## Fast terahertz reflection tomography using block-based compressed sensing |

Optics Express, Vol. 19, Issue 17, pp. 16401-16409 (2011)

http://dx.doi.org/10.1364/OE.19.016401

Acrobat PDF (3583 KB)

### Abstract

In this paper, a new fast terahertz reflection tomography is proposed using block-based compressed sensing. Since measuring the time-domain signal on two-dimensional grid requires excessive time, reducing measurement time is highly demanding in terahertz tomography. The proposed technique directly reduces the number of sampling points in the spatial domain without modulation or transformation of the signal. Compressed sensing in spatial domain suggests a block-based reconstruction, which substantially reduces computational time without degrading the image quality. An overlap-average method is proposed to remove the block artifact in the block-based compressed sensing. Fast terahertz reflection tomography using the block-based compressed sensing is demonstrated with an integrated circuit and parched anchovy as examples.

© 2011 OSA

## 1. Introduction

1. B. B. Hu and M. C. Nuss, “Imaging with terahertz waves,” Opt. Lett. **20**(16), 1716–1718 (1995). [CrossRef] [PubMed]

6. S. J. Oh, J. Choi, I. Maeng, J. Y. Park, K. Lee, Y.-M. Huh, J.-S. Suh, S. Haam, and J.-H. Son, “Molecular imaging with terahertz waves,” Opt. Express **19**(5), 4009–4016 (2011). [CrossRef] [PubMed]

7. D. M. Mittleman, S. Hunsche, L. Boivin, and M. C. Nuss, “T-ray tomography,” Opt. Lett. **22**(12), 904–906 (1997). [CrossRef] [PubMed]

8. S. Wang and X.-C. Zhang, “Pulsed terahertz tomography,” J. Phys. D Appl. Phys. **37**(4), R1–R36 (2004). [CrossRef]

9. Z. Jiang and X. C. Zhang, “Terahertz imaging via electrooptic effect,” IEEE Trans. Microw. Theory Tech. **47**(12), 2644–2650 (1999). [CrossRef]

10. J. Xu and X. C. Zhang, “Terahertz wave reciprocal imaging,” Appl. Phys. Lett. **88**(15), 151107 (2006). [CrossRef]

11. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory **52**(4), 1289–1306 (2006). [CrossRef]

14. K. H. Jin, Y. Kim, D. S. Yee, O. K. Lee, and J. C. Ye, “Compressed sensing pulse-echo mode terahertz reflectance tomography,” Opt. Lett. **34**(24), 3863–3865 (2009). [CrossRef] [PubMed]

12. W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. **93**(12), 121105 (2008). [CrossRef]

## 2. The block-based compressed sensing

*m*is the measured vector of time signal,

*μ*is the image to be reconstructed, and

*A*is the sampling matrix having 1’s for measured indices. In Eq. (1) sparse sampling is applied in a transverse plane, and

*t*denotes the time delay which is equivalent to depth at a given spatial location. Since compressed sensing is applied to the spatial domain, not to the time domain, further description of the algorithm is made for a given

*t*without loss of generality. If the number of measured TDS is

*L*and that of pixels for the image of size

*n*x

*n*is

*N*(=

*n*), then

^{2}*m*and

*μ*are represented as column vectors of size

*L*x

*1*and

*N*x

*1*, and

*A*as a matrix of size

*L*x

*N*. Since

*L*is usually much smaller than

*N*, reconstruction finds a solution to the underdetermined Eq. (1). By utilizing sparseness of the image in a transformed domain, reconstruction may be done by minimizing ||

*Fμ*||

_{1}subject to ||

*m - Aμ*||

_{2}<

*ε*, where

*F*denotes a transform whose transformed coefficients appear sparse, e.g., Fourier transform or discrete cosine transform (DCT), ||

**·**||

_{p}denotes

*L*norm, and

_{p}*ε*is a constant denoting tolerance or error bound for the measured data. Since the solution to minimize

*L*norm is not given in a closed form, an iterative solution by FOCUSS algorithm is used [18

_{1}18. I. F. Gorodnitsky, J. S. George, and B. D. Rao, “Neuromagnetic source imaging with FOCUSS: a recursive weighted minimum norm algorithm,” Electroencephalogr. Clin. Neurophysiol. **95**(4), 231–251 (1995). [CrossRef] [PubMed]

19. H. Jung, J. C. Ye, and E. Y. Kim, “Improved k-t BLAST and k-t SENSE using FOCUSS,” Phys. Med. Biol. **52**(11), 3201–3226 (2007). [CrossRef] [PubMed]

*L*norm minimization is known. The FOCUSS algorithm is a recursive linear estimation based on a weighted pseudo-inverse solution. The weights at each step are derived from the transformed coefficients of the solution (estimated image) of the previous iterative step. Thus, a large transformed coefficient of the solution becomes larger as the number of iterations increases, which enforces the sparseness in the transformed domain. For a fast convergence, the weighting values of small coefficients are truncated with a threshold. The choice of initial image affects the convergence speed of the algorithm. Linear interpolation of the measured data is used as the initial estimate.

_{2}*M*x

*M*if the number of sampling points with the compressed sensing is

*M*. Thus, the complexity of the reconstruction is

*O*(

*M*

^{3}) [20]. Let

*Q*be the ratio of the number of pixels in the entire image to that in a block. Since the number of sampling points in a block is reduced by the same factor of

*Q*, the complexity of the reconstruction for a block is reduced by the factor

*Q*. The number of blocks to be reconstructed is increased by the factor of

^{3}*Q*if the blocks are laid without overlap. By considering an overlap factor

*K*which will be described below, the computational gain using the block-based CS is

*Q*

^{3}/(Q

*·**K) = Q*.

^{2}/ K*K*is defined as the number of different reconstruction blocks in which a pixel belongs. If the overlap blocks are generated by shifting a fixed number of pixels (

*S*) sequentially in both horizontal and vertical directions, the overlap factor is given by

*(B/S)*for a block size of

^{2}*B*x

*B*except some pixels near the boundaries of the image. For instance if a block size of 16x16 is used for the reconstruction, and the overlap blocks are generated by shifting 2 pixels sequentially, the overlap factor is given by (16/2)

^{2}=64.

*K*=64). The block artifacts are observed in (a-c), especially in the CT phantom, while they are mostly removed by the overlap-average method as seen in (d-f). In this example, the computational gain by the block-based CS is 256

^{3}/(256*64) = 1024. Compared to the interpolated images (g-i), the reconstructed images by the block-based CS show higher resolution. For example, the vertical and horizontal lines are seriously degraded in the cubic interpolated images (g, h) when the compression factors are high, while they are mostly resolvable in the block-based CS images (d, e). A quantitative evaluation by the peak-signal-to-noise-ratio is summarized in Table 1 .

## 3. T-ray tomography by block-based CS

7. D. M. Mittleman, S. Hunsche, L. Boivin, and M. C. Nuss, “T-ray tomography,” Opt. Lett. **22**(12), 904–906 (1997). [CrossRef] [PubMed]

22. S.-H. Ding, Q. Li, R. Yao, and Q. Wang, “High-resolution terahertz reflective imaging and image restoration,” Appl. Opt. **49**(36), 6834–6839 (2010). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

1. | B. B. Hu and M. C. Nuss, “Imaging with terahertz waves,” Opt. Lett. |

2. | D. M. Mittleman, M. Gupta, R. Neelamani, R. G. Baraniuk, J. V. Rudd, and M. Koch, “Recent advances in terahertz imaging,” Appl. Phys. B |

3. | J.-H. Son, “Terahertz electromagnetic interactions with biological matter and their applications,” J. Appl. Phys. |

4. | A. J. Fitzgerald, V. P. Wallace, M. Jimenez-Linan, L. Bobrow, R. J. Pye, A. D. Purushotham, and D. D. Arnone, “Terahertz pulsed imaging of human breast tumors,” Radiology |

5. | S. J. Oh, J. Kang, I. Maeng, J.-S. Suh, Y.-M. Huh, S. Haam, and J.-H. Son, “Nanoparticle-enabled terahertz imaging for cancer diagnosis,” Opt. Express |

6. | S. J. Oh, J. Choi, I. Maeng, J. Y. Park, K. Lee, Y.-M. Huh, J.-S. Suh, S. Haam, and J.-H. Son, “Molecular imaging with terahertz waves,” Opt. Express |

7. | D. M. Mittleman, S. Hunsche, L. Boivin, and M. C. Nuss, “T-ray tomography,” Opt. Lett. |

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

9. | Z. Jiang and X. C. Zhang, “Terahertz imaging via electrooptic effect,” IEEE Trans. Microw. Theory Tech. |

10. | J. Xu and X. C. Zhang, “Terahertz wave reciprocal imaging,” Appl. Phys. Lett. |

11. | D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory |

12. | W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. |

13. | Y. C. Shen, L. Gan, M. Stringer, A. Burnett, K. Tych, H. Shen, J. E. Cunningham, E. P. J. Parrott, J. A. Zeitler, L. F. Gladden, E. H. Linfield, and A. G. Davies, “Terahertz pulsed spectroscopic imaging using optimized binary masks,” Appl. Phys. Lett. |

14. | K. H. Jin, Y. Kim, D. S. Yee, O. K. Lee, and J. C. Ye, “Compressed sensing pulse-echo mode terahertz reflectance tomography,” Opt. Lett. |

15. | L. Gan, “Block compressed sensing of natural images,” in Proc. Int. Conf. Digital Signal Processing, pp.403–406, Cardiff, UK (2007). |

16. | S. H. Cho, J.-H. Park, S.-H. Lee, H. Park, J.-H. Son, and C. B. Ahn, “Direct block-based compressed sensing for THz reflection tomography,” in Proc. Int. 2nd THz-Bio Workshop, pp.92–93, Seoul, Korea (2011). |

17. | T. Acharya and P.-S. Tsai, |

18. | I. F. Gorodnitsky, J. S. George, and B. D. Rao, “Neuromagnetic source imaging with FOCUSS: a recursive weighted minimum norm algorithm,” Electroencephalogr. Clin. Neurophysiol. |

19. | H. Jung, J. C. Ye, and E. Y. Kim, “Improved k-t BLAST and k-t SENSE using FOCUSS,” Phys. Med. Biol. |

20. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

21. | S. Qian and D. Chen, |

22. | S.-H. Ding, Q. Li, R. Yao, and Q. Wang, “High-resolution terahertz reflective imaging and image restoration,” Appl. Opt. |

**OCIS Codes**

(110.6795) Imaging systems : Terahertz imaging

(110.6955) Imaging systems : Tomographic imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: April 25, 2011

Revised Manuscript: June 30, 2011

Manuscript Accepted: July 16, 2011

Published: August 11, 2011

**Virtual Issues**

Vol. 6, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Sang-Heum Cho, Sang-Hun Lee, Chan Nam-Gung, Seoung-Jun Oh, Joo-Hiuk Son, Hochong Park, and Chang-Beom Ahn, "Fast terahertz reflection tomography using block-based compressed sensing," Opt. Express **19**, 16401-16409 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16401

Sort: Year | Journal | Reset

### References

- B. B. Hu and M. C. Nuss, “Imaging with terahertz waves,” Opt. Lett. 20(16), 1716–1718 (1995). [CrossRef] [PubMed]
- D. M. Mittleman, M. Gupta, R. Neelamani, R. G. Baraniuk, J. V. Rudd, and M. Koch, “Recent advances in terahertz imaging,” Appl. Phys. B 68(6), 1085–1094 (1999). [CrossRef]
- J.-H. Son, “Terahertz electromagnetic interactions with biological matter and their applications,” J. Appl. Phys. 105(10), 102033 (2009). [CrossRef]
- A. J. Fitzgerald, V. P. Wallace, M. Jimenez-Linan, L. Bobrow, R. J. Pye, A. D. Purushotham, and D. D. Arnone, “Terahertz pulsed imaging of human breast tumors,” Radiology 239(2), 533–540 (2006). [CrossRef] [PubMed]
- S. J. Oh, J. Kang, I. Maeng, J.-S. Suh, Y.-M. Huh, S. Haam, and J.-H. Son, “Nanoparticle-enabled terahertz imaging for cancer diagnosis,” Opt. Express 17(5), 3469–3475 (2009). [CrossRef] [PubMed]
- S. J. Oh, J. Choi, I. Maeng, J. Y. Park, K. Lee, Y.-M. Huh, J.-S. Suh, S. Haam, and J.-H. Son, “Molecular imaging with terahertz waves,” Opt. Express 19(5), 4009–4016 (2011). [CrossRef] [PubMed]
- D. M. Mittleman, S. Hunsche, L. Boivin, and M. C. Nuss, “T-ray tomography,” Opt. Lett. 22(12), 904–906 (1997). [CrossRef] [PubMed]
- S. Wang and X.-C. Zhang, “Pulsed terahertz tomography,” J. Phys. D Appl. Phys. 37(4), R1–R36 (2004). [CrossRef]
- Z. Jiang and X. C. Zhang, “Terahertz imaging via electrooptic effect,” IEEE Trans. Microw. Theory Tech. 47(12), 2644–2650 (1999). [CrossRef]
- J. Xu and X. C. Zhang, “Terahertz wave reciprocal imaging,” Appl. Phys. Lett. 88(15), 151107 (2006). [CrossRef]
- D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). [CrossRef]
- W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008). [CrossRef]
- Y. C. Shen, L. Gan, M. Stringer, A. Burnett, K. Tych, H. Shen, J. E. Cunningham, E. P. J. Parrott, J. A. Zeitler, L. F. Gladden, E. H. Linfield, and A. G. Davies, “Terahertz pulsed spectroscopic imaging using optimized binary masks,” Appl. Phys. Lett. 95(23), 231112 (2009). [CrossRef]
- K. H. Jin, Y. Kim, D. S. Yee, O. K. Lee, and J. C. Ye, “Compressed sensing pulse-echo mode terahertz reflectance tomography,” Opt. Lett. 34(24), 3863–3865 (2009). [CrossRef] [PubMed]
- L. Gan, “Block compressed sensing of natural images,” in Proc. Int. Conf. Digital Signal Processing, pp.403–406, Cardiff, UK (2007).
- S. H. Cho, J.-H. Park, S.-H. Lee, H. Park, J.-H. Son, and C. B. Ahn, “Direct block-based compressed sensing for THz reflection tomography,” in Proc. Int. 2nd THz-Bio Workshop, pp.92–93, Seoul, Korea (2011).
- T. Acharya and P.-S. Tsai, JPEG2000 standard for image compression: concepts, algorithms and VLSI architectures (Wiley-Interscience, 2005).
- I. F. Gorodnitsky, J. S. George, and B. D. Rao, “Neuromagnetic source imaging with FOCUSS: a recursive weighted minimum norm algorithm,” Electroencephalogr. Clin. Neurophysiol. 95(4), 231–251 (1995). [CrossRef] [PubMed]
- H. Jung, J. C. Ye, and E. Y. Kim, “Improved k-t BLAST and k-t SENSE using FOCUSS,” Phys. Med. Biol. 52(11), 3201–3226 (2007). [CrossRef] [PubMed]
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C: the art of scientific computing 2nd ed. (Cambridge University Press, 1992).
- S. Qian and D. Chen, Joint time-frequency analysis: methods and applications (Prentice Hall, 1996).
- S.-H. Ding, Q. Li, R. Yao, and Q. Wang, “High-resolution terahertz reflective imaging and image restoration,” Appl. Opt. 49(36), 6834–6839 (2010). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.