Abstract

We propose three amplitude filters for visualization of phase objects. They interact with the spectra of pure-phase objects in the frequency plane and are based on tangent and error functions as well as antisymmetric combination of square roots. The error function is a normalized form of the Gaussian function. The antisymmetric square-root filter is composed of two square-root filters to widen its spatial frequency spectral range. Their advantage over other known amplitude frequency-domain filters, such as linear or square-root graded ones, is that they allow high-contrast visualization of objects with large variations of phase gradients.

© 2009 Optical Society of America

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References

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  1. R. A. Sprague and B. J. Thompson, “Quantitative visualization of large variation phase objects,” Appl. Opt. 11, 1469-1479 (1972).
    [CrossRef] [PubMed]
  2. G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thomson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, 1989).
    [CrossRef]
  3. M. Pluta, Advanced Light Microscopy Vol. 2--Specialized Methods (Polish Scientific Publishers, 1989).
  4. G. S. Settles, Schlieren and Shadowgraph Techniques (Springer-Verlag, 2001).
    [CrossRef]
  5. F. Zernike, “How I discovered phase contrast,” Science 121, 345-349 (1955).
    [CrossRef] [PubMed]
  6. L. Joannes, F. Dubois, and J. Legros, “Phase-shifting schlieren: high-resolution quantitative schlieren that uses the phase-shifting technique principle,” Appl. Opt. 42, 5046-5053(2003).
    [CrossRef] [PubMed]
  7. B. Zakharin and J. Stricker, “Schlieren systems with coherent illumination for quantitative measurements,” Appl. Opt. 43, 4786-4795 (2004).
    [CrossRef] [PubMed]
  8. T.-C. Poon and K. B. Doh, “On the theory of optical Hilbert transform for incoherent objects,” Opt. Express 15, 3006-3011(2007).
    [CrossRef] [PubMed]
  9. R. Hoffman and L. Gross, “Modulation contrast microscope,” Appl. Opt. 14, 1169-1176 (1975).
    [CrossRef] [PubMed]
  10. B. A. Horwitz, “Phase image differentiation with linear intensity output,” Appl. Opt. 17, 181-186 (1978).
    [CrossRef] [PubMed]
  11. H. Kasprzak, “Differentiation of a noninteger order and its optical implementation,” Appl. Opt. 21, 3287-3291(1982).
    [CrossRef] [PubMed]
  12. J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, and M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461-7464(1997).
    [CrossRef]
  13. E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, and M. Fernandez, “Phase-object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481-490(1997).
    [CrossRef]
  14. T. Szoplik, V. Climent, E. Tajahuerce, J. Lancis, and M. Fernández-Alonso, “Phase-change visualization in two-dimensional phase objects with a semiderivative real filter,” Appl. Opt. 37, 5472-5478 (1998).
    [CrossRef]
  15. A. Sagan, S. Nowicki, R. Buczynski, M. Kowalczyk, and T. Szoplik, “Imaging phase objects with square-root, Foucault, and Hoffman real filters: a comparison,” Appl. Opt. 42, 5816-5824 (2003).
    [CrossRef] [PubMed]
  16. B. Rosa, A. Sagan, K. E. Haman, and T. Szoplik, “Visualization of small scale density fluctuations in the atmosphere using the semiderivative real filter,” Proc. SPIE 5237, 228-237(2004).
    [CrossRef]
  17. H. Furuhashi, K. Matsuda, and C. P. Grover, “Visualization of phase objects by use of a differentiation Filter,” Appl. Opt. 42, 218-226 (2003).
    [CrossRef] [PubMed]
  18. H. Furuhashi, R. Sugiyama, Y. Uchida, K. Matsuda, and C. P. Grover, “Optical differentiation phase measurement using the bias shifting method,” Opt. Rev. 12, 109-114(2005).
    [CrossRef]
  19. A. Sagan, M. Kowalczyk, and T. Szoplik, “Optimized visualization of phase objects with semiderivative real filters,” Proc. SPIE 5182, 103-111 (2004).
    [CrossRef]
  20. R. Kasztelanic, W. Grabowski, A. Sagan, J. Liu, R. Buczyński, A. Waddie, and M. Taghizadeh, “Semi-derivative real filter for quality measurement of microlenses array,” Proc. SPIE 6189, 618916 (2006).
    [CrossRef]
  21. F. Henault, “Wavefront sensor based on varying transmission filters: theory and expected performance,” J. Mod. Opt. 52, 1917-1931 (2005).
    [CrossRef]
  22. J. A. Davis, D. A. Smith, D. E. McNamara, D. M. Cottrell, and J. Campos, “Fractional derivatives-analysis and experimental implementation,” Appl. Opt. 40, 5943-5948(2001).
    [CrossRef]

2007

2006

R. Kasztelanic, W. Grabowski, A. Sagan, J. Liu, R. Buczyński, A. Waddie, and M. Taghizadeh, “Semi-derivative real filter for quality measurement of microlenses array,” Proc. SPIE 6189, 618916 (2006).
[CrossRef]

2005

F. Henault, “Wavefront sensor based on varying transmission filters: theory and expected performance,” J. Mod. Opt. 52, 1917-1931 (2005).
[CrossRef]

H. Furuhashi, R. Sugiyama, Y. Uchida, K. Matsuda, and C. P. Grover, “Optical differentiation phase measurement using the bias shifting method,” Opt. Rev. 12, 109-114(2005).
[CrossRef]

2004

A. Sagan, M. Kowalczyk, and T. Szoplik, “Optimized visualization of phase objects with semiderivative real filters,” Proc. SPIE 5182, 103-111 (2004).
[CrossRef]

B. Rosa, A. Sagan, K. E. Haman, and T. Szoplik, “Visualization of small scale density fluctuations in the atmosphere using the semiderivative real filter,” Proc. SPIE 5237, 228-237(2004).
[CrossRef]

B. Zakharin and J. Stricker, “Schlieren systems with coherent illumination for quantitative measurements,” Appl. Opt. 43, 4786-4795 (2004).
[CrossRef] [PubMed]

2003

2001

1998

1997

J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, and M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461-7464(1997).
[CrossRef]

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, and M. Fernandez, “Phase-object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481-490(1997).
[CrossRef]

1982

1978

1975

1972

1955

F. Zernike, “How I discovered phase contrast,” Science 121, 345-349 (1955).
[CrossRef] [PubMed]

Buczynski, R.

R. Kasztelanic, W. Grabowski, A. Sagan, J. Liu, R. Buczyński, A. Waddie, and M. Taghizadeh, “Semi-derivative real filter for quality measurement of microlenses array,” Proc. SPIE 6189, 618916 (2006).
[CrossRef]

A. Sagan, S. Nowicki, R. Buczynski, M. Kowalczyk, and T. Szoplik, “Imaging phase objects with square-root, Foucault, and Hoffman real filters: a comparison,” Appl. Opt. 42, 5816-5824 (2003).
[CrossRef] [PubMed]

Campos, J.

Climent, V.

Cottrell, D. M.

Davis, J. A.

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thomson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, 1989).
[CrossRef]

Doh, K. B.

Dubois, F.

Fernandez, M.

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, and M. Fernandez, “Phase-object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481-490(1997).
[CrossRef]

Fernández-Alonso, M.

Furuhashi, H.

H. Furuhashi, R. Sugiyama, Y. Uchida, K. Matsuda, and C. P. Grover, “Optical differentiation phase measurement using the bias shifting method,” Opt. Rev. 12, 109-114(2005).
[CrossRef]

H. Furuhashi, K. Matsuda, and C. P. Grover, “Visualization of phase objects by use of a differentiation Filter,” Appl. Opt. 42, 218-226 (2003).
[CrossRef] [PubMed]

Grabowski, W.

R. Kasztelanic, W. Grabowski, A. Sagan, J. Liu, R. Buczyński, A. Waddie, and M. Taghizadeh, “Semi-derivative real filter for quality measurement of microlenses array,” Proc. SPIE 6189, 618916 (2006).
[CrossRef]

Gross, L.

Grover, C. P.

H. Furuhashi, R. Sugiyama, Y. Uchida, K. Matsuda, and C. P. Grover, “Optical differentiation phase measurement using the bias shifting method,” Opt. Rev. 12, 109-114(2005).
[CrossRef]

H. Furuhashi, K. Matsuda, and C. P. Grover, “Visualization of phase objects by use of a differentiation Filter,” Appl. Opt. 42, 218-226 (2003).
[CrossRef] [PubMed]

Haman, K. E.

B. Rosa, A. Sagan, K. E. Haman, and T. Szoplik, “Visualization of small scale density fluctuations in the atmosphere using the semiderivative real filter,” Proc. SPIE 5237, 228-237(2004).
[CrossRef]

Henault, F.

F. Henault, “Wavefront sensor based on varying transmission filters: theory and expected performance,” J. Mod. Opt. 52, 1917-1931 (2005).
[CrossRef]

Hoffman, R.

Horwitz, B. A.

Joannes, L.

Kasprzak, H.

Kasztelanic, R.

R. Kasztelanic, W. Grabowski, A. Sagan, J. Liu, R. Buczyński, A. Waddie, and M. Taghizadeh, “Semi-derivative real filter for quality measurement of microlenses array,” Proc. SPIE 6189, 618916 (2006).
[CrossRef]

Kowalczyk, M.

A. Sagan, M. Kowalczyk, and T. Szoplik, “Optimized visualization of phase objects with semiderivative real filters,” Proc. SPIE 5182, 103-111 (2004).
[CrossRef]

A. Sagan, S. Nowicki, R. Buczynski, M. Kowalczyk, and T. Szoplik, “Imaging phase objects with square-root, Foucault, and Hoffman real filters: a comparison,” Appl. Opt. 42, 5816-5824 (2003).
[CrossRef] [PubMed]

Lancis, J.

Legros, J.

Liu, J.

R. Kasztelanic, W. Grabowski, A. Sagan, J. Liu, R. Buczyński, A. Waddie, and M. Taghizadeh, “Semi-derivative real filter for quality measurement of microlenses array,” Proc. SPIE 6189, 618916 (2006).
[CrossRef]

Matsuda, K.

H. Furuhashi, R. Sugiyama, Y. Uchida, K. Matsuda, and C. P. Grover, “Optical differentiation phase measurement using the bias shifting method,” Opt. Rev. 12, 109-114(2005).
[CrossRef]

H. Furuhashi, K. Matsuda, and C. P. Grover, “Visualization of phase objects by use of a differentiation Filter,” Appl. Opt. 42, 218-226 (2003).
[CrossRef] [PubMed]

McNamara, D. E.

Nowicki, S.

Parrent, G. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thomson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, 1989).
[CrossRef]

Pluta, M.

M. Pluta, Advanced Light Microscopy Vol. 2--Specialized Methods (Polish Scientific Publishers, 1989).

Poon, T.-C.

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thomson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, 1989).
[CrossRef]

Rosa, B.

B. Rosa, A. Sagan, K. E. Haman, and T. Szoplik, “Visualization of small scale density fluctuations in the atmosphere using the semiderivative real filter,” Proc. SPIE 5237, 228-237(2004).
[CrossRef]

Sagan, A.

R. Kasztelanic, W. Grabowski, A. Sagan, J. Liu, R. Buczyński, A. Waddie, and M. Taghizadeh, “Semi-derivative real filter for quality measurement of microlenses array,” Proc. SPIE 6189, 618916 (2006).
[CrossRef]

A. Sagan, M. Kowalczyk, and T. Szoplik, “Optimized visualization of phase objects with semiderivative real filters,” Proc. SPIE 5182, 103-111 (2004).
[CrossRef]

B. Rosa, A. Sagan, K. E. Haman, and T. Szoplik, “Visualization of small scale density fluctuations in the atmosphere using the semiderivative real filter,” Proc. SPIE 5237, 228-237(2004).
[CrossRef]

A. Sagan, S. Nowicki, R. Buczynski, M. Kowalczyk, and T. Szoplik, “Imaging phase objects with square-root, Foucault, and Hoffman real filters: a comparison,” Appl. Opt. 42, 5816-5824 (2003).
[CrossRef] [PubMed]

Settles, G. S.

G. S. Settles, Schlieren and Shadowgraph Techniques (Springer-Verlag, 2001).
[CrossRef]

Smith, D. A.

Sprague, R. A.

Stricker, J.

Sugiyama, R.

H. Furuhashi, R. Sugiyama, Y. Uchida, K. Matsuda, and C. P. Grover, “Optical differentiation phase measurement using the bias shifting method,” Opt. Rev. 12, 109-114(2005).
[CrossRef]

Szoplik, T.

B. Rosa, A. Sagan, K. E. Haman, and T. Szoplik, “Visualization of small scale density fluctuations in the atmosphere using the semiderivative real filter,” Proc. SPIE 5237, 228-237(2004).
[CrossRef]

A. Sagan, M. Kowalczyk, and T. Szoplik, “Optimized visualization of phase objects with semiderivative real filters,” Proc. SPIE 5182, 103-111 (2004).
[CrossRef]

A. Sagan, S. Nowicki, R. Buczynski, M. Kowalczyk, and T. Szoplik, “Imaging phase objects with square-root, Foucault, and Hoffman real filters: a comparison,” Appl. Opt. 42, 5816-5824 (2003).
[CrossRef] [PubMed]

T. Szoplik, V. Climent, E. Tajahuerce, J. Lancis, and M. Fernández-Alonso, “Phase-change visualization in two-dimensional phase objects with a semiderivative real filter,” Appl. Opt. 37, 5472-5478 (1998).
[CrossRef]

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, and M. Fernandez, “Phase-object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481-490(1997).
[CrossRef]

J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, and M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461-7464(1997).
[CrossRef]

Taghizadeh, M.

R. Kasztelanic, W. Grabowski, A. Sagan, J. Liu, R. Buczyński, A. Waddie, and M. Taghizadeh, “Semi-derivative real filter for quality measurement of microlenses array,” Proc. SPIE 6189, 618916 (2006).
[CrossRef]

Tajahuerce, E.

Thompson, B. J.

Thomson, B. J.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thomson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, 1989).
[CrossRef]

Uchida, Y.

H. Furuhashi, R. Sugiyama, Y. Uchida, K. Matsuda, and C. P. Grover, “Optical differentiation phase measurement using the bias shifting method,” Opt. Rev. 12, 109-114(2005).
[CrossRef]

Waddie, A.

R. Kasztelanic, W. Grabowski, A. Sagan, J. Liu, R. Buczyński, A. Waddie, and M. Taghizadeh, “Semi-derivative real filter for quality measurement of microlenses array,” Proc. SPIE 6189, 618916 (2006).
[CrossRef]

Zakharin, B.

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121, 345-349 (1955).
[CrossRef] [PubMed]

Appl. Opt.

R. A. Sprague and B. J. Thompson, “Quantitative visualization of large variation phase objects,” Appl. Opt. 11, 1469-1479 (1972).
[CrossRef] [PubMed]

R. Hoffman and L. Gross, “Modulation contrast microscope,” Appl. Opt. 14, 1169-1176 (1975).
[CrossRef] [PubMed]

B. A. Horwitz, “Phase image differentiation with linear intensity output,” Appl. Opt. 17, 181-186 (1978).
[CrossRef] [PubMed]

H. Kasprzak, “Differentiation of a noninteger order and its optical implementation,” Appl. Opt. 21, 3287-3291(1982).
[CrossRef] [PubMed]

J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, and M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461-7464(1997).
[CrossRef]

L. Joannes, F. Dubois, and J. Legros, “Phase-shifting schlieren: high-resolution quantitative schlieren that uses the phase-shifting technique principle,” Appl. Opt. 42, 5046-5053(2003).
[CrossRef] [PubMed]

B. Zakharin and J. Stricker, “Schlieren systems with coherent illumination for quantitative measurements,” Appl. Opt. 43, 4786-4795 (2004).
[CrossRef] [PubMed]

T. Szoplik, V. Climent, E. Tajahuerce, J. Lancis, and M. Fernández-Alonso, “Phase-change visualization in two-dimensional phase objects with a semiderivative real filter,” Appl. Opt. 37, 5472-5478 (1998).
[CrossRef]

A. Sagan, S. Nowicki, R. Buczynski, M. Kowalczyk, and T. Szoplik, “Imaging phase objects with square-root, Foucault, and Hoffman real filters: a comparison,” Appl. Opt. 42, 5816-5824 (2003).
[CrossRef] [PubMed]

H. Furuhashi, K. Matsuda, and C. P. Grover, “Visualization of phase objects by use of a differentiation Filter,” Appl. Opt. 42, 218-226 (2003).
[CrossRef] [PubMed]

J. A. Davis, D. A. Smith, D. E. McNamara, D. M. Cottrell, and J. Campos, “Fractional derivatives-analysis and experimental implementation,” Appl. Opt. 40, 5943-5948(2001).
[CrossRef]

J. Mod. Opt.

F. Henault, “Wavefront sensor based on varying transmission filters: theory and expected performance,” J. Mod. Opt. 52, 1917-1931 (2005).
[CrossRef]

Opt. Express

Opt. Rev.

H. Furuhashi, R. Sugiyama, Y. Uchida, K. Matsuda, and C. P. Grover, “Optical differentiation phase measurement using the bias shifting method,” Opt. Rev. 12, 109-114(2005).
[CrossRef]

Proc. SPIE

A. Sagan, M. Kowalczyk, and T. Szoplik, “Optimized visualization of phase objects with semiderivative real filters,” Proc. SPIE 5182, 103-111 (2004).
[CrossRef]

R. Kasztelanic, W. Grabowski, A. Sagan, J. Liu, R. Buczyński, A. Waddie, and M. Taghizadeh, “Semi-derivative real filter for quality measurement of microlenses array,” Proc. SPIE 6189, 618916 (2006).
[CrossRef]

B. Rosa, A. Sagan, K. E. Haman, and T. Szoplik, “Visualization of small scale density fluctuations in the atmosphere using the semiderivative real filter,” Proc. SPIE 5237, 228-237(2004).
[CrossRef]

Pure Appl. Opt.

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, and M. Fernandez, “Phase-object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481-490(1997).
[CrossRef]

Science

F. Zernike, “How I discovered phase contrast,” Science 121, 345-349 (1955).
[CrossRef] [PubMed]

Other

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thomson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, 1989).
[CrossRef]

M. Pluta, Advanced Light Microscopy Vol. 2--Specialized Methods (Polish Scientific Publishers, 1989).

G. S. Settles, Schlieren and Shadowgraph Techniques (Springer-Verlag, 2001).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Profile of the phase function ϕ ( ξ , η ) of the test conical lens.

Fig. 2
Fig. 2

Intensity distributions and profiles for selected y values and widths s of a conical lens with a maximum phase shift of 0.5 π and 12.5 π visualized by a linear filter of width (a), (b)  w = 2.56 mm ; (c), (d)  w = 0.64 mm ; and (e), (f)  w = 0.16 mm .

Fig. 3
Fig. 3

Smoothed contrast plots for a linear filter of width w = 2.56 mm for objects of (a)  ϕ max = 0.5 π and (b)  ϕ max = 12.5 π .

Fig. 4
Fig. 4

Plot of K N / ( N K 1 ) normalized contrast ratios obtained for a linear filter of width 2.56 mm . K 1 and K N are contrast functions calculated for a thin and a thick phase object of ϕ max = 0.5 π and ϕ max = 12.5 π , respectively.

Fig. 5
Fig. 5

Examples of central parts of amplitude transmittance profiles of (a) tan filter with m = 16 , (b) erf filter with w = 320 μm , (c) antisymmetric sqrt filter with w = 1.28 mm . All filter windows have the same size of 5.12 mm .

Fig. 6
Fig. 6

Intensity distributions and profiles for selected y coordinate values, where the width of the lenses is different. The lenses used in simulations have maximum phase shifts equal to 0.5 π and 12.5 π . Tangent filter parameters are (a), (b)  m = 4 ; (c), (d)  m = 16 ; and (e), (f)  m = 64 .

Fig. 7
Fig. 7

Plots of K N / ( N K 1 ) normalized contrast ratios calculated for the tangent filter with parameters (a)  m = 4 and (b)  m = 16 for phase objects of thickness ϕ max = 0.5 π ( K 1 ) and ϕ max = 12.5 π ( K N ).

Fig. 8
Fig. 8

Intensity distributions and profiles for selected y coordinate values, where the width of the lenses is different. The lenses used in simulations have maximum phase shifts equal to 0.5 π and 12.5 π . Error function filter parameters are (a), (b)  w = 1.28 mm ; (c), (d)  w = 0.32 mm ; and (e), (f)  w = 0.08 mm .

Fig. 9
Fig. 9

Plots of K N / ( N K 1 ) normalized contrast ratios calculated for the error function filter with parameters (a)  w = 1.28 mm , (b)  w = 0.32 mm for phase objects of thickness ϕ max = 0.5 π ( K 1 ) and ϕ max = 12.5 π ( K N ).

Fig. 10
Fig. 10

Intensity distributions and profiles for selected y coordinate values. The phase objects have maximum phase shifts equal to 0.5 π and 12.5 π . Antisymmetric square-root filter widths are (a), (b)  w = 5.12 mm ; (c), (d)  w = 2.56 mm ; and (e), (f)  w = 1.28 mm .

Fig. 11
Fig. 11

Plots of K N / ( N K 1 ) normalized contrast ratios calculated for the antisymmetric square-root filter with parameters (a)  w = 5.12 mm , (b)  w = 1.28 mm for phase objects of thickness ϕ max = 0.5 π ( K 1 ) and ϕ max = 12.5 π ( K N ).

Equations (13)

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t ( ξ , η ) = exp [ i ϕ ( ξ , η ) ] ,
ϕ ( ξ , η ) = { ϕ max [ 1 ( 2 ξ s ( η ) ) 2 ] for     | ξ | s ( η ) 2 , | η | η 0 2 0 otherwise ,
s ( η ) = η η 0 ( ξ 0 2 ξ 1 2 ) + 1 2 ( ξ 0 2 + ξ 1 2 ) ,
ϕ ( ξ , η ) ξ = { ϕ max 8 ξ [ s ( η ) ] 2 for     | ξ | s ( η ) 2 , | η | η 0 2 0 otherwise .
ϕ max | ϕ ( x , y ) x | max = 4 ξ 1 ϕ max .
t filter ( x , y ) = a + x w ,
I ( x , y ) = A 2 ( x , y ) [ a + λ f 2 π w ϕ ( x , y ) x ] 2 + ( λ f 2 π w ) 2 [ A ( x , y ) x ] 2 ,
I ( x , y ) = A 0 2 [ a + λ f 2 π w ϕ ( x , y ) x ] 2 ,
K ( x , y ) = | I ( x , y ) I ( x , y ) | I 0 .
tan filter t tan ( x , y ) = { 0 for     x < w 2 , 1 2 + 1 π arctan [ m · tan ( π x w ) ] for     x [ w 2 ; w 2 ] , 1 for     x > w 2 ,
erf   filter t erf ( x , y ) = 1 2 + 1 2 erf ( x w ) ,
erf ( x ) = 2 π 0 x e t 2 d t .
t sqrt ( x , y ) = { 0 for     x < w 2 , 1 2 + 1 2 sgn ( x ) | 2 x w | for     x [ w 2 ; w 2 ] , 1 for     x > w 2 ,

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