Abstract

In optical image processing, selective edge enhancement is important when it is preferable to emphasize some edges of an object more than others. We propose a new method for selective edge enhancement of amplitude objects using the anisotropic vortex phase mask by introducing anisotropy in a conventional vortex mask with the help of the sine function. The anisotropy is capable of edge enhancement in the selective region and in the required direction by changing the power and offset angle, respectively, of the sine function.

© 2011 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2007).
  2. S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
    [CrossRef]
  3. J. A. Davis, D. E. Mcnamara, D. M. Cottrel, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25, 99–101 (2000).
    [CrossRef]
  4. R. B. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, 1965).
  5. J. A. Davis, D. E. McNamara, and D. M. Cottrell, “Analysis of the fractional Hilbert transform,” Appl. Opt. 37, 6911–6913(1998).
    [CrossRef]
  6. 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]
  7. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
    [CrossRef] [PubMed]
  8. A. W. Lohmann, E. Tepichín, and J. G. Ramírez, “Optical implementation of the fractional Hilbert transform for two-dimensional objects,” Appl. Opt. 36, 6620–6626 (1997).
    [CrossRef]
  9. G. Situ, G. Pedrini, and W. Osten, “Spiral phase filtering and orientation-selective edge detection/enhancement,” J. Opt. Soc. Am. A 26, 1788–1797 (2009).
    [CrossRef]
  10. A. Papoulis, “Optical systems, singularity functions, complex Hankel transforms,” J. Opt. Soc. Am. A 57, 207–213 (1967).
    [CrossRef]
  11. G.-H. Kim, H. J. Lee, J.-U. Kim, and H. Suk, “Propagation dynamics of optical vortices with anisotropic phase profiles,” J. Opt. Soc. Am. B 20, 351–359 (2003).
    [CrossRef]
  12. N. Baddour, “Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates,” J. Opt. Soc. Am. A 26, 1767–1777 (2009).
    [CrossRef]
  13. C.-S. Guo, Y.-J. Han, J.-B. Xu, and J. Ding, “Radial Hilbert transform with Laguerre–Gaussian spatial filters,” Opt. Lett. 31, 1394–1396 (2006).
    [CrossRef] [PubMed]
  14. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1983).
  15. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13, 689–694 (2005).
    [CrossRef] [PubMed]
  16. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, and V. A. Soifer, “Generation of phase singularities through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22, 849–861 (2005).
    [CrossRef]

2009 (2)

2006 (1)

2005 (2)

2003 (1)

2001 (1)

2000 (1)

1998 (1)

1997 (1)

1996 (1)

1992 (1)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

1967 (1)

A. Papoulis, “Optical systems, singularity functions, complex Hankel transforms,” J. Opt. Soc. Am. A 57, 207–213 (1967).
[CrossRef]

Almazov, A. A.

Baddour, N.

Bernet, S.

Bracewell, R. B.

R. B. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, 1965).

Campos, J.

Cottrel, D. M.

Cottrell, D. M.

Davis, J. A.

Ding, J.

Fürhapter, S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2007).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1983).

Guo, C.-S.

Han, Y.-J.

Jesacher, A.

Khonina, S. N.

Kim, G.-H.

Kim, J.-U.

Kotlyar, V. V.

Lee, H. J.

Lohmann, A. W.

McNamara, D. E.

Mendlovic, D.

Osten, W.

Papoulis, A.

A. Papoulis, “Optical systems, singularity functions, complex Hankel transforms,” J. Opt. Soc. Am. A 57, 207–213 (1967).
[CrossRef]

Pedrini, G.

Ramírez, J. G.

Ritsch-Marte, M.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1983).

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Situ, G.

Smith, D. A.

Soifer, V. A.

Suk, H.

Tepichín, E.

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Xu, J.-B.

Zalevsky, Z.

Appl. Opt. (3)

J. Mod. Opt. (1)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

J. Opt. Soc. Am. A (4)

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

Opt. Express (1)

Opt. Lett. (3)

Other (3)

R. B. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, 1965).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1983).

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2007).

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Figures (13)

Fig. 1
Fig. 1

(a) Phase distribution of isotropic vortex, (b) phase distribution of anisotropic vortex, (c) phase plot for isotropic vortex ( σ = 1 ) and for anisotropic vortex ( σ = 5 ), and (d) rate of change of phase for isotropic and anisotropic vortices.

Fig. 2
Fig. 2

Phase profiles of (a) function S ( n = 30 ) and (b) plot of phase of S ( n = 30 ) as a function of θ.

Fig. 3
Fig. 3

Plot of rate of change of phase of function S for n = 10 and n = 30 as a function of θ.

Fig. 4
Fig. 4

Simulation results for edge enhancement by isotropic vortex function (a) object, (b) isotropic edge enhancement, and (c) 3D view of (b).

Fig. 5
Fig. 5

Plot of sin n ( θ ) for n = 10 and n = 30 .

Fig. 6
Fig. 6

Phase difference between two radially opposite points about the phase singularity for the function S with n = 30 and for an isotropic vortex.

Fig. 7
Fig. 7

Simulation results of selective edge enhancement for a circular aperture using the anisotropic vortex function S when n is 10. Enhanced edges in different orientations of 0, π / 4 , π / 2 , and 3 π / 4 are shown in (a)–(d), respectively.

Fig. 8
Fig. 8

Simulation results to show the effect of increasing n in the function S on the selectivity. (a)–(d) Show that the region of edge enhancement gets narrower when the power n in S is increased by 5, 10, 30, and 50.

Fig. 9
Fig. 9

Simulation results to show the selectivity with increasing n in 3D plots (a)–(c) and show that the region of edge enhancement decreases when n in S is increased by 5, 10, 30, and 50, respectively.

Fig. 10
Fig. 10

Simulation results for selective edge enhancement of the number 4 using the anisotropic vortex function S for n = 10 , and the angle of rotation is (a) 0, (b)  π / 4 , and (c)  π / 2 .

Fig. 11
Fig. 11

Experimental setup: SF, spatial filter; L 1 , collimator; S, sample stage; M, microscopic objective; L 2 , lens to image the FT on the SLM with magnification of 4; SLM, spatial light modulator to display the CGH of the proposed functions in phase mode; L 3 , imaging lens; NDF, neutral density filter; CCD, charge-coupled device to record the images after the filtering process.

Fig. 12
Fig. 12

(1) Experimental result: (a) object (b) isotropic edge enhancement. (2) Experimental results for selective edge enhancement for the number 4, using the anisotropic vortex function S when n is 10, and the orientation selection is done by θ 0 equal to (a) 0, (b)  π / 4 , and (c)  π / 2 .

Fig. 13
Fig. 13

(1) Experimental result: (a) object (b) isotropic edge enhancement. (2) Experimental results for selective edge enhancement for a circular aperture using the sin-anisotropic function S when n is 10, and orientation selection is done by changing θ 0 by (a) 0, (b)  π / 4 , (c)  π / 2 , and (d)  3 π / 2 . (3) Experimental results for selective edge enhancement for a circular aperture using the sin-anisotropic function S at a particular angle θ 0 = 0 and n is (a) 5, (b) 10, (c) 30, and (d) 50.

Equations (23)

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V ˜ i ( x , y ) = x + i y = r exp ( i θ ) ,
V ˜ a ( x , y ) = x + i σ y = r exp ( i ψ ( x , y ) ) ,
ψ ( x , y ) = tan 1 ( σ y x ) = tan 1 ( σ sin θ cos θ ) .
d ψ d θ = σ cos 2 ( θ ) + σ sin 2 ( θ ) .
S ( r , θ ) = exp [ i θ { | sin n ( θ / 2 ) | } ] = exp ( i ψ s ) .
d ψ S d θ = | sin ( n 1 ) ( θ / 2 ) [ sin ( θ / 2 ) + n 2 θ cos ( θ / 2 ) ] | .
f 0 ( ρ , ϕ ) = f i ( ρ , ϕ ) * h ( ρ , ϕ ) ,
h ( ρ , ϕ ) = c p ( i ) p exp ( i p ϕ ) H p { g R ( r ) } ,
c p = 1 2 π 0 2 π g Θ ( θ ) exp ( i p θ ) d θ ,
H p { g R ( r ) } = 2 π 0 r g R ( r ) J 1 ( 2 π ρ r ) d r .
H 1 { H R ( r ) } = 2 π 0 R r H R ( r ) J 1 ( 2 π ρ r / λ f ) d r = 2 π 0 R r J 1 ( 2 π ρ r / λ f ) d r = π R 2 ρ [ J 1 ( x ) H 0 ( x ) J 0 ( x ) H 1 ( x ) ] ,
0 2 π e i ( p 1 ) θ d θ = 2 π δ 1 p .
h ( ρ , ϕ ) = i exp ( i ϕ ) × π R 2 ρ { J 1 ( x ) H 0 ( x ) J 0 ( x ) H 1 ( x ) } .
s ( ρ , ϕ ) = c p ( i ) p exp ( i p ϕ ) H p { S R ( r ) } ,
c p = 1 2 π 0 2 π S Θ ( θ ) exp ( i p θ ) d θ = 1 2 π 0 2 π e i θ [ | sin n ( θ / 2 ) | p ] d θ .
c p = 1 2 π 0 2 π e i θ ( p | sin n ( θ / 2 ) | ) d θ ,
c 0 = 1 2 π 0 θ 1 d θ + 1 2 π θ 2 2 π d θ = 1 2 π ( 2 π + θ 1 θ 2 ) for     0 < θ < θ 1 and θ 2 < θ < 2 π = 0 otherwise ,
c 1 = 1 2 π θ 1 θ 2 d θ = 1 2 π ( θ 2 θ 1 ) for     θ 1 < θ < θ 2 = 0 otherwise .
H 1 { S R ( r ) } = 2 π 0 R r S R ( r ) J 1 ( 2 π ρ r / λ f ) d r = 2 π 0 R r J 1 ( 2 π ρ r / λ f ) d r = π R 2 ρ [ J 1 ( x ) H 0 ( x ) J 0 ( x ) H 1 ( x ) ] ,
H 0 { S R ( r ) } = 2 π 0 R r S R ( r ) J 0 ( 2 π ρ r ) d r = x J 1 ( x ) .
s ( ρ , ϕ ) = c 0 H 0 { S R ( r ) } + c 1 ( i ) exp ( i ϕ ) H 1 { S R ( r ) } .
s ( ρ , ϕ ) = ( 2 π + θ 2 θ 1 ) 2 π x J 1 ( x ) i exp ( i ϕ ) × ( θ 2 θ 1 ) 2 π × π R 2 ρ [ J 1 ( x ) H 0 ( x ) J 0 ( x ) H 1 ( x ) ] ,
S ( r , θ ) = exp [ i ( θ + θ 0 ) { | sin n ( ( θ + θ 0 ) / 2 ) | } ] .

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