## Directional edge enhancement in optical tomography of thin phase objects |

Optics Express, Vol. 19, Issue 3, pp. 2608-2618 (2011)

http://dx.doi.org/10.1364/OE.19.002608

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### 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 4*f* 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

© 2011 OSA

## 1. Introduction

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]

*π*radians as a spatial filter in the frequency space, such as it has been proposed in several investigations [10

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]

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]

## 2. Basic considerations

*λ*and

*ω*are the wavelength and the frequency of light respectively,

**r**is a position vector, and

*φ*is the azimuth rotation angle with respect to

*x*-axis, as is illustrated in Fig. 1 . In rotated coordinates this plane wave can be rewritten aswhere the equations

*z*by traveling on the

*z,*and in particular, the optical path can be expressed in a more convenient form, relating it with the RT [2] by means ofwhere

*φ*in the

*φ*is kept fixed, while the projection coordinate

*p*is varied, so that under these conditions Eq. (3) are considered to be a sample of the RT, and they are known as a profile or a projection at

*φ*angle. As it is well known in tomographic theory, these projection data constitute the basis for the reconstruction of an object slice [1

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]

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]

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

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]

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]

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]

*p*coordinate). A given object slice defines the function

*L*

_{1}performs the FT of the wave leaving the object

*π*phase-shifting layer, which is the step phase, and it is mathematically approximated by the

*signum*function

## 3. Theoretical analysis

*φ*in the range

*φ*, as described in Eq. (3), is denoted by

*φ*,

*p*and

*μ*and

*ν*(the spatial frequencies) at the

*φ*angle. Equation (8) is known as the Fourier slice theorem, which means that the projection spectrum is equal to a sample of the slice function bi-dimensional spectrum along of the line

### 3.1. HT of projections: Proof

*φ*projection angle by using Eq. (6). The

*φ*angle is carried out by rotating the object around the

*z*axis and by considering that the reference system turns together the object, while the optical system is kept fixed; i. e.,

*φ*. The symmetry property states thatbut the projection

**12**(4), 594–601 (2003). [CrossRef]

*φ*∈ (0,

*π*) and negative when

### 3.2. Vertical edge-enhancement: particular case

*φ*)sgn(

*w*), and

*f*(

*x*,

*y*), thus Equation (20) means that when a sign compensation by means of

### 3.2.1. Numerical simulation 1

*p*has 200 data entries and the angular step used is 1.8°, so the projection number is 200 for

### 3.3. Directional edge-enhancement: general case

*α*asNote that if

*α*according with Eq. (22). The FT of Eq. (23) can thus be written as

*α*of the slice function

*α*and the

*φ*projection angle through the cosine function. Therefore, the experimental implementation is viable for any given direction of enhancement by using the same scheme depicted in Fig. 2. For the case of

*α*and the HT of the RT of

### 3.3.1. Numerical simulation 2

*α*angle can be done with the same optical system of Fig. 2 and the same phase-step filter, but with the difference of choosing its initial position according to the angle

*α*. Such a position is the angle value at which the turning of the filter has to be done in order to fulfill the RT symmetry properties.

## 4. Conclusions

*π*radians. In general, the filter has to be rotated 180° once around the optical axis at a certain projecting angle

*α*(in practice, when

*α*. Thus, it is possible to choose the enhancement direction only selecting the projection angle value at which the turn is made. Numerical results for two different slices were presented, using three different values of

*α*. An experimental implementation of this proposal seems thus to be possible and work on it is in progress.

## Acknowledgment

## References and links

1. | G. S. Abdoulaev and A. H. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging |

2. | S. R. Deans, |

3. | D. Yan and S. S. Cha, “Computational and interferometric system for real-time limited-view tomography of flow fields,” Appl. Opt. |

4. | D. Wu and A. He, “Measurement of three-dimensional temperature fields with interferometric tomography,” Appl. Opt. |

5. | I. H. Lira and C. M. Vest, “Refraction correction in holographic interferometry and tomography of transparent objects,” Appl. Opt. |

6. | S. Cha and C. M. Vest, “Tomographic reconstruction of strongly refracting fields and its application to interferometric measurement of boundary layers,” Appl. Opt. |

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 |

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. |

9. | G. Rodríguez-Zurita, C. Meneses-Fabián, J.-S. Pérez-Huerta, and J.-F. Vázquez-Castillo, “ |

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. |

11. | J. A. Davis, D. E. McNamara, and D. M. Cottrell, “Analysis of the fractional hilbert transform,” Appl. Opt. |

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. |

13. | J. A. Davis and M. D. Nowak, “Selective edge enhancement of images with an acousto-optic light modulator,” Appl. Opt. |

14. | F. Zernike, “How I discovered phase contrast,” Science |

15. | J. Glückstad, P. C. Mogensen and R. L. Eriksen, “The generalized phase contrast method and its applications,” DOPS-NYT |

16. | S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express |

**OCIS Codes**

(070.6110) Fourier optics and signal processing : Spatial filtering

(100.1160) Image processing : Analog optical image processing

(170.6960) Medical optics and biotechnology : Tomography

(200.4740) Optics in computing : Optical processing

(110.6955) Imaging systems : Tomographic imaging

**ToC Category:**

Fourier Optics and Signal Processing

**History**

Original Manuscript: November 2, 2010

Revised Manuscript: December 24, 2010

Manuscript Accepted: December 27, 2010

Published: January 27, 2011

**Virtual Issues**

Vol. 6, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Cruz Meneses-Fabian, Areli Montes-Perez, and Gustavo Rodriguez-Zurita, "Directional edge enhancement in optical tomography of thin phase objects," Opt. Express **19**, 2608-2618 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2608

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### References

- 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]
- S. R. Deans, The Radon Transform and Some of its Applications (Wiley, New York. 1983).
- 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]
- 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]
- 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).
- 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]
- 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, 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 and M. D. Nowak, “Selective edge enhancement of images with an acousto-optic light modulator,” Appl. Opt. 41(23), 4835–4839 (2002). [CrossRef] [PubMed]
- F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955). [CrossRef] [PubMed]
- J. Glückstad, P. C. Mogensen and R. L. Eriksen, “The generalized phase contrast method and its applications,” DOPS-NYT 1–2001, 49–54.
- 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]

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