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Virtual Journal for Biomedical Optics

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  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 6, Iss. 4 — May. 4, 2011
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Isotropic edge-enhancement by the Hilbert-transform in optical tomography of phase objects

Areli Montes-Perez, Cruz Meneses-Fabian, and Gustavo Rodriguez-Zurita  »View Author Affiliations


Optics Express, Vol. 19, Issue 6, pp. 5350-5356 (2011)
http://dx.doi.org/10.1364/OE.19.005350


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Abstract

In optical tomography, isotropic edge-enhancement of phase-object slices under the refractionless limit approximation can be reconstructed using spatial filtering techniques. The optical Hilbert-transform of the transmittance function leaving the object at projection angles ϕ ( 0 0 , 360 0 ) , is one of these techniques with some advantages. The corresponding irradiance of the so modified transmittance is considered as projection data, and is proved that they share two properties with the Radon transform: its symmetry property and its zeroth-moment conservation. Accordingly, a modified sinogram able to reconstruct edge-enhanced phase slices is obtained. In this paper, the theoretical model is amply discussed and illustrated both with numerical and experimental results.

© 2011 OSA

1. Introduction

Tomography is a non-invasive technique aimed at obtaining spatial distributions of some physical property lying within a given object. A probe beam, often an input wave, travels across the object under inspection, carrying away information about its interior. At the output, the probe is modified in a distribution known as projection. In general, each projection depends on the angle the probe forms with a direction fixed to the object (projection angle). Optical tomography is a special case of tomography in which the probe is a light beam [1

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

]. Optical tomographic techniques are of particular importance in the medical field, because these techniques can provide non-invasive diagnostic images. When the beam travels through the object interior with no deviation (refractiveless, difractionless, isotropic and without scattering), the probe beam can be considered as composed of straight rays. Being such a beam composed of only parallel rays, it is spoken of parallel ray tomography. The present proposal could be a very particular case of the optical diffraction and/or diffuse optical tomography [2

2. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003). [CrossRef]

]. A feature of this case is that a description with Fourier methods turns out to be very useful. In this context, a parallel projection can be conveniently described with a Radon transform (RT), as it has been done in several fields, including computer aided x-ray tomography, radar imagery, and geophysics. In optical tomography, a lot of research has been carried out to study transparent objects, such as temperature in flames considering axial symmetry [3

3. G. P. Montgomery Jr and D. L. Reuss, “Effects of refraction on axisymmetric flame temperatures measured by holographic interferometry,” Appl. Opt. 21(8), 1373–1380 (1982). [CrossRef] [PubMed]

], or considering error by this approximation [4

4. J. M. Mehta and W. Z. Black, “Errors associated with interferometric measurement of convective heat transfer coefficients,” Appl. Opt. 16(6), 1720–1726 (1977). [CrossRef] [PubMed]

], and also refraction effects as well [5

5. J. M. Mehta and W. M. Worek, “Analysis of refraction errors for interferometric measurements in multicomponent systems,” Appl. Opt. 23(6), 928–933 (1984). [CrossRef] [PubMed]

-6

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]

]. Most of the aforementioned techniques obtain information from the detected interferograms [7

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]

]. These techniques have been oriented to obtaining the distribution of phase object slices. However, in many practical situations, the phase gradient is too large, and conventional interferometry methods are not practicable. In order to overcome or minimize these deficiencies, other approaches have been proposed to obtain some modified object slice instead of the object slice itself, such as obtaining the angular derivative by using speckle interferometry [8

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

], or the directional derivative by using a semiderivative filter in a 4f optical imaging system [9

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).

].

In the present manuscript, the optical HT is implemented by using a 4f optical imaging system, and a phase step of π radians used as a spatial filter at the Fourier plane of the projection detection system [10

10. 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

12. 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]

], in order to obtain the HT of the field leaving the phase object. In this proposal, the irradiance of field at the image plane is considered as a special sinogram and is demonstrated that, with this sinogram, an enhancement of borders of tomographic images is obtained, a technique that could be particularly useful for phase-only objects. A theoretical background to support the method is presented, along with some numerical and experimental results.

2. Basic considerations

We propose to carry out the optical HT of the field leaving the object by means of a 4f L optical imaging system, as it is depicted in Fig. 2
Fig. 2 (color online) 4f L optical imaging system with a sign filter at the Fourier plane to obtain the HT of the projection data leaving the phase object.
, where the field indicated in Eq. (1) is considered as the entrance of the Fourier system. Lens L1 performs the FT of the wave leaving the object, located on the back focal plane (system’s frequency plane). Lens L2 performs the inverse FT after a sign filter is placed at the frequency plane, and then, at the image plane, the resulting field is the HT of entrance field, which is symbolically expressed by Aϕ(p,z)=H{Aϕ(p,z)}, with H{...} indicating the operator of HT, and is described by
Aϕ(p,z)=1{isgn(w)A~ϕ(w,ζ)},
(3)
where 1{} is the operator of the inverse FT, A~ϕ(w,ζ)={Aϕ(p,z)}, sgn(w) is the filter function, and (w,ζ) are spatial frequencies. By applying the convolution property of the FT in Eq. (3) we have
Aϕ(p,z)=δ(z)πpAϕ(p,z),
(4)
meaning that for every z-value, the one-dimensional HT in the p-direction is carried-out. δ(z)/πp=1{isgn(w)} is known as the system’s impulse response. The irradiance of the field at the image plane would thus be written as
Iϕ(p,z)=Aϕ(p,z)Aϕ*(p,z)=|Aϕ(p,z)|2=|δ(z)πpAϕ(p,z)|2,
(5)
where the symbol * indicates the complex conjugate. Now, considering Eq. (5) as the projection data, the symmetry property of RT should be
Iϕ+π(p,z)=|δ(z)πpAϕ+π(p,z)|2=|δ(z)π(p)Aϕ(p,z)|2=Iϕ(p,z),
(6)
where Aϕ+π(p,z)=Aϕ(p,z) is verifiable from Eq. (1) because fϕ+π(p,z)=fϕ(p,z) is a well-known property [1

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

]. For the zero-moment of RT, it could be established that
dpIϕ(p,z)=I~ϕ(w,z)|w=0={isgn(w)A~ϕ(w,z)isgn(w)Aϕ*~(w,z)}|w=0,
(7)
where is assumed that I~ϕ(w,z)={Iϕ(p,z)}={Aϕ(p,z)}{Aϕ*(p,z)}. Substituting the integral definition of convolution, and evaluating at w=0, we obtain
dpIϕ(p,z)=dusgn2(u)A~ϕ(u,z)Aϕ*~(u,z),
(8)
where u is an auxiliary variable of integration. Now, considering sgn2(u)=1, and applying the Parseval theorem, Eq. (8) could be written as
dpIϕ(p,z)=dpAϕ(p,z)Aϕ*(p,z),
(9)
indicating that the total intensity is the same both at the object and image planes, since the used filter is an only-phase one. Substituting Eq. (1) into Eq. (9), we finally have
dpIϕ(p,z)=Γ/2Γ/2dpA2=A2Γ,
(10)
where Γ is the detector width, and the irradiance out of this region is taken to be zero. Considering the irradiance in the image plane as the projection data, the zero-moment of the RT is the total energy at the detector, finite and independent of ϕ. That way, the zero-moment of the RT is satisfied. In addition, as it has been demonstrated, the irradiance at the image plane (Eq. (5) complies both with the symmetry property and the zero-moment of the RT; therefore, it can be considered as a projection of a given object, and therefore its tomographic reconstruction must render a consistent image.

3. Numerical Simulation

In Fig. 4
Fig. 4 (color online) Numerical simulations. (a) Two test slices: uniform ring and non-symmetric object, (b) IH-Sinograms, (c) Zero-moment of RT, d-e) reconstructions showing isotropic edge-enhancement.
, two examples are shown: on row 1, a uniform ring is considered as the object slice, and on row 2, a non-symmetric object slice is considered. On column (a), the object slices are shown; on column (b), the IH-sinograms are shown; on column (c) the zero-moments of the RT are illustrated; finally, on columns (d) and (e), both the reconstructions in gray-levels and on 3-D plots are respectively shown.

4. Experimental results

Two phase objects were used in this experiment: a rectangular slide piece of 1×3×20mm3 in size, and a pipette of 5.6mm in external diameter and 0.25mm in wall thickness, both made of Pyrex glass with a nominal refraction index of 1.52; a photo of each object is depicted in Fig. 5a
Fig. 5 (color online) Experimental results, (a) rectangular slide glass block, (b) pipette.
. Figure 5b shows the IH-sinograms constructed from a slice from each one of the objects; the slices are indicated on the objects in Fig. 5a with a white-dotted line, and Fig. 5c shows the corresponding tomographic reconstructions by using the back-projection algorithm.

5. Conclusion and remarks

The irradiance at the image plane of a 4f L optical imaging system using a phase step filter of π radians has been considered as the projection data in optical tomography of phase objects, and it has been mathematically proved that this irradiance complies with the symmetry property and with the zero moment of the RT. It was shown that, for the all possible projection angles, a modified sinogram can be obtained, and this is called irradiance-Hilbert-sinogram (IH-sinogram). As a consequence of using directly these IH-sinograms, the obtained reconstructions consist of images showing isotropic edge-enhancement for both numerical simulations and experimental results. Thus, the HT filtering approach does not only serve to detect phase projections, but it is also capable to render phase-edge enhanced tomographic images as an extra feature after a usual reconstruction using a routine algorithm. It is important to note that, for the phase objects in the numerical simulation, the obtained projection data were smaller or equal than a wavelength, so that the phase was minor or equal than 2π radians. Therefore, the irradiance did not show interference fringes (Fig. 3-d1 and Fig. 4b), and this condition was sought to be met for the phase objects shown in Fig. 5a. However, some fringes can be observed in the experimental IH-sinograms, as illustrated in Fig. 5b. Maybe due to these diffraction fringes, some discontinuities in the reconstructed images can be observed (Fig. 5c). For thick-phase objects, the interference pattern will be inevitably present, and a new method will be necessary. On the other hand, when thin-phase objects are considered, the transmittance function can be approximated to the projection data, and then a new method will also be necessary. We have taken these two considerations for future work.

Acknowledgments

One of the authors (AMP) appreciates CONACYT’s grant under number 160260/160260. This work was partially supported by PROMEP under grant PROMEP/103.5/09/4544.

References and links

1.

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

2.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003). [CrossRef]

3.

G. P. Montgomery Jr and D. L. Reuss, “Effects of refraction on axisymmetric flame temperatures measured by holographic interferometry,” Appl. Opt. 21(8), 1373–1380 (1982). [CrossRef] [PubMed]

4.

J. M. Mehta and W. Z. Black, “Errors associated with interferometric measurement of convective heat transfer coefficients,” Appl. Opt. 16(6), 1720–1726 (1977). [CrossRef] [PubMed]

5.

J. M. Mehta and W. M. Worek, “Analysis of refraction errors for interferometric measurements in multicomponent systems,” Appl. Opt. 23(6), 928–933 (1984). [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 F. Jose, “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.

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

11.

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]

12.

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]

13.

J. Hsieh, “Computed Tomography: principles, design, artifacts, and recent advances,” (SPIE PRESS, Bellingham, Washington USA, 2003).

OCIS Codes
(070.1170) Fourier optics and signal processing : Analog optical signal processing
(100.2980) Image processing : Image enhancement
(100.6950) Image processing : Tomographic image processing
(070.2615) Fourier optics and signal processing : Frequency filtering

ToC Category:
Image Processing

History
Original Manuscript: November 29, 2010
Revised Manuscript: January 20, 2011
Manuscript Accepted: January 21, 2011
Published: March 7, 2011

Virtual Issues
Vol. 6, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Areli Montes-Perez, Cruz Meneses-Fabian, and Gustavo Rodriguez-Zurita, "Isotropic edge-enhancement by the Hilbert-transform in optical tomography of phase objects," Opt. Express 19, 5350-5356 (2011)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-6-5350


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References

  1. S. R. Deans, “The Radon Transform and Some of its Applications,” (Wiley, New York. 1983).
  2. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003). [CrossRef]
  3. G. P. Montgomery and D. L. Reuss, “Effects of refraction on axisymmetric flame temperatures measured by holographic interferometry,” Appl. Opt. 21(8), 1373–1380 (1982). [CrossRef] [PubMed]
  4. J. M. Mehta and W. Z. Black, “Errors associated with interferometric measurement of convective heat transfer coefficients,” Appl. Opt. 16(6), 1720–1726 (1977). [CrossRef] [PubMed]
  5. J. M. Mehta and W. M. Worek, “Analysis of refraction errors for interferometric measurements in multicomponent systems,” Appl. Opt. 23(6), 928–933 (1984). [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 F. Jose, “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. J. A. Davis, D. E. McNamara, and D. M. Cottrell, “Analysis of the fractional hilbert transform,” Appl. Opt. 37(29), 6911–6913 (1998). [CrossRef]
  11. 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]
  12. 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]
  13. J. Hsieh, “Computed Tomography: principles, design, artifacts, and recent advances,” (SPIE PRESS, Bellingham, Washington USA, 2003).

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