( π α 2 ) 1 / 2 L ] + i S [ ( π α 2 ) 1 / 2 L ] } ,
C ( z ) = ( 2 π ) 1 / 2 0 z cos ( t 2 ) d t , S ( z ) = ( 2 π ) 1 / 0 z sin ( t 2 ) d t .
o ( x , y ) = o x ( x ) o y ( y ) = o x ( x ) rect ( y / L ) .
o y ( y ) = rect ( y / L ) ,
O y ( η ) = L sinc ( L η ) = sin ( π L η ) / π η .
T ( η ) = sin ( π L η ) / π η 2 .
T ^ ( - α ) = 2 0 [ sin ( π L η ) / π η 2 ] exp ( 2 π i α η 2 ) η d η = ( 1 + i ) { C [ ( π 8 α ) 1 / 2 L ] - i S [ ( π 8 α ) 1 / 2 L ] } .
p ( r ) = δ ( r - r 0 ) / 2 r 0 , o ( x , y ) = δ ( x ) rect ( y / L ) ,
p ^ ( α ) = 0 δ ( r - r 0 ) 2 r 0 exp ( - 2 π i α r 2 ) r d r = 1 2 exp ( - 2 π i α r 0 2 ) .
i ( x ) = 2 0 rect ( y / L ) δ [ ( x 2 + y 2 ) 1 / 2 - r o ] 2 r 0 d y = 1 r 0 0 L / 2 δ [ y - ( r 0 2 - x 2 ) 1 / 2 ] y / ( x 2 + y 2 ) 1 / 2 d y = rect [ ( r 0 2 - x 2 ) 1 / 2 L ] ( r 0 2 - x 2 ) 1 / 2 ,
i ^ ( α ) = 0 rect [ ( r 0 2 - x 2 ) 1 / 2 L ] ( r 0 2 - x 2 ) 1 / 2 exp ( - 2 π i α x 2 ) x d x .
i ^ ( α ) = 0 rect ( u L ) u exp [ - 2 π i α ( r 0 2 - u 2 ) ] u d u ,
i ^ ( α ) = exp ( - 2 π i α r 0 2 ) 2 α 1 / 2 { C [ ( π α 2 ) 1 / 2 L ] + i S [ ( π α 2 ) 1 / 2 L ] } .
p ^ ( α ) = i ^ ( α ) t ^ ( - α ) = 1 2 α 1 / 2 exp ( - 2 π i α r 0 2 ) { C [ ( π α 2 ) 1 / 2 L ] + i S [ ( π α 2 ) 1 / 2 L ] } 1 α 1 / 2 { C [ ( π α 2 ) 1 / 2 L ] + i S [ ( π α 2 ) 1 / 2 L ] } = 1 2 exp ( - 2 π i α r 0 2 ) ,
F ( α ) = 0 f ( x ) exp ( - 2 π i α x ) d x , f ( x ) = - F ( α ) exp ( 2 π i α x ) d α .
F ( α ) = 2 0 f ( u 2 ) exp ( - 2 π i α u 2 ) u d u .
g ( u ) = f ( u 2 ) = f ( x ) , g ^ ( α ) = F ( α )
g ^ ( α ) = 2 0 g ( u ) exp ( - 2 π i α u 2 ) y d u , g ( u ) = 2 - g ^ ( α ) exp ( 2 π i α u 2 ) d α .
- exp [ - 2 π i α ( x 2 - a 2 ) ] d α = δ ( x 2 - a 2 ) = δ ( x - a ) + δ ( x + a ) 2 x = δ ( x - a ) 2 x .
Δ ( α ) = 0 exp ( - 2 π i α u ) d u = 2 0 ( - 2 π i α x 2 ) x d x = δ ( α ) 2 + 1 2 π i α .
- g ^ ( β ) Δ ( α - β ) d β = g ^ ( α ) .
- g ^ ( β ) Δ ( α - β ) d β = 2 - 0 0 u g ( u ) exp ( - 2 π i β u 2 ) × exp [ - 2 π i ( α - β ) v ] d u d v d β = 2 0 u g ( u ) 0 exp ( - 2 π i α v ) × - exp [ - 2 π i β ( u 2 - v ) ] d β d v d u = 2 0 u g ( u ) × 0 exp ( - 2 π i α v ) δ ( u 2 - v ) d u d v = 2 0 u g ( u ) exp ( - 2 π i α u 2 ) d u = g ^ ( α ) .
T ^ ( - α ) = 2 0 sin ( π L η ) π η exp ( 2 π i α η 2 ) d η = 1 i - exp ( i π L η ) π η exp ( 2 π i α η 2 ) d η .
T ^ ( - α ) L = - exp [ 2 π i α ( η 2 + L η / 2 α ) ] d η = exp [ - i π ( L 2 / 8 α ) ] - exp [ 2 π i α ( η + L 4 α ) 2 ] d η = exp [ - i π ( L 2 / 8 α ) ] 1 ( 2 π α ) 1 / 2 2 ( 1 + i ) ( 2 π ) 1 / 2 4 .
C ( ) = 1 2 , S ( ) = 1 2 .
T ^ ( - α ) = 1 2 α 1 / 2 ( 1 + i ) 0 L exp ( - i π t 2 / 8 α ) d t .
T ^ ( - α ) = ( 1 + i ) { C [ ( π 8 α ) 1 / 2 L ] - i S [ ( π 8 α ) 1 / 2 L ] } .
T ^ ( - α ) = lim L T ^ ( - α ) = ( 1 + i ) ( 1 2 - 1 2 i ) = 1 ,
P ^ ( α ) = I ^ ( α ) .
T ^ 0 ( - α ) = lim L 0 1 L T ^ ( - α ) = lim L 0 ( 1 + i ) ( π 8 α ) 1 / 2 ( 2 π ) 1 / 2 × [ cos ( π L 2 8 α ) - i sin ( π L 2 8 α ) ] = ( 1 + i ) 2 α 1 / 2 ,
P ^ ( α ) = I ^ 0 ( α ) α 1 / 2 ( 1 - i ) .

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