Abstract

This paper presents a novel optical system for the realization of the Radon transform in a single frame. The optical system is simple, fast and accurate and consists of a 4F system, where in the 2F plane a vortex like optical element is placed. This optical element performs the rotation of the object, which replaces the need for mechanically rotating it, as is done in other common optical realization techniques of the Radon transform. This optical element is realized using a spatial light modulator (SLM) and an amplitude slide. The obtained Radon transform is given in Cartesian coordinates, which can subsequently be transformed using a computer to a polar set. The proposed concept is supported mathematically, numerically and experimentally.

© 2014 Optical Society of America

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References

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  1. J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte Längs gewisser Mannigfaltigkeiten,” Ber. Sächs. Akad. Wiss 69, 262–278 (1917).
  2. M. Nishimura, D. Casasent, and F. Caimi, “Optical inverse Radon transform,” Opt. Commun. 24(3), 276–280 (1978).
    [Crossref]
  3. F. Colonna, G. Easley, K. Guo, and D. Labate, “Radon transform inversion using the shearlet representation,” Appl. Comput. Harmon. Anal. 29(2), 232–250 (2010).
    [Crossref]
  4. S. R. Deans, The Radon Transform and Some of Its Applications (New York: Wiley, 1983).
  5. D. Mendlovic, R. G. Dorsch, A. W. Lohmann, Z. Zalevsky, and C. Ferreira, “Optical illustration of a varied fractional Fourier-transform order and the Radon-Wigner display,” Appl. Opt. 35(20), 3925–3929 (1996).
    [Crossref] [PubMed]
  6. C. G. Ho, R. C. D. Young, C. D. Bradfield, and C. R. Chatwin, “A Fast Hough Transform for the Parametrisation of Straight Lines using Fourier Methods,” Real Time Imaging 6(2), 113–127 (2000).
    [Crossref]
  7. W. Götz and H. Druckmüller, “A fast digital Radon transform—An efficient means for evaluating the Hough transform,” Pattern Recognit. 29(4), 711–718 (1996).
    [Crossref]
  8. Y. Kashter, O. Levi, and A. Stern, “Optical compressive change and motion detection,” Appl. Opt. 51(13), 2491–2496 (2012).
    [Crossref] [PubMed]
  9. V. Farber, Y. August, and A. Stern, “Super-resolution compressive imaging with anamorphic optics,” Opt. Express 21(22), 25851–25863 (2013).
    [Crossref] [PubMed]
  10. W. H. Steier and R. K. Shori, “Optical Hough transform,” Appl. Opt. 25(16), 2734 (1986).
    [Crossref] [PubMed]
  11. S. Woolven, V. M. Ristic, and P. Chevrette, “Hybrid implementation of a real-time Radon-space image-processing system,” Appl. Opt. 32(32), 6556–6561 (1993).
    [Crossref] [PubMed]
  12. H. H. Barrett and W. Swindell, Radiological Imaging: The Theory of Image Formation, Detection, and Processing‏ (Elsevier, 1996).
  13. O. Fixler and Z. Zalevsky, “Geometrically superresolved lensless imaging using a spatial light modulator,” Appl. Opt. 50(29), 5662–5673 (2011).
    [Crossref] [PubMed]

2013 (1)

2012 (1)

2011 (1)

2010 (1)

F. Colonna, G. Easley, K. Guo, and D. Labate, “Radon transform inversion using the shearlet representation,” Appl. Comput. Harmon. Anal. 29(2), 232–250 (2010).
[Crossref]

2000 (1)

C. G. Ho, R. C. D. Young, C. D. Bradfield, and C. R. Chatwin, “A Fast Hough Transform for the Parametrisation of Straight Lines using Fourier Methods,” Real Time Imaging 6(2), 113–127 (2000).
[Crossref]

1996 (2)

W. Götz and H. Druckmüller, “A fast digital Radon transform—An efficient means for evaluating the Hough transform,” Pattern Recognit. 29(4), 711–718 (1996).
[Crossref]

D. Mendlovic, R. G. Dorsch, A. W. Lohmann, Z. Zalevsky, and C. Ferreira, “Optical illustration of a varied fractional Fourier-transform order and the Radon-Wigner display,” Appl. Opt. 35(20), 3925–3929 (1996).
[Crossref] [PubMed]

1993 (1)

1986 (1)

1978 (1)

M. Nishimura, D. Casasent, and F. Caimi, “Optical inverse Radon transform,” Opt. Commun. 24(3), 276–280 (1978).
[Crossref]

1917 (1)

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte Längs gewisser Mannigfaltigkeiten,” Ber. Sächs. Akad. Wiss 69, 262–278 (1917).

August, Y.

Bradfield, C. D.

C. G. Ho, R. C. D. Young, C. D. Bradfield, and C. R. Chatwin, “A Fast Hough Transform for the Parametrisation of Straight Lines using Fourier Methods,” Real Time Imaging 6(2), 113–127 (2000).
[Crossref]

Caimi, F.

M. Nishimura, D. Casasent, and F. Caimi, “Optical inverse Radon transform,” Opt. Commun. 24(3), 276–280 (1978).
[Crossref]

Casasent, D.

M. Nishimura, D. Casasent, and F. Caimi, “Optical inverse Radon transform,” Opt. Commun. 24(3), 276–280 (1978).
[Crossref]

Chatwin, C. R.

C. G. Ho, R. C. D. Young, C. D. Bradfield, and C. R. Chatwin, “A Fast Hough Transform for the Parametrisation of Straight Lines using Fourier Methods,” Real Time Imaging 6(2), 113–127 (2000).
[Crossref]

Chevrette, P.

Colonna, F.

F. Colonna, G. Easley, K. Guo, and D. Labate, “Radon transform inversion using the shearlet representation,” Appl. Comput. Harmon. Anal. 29(2), 232–250 (2010).
[Crossref]

Dorsch, R. G.

Druckmüller, H.

W. Götz and H. Druckmüller, “A fast digital Radon transform—An efficient means for evaluating the Hough transform,” Pattern Recognit. 29(4), 711–718 (1996).
[Crossref]

Easley, G.

F. Colonna, G. Easley, K. Guo, and D. Labate, “Radon transform inversion using the shearlet representation,” Appl. Comput. Harmon. Anal. 29(2), 232–250 (2010).
[Crossref]

Farber, V.

Ferreira, C.

Fixler, O.

Götz, W.

W. Götz and H. Druckmüller, “A fast digital Radon transform—An efficient means for evaluating the Hough transform,” Pattern Recognit. 29(4), 711–718 (1996).
[Crossref]

Guo, K.

F. Colonna, G. Easley, K. Guo, and D. Labate, “Radon transform inversion using the shearlet representation,” Appl. Comput. Harmon. Anal. 29(2), 232–250 (2010).
[Crossref]

Ho, C. G.

C. G. Ho, R. C. D. Young, C. D. Bradfield, and C. R. Chatwin, “A Fast Hough Transform for the Parametrisation of Straight Lines using Fourier Methods,” Real Time Imaging 6(2), 113–127 (2000).
[Crossref]

Kashter, Y.

Labate, D.

F. Colonna, G. Easley, K. Guo, and D. Labate, “Radon transform inversion using the shearlet representation,” Appl. Comput. Harmon. Anal. 29(2), 232–250 (2010).
[Crossref]

Levi, O.

Lohmann, A. W.

Mendlovic, D.

Nishimura, M.

M. Nishimura, D. Casasent, and F. Caimi, “Optical inverse Radon transform,” Opt. Commun. 24(3), 276–280 (1978).
[Crossref]

Radon, J.

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte Längs gewisser Mannigfaltigkeiten,” Ber. Sächs. Akad. Wiss 69, 262–278 (1917).

Ristic, V. M.

Shori, R. K.

Steier, W. H.

Stern, A.

Woolven, S.

Young, R. C. D.

C. G. Ho, R. C. D. Young, C. D. Bradfield, and C. R. Chatwin, “A Fast Hough Transform for the Parametrisation of Straight Lines using Fourier Methods,” Real Time Imaging 6(2), 113–127 (2000).
[Crossref]

Zalevsky, Z.

Appl. Comput. Harmon. Anal. (1)

F. Colonna, G. Easley, K. Guo, and D. Labate, “Radon transform inversion using the shearlet representation,” Appl. Comput. Harmon. Anal. 29(2), 232–250 (2010).
[Crossref]

Appl. Opt. (5)

Ber. Sächs. Akad. Wiss (1)

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte Längs gewisser Mannigfaltigkeiten,” Ber. Sächs. Akad. Wiss 69, 262–278 (1917).

Opt. Commun. (1)

M. Nishimura, D. Casasent, and F. Caimi, “Optical inverse Radon transform,” Opt. Commun. 24(3), 276–280 (1978).
[Crossref]

Opt. Express (1)

Pattern Recognit. (1)

W. Götz and H. Druckmüller, “A fast digital Radon transform—An efficient means for evaluating the Hough transform,” Pattern Recognit. 29(4), 711–718 (1996).
[Crossref]

Real Time Imaging (1)

C. G. Ho, R. C. D. Young, C. D. Bradfield, and C. R. Chatwin, “A Fast Hough Transform for the Parametrisation of Straight Lines using Fourier Methods,” Real Time Imaging 6(2), 113–127 (2000).
[Crossref]

Other (2)

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

H. H. Barrett and W. Swindell, Radiological Imaging: The Theory of Image Formation, Detection, and Processing‏ (Elsevier, 1996).

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Figures (7)

Fig. 1
Fig. 1 The optical setup which consists of a 4F system, where an optical element is placed in the 2F plane.
Fig. 2
Fig. 2 A schematic sketch of the obtained output of the proposed technique. (a) an input object of a slit. (b) The obtained output that is the Radon transform of the input object in Cartesian coordinates.
Fig. 3
Fig. 3 Simulation results. (a) An input object of a Shepp-Logan head phantom. (b) The mathematical Radon transform of the input object. (c) The obtained output image using the proposed setup. (d) The output image transformed to polar coordinates.
Fig. 4
Fig. 4 The experimental setup (right to left). A 4F system that consists of a collimated green laser beam at a wavelength of 532 nm, illuminating an input object. The light propagates through a 2F system, and illuminates an optical element realized using an amplitude slide attached to a SLM. The reflected light from the optical element propagates through a second 2F system and is captured using a standard USB camera.
Fig. 5
Fig. 5 The proposed optical element. (a) The phase of the optical element. (b) The amplitude of the optical element.
Fig. 6
Fig. 6 Experimental results.(a) The input object is a slit of dimensions of 0.25x0.5 mm. (b) The simulated output image using the proposed setup. (c) The experimental output image using the proposed setup. (d) The mathematical Radon transform of the input object. (d) The output image transformed to polar coordinates. (e) The output image of (c) transformed to polar coordinates.
Fig. 7
Fig. 7 Experimental results. (a) The input object, an amplitude slide showing the outline of an airplane with dimensions of 1.5x2.0 mm. (b) The experimental results obtained using the proposed setup. (c) The experimental results transformed to polar coordinates. (d) The reconstruction of the input object using the inverse Radon transform applied on the experimental result in (c).

Equations (9)

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L=xcos( θ )+ysin( θ )
g( L,θ )= f( x,y )δ(xcos( θ )+ysin( θ )L)dxdy
S ˜ (x,y)= s(x',y')exp( 2πi(xx'+yy') λF )dx'dy' = S ˜ (r,θ)
T(r,θ)= exp( 2πiβθr ) r
O(ρ,α)= 0 2π 0 S ˜ (r,θ)T (r,θ)exp( 2πi λF rρcos( θα ) )rdrdθ
O(ρ,α)= 0 2π 0 S ˜ (r,θ) exp( 2πi λF rρcos( θα ) )exp( 2πiβθr )drdθ
O(ρ,α) n O n (ρ,α) = n 0 S ˜ (r,nδθ) exp( 2πi λF rρcos( nδθα ) )exp( 2πiβnδθr )dr
O(ρ,α=nδθ)= 0 S ˜ (r,nδθ) exp( 2πi λF rρ )exp( 2πiβnδθr )dr= S ˜ ˜ ( ρ λF βnδθ,α=nδθ )
O(ρ,α)= n S ˜ ˜ ( ρ λF βnδθ,α=nδθ )

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