Abstract

A mathematical model based on the methods of Fourier optics is presented as a description of the signal-processing characteristics of differential-interference-contrast microscopy. A computerized simulation of this signal processing is described, and some images of abstract objects generated by this simulation are presented. This model and its simulation have implications on the feasibility of image restoration with superresolution by way of differential-interference-contrast microscopy under the classical constraints of finite object size and non-negativity. Such implications are discussed, and the encouraging results of a study into such feasibility are shown.

© 1987 Optical Society of America

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  1. R. D. Allen, G. B. David, G. Nomarski, “The Zeiss-Nomarski Differential Interference Equipment for Transmitted Light Microscopy,” Z. Wiss. Mikrosk. 69, Heft 4, 193 (1969).
    [PubMed]
  2. W. Lang, “Nomarski Differential Interference-Contrast Microscopy,” Zeiss Reprint, Carl Zeiss, 7082 Oberkochen, West Germany.
  3. W. J. Levy, R. Rumpf, T. Spagnolia, D. H. York, “Model for the Study of Individual Mammalian Axons in Vivo, with Anatomical Continuity and Function Maintained,” Neurosurgery 17, 459 (1985).
    [CrossRef] [PubMed]
  4. W. J. Levy, “Use of Video Enhanced Contrast Microscopy to Study Axonal Transport In-Vivo,” in Proceedings Twenty-third Annual Rocky Mountain Bioengineering Symposium, Library of Congress 63-25575 (Instrument Society of America, Research Triangle Park, NC, 1986), pp. 123–125.
  5. R. D. Allen, N. S. Allen, J. L. Travis, “Video-Enhanced Contrast, Differential Interference Contrast (AVEC-DIC) Microscopy: A New Method Capable of Analyzing Microtubule-Related Motility in the Reticulopodial Network of Allogromia Laticollaris,” Cell Motility 1, 275 (1981).
    [CrossRef] [PubMed]
  6. R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).
  7. R. W. Gerchberg, “Super-Resolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
    [CrossRef]
  8. J. R. Fienup, “Phase Retrieval Algorithms: A Comparison,” Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
  9. J. Maeda, “Restoration of Bandlimited Images by an Iterative Damped Least-Squares Method with Adaptive Regularization,” Appl. Opt. 24, 1421 (1985).
    [CrossRef] [PubMed]
  10. W. Galbraith, G. B. David, “An Aid to Understanding Differential Interference Contrast Microscopy: Computer Simulation,” J. Microsc. 108, (1976).
    [CrossRef]
  11. W. Galbraith, “The Image of a Point of Light in Differential Interference Contrast Microscopy: Computer Simulation,” Microsc. Acta 85, 233 (1982).
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  13. E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, MA, 1979).
  14. D. A. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191 (1984).
    [CrossRef]
  15. K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, NJ, 1979).
  16. R. C. Singleton, “An Algorithm for Computing the Mixed Radix Fast Fourier Transform,” in Digital Signal Processing, L. R. Rabiner, C. M. Rader, Eds. (IEEE Press, The Institute of Electrical and Electronic Engineers, Inc., New York, 1972).
  17. T. J. Holmes, “Noise Considerations in Signal Recovery with Differential-Interference-Contrast Microscopy,” manuscript in preparation.

1985 (2)

W. J. Levy, R. Rumpf, T. Spagnolia, D. H. York, “Model for the Study of Individual Mammalian Axons in Vivo, with Anatomical Continuity and Function Maintained,” Neurosurgery 17, 459 (1985).
[CrossRef] [PubMed]

J. Maeda, “Restoration of Bandlimited Images by an Iterative Damped Least-Squares Method with Adaptive Regularization,” Appl. Opt. 24, 1421 (1985).
[CrossRef] [PubMed]

1984 (1)

D. A. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191 (1984).
[CrossRef]

1982 (2)

J. R. Fienup, “Phase Retrieval Algorithms: A Comparison,” Appl. Opt. 21, 2758 (1982).
[CrossRef] [PubMed]

W. Galbraith, “The Image of a Point of Light in Differential Interference Contrast Microscopy: Computer Simulation,” Microsc. Acta 85, 233 (1982).

1981 (1)

R. D. Allen, N. S. Allen, J. L. Travis, “Video-Enhanced Contrast, Differential Interference Contrast (AVEC-DIC) Microscopy: A New Method Capable of Analyzing Microtubule-Related Motility in the Reticulopodial Network of Allogromia Laticollaris,” Cell Motility 1, 275 (1981).
[CrossRef] [PubMed]

1976 (1)

W. Galbraith, G. B. David, “An Aid to Understanding Differential Interference Contrast Microscopy: Computer Simulation,” J. Microsc. 108, (1976).
[CrossRef]

1974 (1)

R. W. Gerchberg, “Super-Resolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).

1969 (1)

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss-Nomarski Differential Interference Equipment for Transmitted Light Microscopy,” Z. Wiss. Mikrosk. 69, Heft 4, 193 (1969).
[PubMed]

Agard, D. A.

D. A. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191 (1984).
[CrossRef]

Allen, N. S.

R. D. Allen, N. S. Allen, J. L. Travis, “Video-Enhanced Contrast, Differential Interference Contrast (AVEC-DIC) Microscopy: A New Method Capable of Analyzing Microtubule-Related Motility in the Reticulopodial Network of Allogromia Laticollaris,” Cell Motility 1, 275 (1981).
[CrossRef] [PubMed]

Allen, R. D.

R. D. Allen, N. S. Allen, J. L. Travis, “Video-Enhanced Contrast, Differential Interference Contrast (AVEC-DIC) Microscopy: A New Method Capable of Analyzing Microtubule-Related Motility in the Reticulopodial Network of Allogromia Laticollaris,” Cell Motility 1, 275 (1981).
[CrossRef] [PubMed]

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss-Nomarski Differential Interference Equipment for Transmitted Light Microscopy,” Z. Wiss. Mikrosk. 69, Heft 4, 193 (1969).
[PubMed]

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, NJ, 1979).

David, G. B.

W. Galbraith, G. B. David, “An Aid to Understanding Differential Interference Contrast Microscopy: Computer Simulation,” J. Microsc. 108, (1976).
[CrossRef]

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss-Nomarski Differential Interference Equipment for Transmitted Light Microscopy,” Z. Wiss. Mikrosk. 69, Heft 4, 193 (1969).
[PubMed]

Fienup, J. R.

Galbraith, W.

W. Galbraith, “The Image of a Point of Light in Differential Interference Contrast Microscopy: Computer Simulation,” Microsc. Acta 85, 233 (1982).

W. Galbraith, G. B. David, “An Aid to Understanding Differential Interference Contrast Microscopy: Computer Simulation,” J. Microsc. 108, (1976).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, “Super-Resolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
[CrossRef]

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, MA, 1979).

Holmes, T. J.

T. J. Holmes, “Noise Considerations in Signal Recovery with Differential-Interference-Contrast Microscopy,” manuscript in preparation.

Lang, W.

W. Lang, “Nomarski Differential Interference-Contrast Microscopy,” Zeiss Reprint, Carl Zeiss, 7082 Oberkochen, West Germany.

Levy, W. J.

W. J. Levy, R. Rumpf, T. Spagnolia, D. H. York, “Model for the Study of Individual Mammalian Axons in Vivo, with Anatomical Continuity and Function Maintained,” Neurosurgery 17, 459 (1985).
[CrossRef] [PubMed]

W. J. Levy, “Use of Video Enhanced Contrast Microscopy to Study Axonal Transport In-Vivo,” in Proceedings Twenty-third Annual Rocky Mountain Bioengineering Symposium, Library of Congress 63-25575 (Instrument Society of America, Research Triangle Park, NC, 1986), pp. 123–125.

Maeda, J.

Nomarski, G.

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss-Nomarski Differential Interference Equipment for Transmitted Light Microscopy,” Z. Wiss. Mikrosk. 69, Heft 4, 193 (1969).
[PubMed]

Rumpf, R.

W. J. Levy, R. Rumpf, T. Spagnolia, D. H. York, “Model for the Study of Individual Mammalian Axons in Vivo, with Anatomical Continuity and Function Maintained,” Neurosurgery 17, 459 (1985).
[CrossRef] [PubMed]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).

Singleton, R. C.

R. C. Singleton, “An Algorithm for Computing the Mixed Radix Fast Fourier Transform,” in Digital Signal Processing, L. R. Rabiner, C. M. Rader, Eds. (IEEE Press, The Institute of Electrical and Electronic Engineers, Inc., New York, 1972).

Spagnolia, T.

W. J. Levy, R. Rumpf, T. Spagnolia, D. H. York, “Model for the Study of Individual Mammalian Axons in Vivo, with Anatomical Continuity and Function Maintained,” Neurosurgery 17, 459 (1985).
[CrossRef] [PubMed]

Travis, J. L.

R. D. Allen, N. S. Allen, J. L. Travis, “Video-Enhanced Contrast, Differential Interference Contrast (AVEC-DIC) Microscopy: A New Method Capable of Analyzing Microtubule-Related Motility in the Reticulopodial Network of Allogromia Laticollaris,” Cell Motility 1, 275 (1981).
[CrossRef] [PubMed]

York, D. H.

W. J. Levy, R. Rumpf, T. Spagnolia, D. H. York, “Model for the Study of Individual Mammalian Axons in Vivo, with Anatomical Continuity and Function Maintained,” Neurosurgery 17, 459 (1985).
[CrossRef] [PubMed]

Zajac, A.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, MA, 1979).

Ann. Rev. Biophys. Bioeng. (1)

D. A. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191 (1984).
[CrossRef]

Appl. Opt. (2)

Cell Motility (1)

R. D. Allen, N. S. Allen, J. L. Travis, “Video-Enhanced Contrast, Differential Interference Contrast (AVEC-DIC) Microscopy: A New Method Capable of Analyzing Microtubule-Related Motility in the Reticulopodial Network of Allogromia Laticollaris,” Cell Motility 1, 275 (1981).
[CrossRef] [PubMed]

J. Microsc. (1)

W. Galbraith, G. B. David, “An Aid to Understanding Differential Interference Contrast Microscopy: Computer Simulation,” J. Microsc. 108, (1976).
[CrossRef]

Microsc. Acta (1)

W. Galbraith, “The Image of a Point of Light in Differential Interference Contrast Microscopy: Computer Simulation,” Microsc. Acta 85, 233 (1982).

Neurosurgery (1)

W. J. Levy, R. Rumpf, T. Spagnolia, D. H. York, “Model for the Study of Individual Mammalian Axons in Vivo, with Anatomical Continuity and Function Maintained,” Neurosurgery 17, 459 (1985).
[CrossRef] [PubMed]

Opt. Acta (1)

R. W. Gerchberg, “Super-Resolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
[CrossRef]

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).

Z. Wiss. Mikrosk. (1)

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss-Nomarski Differential Interference Equipment for Transmitted Light Microscopy,” Z. Wiss. Mikrosk. 69, Heft 4, 193 (1969).
[PubMed]

Other (7)

W. Lang, “Nomarski Differential Interference-Contrast Microscopy,” Zeiss Reprint, Carl Zeiss, 7082 Oberkochen, West Germany.

W. J. Levy, “Use of Video Enhanced Contrast Microscopy to Study Axonal Transport In-Vivo,” in Proceedings Twenty-third Annual Rocky Mountain Bioengineering Symposium, Library of Congress 63-25575 (Instrument Society of America, Research Triangle Park, NC, 1986), pp. 123–125.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, MA, 1979).

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, NJ, 1979).

R. C. Singleton, “An Algorithm for Computing the Mixed Radix Fast Fourier Transform,” in Digital Signal Processing, L. R. Rabiner, C. M. Rader, Eds. (IEEE Press, The Institute of Electrical and Electronic Engineers, Inc., New York, 1972).

T. J. Holmes, “Noise Considerations in Signal Recovery with Differential-Interference-Contrast Microscopy,” manuscript in preparation.

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Figures (13)

Fig. 1
Fig. 1

Schematic illustration of a microscope with Nomarski optics.

Fig. 2
Fig. 2

Signal-processing model of differential-interference microscopy. The forward and reverse complex-valued 2-D Fourier transforms defined in Ref. 12 are indicated by the notation F and F−1, respectively.

Fig. 3
Fig. 3

Coherent transfer functions for (a) a circular pupil and (b) a square pupil.

Fig. 4
Fig. 4

(a) Phase signal φa(x,y) used in the simulation. The bright areas indicate a value of 0.3 rad, and the dark areas indicate a value of 0. (b) Approximate dimensions of (a). Note: The aspect ratio for the images in Figs. 4,5,6,7,8,9,10,12, and 13 is approximately 1.3 so that the rectangle shown in Fig. 6 actually represents a square. The spatial coordinates for Figs. 4, 8, 9, 10, and 12 are such that x represents the horizontal direction, y represents the vertical direction, and (x,y) = (0,0) is at the center of the image. The frequency coordinates for Figs. 5, 6, 7, and 13 are such that u represents the horizontal direction, υ represents the vertical direction, and (u,υ) = (0,0) is at the center of the image.

Fig. 5
Fig. 5

Imaginary part of the Fourier transform of ā(x,y). The brightness range is normalized so that the brightest areas indicated a value of 125.892, and the darkest areas indicate a value of −47.745; see Note in Fig. 4 caption.

Fig. 6
Fig. 6

Coherent transfer function for a square pupil used in the simulation. The frequency domain dimensions are shown with bright areas indicating a value of 1.0 and dark areas indicating a value of 0. The cutoff frequency represented by the edge of the square is 2.38 × 106 cycles/m; see Note in Fig. 4 caption.

Fig. 7
Fig. 7

Imaginary part of the Fourier transform of c ̅ ( x , y ). The brightness range is normalized so that the brightest areas indicate a value of 125.892 and the darkest areas indicate a value of −47.745. The constant gray areas outside the square all represent a value of 0; see Note in Fig. 4 caption.

Fig. 8
Fig. 8

Amplitude signal c(x,y) shown in Fig. 2 as a component output of the objective lens. This image demonstrates the spatial variation of c(x,y). The brightness range is normalized so that the brightest areas indicate a value of 1.028 and the darkest areas indicate a value of 0.989; see Note in Fig. 4 caption.

Fig. 9
Fig. 9

Phase signal φc(x,y). The brightness range is normalized so that the brightest areas indicate a value of 0.333, and the darkest areas indicate a value of 0.127; see Note in Fig. 4 caption.

Fig. 10
Fig. 10

Differential-interference-contrast image with various contrast and brightness settings. The brightest areas in this image indicate a value of 2.507, and the darkest areas indicate a value of 1.484; see Note in Fig. 4 caption.

Fig. 11
Fig. 11

Flow chart of the phase-object image-restoration algorithm extended from the Gerchberg-Saxton algorithm. Notation is defined in the text and as follows: C ̅ ̂ n ( u , υ )= nth iterative estimate of C ̅ ( u , υ ); Ā ̂ k ( u , υ )= kth iterative estimate of Ā(u,υ); ā ̂ k ( x , y )= kth iterative estimate of ā (x,y); a ̂ k ( x , y )= kth iterative estimate of a(x,y); φ ̂ a k ( x , y )= kth iterative estimate of φa(x,y);<== replace with; c ̅ ̂ n ( x , y )= nth iterative estimate of c ̅ ( x , y );m= inner-loop index;mmax= maximum number of inner loops to be executed per outer loop;k= total number of inner loops currently executed;kmax= maximum total number of inner loops to be executed;n= outer-loop index;nmax= maximum number of outer loops to be executed.

Fig. 12
Fig. 12

Estimate of the phase image φa(x,y) resulting from the algorithm outlined in Fig. 11. This was done (a) with nmax = 10 outer loops and mmax = 10 inner loops so that a total of kmax = 100 inner loops was executed. The brightness range is normalized so that the brightest areas indicate a value of 0.464, and the darkest areas indicate a value of 0.0.

Fig. 13
Fig. 13

Imaginary part of the Fourier transform of a exp [ j φ ̂ a ( x , y ) ], where φ ̂ a ( x , y ) represents the estimate shown in Fig. 12. The brightness range is normalized so that the brightest areas indicate a value of 125.892, and the darkest areas indicate a value of −47.745; see Note in Fig. 4 caption.

Equations (54)

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φ b ( x , y ) = φ a ( x + Δ x , y + Δ y ) + Δ φ ,
Δ = Δ x 2 + Δ y 2 ,
( Δ x , Δ y ) = ( Δ x , 0 )
Δ x = Δ .
ā ( x , y ) = a ( x , y ) exp [ j φ a ( x , y ) ]
b ̅ ( x , y ) = b ( x , y ) exp [ j φ b ( x , y ) ] .
a ( x , y ) = b ( x , y ) = a ,
ā ( x , y ) = a exp [ j φ a ( x , y ) ] ,
b ̅ ( x , y ) = a exp [ j φ b ( x , y ) ] ,
b ̅ ( x , y ) = a exp [ j φ a ( x + Δ x , y + Δ y ) + Δ φ ] = a exp [ j φ a ( x + Δ x , y ) ] exp ( j Δ φ ) = ā ( x + Δ x , y ) exp ( j Δ φ ) .
c ̅ ( x , y ) = c ( x , y ) exp [ j φ c ( x , y ) ] ,
d ̅ ( x , y ) = d ( x , y ) exp [ j φ d ( x , y ) ] ,
C ̅ ( u , υ ) = Ā ( u , υ ) H ( u , υ ) ,
D ̅ ( u , υ ) = B ̅ ( u , υ ) H ( u , υ ) ,
c ̅ ( x , y ) = L [ ā ( x , y ) ] = F 1 [ C ̅ ( u , υ ) ] ,
d ̅ ( x , y ) = L [ b ̅ ( x , y ) ] = F 1 [ D ̅ ( u , υ ) ] ,
L [ ā ( x + Δ x , y ) ] = c ̅ ( x + Δ x , y ) .
d ̅ ( x , y ) = L [ b ̅ ( x , y ) ] = L [ ā ( x + Δ x , y ) exp ( j Δ φ ) ] .
d ̅ ( x , y ) = exp ( j Δ φ ) L [ ā ( x + Δ x , y ) ] ,
d ̅ ( x , y ) = exp ( j Δ φ ) c ̅ ( x + Δ x , y ) = c ( x + Δ x , y ) exp { j [ φ c ( x + Δ x , y ) + Δ φ ] } ,
d ( x , y ) = c ( x + Δ x , y ) ,
φ d ( x , y ) = φ c ( x + Δ x , y ) + Δ φ .
H ( u , υ ) = P ( x , y ) | u = λ d i x , υ = λ d i x , = [ 1 , ( x , y ) S , 0 , otherwise , = [ 1 , ( u , υ ) S u υ , 0 , otherwise ,
f c = a 2 λ d i ,
N . A . = a M 2 d i ,
f c = N . A . λ .
I ( x , y ) = c 2 ( x , y ) + d 2 ( x , y ) 2 c ( x , y ) d ( x , y ) cos [ φ c ( x , y ) φ d ( x , y ) ] .
I ( x , y ) = c 2 ( x , y ) + c 2 ( x + Δ x , y ) 2 c ( x , y ) c ( x + Δ x , y ) × cos [ φ c ( x , y ) φ c ( x + Δ x , y ) Δ φ ] .
Δ φ = π / 2 ,
c ( x , y ) c ( x + Δ x , y ) a ,
φ c ( x , y ) φ c ( x + Δ x , y ) Δ x φ c ( x , y ) x ,
φ c ( x , y ) φ c ( x + Δ x , y ) < < π / 4 ,
cos [ φ c ( x , y ) φ c ( x + Δ x , y ) Δ φ ] Δ x φ c ( x , y ) x .
I ( x , y ) c 2 ( x , y ) + c 2 ( x + Δ x , y ) + 2 c ( x , y ) c ( x + Δ x , y ) Δ x φ c ( x , y ) x ,
φ c ( x , y ) 1 Δ x x I ( x , y ) c 2 ( x , y ) c 2 ( x + Δ x , y ) 2 c ( x , y ) c ( x + Δ x , y ) d x .
φ c ( x , y ) 1 Δ x x I ( x , y ) c 2 ( x , y ) c 2 ( x + Δ x , y ) 2 a 2 d x .
φ c ( x , y ) 1 Δ x x [ I ( x , y ) 2 a 2 1 ] d x .
C ̅ ̂ ( u , υ ) = F { a exp [ j φ ̂ c ( x , y ) ] } ,
I ( x , y ) = 2 a 2 [ 1 + Δ x φ c ( x , y ) x ] ,
b ̅ ( x , y ) = a exp ( j Δ φ ) .
B ̅ ( u , υ ) = δ ( u , υ ) exp ( j Δ φ ) ,
D ̅ ( u , υ ) = δ ( u , υ ) exp ( j Δ φ ) = B ̅ ( u , υ ) ,
d ̅ ( x , y ) = b ̅ ( x , y ) = a exp ( j Δ φ ) .
I ( x , y ) = c 2 ( x , y ) + a 2 2 c ( x , y ) a cos [ φ c ( x , y ) Δ φ ] .
cos [ φ c ( x , y ) Δ φ ] = cos [ φ c ( x , y ) + π 2 ] = sin [ φ c ( x , y ) ] φ c ( x , y ) ,
I ( x , y ) 2 a 2 [ 1 + φ c ( x , y ) ] .
Δ x = 0.15 μ m , Δ φ = π / 2 , a = 1.0 , N . A . = 1.25 , λ = 0.525 μ m .
δ = 0.61 λ / N . A .
C ̅ ̂ n ( u , υ )
Ā ̂ k ( u , υ )
ā ̂ k ( x , y )
a ̂ k ( x , y )
φ ̂ a k ( x , y )
c ̅ ̂ n ( x , y )

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