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.

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2012

2011

2010

2009

2008

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

2006

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. 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]

2005

2004

2000

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

1971

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. Theory52(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. Theory52(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. Photonics4(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. Photonics4(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. Photonics4(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. Photonics4(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. Photonics4(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. Theory52(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.

Appl. Phys. Lett.

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.

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

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]

Nat. Photonics

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]

Opt. Eng.

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

Opt. Lett.

Other

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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