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

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2012 (1)

2011 (2)

2010 (3)

2009 (3)

2008 (4)

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]

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]

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]

2007 (1)

2006 (3)

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]

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

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]

2005 (1)

2004 (2)

2000 (2)

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]

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]

1984 (1)

1971 (1)

Alexeenko, I.

Arai, K.

Araki, T.

Balling, P.

Baraniuk, R.

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]

Baraniuk, R. G.

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]

Bhattacharya, N.

Brady, D. J.

Candès, E. J.

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]

Chan, W.

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]

Charan, K.

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]

Choi, K.

Choi, S.

Cui, M.

Cull, C. F.

Davenport, M. A.

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]

Donoho, D. L.

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

Duarte, M. F.

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]

Gaskill, J. D.

Gray, M. B.

Hagihara, Y.

Herrmann, J.

Holzwarth, R.

Horisaki, R.

Hsu, M. T. L.

Inaba, H.

Ishii, Y.

Jones, J. D. C.

Joo, K. N.

Kato, M.

Kelly, K.

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]

Kelly, K. F.

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]

Kim, S. W.

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]

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]

Kim, S.-W.

Kim, Y. J.

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]

Körner, K.

Kren, P.

Kurokawa, T.

Lam, P. S.

Laska, J. N.

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]

Le Floch, S.

Lee, J.

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]

Lee, K.

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]

Lee, S.

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]

Lévêque, S.

Lim, S.

Littler, I. C. M.

MacPherson, W. N.

Maier, R. R. J.

Mait, J. N.

Marks, D. L.

Mašika, P.

Matsumoto, H.

Mattheiss, M.

Minoshima, K.

Mittleman, D.

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]

Moteki, D.

Oh, J. S.

Osten, W.

Pedrini, G.

Reid, D. T.

Salvadé, Y.

Schuhler, N.

Seebacher, S.

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]

Shaddock, D. A.

Shioda, T.

Steinmetz, T.

Sun, T.

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]

Takhar, D.

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]

Tanaka, Y.

Tao, T.

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]

Towers, C. E.

Towers, D. P.

Urbach, H. P.

van den Berg, S. A.

Wada, A.

Wagner, C.

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]

Warrington, R. B.

Wikner, D. A.

Wyant, J. C.

Yamamoto, M.

Yasui, T.

Ye, J.

Yokoyama, S.

Yokoyama, T.

Zeitouny, M. G.

Appl. Opt. (6)

Appl. Phys. Lett. (1)

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]

IEEE Signal Process. Mag. (1)

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]

IEEE Trans. Inf. Theory (2)

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

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]

Nat. Photonics (1)

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]

Opt. Eng. (1)

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]

Opt. Express (6)

Opt. Lett. (6)

Other (2)

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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Figures (6)

Fig. 1
Fig. 1

Experimental setup. BPF, band pass filter; BS, beam splitter; PBS, polarization beam splitter; M, reflecting mirror; L, lens; SLM, spatial light modulator; QWP, quarter wave plate; DS, DR, fast photo-receivers; FSS, frequency selection system; PDC, PDS, phase detectors; M, mirror; PS, phase shifter.

Fig. 2
Fig. 2

Pseudorandom patterns of (a) 4 × 4 elements and (b) the corresponding shifted version, (c) 6 × 6 elements and (d) the corresponding shifted version, and (e) 10 × 10 elements and (f) the corresponding shifted version used to encode the object wave.

Fig. 3
Fig. 3

The accuracy of the proposed method, measured with (a) 4 × 4, (b) 6 × 6 and (c) 10 × 10 pseudorandom patterns.

Fig. 4
Fig. 4

The accuracy of the proposed method with different types of pseudorandom patterns was investigated.

Fig. 5
Fig. 5

Actual object profiles measured with (a) 4 × 4 and (b) 6 × 6 pseudorandom patterns.

Fig. 6
Fig. 6

Actual object profile measured with 10 × 10 pseudorandom patterns.

Equations (22)

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U n (t)= y=1 Y x=1 X R n (x,y,l) A n (x,y) e j[ w n t+ φ n (x,y)] ,
U TT (t)= n=1 N max U n (x,y,t) .
U TN (t)= n=1 N U n (x,y,t) .
U RTN (t)= n=1 N A Rn e j( w n t+ φ Rn ) .
U T (t)= y=1 Y x=1 X R k (x,y,l) A k (x,y) e j[ w k t+ φ k (x,y)]
P c (l)=real[ U * RT (t)]real[ U T (t)] = y=1 Y x=1 X α R k (x,y,l) A Rk A k (x,y){cos[Δ φ k (x,y)] +cos[2 w k t+ φ k (x,y)+ φ Rk ]} ,
P c (l)= y=1 Y x=1 X R k (x,y,l) E ck (x,y) ,
P s (l)= y=1 Y x=1 X R k (x,y,l) E sk (x,y) ,
Δ φ k (x,y)=arctan[ E sk (x,y) E ck (x,y) × β α ].
ΔD(x,y)= c[2πp+Δ φ k (x,y)] 4π f k n g ,
{ R sk (x,y,l)=0 if x=1 and y=1 R sk (x,y,l)= R k (X,y1,l) if x=1 and y>1 R sk (x,y,l)= R k (x1,y,l) if x>1 .
P sc (l)= y=1 Y x=1 X R sk (x,y,l) E ck (x,y) ,
P ss (l)= y=1 Y x=1 X R sk (x,y,l) E sk (x,y) .
Δ P c (l)= y=1 Y x=1 X R k (x,y,l)Δ E ck (x,y) ,
{ Δ E ck (x,y)= E ck (x,y) E ck (x+1,y) Δ E ck (x,y)= E ck (x,y) E ck (1,y+1) if x=X Δ E ck (x,y)= E ck (x,y) if x=X and y=Y .
Δ P s (l)= y=1 Y x=1 X R k (x,y,l)Δ E sk (x,y) ,
{ Δ E sk (x,y)= E sk (x,y) E sk (x+1,y) Δ E sk (x,y)= E sk (x,y) E sk (1,y+1) if x=X Δ E sk (x,y)= E sk (x,y) if x=X and y=Y .
M=O(2mlog(XY/m)).
Δ P c = Φ R Δ E c ,
Δ P s = Φ R Δ E s ,
minΔ E c l1 subject to Φ R Δ E c Δ P c 2 ε,
minΔ E s l1 subject to Φ R Δ E s Δ P s 2 ε,

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