## Investigation on reconstruction methods applied to 3D terahertz computed tomography |

Optics Express, Vol. 19, Issue 6, pp. 5105-5117 (2011)

http://dx.doi.org/10.1364/OE.19.005105

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### Abstract

3D terahertz computed tomography has been performed using a monochromatic millimeter wave imaging system coupled with an infrared temperature sensor. Three different reconstruction methods (standard back-projection algorithm and two iterative analysis) have been compared in order to reconstruct large size 3D objects. The quality (intensity, contrast and geometric preservation) of reconstructed cross-sectional images has been discussed together with the optimization of the number of projections. Final demonstration to real-life 3D objects has been processed to illustrate the potential of the reconstruction methods for applied terahertz tomography.

© 2011 Optical Society of America

## 1. Introduction

1. W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. **70**, 1325–1379 (2007). [CrossRef]

2. K. Kawase, Y. Ogawa, Y. Watanabe, and H. Inoue, “Non-destructive terahertz imaging of illicit drugs using spectral fingerprints,” Opt. Express **11**, 2549–2554 (2003). [CrossRef] [PubMed]

3. Y. C. Shen, T. Lo, P. F. Taday, B. E. Cole, W. R. Tribe, and M. C. Kemp, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. **86**, 241116 (2005). [CrossRef]

4. K. Fukunaga and M. Picollo, “Terahertz spectroscopy applied to the analysis of artists materials,” Appl. Phys. A **100**, 591–597 (2010). [CrossRef]

5. E. Abraham, A. Younus, J.-C. Delagnes, and P. Mounaix, “Non-invasive investigation of art paintings by terahertz imaging,” Appl. Phys. A **100**, 585–590 (2010). [CrossRef]

6. S. Y. Huang, Y. X. J. Wang, D. K. W. Yeung, A. T. Ahuja, Y. T. Zhang, and E. Pickwell-MacPherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. **54**, 149–160 (2009). [CrossRef]

7. J. Takayanagi, H. Jinno, S. Ichino, K. Suizu, M. Yamashita, T. Ouchi, S. Kasai, H. Ohtake, H. Uchida, N. Nishizawa, and K. Kawase, “High-resolution time-of-flight terahertz tomography using a femtosecond fiber laser,” Opt. Express **17**, 7549–7555 (2009). [CrossRef] [PubMed]

8. K. Iwaszczuk, H. Heiselberg, and P. U. Jepsen, “Terahertz radar cross section measurements,” Opt. Express **18**, 26399–26408 (2010). [CrossRef] [PubMed]

9. B. Ferguson, S. Wang, D. Gray, D. Abbot, and X. C. Zhang, “T-ray computed tomography,” Opt. Lett. **27**, 1312–1314 (2002). [CrossRef]

*et al.*demonstrated that cross-sectional images can be obtained by measuring the transmitted amplitude and phase of broadband THz pulses at multiple projection angles. However, it has been emphasized that several peaks in the THz waveform obtained with a time-domain spectrometer strongly complicate the signal analysis [10

10. E. Abraham, A. Younus, C. Aguerre, P. Desbarats, and P. Mounaix, “Refraction losses in terahertz computed tomography,” Opt. Commun. **283**, 2050–2055 (2010). [CrossRef]

9. B. Ferguson, S. Wang, D. Gray, D. Abbot, and X. C. Zhang, “T-ray computed tomography,” Opt. Lett. **27**, 1312–1314 (2002). [CrossRef]

11. S. Wang, B. Ferguson, D. Abbott, and X. C. Zhang, “T-ray imaging and tomography,” J. Biol. Phys. **29**, 247–256 (2003). [CrossRef]

12. S. Wang and X. C. Zhang, “Pulsed terahertz tomography,” J. Phys. D: Appl. Phys. **37**, R1–R36 (2004). [CrossRef]

13. M. M. Awad and R. A. Cheville, “Transmission terahertz waveguide-based imaging below the diffraction limit,” Appl. Phys. Lett. **86**, 221107 (2005). [CrossRef]

14. X. Yin, B. W. H. Ng, B. Ferguson, and D. Abbott, “Wavelet based local tomographic image using terahertz techniques,” Digit. Signal Process. **19**, 750–763 (2009). [CrossRef]

15. A. Brahm, M. Kunz, S. Riehemann, G. Notni, and A. Tünnermann, “Volumetric spectral analysis of materials using terahertz-tomography techniques,” Appl. Phys. B **100**, 151–158 (2010). [CrossRef]

9. B. Ferguson, S. Wang, D. Gray, D. Abbot, and X. C. Zhang, “T-ray computed tomography,” Opt. Lett. **27**, 1312–1314 (2002). [CrossRef]

**27**, 1312–1314 (2002). [CrossRef]

22. A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging **6**, 81–94 (1984). [CrossRef] [PubMed]

23. L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” *IEEE Trans. Med. Imaging*1, 113–122 (1982). [CrossRef] [PubMed]

24. H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging **13**, 601–609 (1994). [CrossRef] [PubMed]

## 2. Experimental setup

*f*′ = 150 mm). Two different Gunn diodes have been used: a 80 GHz diode coupled to a frequency tripler delivering 0.3 mW at 240 GHz and a 110 GHz diode delivering 20 mW. The advantage of the 240 GHz source is essentially the better spatial resolution (wavelength 1.25 mm), whereas the second source has a much larger output power. The THz beam is then focused with a Teflon lens L (

*f*′ = 60 mm) on the sample S which is positioned on three-axes XY-

*θ*motorized stages, the angle

*θ*corresponding to a rotation around the vertical Y-axis. For the 110 GHz source, Fig.1(b) shows the 2D transversal profile of the THz beam at the beam waist visualized using a photothermal THz convertor (Teracam-Alphanov, [25

25. C. Pradere, J.-P. Caumes, D. Balageas, S. Salort, E. Abraham, B. Chassagne, and J.-C. Batsale, “Photothermal converters for quantitative 2D and 3D real-time terahertz imaging,” Quant. Infrared Thermog. **7**, 217–235 (2010). [CrossRef]

^{3}. Moreover, to properly apply THz CT reconstruction algorithms and avoid depth resolution degradation, the target’s lateral extent in the direction of the THz wave vector has to be in the order of the confocal parameter of the THz beam [11

11. S. Wang, B. Ferguson, D. Abbott, and X. C. Zhang, “T-ray imaging and tomography,” J. Biol. Phys. **29**, 247–256 (2003). [CrossRef]

17. A. Younus, S. Salort, B. Recur, P. Desbarats, P. Mounaix, J-P. Caumes, and E. Abraham, “3D millimeter wave tomographic scanner for large size opaque object inspection with different refractive index contrasts,” in Millimetre Wave and Terahertz Sensors and Technology III , K.A. Krapels and N.A. Salmon, eds., Proc. SPIE **7837**, 783709 (2010).

*dθ*in order to provide a different visualization of the object. The operation is repeated

*N*times from

_{θ}*θ*= 0° to

*θ*= 180° and we finally get a set of

*N*projections of the sample corresponding to the different angles of visualization. For instance, if

_{θ}*N*= 18,

_{θ}*dθ*= 10° and the total acquisition time is about 9 hours for the complete 3D visualization of the sample. However, to visualize only a single cross-sectional image of the sample, corresponding to a 100 pixels horizontal scan associated to 18 different projections, this acquisition time is reduced to approximately 6 minutes. From these projection data, we are able to construct the sinogram of the object which represents, for a given horizontal slice, the evolution of the transmitted THz amplitude as a function of the rotation angle. Finally we apply the BFP, SART or OSEM algorithms to reconstruct the final 3D volume of the sample.

## 3. Tomographic reconstruction methods

*f*(

*x*,

*y*) into a 1D projection along an angle

*θ*and a module

*ρ*. It is defined by the following formula: where

*θ*and

*ρ*are respectively the angular and radial coordinates of the projection line (

*θ*,

*ρ*) and

*δ*is the Dirac impulse. Then, ℛ

*(*

_{θ}*ρ*) represents the absorption sum undergone by the ray along the line. As an illustration, let us consider a black foam parallelepiped (absorbance

*α*= 0.02 cm

^{−1}, refractive index 1.1) with a (41 × 49)mm

^{2}cross section. The parallelepiped is drilled by two holes (15 mm diameter): one hole with air and another hole filled by a Teflon cylinder (absorbance

*α*< 1 cm

^{−1}, refractive index 1.45) having a small cylindrical air hole inside (6 mm diameter) (Fig. 2(a)). For one horizontal cross-section, the acquisition along several angles gives a sinogram composed with

*N*lines, corresponding to the number of projections, and

_{θ}*N*columns corresponding to the number of pixels in the horizontal direction. For instance, Fig. 2(b) represents the sinogram acquired with the 110 GHz source with

_{ρ}*N*= 72 projections (

_{θ}*dθ*= 2.5°) and

*N*= 128 horizontal pixels (0.5 mm scan step). The black color corresponds to a maximum transmitted signal whereas the white color indicates a decrease of the transmitted signal. Similar color coding has been applied throughout the paper. Low signal intensity can be due to either absorption either deviation of the THz beam due to refraction. Here, owing to the low absorbance of Teflon, mainly refraction losses have to be considered since the Teflon cylinder acts as a lens. Before the application of any reconstruction method, this sinogram already reflects the global geometric structure of the sample cross-section with especially the presence of the refracting Teflon material. A more precise description of the THz imaging of this sample will be presented in section 4.

_{ρ}*(*

_{θ}*ρ*) values (

*θ*∈ [0,

*π*[ and

*ρ*∈ ℝ), the discrete inverse Radon transform computes each image pixel as follows: where:

*in the Fourier domain with the ramp filter |*

_{θ}*ν*| to increase geometric details. Second, it computes the pixel values from the filtered projections

*W*

_{θ(iθ)}. This method is thus denoted back-projection of filtered projections (BFP). As already pointed out in the introduction, this technique is widely used in THz CT imaging since it is proposed in most of CT software tools.

*N*. If this number is too low, beam hardening artifacts will be enhanced. This number

_{θ}*N*is also very critical because it acts on the global acquisition time, which is particularly important in THz CT imaging. Consequently, in this paper, we also investigated other reconstruction methods allowing to reduce the projection number (i.e. the acquisition time) while preserving reconstruction quality.

_{θ}22. A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging **6**, 81–94 (1984). [CrossRef] [PubMed]

24. H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging **13**, 601–609 (1994). [CrossRef] [PubMed]

*I*=

*A*, where

^{T}R*I*is the image,

*R*is the sinogram and

*A*is the weight-matrix [28

28. R. Gordon, R. Bender, and G. T. Herman, “Algebraic Reconstruction Techniques (ART) for Three-dimensional Electron Microscopy and X-ray Photography,” J. Theor. Biol. **29**, 471–481 (1970). [CrossRef] [PubMed]

*k*∈ [0 ⋯

*N*[. Each sub-iteration

_{iter}*s*, 0 ≤

*s*<

*N*, updates each pixel of the image

_{θ}*I*

^{k,s}by comparing the original projection ℛ

_{θs}with

*I*

^{k,s−1}by using direct Radon transform). A super-iteration

*k*is over when all the projections have been used. Consequently, pixel update using SART is computed as follows: where

*θ*,

_{s}*ρ*) crossing the image and (

*W*×

*H*) is the image size. Iterations are performed until the convergence of the solution. The initial

*I*

^{0,0}image is usually an uniform image.

23. L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” *IEEE Trans. Med. Imaging*1, 113–122 (1982). [CrossRef] [PubMed]

## 4. Quality and accuracy according to the projection number and the reconstruction method

*N*= 12. This behavior is well known in tomography since BFP is very sensitive to the number of projections. For SART, this problem is strongly reduced even if some background artifacts are still noticeable for

_{θ}*N*= 12. For OSEM, it seems that the reconstruction quality is constant whatever is the number of projection in the range 12 to 72. However, as pointed out previously with the two metallic bars, we can observe a small degradation of the reconstruction accuracy for OSEM compared with BFP and SART with blurred contours of the foam parallelepiped. This degradation is directly connected to the OSEM algorithm which uses a subset of several projections at once with a multiplicative error correction.

_{θ}*I*and a transformed image

*I*. The SSIM parameter is given by [30

^{t}30. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. **13**600–612 (2004). [CrossRef] [PubMed]

*l*(

*I*,

*I*),

^{t}*c*(

*I*,

*I*) and

^{t}*r*(

*I*,

*I*) are the intensity, contrast and geometric equivalence rates between both images, respectively. The image

^{t}*I*is the reference corresponding to the 72 projection sinogram, whereas the images

*I*are reconstructed from

^{t}*N*= 12 to

_{θ}*N*= 36. Table 1 details the three equivalence rates obtained by BFP (Table 1(a)), SART (Table 1(b)) and OSEM (Table 1(c)) methods according to the projection number. Here, it is important to notice that these tables are independent and cannot be compared each others since, for each table, the reconstructed image obtained with

_{θ}*N*= 72 is considered as the reference with all metric values set to unity. However, for each reconstruction method and depending on the projection number, the tables indicate the image degradation with respect to the intensity, contrast and geometric preservation. We can notice that all metric values slightly decrease whatever the method, but

_{θ}*r*(

*I*,

*I*) especially decreases significantly with BFP (indicated in red color in Table 1(a)). Then, the SSIM parameter of BFP is mainly deteriorated because the geometrical aspect of the image is not well preserved. For SART, the global image quality is already optimized with

^{t}*N*= 18. For SART, it appears that the method is almost insensitive to the projection number. However, as pointed out previously, we have to keep in mind that this method also provides a general vagueness in the final reconstructed image.

_{θ}*N*in Fig. 7. Independently from each others, the curves illustrate the reconstruction quality losses depending on the number of projections. As previously noticed in Table 1, we can remark that the SART and OSEM algorithms compute almost without quality loss whatever the projection number. Inversely, the BFP quality decreases if the projection number is less than 25. This observation is essential in case of THz CT since the projection number is limited owing to the long acquisition time. Here, we clearly point out that, for a limited number of projection data (typically less than 20), SART and OSEM methods are more appropriated for efficient THz CT.

_{θ}## 5. 3D reconstructions

*α*= 0.1 cm

^{−1}, refractive index 1.15) drilled by two oblique metallic bars (6 mm diameter). The experimental data obtained with the 240 GHz source correspond to 2D transmission images with a 1 mm step size both in the horizontal and vertical directions. The tomography is achieved with a set of 18 projections (rotation angle step

*dθ*= 10°) representing an acquisition time of nearly 9 hours. Fig. 8 represents the THz 3D video recordings of the sample volume reconstruction obtained by BFP (Fig. 8(b)), SART (Fig. 8(c)) and OSEM (Fig. 8(d)), respectively. The videos correspond to a full 360° rotation of the sample. As revealed in the previous section, since the BFP reconstruction suffers by the lack of projection data, the background surrounding the sample is not uniform with multiple artifacts even if the contours of the parallelepiped and the presence of the two oblique bars are precise. With the SART method, the background is more uniform even if some high intensity spots are still visible as artifacts. Contour quality is similar to BFP. Finally, as explained in the previous section, OSEM reconstruction is excellent even if the number of projection is limited but the method provides a final image which is more blurred with respect to the final image quality.

*Matriochka*(total height 160 mm, Fig. 9(a)). The THz 3D video recordings of the sample volume reconstruction (Fig. 9(b) for BFP ( Media 4), Fig. 9(c) for SART ( Media 5), Fig. 9(d) for OSEM ( Media 6)) clearly reveal the shape of the outer doll and the presence of the smaller doll inside the main doll (measured size 95 mm). This last example of 3D reconstructions of complex objects emphasize the potential of the SART method which provides high quality contour and reasonable artifacts despite the insufficient number of projections.

## 6. Conclusion

## References and links

1. | W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. |

2. | K. Kawase, Y. Ogawa, Y. Watanabe, and H. Inoue, “Non-destructive terahertz imaging of illicit drugs using spectral fingerprints,” Opt. Express |

3. | Y. C. Shen, T. Lo, P. F. Taday, B. E. Cole, W. R. Tribe, and M. C. Kemp, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. |

4. | K. Fukunaga and M. Picollo, “Terahertz spectroscopy applied to the analysis of artists materials,” Appl. Phys. A |

5. | E. Abraham, A. Younus, J.-C. Delagnes, and P. Mounaix, “Non-invasive investigation of art paintings by terahertz imaging,” Appl. Phys. A |

6. | S. Y. Huang, Y. X. J. Wang, D. K. W. Yeung, A. T. Ahuja, Y. T. Zhang, and E. Pickwell-MacPherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. |

7. | J. Takayanagi, H. Jinno, S. Ichino, K. Suizu, M. Yamashita, T. Ouchi, S. Kasai, H. Ohtake, H. Uchida, N. Nishizawa, and K. Kawase, “High-resolution time-of-flight terahertz tomography using a femtosecond fiber laser,” Opt. Express |

8. | K. Iwaszczuk, H. Heiselberg, and P. U. Jepsen, “Terahertz radar cross section measurements,” Opt. Express |

9. | B. Ferguson, S. Wang, D. Gray, D. Abbot, and X. C. Zhang, “T-ray computed tomography,” Opt. Lett. |

10. | E. Abraham, A. Younus, C. Aguerre, P. Desbarats, and P. Mounaix, “Refraction losses in terahertz computed tomography,” Opt. Commun. |

11. | S. Wang, B. Ferguson, D. Abbott, and X. C. Zhang, “T-ray imaging and tomography,” J. Biol. Phys. |

12. | S. Wang and X. C. Zhang, “Pulsed terahertz tomography,” J. Phys. D: Appl. Phys. |

13. | M. M. Awad and R. A. Cheville, “Transmission terahertz waveguide-based imaging below the diffraction limit,” Appl. Phys. Lett. |

14. | X. Yin, B. W. H. Ng, B. Ferguson, and D. Abbott, “Wavelet based local tomographic image using terahertz techniques,” Digit. Signal Process. |

15. | A. Brahm, M. Kunz, S. Riehemann, G. Notni, and A. Tünnermann, “Volumetric spectral analysis of materials using terahertz-tomography techniques,” Appl. Phys. B |

16. | K. L. Nguyen, M. L. Johns, L. F. Gladden, C. H. Worral, P. Alexander, H. E. Beere, M. Pepper, D. A. Ritchie, J. Alton, S. Barbieri, and E. H. Linfield, “Three-dimensional imaging with a terahertz quantum cascade laser,” Opt. Express |

17. | A. Younus, S. Salort, B. Recur, P. Desbarats, P. Mounaix, J-P. Caumes, and E. Abraham, “3D millimeter wave tomographic scanner for large size opaque object inspection with different refractive index contrasts,” in Millimetre Wave and Terahertz Sensors and Technology III , K.A. Krapels and N.A. Salmon, eds., Proc. SPIE |

18. | N. Sunaguchi, Y. Sasaki, N. Maikusa, M. Kawai, T. Yuasa, and C. Otani, “Depth-resolving terahertz imaging with tomosynthesis,” Opt. Express |

19. | T. Yasuda, T. Yasui, T. Araki, and E. Abraham, “Real-time two-dimensional terahertz tomography of moving objects,” Opt. Commun. |

20. | T. Yasui, K. Sawanaka, A. Ihara, E. Abraham, M. Hashimoto, and T. Araki, “Real-time terahertz color scanner for moving objects,” Opt. Express |

21. | G. T. Herman, |

22. | A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging |

23. | L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” |

24. | H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging |

25. | C. Pradere, J.-P. Caumes, D. Balageas, S. Salort, E. Abraham, B. Chassagne, and J.-C. Batsale, “Photothermal converters for quantitative 2D and 3D real-time terahertz imaging,” Quant. Infrared Thermog. |

26. | P. Toft, “The Radon Transform : Theory and Implementation,” Ph.D. thesis, Department of Mathematical Modelling, Section for Digital Signal Processing, Technical University of Denmark (1996). |

27. | J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten.” Ber. Ver. Sachs. Akad. Wiss. Leipzig, Math-Phys. Kl |

28. | R. Gordon, R. Bender, and G. T. Herman, “Algebraic Reconstruction Techniques (ART) for Three-dimensional Electron Microscopy and X-ray Photography,” J. Theor. Biol. |

29. | B. Recur, “Qualité et Précision en Reconstruction Tomographique : Algorithmes et Applications,” Ph.D. thesis, LaBRI, Bordeaux 1 University (2010). |

30. | Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. |

**OCIS Codes**

(100.6890) Image processing : Three-dimensional image processing

(120.5800) Instrumentation, measurement, and metrology : Scanners

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(110.6795) Imaging systems : Terahertz imaging

(110.6955) Imaging systems : Tomographic imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: January 19, 2011

Revised Manuscript: February 16, 2011

Manuscript Accepted: February 16, 2011

Published: March 2, 2011

**Virtual Issues**

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

**Citation**

B. Recur, A. Younus, S. Salort, P. Mounaix, B. Chassagne, P. Desbarats, J.-P. Caumes, and E. Abraham, "Investigation on reconstruction methods applied to 3D terahertz computed tomography," Opt. Express **19**, 5105-5117 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-6-5105

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### References

- W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70, 1325–1379 (2007). [CrossRef]
- K. Kawase, Y. Ogawa, Y. Watanabe, and H. Inoue, “Non-destructive terahertz imaging of illicit drugs using spectral fingerprints,” Opt. Express 11, 2549–2554 (2003). [CrossRef] [PubMed]
- Y. C. Shen, T. Lo, P. F. Taday, B. E. Cole, W. R. Tribe, and M. C. Kemp, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005). [CrossRef]
- K. Fukunaga and M. Picollo, “Terahertz spectroscopy applied to the analysis of artists materials,” Appl. Phys., A Mater. Sci. Process. 100, 591–597 (2010). [CrossRef]
- E. Abraham, A. Younus, J.-C. Delagnes, and P. Mounaix, “Non-invasive investigation of art paintings by terahertz imaging,” Appl. Phys., A Mater. Sci. Process. 100, 585–590 (2010). [CrossRef]
- S. Y. Huang, Y. X. J. Wang, D. K. W. Yeung, A. T. Ahuja, Y. T. Zhang, and E. Pickwell-MacPherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54, 149–160 (2009). [CrossRef]
- J. Takayanagi, H. Jinno, S. Ichino, K. Suizu, M. Yamashita, T. Ouchi, S. Kasai, H. Ohtake, H. Uchida, N. Nishizawa, and K. Kawase, “High-resolution time-of-flight terahertz tomography using a femtosecond fiber laser,” Opt. Express 17, 7549–7555 (2009). [CrossRef] [PubMed]
- K. Iwaszczuk, H. Heiselberg, and P. U. Jepsen, “Terahertz radar cross section measurements,” Opt. Express 18, 26399–26408 (2010). [CrossRef] [PubMed]
- B. Ferguson, S. Wang, D. Gray, D. Abbot, and X. C. Zhang, “T-ray computed tomography,” Opt. Lett. 27, 1312–1314 (2002). [CrossRef]
- E. Abraham, A. Younus, C. Aguerre, P. Desbarats, and P. Mounaix, “Refraction losses in terahertz computed tomography,” Opt. Commun. 283, 2050–2055 (2010). [CrossRef]
- S. Wang, B. Ferguson, D. Abbott, and X. C. Zhang, “T-ray imaging and tomography,” J. Biol. Phys. 29, 247–256 (2003). [CrossRef]
- S. Wang and X. C. Zhang, “Pulsed terahertz tomography,” J. Phys. D Appl. Phys. 37, R1–R36 (2004). [CrossRef]
- M. M. Awad and R. A. Cheville, “Transmission terahertz waveguide-based imaging below the diffraction limit,” Appl. Phys. Lett. 86, 221107 (2005). [CrossRef]
- X. Yin, B. W. H. Ng, B. Ferguson, and D. Abbott, “Wavelet based local tomographic image using terahertz techniques,” Digit. Signal Process. 19, 750–763 (2009). [CrossRef]
- A. Brahm, M. Kunz, S. Riehemann, G. Notni, and A. Tünnermann, “Volumetric spectral analysis of materials using terahertz-tomography techniques,” Appl. Phys. B 100, 151–158 (2010). [CrossRef]
- K. L. Nguyen, M. L. Johns, L. F. Gladden, C. H. Worral, P. Alexander, H. E. Beere, M. Pepper, D. A. Ritchie, J. Alton, S. Barbieri, and E. H. Linfield, “Three-dimensional imaging with a terahertz quantum cascade laser,” Opt. Express 14, 2123–2129 (2006). [CrossRef] [PubMed]
- A. Younus, S. Salort, B. Recur, P. Desbarats, P. Mounaix, J.-P. Caumes, and E. Abraham, “3D millimeter wave tomographic scanner for large size opaque object inspection with different refractive index contrasts,” in Millimetre Wave and Terahertz Sensors and Technology III, K.A. Krapels and N.A. Salmon, eds., Proc. SPIE 7837, 783709 (2010).
- N. Sunaguchi, Y. Sasaki, N. Maikusa, M. Kawai, T. Yuasa, and C. Otani, “Depth-resolving terahertz imaging with tomosynthesis,” Opt. Express 17, 9558–9570 (2009). [CrossRef] [PubMed]
- T. Yasuda, T. Yasui, T. Araki, and E. Abraham, “Real-time two-dimensional terahertz tomography of moving objects,” Opt. Commun. 267, 128–136 (2006). [CrossRef]
- T. Yasui, K. Sawanaka, A. Ihara, E. Abraham, M. Hashimoto, and T. Araki, “Real-time terahertz color scanner for moving objects,” Opt. Express 16, 1208–1221 (2008). [CrossRef] [PubMed]
- G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography (Academic Press Inc., 1980).
- A. H. Andersen, and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984). [CrossRef] [PubMed]
- L. A. Shepp, and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982). [CrossRef] [PubMed]
- H. M. Hudson, and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13, 601–609 (1994). [CrossRef] [PubMed]
- C. Pradere, J.-P. Caumes, D. Balageas, S. Salort, E. Abraham, B. Chassagne, and J.-C. Batsale, “Photothermal converters for quantitative 2D and 3D real-time terahertz imaging,” Quant. Infrared Thermog. 7, 217–235 (2010). [CrossRef]
- P. Toft, “The Radon Transform: Theory and Implementation,” Ph.D. thesis, Department of Mathematical Modelling, Section for Digital Signal Processing, Technical University of Denmark (1996).
- J. Radon, “¨Uber die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten,” Ber. Ver. Sachs. Akad.Wiss. Leipzig, Math-Phys. Kl 69, 262–277 (1917). In German. An english translation can be found in S. R. Deans: The Radon Transform and Some of Its Applications.
- R. Gordon, R. Bender, and G. T. Herman, “Algebraic Reconstruction Techniques (ART) for Three-dimensional Electron Microscopy and X-ray Photography,” J. Theor. Biol. 29, 471–481 (1970). [CrossRef] [PubMed]
- B. Recur, “Qualité et Précision en Reconstruction Tomographique: Algorithmes et Applications,” Ph.D. thesis, LaBRI, Bordeaux 1 University (2010).
- Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004). [CrossRef] [PubMed]

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