## Isotropic edge-enhancement by the Hilbert-transform in optical tomography of phase objects |

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

© 2011 OSA

## 1. Introduction

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]

*f*optical imaging system [9].

*f*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. 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

*f*

_{L}optical imaging system, as it is depicted in Fig. 2 , where the field indicated in Eq. (1) is considered as the entrance of the Fourier system. Lens

**H**{...} indicating the operator of HT, and is described bywhere

*z*-value, the one-dimensional HT in the

*p*-direction is carried-out.

*u*is an auxiliary variable of integration. Now, considering

*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

*z*kept constant. The images are presented in 8-bit gray levels. Figure 3-a1 shows an object slice

*p*has 200 data, and the number of projections is 200 for

## 4. Experimental results

*f*

_{L}optical system depicted in Fig. 2. The positive lenses

*π*radians; and

*z*-axis, and it is put just before the entrance plane of the imaging system. The step motor has a nominal step of 1.8°, and it is driven by a visual interface in the computer via an electronic driver (not shown). A CCD camera is placed at the image plane to capture the projection data. The visual interface controls and synchronizes projection data acquisition with the step motor. Then, to obtain an experimental IH-sinogram, a data line is selected from the actual image. This line defines an object slice and is fixed for all projections. This data row is placed on a new image as the first row to begin to build the IH-sinogram. The step motor turns a step, the next projection angle is generated, and a new image is obtained; then, the aforementioned procedure is applied to obtain the second row at the IH-sinogram. This process is repeated until the object has turned 360°, and therefore an IH-sinogram of the 200 projections is generated. During reconstruction, due to the symmetry property of parallel projections for the reconstructions only are used the projections in the range

*mm*in external diameter and 0.25

*mm*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 . 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

*f*

_{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

## Acknowledgments

## References and links

1. | S. R. Deans, “ |

2. | A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. |

3. | G. P. Montgomery Jr and D. L. Reuss, “Effects of refraction on axisymmetric flame temperatures measured by holographic interferometry,” Appl. Opt. |

4. | J. M. Mehta and W. Z. Black, “Errors associated with interferometric measurement of convective heat transfer coefficients,” Appl. Opt. |

5. | J. M. Mehta and W. M. Worek, “Analysis of refraction errors for interferometric measurements in multicomponent systems,” 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 F. Jose, “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. | J. A. Davis, D. E. McNamara, and D. M. Cottrell, “Analysis of the fractional hilbert transform,” Appl. Opt. |

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. |

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

13. | J. Hsieh, “ |

**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

- S. R. Deans, “The Radon Transform and Some of its Applications,” (Wiley, New York. 1983).
- 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]
- 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]
- 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]
- 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]
- 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 F. Jose, “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).
- 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]
- J. Hsieh, “Computed Tomography: principles, design, artifacts, and recent advances,” (SPIE PRESS, Bellingham, Washington USA, 2003).

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