## Optical frequency comb interference profilometry using compressive sensing |

Optics Express, Vol. 21, Issue 16, pp. 19003-19011 (2013)

http://dx.doi.org/10.1364/OE.21.019003

Acrobat PDF (1293 KB)

### Abstract

We describe a new optical system using an ultra-stable mode-locked frequency comb femtosecond laser and compressive sensing to measure an object’s surface profile. The ultra-stable frequency comb laser was used to precisely measure an object with a large depth, over a wide dynamic range. The compressive sensing technique was able to obtain the spatial information of the object with two single-pixel fast photo-receivers, with no mechanical scanning and fewer measurements than the number of sampling points. An optical experiment was performed to verify the advantages of the proposed method.

© 2013 OSA

## 1. Introduction

1. J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt. **10**(9), 2113–2118 (1971). [CrossRef] [PubMed]

2. P. S. Lam, J. D. Gaskill, and J. C. Wyant, “Two-wavelength holographic interferometer,” Appl. Opt. **23**(18), 3079–3081 (1984). [CrossRef] [PubMed]

1. J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt. **10**(9), 2113–2118 (1971). [CrossRef] [PubMed]

2. P. S. Lam, J. D. Gaskill, and J. C. Wyant, “Two-wavelength holographic interferometer,” Appl. Opt. **23**(18), 3079–3081 (1984). [CrossRef] [PubMed]

3. C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. **39**(1), 79–85 (2000). [CrossRef]

4. A. Wada, M. Kato, and Y. Ishii, “Multiple-wavelength digital holographic interferometry using tunable laser diodes,” Appl. Opt. **47**(12), 2053–2060 (2008). [CrossRef] [PubMed]

4. A. Wada, M. Kato, and Y. Ishii, “Multiple-wavelength digital holographic interferometry using tunable laser diodes,” Appl. Opt. **47**(12), 2053–2060 (2008). [CrossRef] [PubMed]

5. K. N. Joo and S. W. Kim, “Refractive index measurement by spectrally resolved interferometry using a femtosecond pulse laser,” Opt. Lett. **32**(6), 647–649 (2007). [CrossRef] [PubMed]

6. K. Minoshima, K. Arai, and H. Inaba, “Two-Color Interferometry using frequency combs for high-accuracy self-correction of air refractive index,” Opt. Express **19**, 26095–26105 (2011). [CrossRef] [PubMed]

7. C. E. Towers, D. P. Towers, D. T. Reid, W. N. MacPherson, R. R. J. Maier, and J. D. C. Jones, “Fiber interferometer for simultaneous multiwavelength phase measurement with a broadband femtosecond laser,” Opt. Lett. **29**(23), 2722–2724 (2004). [CrossRef] [PubMed]

8. J. S. Oh and S.-W. Kim, “Femtosecond laser pulses for surface-profile metrology,” Opt. Lett. **30**(19), 2650–2652 (2005). [CrossRef] [PubMed]

10. S. Choi, M. Yamamoto, D. Moteki, T. Shioda, Y. Tanaka, and T. Kurokawa, “Frequency-comb-based interferometer for profilometry and tomography,” Opt. Lett. **31**(13), 1976–1978 (2006). [CrossRef] [PubMed]

11. P. Balling, P. Křen, P. Mašika, and S. A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express **17**(11), 9300–9313 (2009). [CrossRef] [PubMed]

13. M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, and H. P. Urbach, “Long distance measurement with femtosecond pulses using a dispersive interferometer,” Opt. Express **19**(7), 6549–6562 (2011). [CrossRef] [PubMed]

14. J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. **29**(10), 1153–1155 (2004). [CrossRef] [PubMed]

15. J. Lee, Y. J. Kim, K. Lee, S. Lee, and S. W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics **4**(10), 716–720 (2010). [CrossRef]

14. J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. **29**(10), 1153–1155 (2004). [CrossRef] [PubMed]

16. Y. Salvadé, N. Schuhler, S. Lévêque, and S. Le Floch, “High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source,” Appl. Opt. **47**(14), 2715–2720 (2008). [CrossRef] [PubMed]

17. S. Yokoyama, T. Yokoyama, Y. Hagihara, T. Araki, and T. Yasui, “A distance meter using a terahertz intermode beat in an optical frequency comb,” Opt. Express **17**(20), 17324–17337 (2009). [CrossRef] [PubMed]

18. K. Minoshima and H. Matsumoto, “High-accuracy measurement of 240-m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt. **39**(30), 5512–5517 (2000). [CrossRef] [PubMed]

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

20. E. J. Candès and T. Tao, “Near optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory **52**(12), 5406–5425 (2006). [CrossRef]

21. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. **25**(2), 83–91 (2008). [CrossRef]

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

21. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. **25**(2), 83–91 (2008). [CrossRef]

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

23. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express **17**(15), 13040–13049 (2009). [CrossRef] [PubMed]

24. C. F. Cull, D. A. Wikner, J. N. Mait, M. Mattheiss, and D. J. Brady, “Millimeter-wave compressive holography,” Appl. Opt. **49**(19), E67–E82 (2010). [CrossRef] [PubMed]

25. R. Berinde, P. Indyk, and M. Ruzic, “Practical near-optimal sparse recovery in the L1 norm,” Communication, Control, and Computing, 2008 46th Annual Allerton Conference, 198–205, 23–26 Sept. (2008). [CrossRef]

26. E. J. Candès and J. Romberg, “Signal recovery from random projections,” in Computational Imaging III, Proc. SPIE Conf. 5674, 76–86, 31 March. (2005). [CrossRef]

## 2. Principle

### 2.1 Interference imaging with optical mode-locked frequency comb femtosecond laser

*f*and

_{R}*f*, respectively, the harmonic frequency of order

_{O}*n*is given by

*f*=

_{n}*f*

_{O}+

*nf*

_{R}, where 0 ≤

*n*≤

*N*and

_{max}*N*is the maximum harmonic frequency order of the frequency comb laser.

_{max}*N*; therefore, the interference pattern in the sensor image is the summation of the monochromatic interference patterns of all optical frequencies, expressed byThe interference pattern is sampled by D

_{max}_{S}with a maximum response sampling frequency of

*f*. Suppose that the harmonic frequency order

_{smax}*N*closest to

*f*satisfies the condition

_{smax}*f*≤

_{N}*f*<

_{smax}*f*

_{N+}_{1}.This means that the high-frequency component of

*U*(

_{TT}*t*) will be cut-off after passing through D

_{S}. The signal output from D

_{S}is now rewritten in the formSimilarly, the signal passing through D

_{R}is simply described byThe output signals from D

_{S}and D

_{R}are led into a frequency selectable system (FSS) [23

23. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express **17**(15), 13040–13049 (2009). [CrossRef] [PubMed]

*f*corresponding to the harmonic frequency order

_{k}*k*(0 <

*k*≤

*N*) satisfies the condition of the FSS are passed through. Equations (3) and (4) are rewritten asand

*U*(

_{RT}*t*) =

*A*[-

_{Rk}exp*j*(

*w*+

_{k}t*φ*)], respectively.

_{Rk}*U*(

_{T}*t*) and

*U*(t) are each separated into two components to measure the phases, as shown in Fig. 1. The output signal from phase-detector PD

_{RT}_{C}is described by

*U*(

_{RT}**t*) is

*U*(

_{RT}*t*) with the phase inverted because of characteristics of the phase detector,

*∆φ*(

_{k}*x*,

*y*) =

*φ*(

_{k}*x*,

*y*) −

*φ*is the phase difference between the coded object wave and the reference wave, and constant

_{Rk}*α*characterizes the effect of the power splitter and the relation between the amplitudes of the output and input signals of the phase detector. The high-frequency component in Eq. (6) will be filtered by a low pass filter; thus, the output signal from the phase detector is simply described bywhere

*E*(

_{ck}*x*,

*y*) =

*αA*(

_{RK}A_{K}*x*,

*y*)cos[

*∆φ*(

_{k}*x*,

*y*)]. Before being led into the phase-detector PD

_{S}, the phase of the reference wave is shifted by π/2 by a phase shifter (PS), and the output signal from the phase detector is expressed bywhere

*E*(

_{sk}*x*,

*y*) =

*βA*(

_{RK}A_{K}*x*,

*y*)sin[

*∆φ*(

_{k}*x*,

*y*)], and

*β*is a different constant from

*α*because of the attenuation caused by the phase shifter. When

*E*(

_{ck}*x*,

*y*) and

*E*(

_{sk}*x*,

*y*) are specified,

*∆φ*(

_{k}*x*,

*y*) is easily obtained by the following equation,The depth difference

*∆D*(

*x*,

*y*) between the object and reference points is calculated bywhere

*p*is the integer part of the phase difference,

*c*is the velocity of light, and

*n*is the group refractive index of the medium. If the object depth is assumed to be smaller than half the wavelength of the waves selected by the FSS, then

_{g}*p*= 0, and thus, the object profile can be specified without any 2π ambiguity.

### 2.2 Applying compressive sensing

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

20. E. J. Candès and T. Tao, “Near optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory **52**(12), 5406–5425 (2006). [CrossRef]

*R*(

_{k}*x*,

*y,l*) is constructed by a pseudorandom pattern consisting of random binary elements of a Bernoulli matrix. Elements “0” and “1” correspond to two states, which block or pass light; the number of “1” and “0” elements in each pseudorandom pattern is the same. For each pseudorandom pattern

*R*(

_{k}*x*,

*y,l*), a shifted version denoted by

*R*(

_{sk}*x*,

*y,l*) is also generated as follows:From Eqs. (7) and (8), when the shifted version of

*R*(

_{k}*x*,

*y,l*) is displayed on the SLM, the outputs from PD

_{C}and PD

_{S}are expressed by From Eqs. (7) and (12), the difference

*∆P*

_{c}(

*l*) obtained by subtracting

*P*(

_{cs}*l*) from

*P*(

_{c}*l*) iswhereSimilarly, from Eqs. (8) and (13), the difference

*∆P*

_{s}(

*l*) obtained by subtracting

*P*(

_{ss}*l*) from

*P*(

_{s}*l*) iswhereFrom Eqs. (15) and (17), it can be seen that ∆

*E*(

_{ck}*x*,

*y*) and

*∆E*(

_{sk}*x*,

*y*) are equivalent representation models of

*E*(

_{ck}*x*,

*y*) and

*E*(

_{sk}*x*,

*y*), respectively. Because the consecutive points in a line on the object often have similar depth and intensity, on some areas of the object, ∆

*E*(

_{ck}*x*,

*y*) and

*∆E*(

_{sk}*x*,

*y*) are approximately equal to 0. This means that ∆

*E*(

_{ck}*x*,

*y*) and ∆

*E*(

_{sk}*x*,

*y*) can be considered as sparse signals which can be obtained and reconstructed by CS [19

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

20. E. J. Candès and T. Tao, “Near optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory **52**(12), 5406–5425 (2006). [CrossRef]

*E*(

_{ck}*x*,

*y*) and ∆

*E*(

_{sk}*x*,

*y*) are the same

*m*-sparse signal, and thus the minimum number of measurements required to exactly reconstruct the original signal is calculated byThis equation is also considered as a condition of CS [19

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

**52**(12), 5406–5425 (2006). [CrossRef]

*M*pseudorandom patterns of

*R*(

_{k}*x*,

*y,l*) and

*R*(

_{ks}*x*,

*y,l*) are used, from Eqs. (14) and (16), the measurement process can be described by the equationsandwhere

*∆P*and

_{c}*∆P*are

_{s}*M*× 1 column vectors of measurements, ∆

*E*and ∆

_{c}*E*are column vectors with

_{s}*X*×

*Y*pixels of ∆

*E*(

_{ck}*x*,

*y*) and

*∆E*(

_{sk}*x*,

*y*) ordered in an

*XY*× 1 vector, and the measurement matrix

*Φ*is

_{R}*M*×

*XY*.

*E*(

_{ck}*x*,

*y*) and ∆

*E*(

_{sk}*x*,

*y*), the

*l*1 minimization method is selected. It is summarized by [25

25. R. Berinde, P. Indyk, and M. Ruzic, “Practical near-optimal sparse recovery in the L1 norm,” Communication, Control, and Computing, 2008 46th Annual Allerton Conference, 198–205, 23–26 Sept. (2008). [CrossRef]

26. E. J. Candès and J. Romberg, “Signal recovery from random projections,” in Computational Imaging III, Proc. SPIE Conf. 5674, 76–86, 31 March. (2005). [CrossRef]

*ɛ*is a tolerance that determines the accuracy of the reconstructed signal [19

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

**52**(12), 5406–5425 (2006). [CrossRef]

*E*(

_{c}*x*,

*y*) and ∆

*E*(

_{s}*x*,

*y*) are specified by the reconstruction process in Eqs. (21) and (22),

*E*(

_{ck}*x*,

*y*) and

*E*(

_{sk}*x*,

*y*) are calculated by Eqs. (15) and (17), so that the profile of the object is measured by applying Eqs. (9) and (10).

## 3. Experiment

*β*/

*α =*0.8, was measured from the ratio between the amplitudes of the input and output signals of the phase shifter.

## 4. Conclusions

18. K. Minoshima and H. Matsumoto, “High-accuracy measurement of 240-m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt. **39**(30), 5512–5517 (2000). [CrossRef] [PubMed]

## References and links

1. | J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt. |

2. | P. S. Lam, J. D. Gaskill, and J. C. Wyant, “Two-wavelength holographic interferometer,” Appl. Opt. |

3. | C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. |

4. | A. Wada, M. Kato, and Y. Ishii, “Multiple-wavelength digital holographic interferometry using tunable laser diodes,” Appl. Opt. |

5. | K. N. Joo and S. W. Kim, “Refractive index measurement by spectrally resolved interferometry using a femtosecond pulse laser,” Opt. Lett. |

6. | K. Minoshima, K. Arai, and H. Inaba, “Two-Color Interferometry using frequency combs for high-accuracy self-correction of air refractive index,” Opt. Express |

7. | C. E. Towers, D. P. Towers, D. T. Reid, W. N. MacPherson, R. R. J. Maier, and J. D. C. Jones, “Fiber interferometer for simultaneous multiwavelength phase measurement with a broadband femtosecond laser,” Opt. Lett. |

8. | J. S. Oh and S.-W. Kim, “Femtosecond laser pulses for surface-profile metrology,” Opt. Lett. |

9. | K. Körner, G. Pedrini, I. Alexeenko, T. Steinmetz, R. Holzwarth, and W. Osten, “Short temporal coherence digital holography with a femtosecond frequency comb laser for multi-level optical sectioning,” Opt. Express |

10. | S. Choi, M. Yamamoto, D. Moteki, T. Shioda, Y. Tanaka, and T. Kurokawa, “Frequency-comb-based interferometer for profilometry and tomography,” Opt. Lett. |

11. | P. Balling, P. Křen, P. Mašika, and S. A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express |

12. | M. T. L. Hsu, I. C. M. Littler, D. A. Shaddock, J. Herrmann, R. B. Warrington, and M. B. Gray, “Subpicometer length measurement using heterodyne laser interferometry and all-digital rf phase meters,” Opt. Lett. |

13. | M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, and H. P. Urbach, “Long distance measurement with femtosecond pulses using a dispersive interferometer,” Opt. Express |

14. | J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. |

15. | J. Lee, Y. J. Kim, K. Lee, S. Lee, and S. W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics |

16. | Y. Salvadé, N. Schuhler, S. Lévêque, and S. Le Floch, “High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source,” Appl. Opt. |

17. | S. Yokoyama, T. Yokoyama, Y. Hagihara, T. Araki, and T. Yasui, “A distance meter using a terahertz intermode beat in an optical frequency comb,” Opt. Express |

18. | K. Minoshima and H. Matsumoto, “High-accuracy measurement of 240-m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt. |

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

20. | E. J. Candès and T. Tao, “Near optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory |

21. | M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. |

22. | W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A Single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. |

23. | D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express |

24. | C. F. Cull, D. A. Wikner, J. N. Mait, M. Mattheiss, and D. J. Brady, “Millimeter-wave compressive holography,” Appl. Opt. |

25. | R. Berinde, P. Indyk, and M. Ruzic, “Practical near-optimal sparse recovery in the L1 norm,” Communication, Control, and Computing, 2008 46th Annual Allerton Conference, 198–205, 23–26 Sept. (2008). [CrossRef] |

26. | E. J. Candès and J. Romberg, “Signal recovery from random projections,” in Computational Imaging III, Proc. SPIE Conf. 5674, 76–86, 31 March. (2005). [CrossRef] |

**OCIS Codes**

(110.6880) Imaging systems : Three-dimensional image acquisition

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(110.1758) Imaging systems : Computational imaging

(100.3175) Image processing : Interferometric imaging

(070.6120) Fourier optics and signal processing : Spatial light modulators

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 1, 2013

Revised Manuscript: July 12, 2013

Manuscript Accepted: July 19, 2013

Published: August 2, 2013

**Citation**

Quang Duc Pham and Yoshio Hayasaki, "Optical frequency comb interference profilometry using compressive sensing," Opt. Express **21**, 19003-19011 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-16-19003

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### References

- J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt.10(9), 2113–2118 (1971). [CrossRef] [PubMed]
- P. S. Lam, J. D. Gaskill, and J. C. Wyant, “Two-wavelength holographic interferometer,” Appl. Opt.23(18), 3079–3081 (1984). [CrossRef] [PubMed]
- C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng.39(1), 79–85 (2000). [CrossRef]
- A. Wada, M. Kato, and Y. Ishii, “Multiple-wavelength digital holographic interferometry using tunable laser diodes,” Appl. Opt.47(12), 2053–2060 (2008). [CrossRef] [PubMed]
- K. N. Joo and S. W. Kim, “Refractive index measurement by spectrally resolved interferometry using a femtosecond pulse laser,” Opt. Lett.32(6), 647–649 (2007). [CrossRef] [PubMed]
- K. Minoshima, K. Arai, and H. Inaba, “Two-Color Interferometry using frequency combs for high-accuracy self-correction of air refractive index,” Opt. Express19, 26095–26105 (2011). [CrossRef] [PubMed]
- C. E. Towers, D. P. Towers, D. T. Reid, W. N. MacPherson, R. R. J. Maier, and J. D. C. Jones, “Fiber interferometer for simultaneous multiwavelength phase measurement with a broadband femtosecond laser,” Opt. Lett.29(23), 2722–2724 (2004). [CrossRef] [PubMed]
- J. S. Oh and S.-W. Kim, “Femtosecond laser pulses for surface-profile metrology,” Opt. Lett.30(19), 2650–2652 (2005). [CrossRef] [PubMed]
- K. Körner, G. Pedrini, I. Alexeenko, T. Steinmetz, R. Holzwarth, and W. Osten, “Short temporal coherence digital holography with a femtosecond frequency comb laser for multi-level optical sectioning,” Opt. Express20(7), 7237–7242 (2012). [CrossRef] [PubMed]
- S. Choi, M. Yamamoto, D. Moteki, T. Shioda, Y. Tanaka, and T. Kurokawa, “Frequency-comb-based interferometer for profilometry and tomography,” Opt. Lett.31(13), 1976–1978 (2006). [CrossRef] [PubMed]
- P. Balling, P. Křen, P. Mašika, and S. A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express17(11), 9300–9313 (2009). [CrossRef] [PubMed]
- M. T. L. Hsu, I. C. M. Littler, D. A. Shaddock, J. Herrmann, R. B. Warrington, and M. B. Gray, “Subpicometer length measurement using heterodyne laser interferometry and all-digital rf phase meters,” Opt. Lett.35(24), 4202–4204 (2010). [CrossRef] [PubMed]
- M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, and H. P. Urbach, “Long distance measurement with femtosecond pulses using a dispersive interferometer,” Opt. Express19(7), 6549–6562 (2011). [CrossRef] [PubMed]
- J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett.29(10), 1153–1155 (2004). [CrossRef] [PubMed]
- J. Lee, Y. J. Kim, K. Lee, S. Lee, and S. W. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics4(10), 716–720 (2010). [CrossRef]
- Y. Salvadé, N. Schuhler, S. Lévêque, and S. Le Floch, “High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source,” Appl. Opt.47(14), 2715–2720 (2008). [CrossRef] [PubMed]
- S. Yokoyama, T. Yokoyama, Y. Hagihara, T. Araki, and T. Yasui, “A distance meter using a terahertz intermode beat in an optical frequency comb,” Opt. Express17(20), 17324–17337 (2009). [CrossRef] [PubMed]
- K. Minoshima and H. Matsumoto, “High-accuracy measurement of 240-m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt.39(30), 5512–5517 (2000). [CrossRef] [PubMed]
- D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory52(4), 1289–1306 (2006). [CrossRef]
- E. J. Candès and T. Tao, “Near optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory52(12), 5406–5425 (2006). [CrossRef]
- M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag.25(2), 83–91 (2008). [CrossRef]
- W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A Single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett.93(12), 121105 (2008). [CrossRef]
- D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express17(15), 13040–13049 (2009). [CrossRef] [PubMed]
- C. F. Cull, D. A. Wikner, J. N. Mait, M. Mattheiss, and D. J. Brady, “Millimeter-wave compressive holography,” Appl. Opt.49(19), E67–E82 (2010). [CrossRef] [PubMed]
- R. Berinde, P. Indyk, and M. Ruzic, “Practical near-optimal sparse recovery in the L1 norm,” Communication, Control, and Computing, 2008 46th Annual Allerton Conference, 198–205, 23–26 Sept. (2008). [CrossRef]
- E. J. Candès and J. Romberg, “Signal recovery from random projections,” in Computational Imaging III, Proc. SPIE Conf. 5674, 76–86, 31 March. (2005). [CrossRef]

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