Abstract

In this work we analyze the spatial frequency response, the spatial distribution and continuity of the recovered phase in Lateral Shearing Interferometry (LSI). The frequency content and the spatial topology of the recovered phase is analyzed for the forward LSI operator as well as its inverse LSI operator using one, two, or n two-dimensional sheared interferograms. The wavefront’s spatial frequency response of the lateral shearing interferometer is well known and for the reader’s convenience, it is briefly revisited in a new perspective. It is however less well-known and more interesting to analyze the spatial distribution of the lateral sheared data as well as the spatial topology of the recovered phase produced by some inverse LSI operators. Also we define a useful space of functions S with the property that any sheared data available, along any direction, may be used to recovered a smooth continuous phase with the bonus property of fully covering the pupil of the wavefront being tested. These combined aspects allow us to find the best possible wave-front reconstruction from the available sheared data using one, two or n sheared interferograms.

©2007 Optical Society of America

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References

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  1. G. Paez, M. Strojnik, and M. MantravadiD. Malacara, “Shearing Interferometry,” Optical Shop Testing, ed., (2007).
  2. C. Falldorf, Y. Heimbach, C. von Kopylow, and W. Juptner, “Efficient reconstruction of spatially limited phase distributions from their sheared representation,” Appl. Opt. 46, 5038–5043 (2007).
    [Crossref] [PubMed]
  3. P. Y. Liang, J. P. Ding, Z. Jin, C. S. Guo, and H. T. Wang, “Two-dimensional wave-front reconstruction from lateral shearing interferograms,” Opt. Express 14, 625–634 (2006).
    [Crossref] [PubMed]
  4. G. Garcia-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. G. Alvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
    [Crossref]
  5. F. J. Casillas, A. Davila, S. J. Rothberg, and G. Garnica, “Small amplitude estimation of mechanical vibrations using electronic speckle shearing pattern interferometry,” Opt. Eng. 43, 880–887 (2004).
    [Crossref]
  6. T. Nomura, S. Okuda, K. Kamiya, H. Tashiro, and K. Yoshikawa, “Improved Saunders method for the analysis of lateral shearing interferograms,” Appl. Opt. 41, 1954–1961 (2002).
    [Crossref] [PubMed]
  7. P. Ferraro, S. De Nicola, A. Finizio, and G. Pierattini, “Reflective grating interferometer: a folded reversal and shearing wave-front interferometer,” Appl. Opt. 41, 342–347 (2002).
    [Crossref] [PubMed]
  8. J. Villa, G. Garcia, and G. Gomez, “Wavefront recovery in shearing interferometry with variable magnitude and direction shear,” Opt. Commun. 195, 85–91(2001).
    [Crossref]
  9. S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Two-beam interferometer for measuring aberrations of optical components with axial symmetry,” Appl. Opt. 40, 1631–1636 (2001)
    [Crossref]
  10. F. Chen, “Digital shearography: state of the art and some applications,” J. Electron. Imaging 10, 240–250 (2001).
    [Crossref]
  11. C. Elster, “Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears,” Appl. Opt. 39, 5353–5359 (2000).
    [Crossref]
  12. S. Okuda, T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, and H. Tashiro, “High-precision analysis of a lateral shearing interferogram by use of the integration method and polynomials,” Appl. Opt. 39, 5179–5186 (2000).
    [Crossref]
  13. A. Davila, M. Servin, and M. Facchini, “Fast phase-map recovery from large shears in an electronic speckle shearing pattern interferometer using a Fourier least-squares estimation,” Opt. Eng. 39, 2487–2494 (2000).
    [Crossref]
  14. C. Elster and I. Weingartner, “Exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A 16, 2281–2285 (1999).
    [Crossref]
  15. C. Elster and I. Weingartner, “Solution to the shearing problem,” Appl. Opt. 38, 5024–5031 (1999).
    [Crossref]
  16. H. J. Lee and S. W. Kim, “Precision profile measurement of aspheric surfaces by improved Ronchi test,” Opt. Eng. 6, 1041–1047 (1999).
    [Crossref]
  17. J. A. Quiroga and A. Gonzalez-Cano, “Stress separation from photoelastic data by a multigrid method,” Meas. Sci. Technol. 9, 1204–1210 (1998).
    [Crossref]
  18. S. W. Kim, W. J. Cho, and B. C. Kim, “Lateral-shearing interferometer using square prisms for optical testing of aspheric lenses,” Meas. Sci. Technol. 9, 1129–1136 (1998).
    [Crossref]
  19. S. Loheide, “Innovative evaluation method for shearing interferograms,” Opt. Commun. 141, 254–258 (1997).
    [Crossref]
  20. M. Servin, D. Malacara, and J. L. Marroquin, “Wavefront recovery from two orthogonal sheared interferometers,” Appl. Opt. 35, 4343–4348 (1996).
    [Crossref] [PubMed]

2007 (1)

2006 (2)

G. Garcia-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. G. Alvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[Crossref]

P. Y. Liang, J. P. Ding, Z. Jin, C. S. Guo, and H. T. Wang, “Two-dimensional wave-front reconstruction from lateral shearing interferograms,” Opt. Express 14, 625–634 (2006).
[Crossref] [PubMed]

2004 (1)

F. J. Casillas, A. Davila, S. J. Rothberg, and G. Garnica, “Small amplitude estimation of mechanical vibrations using electronic speckle shearing pattern interferometry,” Opt. Eng. 43, 880–887 (2004).
[Crossref]

2002 (2)

2001 (3)

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Two-beam interferometer for measuring aberrations of optical components with axial symmetry,” Appl. Opt. 40, 1631–1636 (2001)
[Crossref]

J. Villa, G. Garcia, and G. Gomez, “Wavefront recovery in shearing interferometry with variable magnitude and direction shear,” Opt. Commun. 195, 85–91(2001).
[Crossref]

F. Chen, “Digital shearography: state of the art and some applications,” J. Electron. Imaging 10, 240–250 (2001).
[Crossref]

2000 (3)

1999 (3)

1998 (2)

J. A. Quiroga and A. Gonzalez-Cano, “Stress separation from photoelastic data by a multigrid method,” Meas. Sci. Technol. 9, 1204–1210 (1998).
[Crossref]

S. W. Kim, W. J. Cho, and B. C. Kim, “Lateral-shearing interferometer using square prisms for optical testing of aspheric lenses,” Meas. Sci. Technol. 9, 1129–1136 (1998).
[Crossref]

1997 (1)

S. Loheide, “Innovative evaluation method for shearing interferograms,” Opt. Commun. 141, 254–258 (1997).
[Crossref]

1996 (1)

Alvarez, A. G.

G. Garcia-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. G. Alvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[Crossref]

Casillas, F. J.

F. J. Casillas, A. Davila, S. J. Rothberg, and G. Garnica, “Small amplitude estimation of mechanical vibrations using electronic speckle shearing pattern interferometry,” Opt. Eng. 43, 880–887 (2004).
[Crossref]

Chen, F.

F. Chen, “Digital shearography: state of the art and some applications,” J. Electron. Imaging 10, 240–250 (2001).
[Crossref]

Cho, W. J.

S. W. Kim, W. J. Cho, and B. C. Kim, “Lateral-shearing interferometer using square prisms for optical testing of aspheric lenses,” Meas. Sci. Technol. 9, 1129–1136 (1998).
[Crossref]

Davila, A.

F. J. Casillas, A. Davila, S. J. Rothberg, and G. Garnica, “Small amplitude estimation of mechanical vibrations using electronic speckle shearing pattern interferometry,” Opt. Eng. 43, 880–887 (2004).
[Crossref]

A. Davila, M. Servin, and M. Facchini, “Fast phase-map recovery from large shears in an electronic speckle shearing pattern interferometer using a Fourier least-squares estimation,” Opt. Eng. 39, 2487–2494 (2000).
[Crossref]

De Nicola, S.

Ding, J. P.

Elster, C.

Facchini, M.

A. Davila, M. Servin, and M. Facchini, “Fast phase-map recovery from large shears in an electronic speckle shearing pattern interferometer using a Fourier least-squares estimation,” Opt. Eng. 39, 2487–2494 (2000).
[Crossref]

Falldorf, C.

Ferraro, P.

Finizio, A.

Flores, J. L.

G. Garcia-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. G. Alvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[Crossref]

Garcia, G.

J. Villa, G. Garcia, and G. Gomez, “Wavefront recovery in shearing interferometry with variable magnitude and direction shear,” Opt. Commun. 195, 85–91(2001).
[Crossref]

Garcia-Torales, G.

G. Garcia-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. G. Alvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[Crossref]

Garnica, G.

F. J. Casillas, A. Davila, S. J. Rothberg, and G. Garnica, “Small amplitude estimation of mechanical vibrations using electronic speckle shearing pattern interferometry,” Opt. Eng. 43, 880–887 (2004).
[Crossref]

Gomez, G.

J. Villa, G. Garcia, and G. Gomez, “Wavefront recovery in shearing interferometry with variable magnitude and direction shear,” Opt. Commun. 195, 85–91(2001).
[Crossref]

Gonzalez-Cano, A.

J. A. Quiroga and A. Gonzalez-Cano, “Stress separation from photoelastic data by a multigrid method,” Meas. Sci. Technol. 9, 1204–1210 (1998).
[Crossref]

Guo, C. S.

Heimbach, Y.

Jin, Z.

Juptner, W.

Kamiya, K.

Kim, B. C.

S. W. Kim, W. J. Cho, and B. C. Kim, “Lateral-shearing interferometer using square prisms for optical testing of aspheric lenses,” Meas. Sci. Technol. 9, 1129–1136 (1998).
[Crossref]

Kim, S. W.

H. J. Lee and S. W. Kim, “Precision profile measurement of aspheric surfaces by improved Ronchi test,” Opt. Eng. 6, 1041–1047 (1999).
[Crossref]

S. W. Kim, W. J. Cho, and B. C. Kim, “Lateral-shearing interferometer using square prisms for optical testing of aspheric lenses,” Meas. Sci. Technol. 9, 1129–1136 (1998).
[Crossref]

Lee, H. J.

H. J. Lee and S. W. Kim, “Precision profile measurement of aspheric surfaces by improved Ronchi test,” Opt. Eng. 6, 1041–1047 (1999).
[Crossref]

Liang, P. Y.

Loheide, S.

S. Loheide, “Innovative evaluation method for shearing interferograms,” Opt. Commun. 141, 254–258 (1997).
[Crossref]

Malacara, D.

Mantravadi, M.

G. Paez, M. Strojnik, and M. MantravadiD. Malacara, “Shearing Interferometry,” Optical Shop Testing, ed., (2007).

Marroquin, J. L.

Miyashiro, H.

Nomura, T.

Okuda, S.

Paez, G.

G. Garcia-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. G. Alvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[Crossref]

G. Paez, M. Strojnik, and M. MantravadiD. Malacara, “Shearing Interferometry,” Optical Shop Testing, ed., (2007).

Pierattini, G.

Quiroga, J. A.

J. A. Quiroga and A. Gonzalez-Cano, “Stress separation from photoelastic data by a multigrid method,” Meas. Sci. Technol. 9, 1204–1210 (1998).
[Crossref]

Rothberg, S. J.

F. J. Casillas, A. Davila, S. J. Rothberg, and G. Garnica, “Small amplitude estimation of mechanical vibrations using electronic speckle shearing pattern interferometry,” Opt. Eng. 43, 880–887 (2004).
[Crossref]

Servin, M.

A. Davila, M. Servin, and M. Facchini, “Fast phase-map recovery from large shears in an electronic speckle shearing pattern interferometer using a Fourier least-squares estimation,” Opt. Eng. 39, 2487–2494 (2000).
[Crossref]

M. Servin, D. Malacara, and J. L. Marroquin, “Wavefront recovery from two orthogonal sheared interferometers,” Appl. Opt. 35, 4343–4348 (1996).
[Crossref] [PubMed]

Strojnik, M.

G. Garcia-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. G. Alvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[Crossref]

G. Paez, M. Strojnik, and M. MantravadiD. Malacara, “Shearing Interferometry,” Optical Shop Testing, ed., (2007).

Tashiro, H.

Villa, J.

G. Garcia-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. G. Alvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[Crossref]

J. Villa, G. Garcia, and G. Gomez, “Wavefront recovery in shearing interferometry with variable magnitude and direction shear,” Opt. Commun. 195, 85–91(2001).
[Crossref]

von Kopylow, C.

Wang, H. T.

Weingartner, I.

Yoshikawa, K.

Appl. Opt. (8)

C. Elster and I. Weingartner, “Solution to the shearing problem,” Appl. Opt. 38, 5024–5031 (1999).
[Crossref]

M. Servin, D. Malacara, and J. L. Marroquin, “Wavefront recovery from two orthogonal sheared interferometers,” Appl. Opt. 35, 4343–4348 (1996).
[Crossref] [PubMed]

C. Elster, “Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears,” Appl. Opt. 39, 5353–5359 (2000).
[Crossref]

S. Okuda, T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, and H. Tashiro, “High-precision analysis of a lateral shearing interferogram by use of the integration method and polynomials,” Appl. Opt. 39, 5179–5186 (2000).
[Crossref]

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Two-beam interferometer for measuring aberrations of optical components with axial symmetry,” Appl. Opt. 40, 1631–1636 (2001)
[Crossref]

P. Ferraro, S. De Nicola, A. Finizio, and G. Pierattini, “Reflective grating interferometer: a folded reversal and shearing wave-front interferometer,” Appl. Opt. 41, 342–347 (2002).
[Crossref] [PubMed]

T. Nomura, S. Okuda, K. Kamiya, H. Tashiro, and K. Yoshikawa, “Improved Saunders method for the analysis of lateral shearing interferograms,” Appl. Opt. 41, 1954–1961 (2002).
[Crossref] [PubMed]

C. Falldorf, Y. Heimbach, C. von Kopylow, and W. Juptner, “Efficient reconstruction of spatially limited phase distributions from their sheared representation,” Appl. Opt. 46, 5038–5043 (2007).
[Crossref] [PubMed]

J. Electron. Imaging (1)

F. Chen, “Digital shearography: state of the art and some applications,” J. Electron. Imaging 10, 240–250 (2001).
[Crossref]

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (2)

J. A. Quiroga and A. Gonzalez-Cano, “Stress separation from photoelastic data by a multigrid method,” Meas. Sci. Technol. 9, 1204–1210 (1998).
[Crossref]

S. W. Kim, W. J. Cho, and B. C. Kim, “Lateral-shearing interferometer using square prisms for optical testing of aspheric lenses,” Meas. Sci. Technol. 9, 1129–1136 (1998).
[Crossref]

Opt. Commun. (3)

S. Loheide, “Innovative evaluation method for shearing interferograms,” Opt. Commun. 141, 254–258 (1997).
[Crossref]

G. Garcia-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. G. Alvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[Crossref]

J. Villa, G. Garcia, and G. Gomez, “Wavefront recovery in shearing interferometry with variable magnitude and direction shear,” Opt. Commun. 195, 85–91(2001).
[Crossref]

Opt. Eng. (3)

F. J. Casillas, A. Davila, S. J. Rothberg, and G. Garnica, “Small amplitude estimation of mechanical vibrations using electronic speckle shearing pattern interferometry,” Opt. Eng. 43, 880–887 (2004).
[Crossref]

A. Davila, M. Servin, and M. Facchini, “Fast phase-map recovery from large shears in an electronic speckle shearing pattern interferometer using a Fourier least-squares estimation,” Opt. Eng. 39, 2487–2494 (2000).
[Crossref]

H. J. Lee and S. W. Kim, “Precision profile measurement of aspheric surfaces by improved Ronchi test,” Opt. Eng. 6, 1041–1047 (1999).
[Crossref]

Opt. Express (1)

Other (1)

G. Paez, M. Strojnik, and M. MantravadiD. Malacara, “Shearing Interferometry,” Optical Shop Testing, ed., (2007).

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Figures (10)

Fig. 1.
Fig. 1. Schematics of a lateral shear interferometer. The incoming wavefront is divided in two and shifted laterally. Their common area produces the shearing fringes over a CCD camera.

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Fig. 2.
Fig. 2. Graph of the inverse shearing filter

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Fig. 3.
Fig. 3. Graph of the regularized inverse shearing filter.

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Fig. 4.
Fig. 4. Spatial distribution of: a) the pupil, b) the shearing fringes, c) the regularized least squares estimated phase, d) the simply integrated estimated phase.

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Fig. 5.
Fig. 5. Spatial distribution or support of: a) the wavefront’s pupil supp ϕ(x, y), b) the sheared data area along x, c) the shearing data along y, d) the recovered area using only the x shearing data, e) the recovered area using only the y shearing data, f) the recovered support supp ϕ ^ (x, y) using both, the x and y shearing data.

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Fig. 6.
Fig. 6. Spatial distribution or support of: a) the wavefront’s pupil supp ϕ(x, y), b) the sheared data area along x, c) the shearing data along y, d) the recovered area using only the x shearing data, e) the recovered area using only the y shearing data, f) the recovered phase support supp ϕ ^ (x, y) using, the x and y data. The red regions are where both sheared data along the x and y are present simultaneously.

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Fig. 7.
Fig. 7. (a). Gray level of the phase being tested ϕ(x, y), 24 radians is white, 0 radians is black, (b) fringes for the shearing along x, (c) fringes for the shearing along y, (d) recovered phase ϕ ^ (x, y) using the shearing along x and y. Note the smooth blend of the data.

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Fig. 8.
Fig. 8. (a). Gray level of the phase being tested ϕ(x, y), 24 radians is white, 0 radians is black, (b) fringes for the shearing along x, (c) fringes for the shearing along y, (d) recovered phase ϕ ^ (x, y) using the shearing along x and y. Note the smooth blend of the data to obtain ϕ ^ (x, y).

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Fig. 9.
Fig. 9. (a). Surface graph of the recovered phase shown in Fig. 7(d), b) difference between the original wavefront (panel 7(a)) and the estimated one (panel 7(d)).

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Fig. 10.
Fig. 10. (a). Surface graph of the recovered phase ϕ ^ (x, y) shown in Fig. 8(d), (b) difference between the original phase (panel 8(a)) and the estimated one (panel 8(d)).

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Equations (31)

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Δ x ( x , y ) = ( 2 π ⁄ λ ) [ W ( x + d x , y ) − W ( x − d x , y ) ] .
I ( x , y ) = a ( x , y ) + b ( x , y ) cos ⁡ [ Δ x ( x , y ) ] .
L x [ ϕ ( x , y ) ] = ϕ ( x − d x , y ) − ϕ ( x + d x , y ) ≈ 2 d x ∂ ϕ ( x , y ) ∂ x .
Δ x ( x , y ) ≈ 2 d x ∂ ϕ ( x , y ) ∂ x [ P ( x + d x , y ) ∩ P ( x − d x , y ) ] .
ϕ ( x , y ) = [ f ( x ) + g ( y ) + c ] P ( x , y ) .
Δ x ( x , y ) = 2 d x ∂ ϕ ( x , y ) ∂ x = 2 d x { [ f ( x ) + g ( y ) + c ] ∂ P ( x , y ) ∂ x + ∂ f ( x ) ∂ x P ( x , y ) } .
Δ x ( x , y ) = 2 d x ∂ f ( x ) ∂ x [ P ( x + d x , y ) ∩ P ( x − d x , y ) ] .
F [ Δ x ( x , y ) ] = Φ ( ω x , ω y ) [ e i d x ω x − e − i d x ω x ] = 2 i sin ⁡ ( d x ω x ) Φ ( ω x , ω y ) .
Φ ̂ ( ω x , ω y ) = H − 1 ( ω x , ω y ) F [ Δ x ( x , y ) ] = − i 2 sin ⁡ ( d x ω x ) F [ Δ x ( x , y ) ] .
U ( ϕ ̂ ) = ∑ ( x , y ) ∈ P ( x , y ) [ ϕ ̂ ( x + d x , y ) − ϕ ̂ ( x − d x , y ) − Δ x ( x , y ) ] 2 .
∂ U ( ϕ ̂ ) ∂ ϕ ̂ = ϕ ̂ ( x + 2 d x , y ) − 2 ϕ ̂ ( x , y ) + ϕ ̂ ( x − 2 d x , y ) − Δ x ( x + d x , y ) + Δ x ( x − d x , y ) = 0 .
A ϕ ^ ( x , y ) − b = 0.
U ( ϕ ̂ ) = ∑ P ( x , y ) { [ ϕ ̂ ( x + d x , y ) − ϕ ̂ ( x − d x , y ) − Δ x ( x , y ) ] 2 + η [ ϕ ̂ ( x , y ) − ϕ ̂ ( x − 1 , y ) ] 2 } .
H − 1 ( ω x ) = i sin ⁡ ( d x ω x ) 1 + η − cos ⁡ ( 2 d x ω x ) − η cos ⁡ ( ω x ) .
Δ x ( x , y ) = L x [ ϕ ( x , y ) ] = [ ϕ ( x + d x , y ) − ϕ ( x − d x , y ) ] [ P ( x + d x , y ) ∩ P ( x − d x , y ) ] .
ϕ ̂ ( x , y ) = L x − 1 [ Δ x ( x , y ) ] .
supp ϕ ^ ( x , y ) = { [ P ( x + 2 d x , y ) ∩ P ( x , y ) ] ∪ [ P ( x , y ) ∩ P ( x − 2 d x , y ) ] } ⊂ P ( x , y ) .
supp ϕ ̂ ( x , y ) = supp L x − 1 [ L x [ ϕ ( x , y ) ] ] = P ( x , y ) ∩ [ P ( x + 2 d x , y ) ∪ P ( x − 2 d x , y ) ] .
supp ϕ ^ ( x , y ) = supp ∫ Δ x ( x , y ) d x = P ( x + d x , y ) ∩ P ( x − d x , y ) .
U ( ϕ ̂ ) = ∑ P ( x , y ) { [ ϕ ̂ ( x + d x , y ) − ϕ ̂ ( x − d x , y ) − Δ x ] 2 + [ ϕ ̂ ( x , y + d y ) − ϕ ̂ ( x , y − d y ) − Δ y ] 2
η [ ϕ ̂ ( x , y ) − ϕ ̂ ( x − 1 , y ) ] 2 + η [ ϕ ̂ ( x , y ) − ϕ ̂ ( x , y − 1 ) ] 2 } .
supp ϕ ^ = { P ( x , y ) ∩ [ P ( x + 2 d x , y ) ∪ P ( x − 2 d x , y ) ] } ∪ { P ( x , y ) ∩ [ P ( x , y + 2 d y ) ∪ P ( x , y − 2 d y ) ] }
supp ϕ ̂ = P ( x , y ) ∩ [ P ( x + 2 d x , y ) ∪ P ( x − 2 d x , y ) ∪ P ( x , y + 2 d y ) ∪ P ( x , y − 2 d y ) ] ⊆ P ( x , y ) .
S = { ϕ : R 2 → R | ∃ ( x , y ) ∈ P ( x , y ) ,         L x − 1 [ L x ( ϕ ) ] = ϕ       and        L y − 1 [ L y ( ϕ ) ] = ϕ } .
ϕ ( x , y ) = [ f ( x ) + g ( y ) ] P ( x , y ) , ϕ ( x , y ) ∉ S .
exp ⁡ [ − ( x 2 + y 2 ) / σ 2 ] ,           sin ⁡ ( x 2 + y 2 ) ,        x y ,        J 0 ( x 2 + y 2 ) .
⌊ ϕ ^ ( x , y ) g r e e n + ϕ ^ ( x , y ) b l u e + ϕ ^ ( x , y ) r e d ⌋ P ( x , y ) = ϕ ^ ( x , y ) P ( x , y ) ,             ϕ ^ ( x , y ) ∈ S .
Δ i ( r ) = L i [ ϕ ( r ) ] = [ ϕ ( r ) − ϕ ( r − d i ) ] P ( r ) ∩ P ( r − d i ) ,     i ∈ { 0 , 1 , ... , n − 1 } .
I i ( r ) = a ( r ) + b ( r ) cos ⁡ [ Δ i ( r ) ] .
S = { ϕ : R 2 → R | ∃ r ∈ P ( r ) ,    L 0 − 1 [ Δ 0 ( r ) ] = L 1 − 1 [ Δ 1 ( r ) ] = , ... , = L n − 1 − 1 [ Δ n − 1 ( r ) ] = ϕ } .
ϕ ^ ( r ) = L i − 1 [ Δ i ( r ) ] ≈ ϕ ( r ) .

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