Abstract

In this paper, we make a proposal to obtain the Hilbert-transform for each entry of the projection data leaving the slice of a thin phase object. These modified projections are stacked in such a way that they form a modified sinogram called Hilbert-sinogram. We prove that the inverse Radon-transform of this sinogram is the directional Hilbert-transform of the slice function, and the reconstructed image is the directional edge enhancement of the distribution function on the slice. The Hilbert-transform is implemented by a 4f optical Fourier-transform correlator and a spatial filter consisting of a phase step of π radians. One important feature of this proposal is to perform a turn of 180° in the spatial filter at a certain value of the projection angle within the range [0°, 360°]. The desired direction of enhancement can be chosen by the proper selection of such turning angle. We present both the mathematical modeling and numerical results.

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References

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  1. G. S. Abdoulaev and A. H. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging 12(4), 594–601 (2003).
    [CrossRef]
  2. S. R. Deans, The Radon Transform and Some of its Applications (Wiley, New York. 1983).
  3. D. Yan and S. S. Cha, “Computational and interferometric system for real-time limited-view tomography of flow fields,” Appl. Opt. 37(7), 1159–1164 (1998).
    [CrossRef]
  4. D. Wu and A. He, “Measurement of three-dimensional temperature fields with interferometric tomography,” Appl. Opt. 38(16), 3468–3473 (1999).
    [CrossRef]
  5. I. H. Lira and C. M. Vest, “Refraction correction in holographic interferometry and tomography of transparent objects,” Appl. Opt. 26(18), 3919–3928 (1987).
    [CrossRef] [PubMed]
  6. S. Cha and C. M. Vest, “Tomographic reconstruction of strongly refracting fields and its application to interferometric measurement of boundary layers,” Appl. Opt. 20(16), 2787–2794 (1981).
    [CrossRef] [PubMed]
  7. C. Meneses-Fabian, G. Rodriguez-Zurita, and V. Arrizón, “Optical tomography of transparent objects with phase-shifting interferometry and stepwise-shifted Ronchi ruling,” J. Opt. Soc. Am. A 23(2), 298–305 (2006).
    [CrossRef]
  8. C. Meneses-Fabian, G. Rodriguez-Zurita, R. Rodriguez-Vera, and J. F. Vazquez-Castillo, “Optical tomography with parallel projection differences and Electronic Speckle Pattern Interferometry,” Opt. Commun. 228(4-6), 201–210 (2003).
    [CrossRef]
  9. G. Rodríguez-Zurita, C. Meneses-Fabián, J.-S. Pérez-Huerta, and J.-F. Vázquez-Castillo, ““Tomographic directional derivative of phase objects slices using 1-D derivative spatial filtering of fractional order ½,” ICO20,” Proc. SPIE 6027, 410–416 (2006).
  10. K. Sendhil, C. Vijayan, and M. P. Kothiyal, “Spatial phase filtering with a porphyrin derivative as phase filter in an optical image processor,” Opt. Commun. 251(4-6), 292–298 (2005).
    [CrossRef]
  11. J. A. Davis, D. E. McNamara, and D. M. Cottrell, “Analysis of the fractional hilbert transform,” Appl. Opt. 37(29), 6911–6913 (1998).
    [CrossRef]
  12. J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
    [CrossRef]
  13. J. A. Davis and M. D. Nowak, “Selective edge enhancement of images with an acousto-optic light modulator,” Appl. Opt. 41(23), 4835–4839 (2002).
    [CrossRef] [PubMed]
  14. F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955).
    [CrossRef] [PubMed]
  15. J. Glückstad, P. C. Mogensen and R. L. Eriksen, “The generalized phase contrast method and its applications,” DOPS-NYT 1–2001, 49–54.
  16. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13(3), 689–694 (2005).
    [CrossRef] [PubMed]

2006 (2)

C. Meneses-Fabian, G. Rodriguez-Zurita, and V. Arrizón, “Optical tomography of transparent objects with phase-shifting interferometry and stepwise-shifted Ronchi ruling,” J. Opt. Soc. Am. A 23(2), 298–305 (2006).
[CrossRef]

G. Rodríguez-Zurita, C. Meneses-Fabián, J.-S. Pérez-Huerta, and J.-F. Vázquez-Castillo, ““Tomographic directional derivative of phase objects slices using 1-D derivative spatial filtering of fractional order ½,” ICO20,” Proc. SPIE 6027, 410–416 (2006).

2005 (2)

K. Sendhil, C. Vijayan, and M. P. Kothiyal, “Spatial phase filtering with a porphyrin derivative as phase filter in an optical image processor,” Opt. Commun. 251(4-6), 292–298 (2005).
[CrossRef]

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13(3), 689–694 (2005).
[CrossRef] [PubMed]

2003 (2)

C. Meneses-Fabian, G. Rodriguez-Zurita, R. Rodriguez-Vera, and J. F. Vazquez-Castillo, “Optical tomography with parallel projection differences and Electronic Speckle Pattern Interferometry,” Opt. Commun. 228(4-6), 201–210 (2003).
[CrossRef]

G. S. Abdoulaev and A. H. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging 12(4), 594–601 (2003).
[CrossRef]

2002 (1)

J. A. Davis and M. D. Nowak, “Selective edge enhancement of images with an acousto-optic light modulator,” Appl. Opt. 41(23), 4835–4839 (2002).
[CrossRef] [PubMed]

2000 (1)

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
[CrossRef]

1999 (1)

D. Wu and A. He, “Measurement of three-dimensional temperature fields with interferometric tomography,” Appl. Opt. 38(16), 3468–3473 (1999).
[CrossRef]

1998 (2)

D. Yan and S. S. Cha, “Computational and interferometric system for real-time limited-view tomography of flow fields,” Appl. Opt. 37(7), 1159–1164 (1998).
[CrossRef]

J. A. Davis, D. E. McNamara, and D. M. Cottrell, “Analysis of the fractional hilbert transform,” Appl. Opt. 37(29), 6911–6913 (1998).
[CrossRef]

1987 (1)

I. H. Lira and C. M. Vest, “Refraction correction in holographic interferometry and tomography of transparent objects,” Appl. Opt. 26(18), 3919–3928 (1987).
[CrossRef] [PubMed]

1981 (1)

S. Cha and C. M. Vest, “Tomographic reconstruction of strongly refracting fields and its application to interferometric measurement of boundary layers,” Appl. Opt. 20(16), 2787–2794 (1981).
[CrossRef] [PubMed]

1955 (1)

F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955).
[CrossRef] [PubMed]

Abdoulaev, G. S.

G. S. Abdoulaev and A. H. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging 12(4), 594–601 (2003).
[CrossRef]

Arrizón, V.

C. Meneses-Fabian, G. Rodriguez-Zurita, and V. Arrizón, “Optical tomography of transparent objects with phase-shifting interferometry and stepwise-shifted Ronchi ruling,” J. Opt. Soc. Am. A 23(2), 298–305 (2006).
[CrossRef]

Bernet, S.

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13(3), 689–694 (2005).
[CrossRef] [PubMed]

Campos, J.

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
[CrossRef]

Cha, S.

S. Cha and C. M. Vest, “Tomographic reconstruction of strongly refracting fields and its application to interferometric measurement of boundary layers,” Appl. Opt. 20(16), 2787–2794 (1981).
[CrossRef] [PubMed]

Cha, S. S.

D. Yan and S. S. Cha, “Computational and interferometric system for real-time limited-view tomography of flow fields,” Appl. Opt. 37(7), 1159–1164 (1998).
[CrossRef]

Cottrell, D. M.

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
[CrossRef]

J. A. Davis, D. E. McNamara, and D. M. Cottrell, “Analysis of the fractional hilbert transform,” Appl. Opt. 37(29), 6911–6913 (1998).
[CrossRef]

Davis, J. A.

J. A. Davis and M. D. Nowak, “Selective edge enhancement of images with an acousto-optic light modulator,” Appl. Opt. 41(23), 4835–4839 (2002).
[CrossRef] [PubMed]

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
[CrossRef]

J. A. Davis, D. E. McNamara, and D. M. Cottrell, “Analysis of the fractional hilbert transform,” Appl. Opt. 37(29), 6911–6913 (1998).
[CrossRef]

Fürhapter, S.

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13(3), 689–694 (2005).
[CrossRef] [PubMed]

He, A.

D. Wu and A. He, “Measurement of three-dimensional temperature fields with interferometric tomography,” Appl. Opt. 38(16), 3468–3473 (1999).
[CrossRef]

Hielscher, A. H.

G. S. Abdoulaev and A. H. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging 12(4), 594–601 (2003).
[CrossRef]

Jesacher, A.

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13(3), 689–694 (2005).
[CrossRef] [PubMed]

Kothiyal, M. P.

K. Sendhil, C. Vijayan, and M. P. Kothiyal, “Spatial phase filtering with a porphyrin derivative as phase filter in an optical image processor,” Opt. Commun. 251(4-6), 292–298 (2005).
[CrossRef]

Lira, I. H.

I. H. Lira and C. M. Vest, “Refraction correction in holographic interferometry and tomography of transparent objects,” Appl. Opt. 26(18), 3919–3928 (1987).
[CrossRef] [PubMed]

McNamara, D. E.

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
[CrossRef]

J. A. Davis, D. E. McNamara, and D. M. Cottrell, “Analysis of the fractional hilbert transform,” Appl. Opt. 37(29), 6911–6913 (1998).
[CrossRef]

Meneses-Fabian, C.

C. Meneses-Fabian, G. Rodriguez-Zurita, and V. Arrizón, “Optical tomography of transparent objects with phase-shifting interferometry and stepwise-shifted Ronchi ruling,” J. Opt. Soc. Am. A 23(2), 298–305 (2006).
[CrossRef]

C. Meneses-Fabian, G. Rodriguez-Zurita, R. Rodriguez-Vera, and J. F. Vazquez-Castillo, “Optical tomography with parallel projection differences and Electronic Speckle Pattern Interferometry,” Opt. Commun. 228(4-6), 201–210 (2003).
[CrossRef]

Meneses-Fabián, C.

G. Rodríguez-Zurita, C. Meneses-Fabián, J.-S. Pérez-Huerta, and J.-F. Vázquez-Castillo, ““Tomographic directional derivative of phase objects slices using 1-D derivative spatial filtering of fractional order ½,” ICO20,” Proc. SPIE 6027, 410–416 (2006).

Nowak, M. D.

J. A. Davis and M. D. Nowak, “Selective edge enhancement of images with an acousto-optic light modulator,” Appl. Opt. 41(23), 4835–4839 (2002).
[CrossRef] [PubMed]

Pérez-Huerta, J.-S.

G. Rodríguez-Zurita, C. Meneses-Fabián, J.-S. Pérez-Huerta, and J.-F. Vázquez-Castillo, ““Tomographic directional derivative of phase objects slices using 1-D derivative spatial filtering of fractional order ½,” ICO20,” Proc. SPIE 6027, 410–416 (2006).

Ritsch-Marte, M.

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13(3), 689–694 (2005).
[CrossRef] [PubMed]

Rodriguez-Vera, R.

C. Meneses-Fabian, G. Rodriguez-Zurita, R. Rodriguez-Vera, and J. F. Vazquez-Castillo, “Optical tomography with parallel projection differences and Electronic Speckle Pattern Interferometry,” Opt. Commun. 228(4-6), 201–210 (2003).
[CrossRef]

Rodriguez-Zurita, G.

C. Meneses-Fabian, G. Rodriguez-Zurita, and V. Arrizón, “Optical tomography of transparent objects with phase-shifting interferometry and stepwise-shifted Ronchi ruling,” J. Opt. Soc. Am. A 23(2), 298–305 (2006).
[CrossRef]

C. Meneses-Fabian, G. Rodriguez-Zurita, R. Rodriguez-Vera, and J. F. Vazquez-Castillo, “Optical tomography with parallel projection differences and Electronic Speckle Pattern Interferometry,” Opt. Commun. 228(4-6), 201–210 (2003).
[CrossRef]

Rodríguez-Zurita, G.

G. Rodríguez-Zurita, C. Meneses-Fabián, J.-S. Pérez-Huerta, and J.-F. Vázquez-Castillo, ““Tomographic directional derivative of phase objects slices using 1-D derivative spatial filtering of fractional order ½,” ICO20,” Proc. SPIE 6027, 410–416 (2006).

Sendhil, K.

K. Sendhil, C. Vijayan, and M. P. Kothiyal, “Spatial phase filtering with a porphyrin derivative as phase filter in an optical image processor,” Opt. Commun. 251(4-6), 292–298 (2005).
[CrossRef]

Vazquez-Castillo, J. F.

C. Meneses-Fabian, G. Rodriguez-Zurita, R. Rodriguez-Vera, and J. F. Vazquez-Castillo, “Optical tomography with parallel projection differences and Electronic Speckle Pattern Interferometry,” Opt. Commun. 228(4-6), 201–210 (2003).
[CrossRef]

Vázquez-Castillo, J.-F.

G. Rodríguez-Zurita, C. Meneses-Fabián, J.-S. Pérez-Huerta, and J.-F. Vázquez-Castillo, ““Tomographic directional derivative of phase objects slices using 1-D derivative spatial filtering of fractional order ½,” ICO20,” Proc. SPIE 6027, 410–416 (2006).

Vest, C. M.

I. H. Lira and C. M. Vest, “Refraction correction in holographic interferometry and tomography of transparent objects,” Appl. Opt. 26(18), 3919–3928 (1987).
[CrossRef] [PubMed]

S. Cha and C. M. Vest, “Tomographic reconstruction of strongly refracting fields and its application to interferometric measurement of boundary layers,” Appl. Opt. 20(16), 2787–2794 (1981).
[CrossRef] [PubMed]

Vijayan, C.

K. Sendhil, C. Vijayan, and M. P. Kothiyal, “Spatial phase filtering with a porphyrin derivative as phase filter in an optical image processor,” Opt. Commun. 251(4-6), 292–298 (2005).
[CrossRef]

Wu, D.

D. Wu and A. He, “Measurement of three-dimensional temperature fields with interferometric tomography,” Appl. Opt. 38(16), 3468–3473 (1999).
[CrossRef]

Yan, D.

D. Yan and S. S. Cha, “Computational and interferometric system for real-time limited-view tomography of flow fields,” Appl. Opt. 37(7), 1159–1164 (1998).
[CrossRef]

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955).
[CrossRef] [PubMed]

Appl. Opt. (6)

D. Yan and S. S. Cha, “Computational and interferometric system for real-time limited-view tomography of flow fields,” Appl. Opt. 37(7), 1159–1164 (1998).
[CrossRef]

D. Wu and A. He, “Measurement of three-dimensional temperature fields with interferometric tomography,” Appl. Opt. 38(16), 3468–3473 (1999).
[CrossRef]

I. H. Lira and C. M. Vest, “Refraction correction in holographic interferometry and tomography of transparent objects,” Appl. Opt. 26(18), 3919–3928 (1987).
[CrossRef] [PubMed]

S. Cha and C. M. Vest, “Tomographic reconstruction of strongly refracting fields and its application to interferometric measurement of boundary layers,” Appl. Opt. 20(16), 2787–2794 (1981).
[CrossRef] [PubMed]

J. A. Davis, D. E. McNamara, and D. M. Cottrell, “Analysis of the fractional hilbert transform,” Appl. Opt. 37(29), 6911–6913 (1998).
[CrossRef]

J. A. Davis and M. D. Nowak, “Selective edge enhancement of images with an acousto-optic light modulator,” Appl. Opt. 41(23), 4835–4839 (2002).
[CrossRef] [PubMed]

J. Electron. Imaging (1)

G. S. Abdoulaev and A. H. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging 12(4), 594–601 (2003).
[CrossRef]

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

C. Meneses-Fabian, G. Rodriguez-Zurita, and V. Arrizón, “Optical tomography of transparent objects with phase-shifting interferometry and stepwise-shifted Ronchi ruling,” J. Opt. Soc. Am. A 23(2), 298–305 (2006).
[CrossRef]

Opt. Commun. (2)

C. Meneses-Fabian, G. Rodriguez-Zurita, R. Rodriguez-Vera, and J. F. Vazquez-Castillo, “Optical tomography with parallel projection differences and Electronic Speckle Pattern Interferometry,” Opt. Commun. 228(4-6), 201–210 (2003).
[CrossRef]

K. Sendhil, C. Vijayan, and M. P. Kothiyal, “Spatial phase filtering with a porphyrin derivative as phase filter in an optical image processor,” Opt. Commun. 251(4-6), 292–298 (2005).
[CrossRef]

Opt. Express (1)

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13(3), 689–694 (2005).
[CrossRef] [PubMed]

Opt. Lett. (1)

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
[CrossRef]

Proc. SPIE (1)

G. Rodríguez-Zurita, C. Meneses-Fabián, J.-S. Pérez-Huerta, and J.-F. Vázquez-Castillo, ““Tomographic directional derivative of phase objects slices using 1-D derivative spatial filtering of fractional order ½,” ICO20,” Proc. SPIE 6027, 410–416 (2006).

Science (1)

F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955).
[CrossRef] [PubMed]

Other (2)

J. Glückstad, P. C. Mogensen and R. L. Eriksen, “The generalized phase contrast method and its applications,” DOPS-NYT 1–2001, 49–54.

S. R. Deans, The Radon Transform and Some of its Applications (Wiley, New York. 1983).

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Figures (4)

Fig. 1
Fig. 1

(Color online) A plane wave crosses a phase object in a rotated 3-D reference system: (a) object in 3D rotating around z axis to generate the projection angles and (b) an object slice at z constant.

Fig. 2
Fig. 2

(Color online) Experimental setup to obtain the HT of the projection data. The slice function f ( x , y ) is obtained at z constant and this is rotated around z axis to obtain the projection angle.

Fig. 3
Fig. 3

(Color online) Numerical simulation. Edge-enhancement along the vertical direction of a unitary rectangle used as slice function of a thin phase object: (a) slice, (b) sinogram, (c) Hilbert-sinogram, (d-e) Vertical edge enhancement reconstruction. Second row shows a line or column data corresponding to each image at first row as it is indicated with dotted-yellow line.

Fig. 4
Fig. 4

(Color online) Numerical simulations. (a) Two test slices: a phantom and a uniform ring, (b) Sinograms, (c, e, g) Hilbert-sinograms obtained under use of Eq. (13) with α = 0 ° ,   45 ° ,   90 ° , respectively. (d, f, h): reconstructions showing directional edge-enhancement using Hilbert-sinograms in (c, e, g) respectively.

Equations (29)

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

A ( p , z ) exp [ i ( 2 π λ p ω t ) ]
A φ ( p , z ) = A ( p , z ) exp [ i ϕ φ ( p , z ) ]
φ { f ( x , y ) } = f φ ( p ) = d ξ d η f ( ξ , η ) δ ( p ξ cos φ η sin φ )
A φ ( p ) 1 + i ϕ φ ( p ) = 1 + i 2 π λ f φ ( p )
A φ ( p ) = A φ ( p ) h φ ( p ) = A φ ( p ) 1 π p = H { A φ ( p ) }
A φ ( p ) = H { 1 + i 2 π λ f φ ( p ) } = i 2 π λ f φ ( p )
φ { f ( x , y ) } = f φ ( p ) = f ( x , y ) δ ( p )
1 D { φ { f ( x , y ) } } = f ~ φ ( w ) = f ~ ( μ , ν ) δ ( w ) ,
H { f φ ( p ) } = f φ ( p ) = f φ ( p ) 1 π p
f φ + π ( p ) = f φ + π ( p ) 1 π p
f φ + π ( p ) = f φ ( p ) 1 π ( p ) = f φ ( p )
sgn ( sin φ ) f φ ( p ) = sgn ( sin φ ) f φ ( p ) 1 π p
d p sgn ( sin φ ) f φ ( p ) = sgn ( sin φ ) d p f φ ( p ) 1 π p
d p sgn ( sin φ ) f φ ( p ) = 1 π sgn ( sin φ ) d p d q f φ ( q ) 1 p q
d p sgn ( sin φ ) f φ ( p ) = sgn ( sin φ ) d q f φ ( q ) 1 π d p q p = 0
sgn ( sin φ ) f φ ( p ) = 1 { i sgn ( w sin φ ) f φ ~ ( w ) }
sgn ( sin φ ) f φ ( p ) = 1 { i sgn ( ν ) f ~ ( μ , ν ) δ ( w ) }
sgn ( sin φ ) f φ ( p ) = [ δ ( x ) π y f ( x , y ) ] δ ( p )
sgn ( sin φ ) f φ ( p ) = φ { f π / 2 ( x , y ) } = f φ π / 2 ( p )
sgn ( sin φ ) H { φ { f ( x , y ) } } = φ { H π / 2 { f ( x , y ) } }
h ~ φ ( w ) = i sgn ( w sin φ ) = { i sgn ( w ) ,        φ ( 0 , π ) + i sgn ( w ) ,     φ ( π , 2 π )
H α { f ( x , y ) } = f α ( x , y ) = δ ( x sin α + y cos α ) π ( x cos α + y sin α ) f ( x , y )
f φ α ( p ) = f α ( x , y ) δ ( p )
f φ ~ α ( w ) = [ i sgn ( μ cos α + ν sin α ) f ~ ( μ , ν ) ] δ ( w )
f φ ~ α ( w ) = h ~ φ ( w ) f φ ~ ( w )
h ~ φ ( w ) = i sgn ( w ) sgn [ cos ( φ α ) ] = i sgn [ w cos ( φ α ) ]
h ~ φ ( w ) = i sgn ( w cos φ ) = { i sgn ( w ) ,           φ ( 0 , π / 2 ) + i sgn ( w ) ,     φ ( π / 2 , 3 π / 2 ) i sgn ( w ) ,       φ ( 3 π / 2 , 2 π )
f φ α ( p ) = sgn [ cos ( φ α ) ] f φ ( p )
φ { H α { f ( x , y ) } } = sgn [ cos ( φ α ) ] H { φ { f ( x , y ) } }

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