Abstract

X-ray Talbot moiré interferometers can now simultaneously generate two differential phase images of a specimen. The conventional approach to integrating differential phase is unstable and often leads to images with loss of visible detail. We propose a new reconstruction method based on the inverse Riesz transform. The Riesz approach is stable and the final image retains visibility of high resolution detail without directional bias. The outline Riesz theory is developed and an experimentally acquired X-ray differential phase data set is presented for qualitative visual appraisal. The inverse Riesz phase image is compared with two alternatives: the integrated (quantitative) phase and the modulus of the gradient of the phase. The inverse Riesz transform has the computational advantages of a unitary linear operator, and is implemented directly as a complex multiplication in the Fourier domain also known as the spiral phase transform.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc.214(1), 7–12 (2004).
    [CrossRef] [PubMed]
  2. M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, “Using the Hilbert transform for 3D visualization of differential interference contrast microscope images,” J. Microsc.199(1), 79–84 (2000).
    [CrossRef] [PubMed]
  3. H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase-contrast imaging using Fourier transform phase retrieval,” Opt. Express19(4), 3339–3346 (2011).
    [CrossRef] [PubMed]
  4. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 2, 42, 866–868 (2003).
  5. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys.2(4), 258–261 (2006).
    [CrossRef]
  6. D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping (John Wiley and Sons, New York, 1998).
  7. R. N. Bracewell, The Fourier Transform and its Applications, McGraw-Hill Electrical and Electronic Engineering Series (McGraw Hill, New York, 1978).
  8. K. R. Freischlad and C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A3(11), 1852–1861 (1986).
    [CrossRef]
  9. R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell.10(4), 439–451 (1988).
    [CrossRef]
  10. D. C. Ghiglia and L. A. Romero, “Robust Two-Dimensional Weighted and Unweighted Phase Unwrapping That Uses Fast Transforms and Iterative Methods,” J. Opt. Soc. Am. A11(1), 107–117 (1994).
    [CrossRef]
  11. S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett.30(3), 245–247 (2005).
    [CrossRef] [PubMed]
  12. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton mathematical series (Princeton University Press, Princeton, N.J., 1970).
  13. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A18(8), 1862–1870 (2001).
    [CrossRef] [PubMed]
  14. M. Felsberg and G. Sommer, “The Monogenic Signal,” IEEE Trans. Signal Process.49(12), 3136–3144 (2001).
    [CrossRef]
  15. R. S. Strichartz, “Radon inversion - variations on a theme,” Am. Math. Mon.89(6), 377–384 (1982).
    [CrossRef]
  16. K. T. Smith and F. Keinert, “Mathematical foundations of computed tomography,” Appl. Opt.24(23), 3950–3957 (1985).
    [CrossRef] [PubMed]
  17. A. Faridani, E. L. Ritman, and K. T. Smith, “Local tomography,” SIAM J. Appl. Math.52(2), 459–484 (1992).
    [CrossRef]
  18. M. Reisert and H. Burkhardt, “Complex derivative filters,” IEEE Trans. Image Process.17(12), 2265–2274 (2008).
    [CrossRef] [PubMed]
  19. M. Unser and N. Chenouard, “A Unifying Parametric Framework for 2D Steerable Wavelet Transforms,” SIAM J. Imaging Sciences6(1), 102–135 (2013).
    [CrossRef]
  20. K. G. Larkin and P. A. Fletcher, “A coherent framework for fingerprint analysis: are fingerprints Holograms?” Opt. Express15(14), 8667–8677 (2007).
    [CrossRef] [PubMed]
  21. K. Nagai, H. Itoh, G. Sato, T. Nakamura, K. Yamaguchi, T. Kondoh, and S. H. T. Den, “New phase retrieval method for single-shot x-ray Talbot imaging using windowed Fourier transform,” Optical Modeling and Performance Predictions V, Proc. SPIE 8127, San Diego, California, August 21, 2011.
  22. D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt.25(10), 1653–1660 (1986).
    [CrossRef] [PubMed]
  23. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am.72(1), 156–160 (1982).
    [CrossRef]
  24. M. W. Levine and J. M. Shefner, Fundamentals of Sensation and Perception (Pacific Grove, CA: Brooks/Cole, 1991) p. 675.

2013 (1)

M. Unser and N. Chenouard, “A Unifying Parametric Framework for 2D Steerable Wavelet Transforms,” SIAM J. Imaging Sciences6(1), 102–135 (2013).
[CrossRef]

2011 (1)

2008 (1)

M. Reisert and H. Burkhardt, “Complex derivative filters,” IEEE Trans. Image Process.17(12), 2265–2274 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys.2(4), 258–261 (2006).
[CrossRef]

2005 (1)

2004 (1)

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc.214(1), 7–12 (2004).
[CrossRef] [PubMed]

2003 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 2, 42, 866–868 (2003).

2001 (2)

2000 (1)

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, “Using the Hilbert transform for 3D visualization of differential interference contrast microscope images,” J. Microsc.199(1), 79–84 (2000).
[CrossRef] [PubMed]

1994 (1)

1992 (1)

A. Faridani, E. L. Ritman, and K. T. Smith, “Local tomography,” SIAM J. Appl. Math.52(2), 459–484 (1992).
[CrossRef]

1988 (1)

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell.10(4), 439–451 (1988).
[CrossRef]

1986 (2)

1985 (1)

1982 (2)

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc.214(1), 7–12 (2004).
[CrossRef] [PubMed]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, “Using the Hilbert transform for 3D visualization of differential interference contrast microscope images,” J. Microsc.199(1), 79–84 (2000).
[CrossRef] [PubMed]

Bachor, H.-A.

Bone, D. J.

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys.2(4), 258–261 (2006).
[CrossRef]

Burkhardt, H.

M. Reisert and H. Burkhardt, “Complex derivative filters,” IEEE Trans. Image Process.17(12), 2265–2274 (2008).
[CrossRef] [PubMed]

Chellappa, R.

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell.10(4), 439–451 (1988).
[CrossRef]

Chenouard, N.

M. Unser and N. Chenouard, “A Unifying Parametric Framework for 2D Steerable Wavelet Transforms,” SIAM J. Imaging Sciences6(1), 102–135 (2013).
[CrossRef]

Cogswell, C. J.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc.214(1), 7–12 (2004).
[CrossRef] [PubMed]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, “Using the Hilbert transform for 3D visualization of differential interference contrast microscope images,” J. Microsc.199(1), 79–84 (2000).
[CrossRef] [PubMed]

Cohen, M.

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys.2(4), 258–261 (2006).
[CrossRef]

Den, T.

Faridani, A.

A. Faridani, E. L. Ritman, and K. T. Smith, “Local tomography,” SIAM J. Appl. Math.52(2), 459–484 (1992).
[CrossRef]

Fekete, P. W.

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, “Using the Hilbert transform for 3D visualization of differential interference contrast microscope images,” J. Microsc.199(1), 79–84 (2000).
[CrossRef] [PubMed]

Felsberg, M.

M. Felsberg and G. Sommer, “The Monogenic Signal,” IEEE Trans. Signal Process.49(12), 3136–3144 (2001).
[CrossRef]

Fletcher, P. A.

Frankot, R. T.

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell.10(4), 439–451 (1988).
[CrossRef]

Freischlad, K. R.

Ghiglia, D. C.

Guérineau, N.

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 2, 42, 866–868 (2003).

Ina, H.

Itoh, H.

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 2, 42, 866–868 (2003).

Keinert, F.

Kobayashi, S.

Koliopoulos, C. L.

Kondoh, T.

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 2, 42, 866–868 (2003).

Larkin, K. G.

K. G. Larkin and P. A. Fletcher, “A coherent framework for fingerprint analysis: are fingerprints Holograms?” Opt. Express15(14), 8667–8677 (2007).
[CrossRef] [PubMed]

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc.214(1), 7–12 (2004).
[CrossRef] [PubMed]

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A18(8), 1862–1870 (2001).
[CrossRef] [PubMed]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, “Using the Hilbert transform for 3D visualization of differential interference contrast microscope images,” J. Microsc.199(1), 79–84 (2000).
[CrossRef] [PubMed]

Momose, A.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 2, 42, 866–868 (2003).

Nagai, K.

Nakamura, T.

Oldfield, M. A.

Ouchi, C.

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys.2(4), 258–261 (2006).
[CrossRef]

Primot, J.

Reisert, M.

M. Reisert and H. Burkhardt, “Complex derivative filters,” IEEE Trans. Image Process.17(12), 2265–2274 (2008).
[CrossRef] [PubMed]

Ritman, E. L.

A. Faridani, E. L. Ritman, and K. T. Smith, “Local tomography,” SIAM J. Appl. Math.52(2), 459–484 (1992).
[CrossRef]

Romero, L. A.

Sandeman, R. J.

Sato, G.

Setomoto, Y.

Sheppard, C. J. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc.214(1), 7–12 (2004).
[CrossRef] [PubMed]

Smith, K. T.

A. Faridani, E. L. Ritman, and K. T. Smith, “Local tomography,” SIAM J. Appl. Math.52(2), 459–484 (1992).
[CrossRef]

K. T. Smith and F. Keinert, “Mathematical foundations of computed tomography,” Appl. Opt.24(23), 3950–3957 (1985).
[CrossRef] [PubMed]

Smith, N. I.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc.214(1), 7–12 (2004).
[CrossRef] [PubMed]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, “Using the Hilbert transform for 3D visualization of differential interference contrast microscope images,” J. Microsc.199(1), 79–84 (2000).
[CrossRef] [PubMed]

Sommer, G.

M. Felsberg and G. Sommer, “The Monogenic Signal,” IEEE Trans. Signal Process.49(12), 3136–3144 (2001).
[CrossRef]

Strichartz, R. S.

R. S. Strichartz, “Radon inversion - variations on a theme,” Am. Math. Mon.89(6), 377–384 (1982).
[CrossRef]

Suzuki, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 2, 42, 866–868 (2003).

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 2, 42, 866–868 (2003).

Takeda, M.

Teshima, T.

Unser, M.

M. Unser and N. Chenouard, “A Unifying Parametric Framework for 2D Steerable Wavelet Transforms,” SIAM J. Imaging Sciences6(1), 102–135 (2013).
[CrossRef]

Velghe, S.

Wattellier, B.

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys.2(4), 258–261 (2006).
[CrossRef]

Yamaguchi, K.

Am. Math. Mon. (1)

R. S. Strichartz, “Radon inversion - variations on a theme,” Am. Math. Mon.89(6), 377–384 (1982).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Image Process. (1)

M. Reisert and H. Burkhardt, “Complex derivative filters,” IEEE Trans. Image Process.17(12), 2265–2274 (2008).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell.10(4), 439–451 (1988).
[CrossRef]

IEEE Trans. Signal Process. (1)

M. Felsberg and G. Sommer, “The Monogenic Signal,” IEEE Trans. Signal Process.49(12), 3136–3144 (2001).
[CrossRef]

J. Microsc. (2)

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc.214(1), 7–12 (2004).
[CrossRef] [PubMed]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, “Using the Hilbert transform for 3D visualization of differential interference contrast microscope images,” J. Microsc.199(1), 79–84 (2000).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys., Part 2 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 2, 42, 866–868 (2003).

Nat. Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys.2(4), 258–261 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

SIAM J. Appl. Math. (1)

A. Faridani, E. L. Ritman, and K. T. Smith, “Local tomography,” SIAM J. Appl. Math.52(2), 459–484 (1992).
[CrossRef]

SIAM J. Imaging Sciences (1)

M. Unser and N. Chenouard, “A Unifying Parametric Framework for 2D Steerable Wavelet Transforms,” SIAM J. Imaging Sciences6(1), 102–135 (2013).
[CrossRef]

Other (5)

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton mathematical series (Princeton University Press, Princeton, N.J., 1970).

D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping (John Wiley and Sons, New York, 1998).

R. N. Bracewell, The Fourier Transform and its Applications, McGraw-Hill Electrical and Electronic Engineering Series (McGraw Hill, New York, 1978).

K. Nagai, H. Itoh, G. Sato, T. Nakamura, K. Yamaguchi, T. Kondoh, and S. H. T. Den, “New phase retrieval method for single-shot x-ray Talbot imaging using windowed Fourier transform,” Optical Modeling and Performance Predictions V, Proc. SPIE 8127, San Diego, California, August 21, 2011.

M. W. Levine and J. M. Shefner, Fundamentals of Sensation and Perception (Pacific Grove, CA: Brooks/Cole, 1991) p. 675.

Supplementary Material (7)

» Media 1: JPG (736 KB)     
» Media 2: JPG (593 KB)     
» Media 3: JPG (578 KB)     
» Media 4: JPG (191 KB)     
» Media 5: JPG (706 KB)     
» Media 6: JPG (706 KB)     
» Media 7: JPG (588 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

(a) Simulation of a deeply phase modulation moiré pattern f(x,y), and (b) its Fourier transform magnitude |F(u,v)| clearly showing the DC lobe and eight side-lobes.

Fig. 2
Fig. 2

Flowchart for inverse Riesz transformation of gradient components

Fig. 3
Fig. 3

The test specimen is a plastic chess piece (a knight).

Fig. 4
Fig. 4

One of sixteen phase-shifted moiré interferograms, f(x,y), from an experimental X-ray Talbot grating interferometer . The specimen is a plastic chess piece (a knight). The image is 1024 x 1024 pixels, pixel size 48μm.

Fig. 5
Fig. 5

(a) Fourier transform magnitude |F(u,v)|, (log-scale). (b) Absorption image b00 (x,y) (Media 1)

Fig. 6
Fig. 6

(a) X-differential phase image ψx(x,y) (Media 2), (b) Y-differential phase image ψy(x,y) (Media 3)

Fig. 7
Fig. 7

(a) Integrated phase image ψ(x,y) (Media 4), (b) negated modulus of the vector gradient image -|gradψ(x,y)| (Media 5), (c) modulus of the vector gradient image |gradψ(x,y)| (Media 6), (d) inverse Riesz transform image Λψ(x,y) (Media 7).

Fig. 8
Fig. 8

The inverse Riesz discrepancy image. Mid-gray represents zero and dominates the image. Note that large signal variations are concentrated at the edges of the specimen.

Tables (1)

Tables Icon

Table 1 Comparative properties of imaging modes*

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

( x,y ),( r,θ ),x=rcosθ,y=rsinθ ( u,v ),( q,ϕ ),u=qcosϕ,v=qsinϕ }
f( x,y )= n=1 +1 m=1 +1 b mn ( x,y ) exp[ 2πi( m u 0 +n v 0 )+i[ m ψ x ( x,y )+n ψ y ( x,y ) ] ]
F( u,v )= + + f( x,y ) exp[ 2πi( ux+vy ) ]dxdy f( x,y )= + + F( u,v ) exp[ +2πi( ux+vy ) ]dudv }
F( u,v )= n=1 +1 m=1 +1 B mn ( u,v ) P m x ( um u 0 ,v ) P n y ( u,vn v 0 )
P m x ( um u 0 ,v ) FT exp[ 2πim u 0 +im ψ x ( x,y ) ] P n y ( u,vn v 0 ) FT exp[ 2πin v 0 +in ψ y ( x,y ) ] }
B 10 ( u,v ) P 1 x ( u u 0 ,v ) FT b 10 ( x,y ).exp[ 2πi u 0 +i ψ x ( x,y ) ] B 01 ( u,v ) P 1 y ( u,v v 0 ) FT b 01 ( x,y ).exp[ 2πi v 0 +i ψ y ( x,y ) ] }
B 10 ( u,v ) P 1 x ( u,v ) FT b 10 ( x,y ).exp[ i ψ x ( x,y ) ] B 01 ( u,v ) P 1 y ( u,v ) FT b 01 ( x,y ).exp[ i ψ y ( x,y ) ] }
ψ x ( x,y )=Arg{ b 10 ( x,y ).exp[ i ψ x ( x,y ) ] }+2kπ ψ y ( x,y )=Arg{ b 01 ( x,y ).exp[ i ψ y ( x,y ) ] }+2lπ }
ψ=i ψ x +j ψ y
ψ( x,y ) FT Ψ( u,v )ψ FT 2πi[ ui+vj ]Ψ( u,v )
x f( x,y )= + + F( u,v ) 2πiuexp[ +2πi( ux+vy ) ]dudv y f( x,y )= + + F( u,v ) 2πivexp[ +2πi( ux+vy ) ]dudv }
Δ= 2 x 2 + 2 y 2 Δf( x,y )= + + F( u,v ) [ ( 2πiu ) 2 + ( 2πiv ) 2 ]exp[ +2πi( ux+vy ) ]du }
Δf( x,y ) FT ( 2π ) 2 ( u 2 + v 2 )F( u,v )= ( 2πq ) 2 F( u,v )
( Δ ) 1 2 f( x,y )=Λf( x,y ) FT 2πqF( u,v )
ψ=ΛRψ FT 2πi u 2 + v 2 [ ui+vj ] u 2 + v 2 Ψ( u,v )
Rψ= ( Δ ) 1 2 ψ FT i [ ui+vj ] u 2 + v 2 Ψ( u,v )=i( icosϕ+jsinϕ )Ψ( u,v )
R{ ψ } FT ( u+iv ) | u+iv | Ψ=exp( iϕ ).Ψ
D{ ψ }=( x +i y )ψ FT 2πi( u+iv ).Ψ=2πiqexp( iϕ ).Ψ
R 1 { ψ } FT exp( iϕ ).Ψ
D{ ψ }=2πiRΛ{ ψ } 1 2πi R 1 D{ ψ }=Λ{ ψ } }

Metrics