Abstract

With the advent of modern-day computational imagers, the phase of the optical transfer function may no longer be summarily ignored. This study discusses some important properties of the phase transfer function (PTF) of digital incoherent imaging systems and their implications on the performance and characterization of these systems. The effects of aliasing and sub-pixel image shifts on the phase of the complex frequency response of these sampled systems are described, including an examination of the specific case of moderate aliasing. Key properties of this function in aliased imaging systems are derived and their potential treatment to a range of diverse applications encompassing traditional and computational imaging systems is discussed.

©2011 Optical Society of America

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References

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  1. H. H. Hopkins, “Image shift, phase distortion and the optical transfer function,” Opt. Acta (Lond.) 31(3), 345–368 (1984).
    [Crossref]
  2. K.-J. Rosenbruch and R. Gerschler, “The Meaning of the Phase Transfer Function and the Modular Transfer Function in Using OTF as a Criterion for Image Quality,” Optik (Stuttg.) 55(2), 173–182 (1980) (In German).
  3. C. S. Williams and O. A. Becklund, Introduction to the Optical Transfer Function (Wiley, 1988), pp. 207–208.
  4. J. Mait, R. Athale, and J. van der Gracht, “Evolutionary paths in imaging and recent trends,” Opt. Express 11(18), 2093–2101 (2003).
    [Crossref] [PubMed]
  5. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995).
    [Crossref] [PubMed]
  6. W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems, Volume 2, Physical Image Formation, 1st ed. (Wiley-VCH, November 7, 2005). pp. 446.
  7. M. Somayaji and M. P. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using a cubic phase mask,” Appl. Opt. 45(13), 2911–2923 (2006).
    [Crossref] [PubMed]
  8. M. Demenikov and A. R. Harvey, “Image artifacts in hybrid imaging systems with a cubic phase mask,” Opt. Express 18(8), 8207–8212 (2010).
    [Crossref] [PubMed]
  9. J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), pp. 60–62.
  10. R. Barakat and A. Houston, “Transfer Function of an Optical System in the Presence of Off-Axis Aberrations,” J. Opt. Soc. Am. 55(9), 1142–1148 (1965).
    [Crossref]
  11. R. E. Hufnagel, “Significance of the Phase of Optical Transfer Functions,” J. Opt. Soc. Am. 58(11), 1505–1506 (1968).
    [Crossref]
  12. W. Wittenstein, J. C. Fontanella, A. R. Newbery, and J. Baars, “The Definition of the OTF and the Measurement of Aliasing for Sampled Imaging Systems,” Opt. Act. 29(1), 41–50 (1982).
    [Crossref]
  13. R. H. Vollmerhausen, D. A. Reago, and R. G. Driggers, Analysis and Evaluation of Sampled Imaging Systems (SPIE Press, 2010), Ch-3, PP 68.
  14. R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. 38(7), 1229–1240 (1999).
    [Crossref]
  15. B. Tatian, “Method for obtaining the transfer function from the edge response function,” J. Opt. Soc. Am. 55(8), 1014–1019 (1965).
  16. D. Williams and P. Burns, “Low-frequency MTF estimation for digital imaging devices using slanted-edge analysis,” Proc. SPIE 5294, 93–101 (2003).
    [Crossref]
  17. V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Phase Transfer Function of Sampled Imaging Systems,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuB1.
  18. H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17(1), 33–54 (1966).
    [Crossref]
  19. V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Image-Based Measurement of Phase Transfer Function,” in Digital Image Processing and Analysis, OSA Technical Digest (CD) (Optical Society of America, 2010), paper DMD1.

2010 (1)

2006 (1)

2003 (2)

J. Mait, R. Athale, and J. van der Gracht, “Evolutionary paths in imaging and recent trends,” Opt. Express 11(18), 2093–2101 (2003).
[Crossref] [PubMed]

D. Williams and P. Burns, “Low-frequency MTF estimation for digital imaging devices using slanted-edge analysis,” Proc. SPIE 5294, 93–101 (2003).
[Crossref]

1999 (1)

R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. 38(7), 1229–1240 (1999).
[Crossref]

1995 (1)

1984 (1)

H. H. Hopkins, “Image shift, phase distortion and the optical transfer function,” Opt. Acta (Lond.) 31(3), 345–368 (1984).
[Crossref]

1982 (1)

W. Wittenstein, J. C. Fontanella, A. R. Newbery, and J. Baars, “The Definition of the OTF and the Measurement of Aliasing for Sampled Imaging Systems,” Opt. Act. 29(1), 41–50 (1982).
[Crossref]

1980 (1)

K.-J. Rosenbruch and R. Gerschler, “The Meaning of the Phase Transfer Function and the Modular Transfer Function in Using OTF as a Criterion for Image Quality,” Optik (Stuttg.) 55(2), 173–182 (1980) (In German).

1968 (1)

1966 (1)

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17(1), 33–54 (1966).
[Crossref]

1965 (2)

Athale, R.

Baars, J.

W. Wittenstein, J. C. Fontanella, A. R. Newbery, and J. Baars, “The Definition of the OTF and the Measurement of Aliasing for Sampled Imaging Systems,” Opt. Act. 29(1), 41–50 (1982).
[Crossref]

Barakat, R.

Burns, P.

D. Williams and P. Burns, “Low-frequency MTF estimation for digital imaging devices using slanted-edge analysis,” Proc. SPIE 5294, 93–101 (2003).
[Crossref]

Cathey, W. T.

Christensen, M. P.

Demenikov, M.

Dowski, E. R.

Fiete, R. D.

R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. 38(7), 1229–1240 (1999).
[Crossref]

Fontanella, J. C.

W. Wittenstein, J. C. Fontanella, A. R. Newbery, and J. Baars, “The Definition of the OTF and the Measurement of Aliasing for Sampled Imaging Systems,” Opt. Act. 29(1), 41–50 (1982).
[Crossref]

Gerschler, R.

K.-J. Rosenbruch and R. Gerschler, “The Meaning of the Phase Transfer Function and the Modular Transfer Function in Using OTF as a Criterion for Image Quality,” Optik (Stuttg.) 55(2), 173–182 (1980) (In German).

Harvey, A. R.

Hopkins, H. H.

H. H. Hopkins, “Image shift, phase distortion and the optical transfer function,” Opt. Acta (Lond.) 31(3), 345–368 (1984).
[Crossref]

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17(1), 33–54 (1966).
[Crossref]

Houston, A.

Hufnagel, R. E.

Mait, J.

Newbery, A. R.

W. Wittenstein, J. C. Fontanella, A. R. Newbery, and J. Baars, “The Definition of the OTF and the Measurement of Aliasing for Sampled Imaging Systems,” Opt. Act. 29(1), 41–50 (1982).
[Crossref]

Rosenbruch, K.-J.

K.-J. Rosenbruch and R. Gerschler, “The Meaning of the Phase Transfer Function and the Modular Transfer Function in Using OTF as a Criterion for Image Quality,” Optik (Stuttg.) 55(2), 173–182 (1980) (In German).

Somayaji, M.

Tatian, B.

Tiziani, H. J.

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17(1), 33–54 (1966).
[Crossref]

van der Gracht, J.

Williams, D.

D. Williams and P. Burns, “Low-frequency MTF estimation for digital imaging devices using slanted-edge analysis,” Proc. SPIE 5294, 93–101 (2003).
[Crossref]

Wittenstein, W.

W. Wittenstein, J. C. Fontanella, A. R. Newbery, and J. Baars, “The Definition of the OTF and the Measurement of Aliasing for Sampled Imaging Systems,” Opt. Act. 29(1), 41–50 (1982).
[Crossref]

Appl. Opt. (2)

Br. J. Appl. Phys. (1)

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17(1), 33–54 (1966).
[Crossref]

J. Opt. Soc. Am. (3)

Opt. Act. (1)

W. Wittenstein, J. C. Fontanella, A. R. Newbery, and J. Baars, “The Definition of the OTF and the Measurement of Aliasing for Sampled Imaging Systems,” Opt. Act. 29(1), 41–50 (1982).
[Crossref]

Opt. Acta (Lond.) (1)

H. H. Hopkins, “Image shift, phase distortion and the optical transfer function,” Opt. Acta (Lond.) 31(3), 345–368 (1984).
[Crossref]

Opt. Eng. (1)

R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. 38(7), 1229–1240 (1999).
[Crossref]

Opt. Express (2)

Optik (Stuttg.) (1)

K.-J. Rosenbruch and R. Gerschler, “The Meaning of the Phase Transfer Function and the Modular Transfer Function in Using OTF as a Criterion for Image Quality,” Optik (Stuttg.) 55(2), 173–182 (1980) (In German).

Proc. SPIE (1)

D. Williams and P. Burns, “Low-frequency MTF estimation for digital imaging devices using slanted-edge analysis,” Proc. SPIE 5294, 93–101 (2003).
[Crossref]

Other (6)

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Phase Transfer Function of Sampled Imaging Systems,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuB1.

C. S. Williams and O. A. Becklund, Introduction to the Optical Transfer Function (Wiley, 1988), pp. 207–208.

R. H. Vollmerhausen, D. A. Reago, and R. G. Driggers, Analysis and Evaluation of Sampled Imaging Systems (SPIE Press, 2010), Ch-3, PP 68.

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Image-Based Measurement of Phase Transfer Function,” in Digital Image Processing and Analysis, OSA Technical Digest (CD) (Optical Society of America, 2010), paper DMD1.

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems, Volume 2, Physical Image Formation, 1st ed. (Wiley-VCH, November 7, 2005). pp. 446.

J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), pp. 60–62.

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Figures (7)

Fig. 1
Fig. 1 MTF and PTF plots of a cubic phase mask wavefront coding imager with considerable defocus. The cubic phase mask strength is α = 25 and the defocus is |ψ| = |kW20| = 30. The usable spatial frequency calculation is based on the analysis in [7].

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Fig. 2
Fig. 2 Schematic representation of the forward image capture model of a sampled imaging system. The green grid on the right hand side represents the detector pixels, while the blue grid on the left hand side denotes the corresponding object pixels. The centers of each of these pixels are marked by the associated dots on these planes. The red dots in the object and image planes represent a point object and its image respectively.

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Fig. 3
Fig. 3 Variation in the sampled PTF of a digital incoherent imaging system for various values of sampling phase Δx. The black curve represents the pre-sampled PTF of this system and the colored lines denote the sampled PTFs for different values of Δx. In this example, uo = 180 cyc/mm and un = 200 cyc/mm.

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Fig. 4
Fig. 4 An example of the sampled PTFs Θs(u) in the event of moderate aliasing and zero sampling phase Δx. The black curve represents the pre-sampled PTF Θ(u) and the dashed red curve identifies Θs(u). For this example, uo = 180 cyc/mm and un = 125 cyc/mm.

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Fig. 5
Fig. 5 An example of the sampled PTFs Θs(u) with various values of the sampling phase Δx for an imager with a linear optical PTF Θ(u) as represented by the thick black line. This data was generated for moderate aliasing, with uo = 180 cyc/mm and un = 125 cyc/mm.

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Fig. 6
Fig. 6 First derivatives Θ′s(u) of the sampled PTF for different values of sampling phase, based on the pre-sampled PTF shown in Fig. 3 and Fig. 4. Moderate aliasing is assumed with uo = 180 cyc/mm and un = 125 cyc/mm so that ua = 70 cyc/mm. The plots are shown to extend beyond u = un as a purely theoretical exercise intended to illustrate behavior at Nyquist.

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Fig. 7
Fig. 7 Second derivatives Θ′′s(u) of the sampled PTF based on the pre-sampled PTF shown in Fig. 3 and Fig. 4, for different values of sampling phase and moderate aliasing. Here uo = 180 cyc/mm and un = 125 cyc/mm so that ua = 70 cyc/mm. Extended data points beyond u = un are shown as a purely theoretical exercise to illustrate behavior at the Nyquist frequency.

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Equations (42)

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( x',y' )=( x M , y M ); ( Δx',Δy' )=( Δx M , Δy M ).
i( x s −Δx, y s −Δy )=[ o( x ′ s −Δx' M , y ′ s −Δy' M )⊗ h o ( x,y )⊗ h d ( x,y ) ]×comb( x p , y p ),
( Φ x , Φ y )=( 2πΔx p , 2πΔy p ),
i( x s −Δx )=[ o( x s −Δx )⊗ h o ( x )⊗ h d ( x ) ]×comb( x p ).
comb( x p )=p ∑ k=−∞ ∞ δ( x−kp ) ,
i( x s −Δx )=h( x s −Δx )=[ h o ( x s −Δx )⊗ h d ( x ) ]×comb( x p ).
h s ( x s )=[ h o ( x s −Δx )⊗ h d ( x ) ]×comb( x p ).
H s ( u )=[ H o ( u ) e −j2πΔxu × H d ( u ) ]⊗ ∑ k=−∞ ∞ δ( u−2k u n ) ,
H s ( u )= ∑ k=−∞ ∞ H( u−2k u n ) e −j2π( u−2k u n )Δx ,
H s ( u )=M( u ) e Θ( u )−j2πuΔx ,
Θ s ( u )=Θ( u )−2πuΔx.
H s ( u )= ∑ k=−∞ ∞ M( u−2k u n ) e j[ Θ( u−2k u n )−2π( u−2k u n )Δx ] .
H s ( u )= ∑ k=−∞ ∞ M( u−2k u n )cos{ Θ( u−2k u n )−2π( u−2k u n )Δx } +j ∑ k=−∞ ∞ M( u−2k u n )sin{ Θ( u−2k u n )−2π( u−2k u n )Δx } .
M s ( u )= [ ( ∑ k=−∞ ∞ M( u−2k u n )cos{ Θ( u−2k u n )−2π( u−2k u n )Δx } ) 2 + ( ∑ k=−∞ ∞ M( u−2k u n )sin{ Θ( u−2k u n )−2π( u−2k u n )Δx } ) 2 ] 1 2 .
Θ s ( u )= tan −1 ( ∑ k=−∞ ∞ M( u−2k u n )sin{ Θ( u−2k u n )−2π( u−2k u n )Δx } ∑ k=−∞ ∞ M( u−2k u n )cos{ Θ( u−2k u n )−2π( u−2k u n )Δx } ).
H s ( u )=H( u ) e −j2πuΔx +H( u−2 u n ) e −j2π( u−2 u n )Δx +H( u+2 u n ) e −j2π( u+2 u n )Δx .
M s ( u )= [ M 2 ( u ) + M 2 ( u−2 u n )+ M 2 ( u+2 u n ) + 2M( u )M( u−2 u n )cos{ 4π u n Δx−Θ( u )+Θ( u−2 u n ) } + 2M( u )M( u+2 u n )cos{ 4π u n Δx+Θ( u )−Θ( u+2 u n ) } + 2M( u−2 u n )M( u+2 u n )cos{ 8π u n Δx+Θ( u−2 u n )−Θ( u+2 u n ) } ] ½ .
Θ s ( u )= tan −1 ( [ M( u )sin{ Θ( u )−2πuΔx } +M( u−2 u n )sin{ Θ( u−2 u n )−2π( u−2 u n )Δx } +M( u+2 u n )sin{ Θ( u+2 u n )−2π( u+2 u n )Δx } ] [ M( u )cos{ Θ( u )−2πuΔx } +M( u−2 u n )cos{ Θ( u−2 u n )−2π( u−2 u n )Δx } +M( u+2 u n )cos{ Θ( u+2 u n )−2π( u+2 u n )Δx } ] ).
H s ( −u )= ∑ k=−∞ ∞ H( −u−2k u n ) e −j2π( −u−2k u n )Δx .
H s ( −u )= ∑ l=−∞ ∞ H( −( u−2l u n ) ) e +j2π( u−2l u n )Δx = ∑ l=−∞ ∞ H ∗ ( u−2l u n ) e +j2π( u−2l u n )Δx = H s ∗ ( u ),
H s ∗ ( u n )= ∑ k=−∞ ∞ H ∗ ( ( 1−2k ) u n ) e +j2π( ( 1−2k ) u n )Δx .
H ∗ ( ( 1−2k ) u n )=H( −( 1−2k ) u n ).
H s ∗ ( u n )= ∑ k=−∞ ∞ H( −( 1−2k ) u n ) e +j2π( ( 1−2k ) u n )Δx .
H s ∗ ( u n )= ∑ l=−∞ ∞ H( ( 1−2l ) u n ) e −j2π( ( 1−2l ) u n )Δx = H s ( u n ).
α s ( u )= ∑ k=−∞ ∞ M( u−2k u n )cos{ Θ( u−2k u n )−2π( u−2k u n )Δx } , β s ( u )= ∑ k=−∞ ∞ M( u−2k u n )sin{ Θ( u−2k u n )−2π( u−2k u n )Δx } .
M s ( u )= [ α s 2 ( u )+ β s 2 ( u ) ] ½ , Θ s ( u )= tan −1 ( β s ( u ) α s ( u ) ).
d du Θ s ( u ) = d du tan -1 ( β s ( u ) α s ( u ) ) = α s ( u ) β ′ s ( u )− α ′ s ( u ) β s ( u ) α s 2 ( u )+ β s 2 ( u ) ,
Θ ′ s ( 0 ) = β ′ s ( 0 ) α s ( 0 ) .
β ′ s ( u )= ∑ k=−∞ ∞ [ M( u−2k u n ){ Θ ′ ( u−2k u n )−2πΔx } ×cos{ Θ( u−2k u n )−2π( u−2k u n )Δx } ] + ∑ k=−∞ ∞ M ′ ( u−2k u n )sin{ Θ( u−2k u n )−2π( u−2k u n )Δx } .
M( −u ) = M( u ), Θ( −u ) = −Θ( u ), M ′ ( −u ) = − M ′ ( u ), Θ ′ ( −u ) = Θ ′ ( u ).
Θ ′ s ( 0 )= 1 α s ( 0 ) [ −2πΔx ∑ k=−∞ ∞ M( −2k u n )cos{ Θ( −2k u n )+4πk u n Δx } + ∑ k=−∞ ∞ M( −2k u n ) Θ ′ ( −2k u n )cos{ Θ( −2k u n )+4πk u n Δx } + ∑ k=−∞ ∞ M ′ ( −2k u n )sin{ Θ( −2k u n )+4πk u n Δx } ].
Θ ′ s ( 0 ) = −2πΔx + Θ ′ ( 0 )M( 0 ) α s ( 0 ) + 2 α s ( 0 ) ∑ k=1 ∞ M( 2k u n ) Θ ′ ( 2k u n )cos{ Θ( 2k u n )−4πk u n Δx } + 2 α s ( 0 ) ∑ k=1 ∞ M ′ ( 2k u n )sin{ Θ( 2k u n )−4πk u n Δx },
Θ ′ s ( 0 ) = Θ ′ ( 0 )−2πΔx ; ∀ u o : 0 < u o ≤ 2 u n ,
Θ ′ s ( u n ) = β ′ s ( u n ) α s ( u n ) .
g 1 ( ( 1−2k ) u n )=M( ( 1−2k ) u n ){ Θ ′ ( ( 1−2k ) u n )−2πΔx } ×cos{ Θ( ( 1−2k ) u n )−2π( 1−2k ) u n Δx }, g 2 ( ( 1−2k ) u n )= M ′ ( ( 1−2k ) u n )sin{ Θ( ( 1−2k ) u n )−2π( 1−2k ) u n Δx },
β s ( u n ) = ∑ k=−∞ ∞ [ g 1 ( ( 1−2k ) u n )+ g 2 ( ( 1−2k ) u n ) ] .
β s ( u n ) = ∑ k=−∞ 0 [ g 1 ( ( 1−2k ) u n )+ g 2 ( ( 1−2k ) u n ) ] + ∑ k=1 ∞ [ g 1 ( ( 1−2k ) u n )+ g 2 ( ( 1−2k ) u n ) ] .
β s ( u n ) = ∑ l=1 ∞ [ g 1 ( −( 1−2l ) u n )+ g 2 ( −( 1−2l ) u n ) ] + ∑ k=1 ∞ [ g 1 ( ( 1−2k ) u n )+ g 2 ( ( 1−2k ) u n ) ] .
β s ( u n ) = 2 ∑ k=1 ∞ [ g 1 ( ( 1−2k ) u n )+ g 2 ( ( 1−2k ) u n ) ] .
Θ ′ s ( u n )= 2 α s ( u n ) ∑ k=1 ∞ [ M( ( 1−2k ) u n ){ Θ ′ ( ( 1−2k ) u n )−2πΔx } ×cos{ Θ( ( 1−2k ) u n )−2π( 1−2k ) u n Δx }0 + M ′ ( ( 1−2k ) u n )sin{ Θ( ( 1−2k ) u n )−2π( 1−2k ) u n Δx } ].
α s ( u n ) = 2M( u n )cos{ Θ( u n )−2π u n Δx },
Θ ′ s ( u n ) = Θ ′ ( u n )−2πΔx + M ′ ( u n ) M( u n ) tan{ Θ( u n )−2π u n Δx } ; ∀ u o : u n < u o ≤2 u n ,

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