Abstract

Three modifications are shown to improve resolution and reduce noise amplification in endoscopic imaging through multi-mode fiber using optimization-based reconstruction (OBR). First, random sampling patterns are replaced by sampling patterns designed to have more nearly equal singular values. Second, the OBR algorithm uses a point-spread function based on the estimated spatial frequency spectrum of the object. Third, the OBR algorithm gives less weight to modes having smaller singular values. In simulations for a step-index fiber supporting 522 spatial modes, the modifications yield a 20% reduction in image error (l2 norm) in the noiseless case, and a further 33% reduction in image error at a 22-dB shot noise-limited SNR as compared to the original method using random sampling patterns and OBR.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
    [CrossRef]
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  21. A. J. Welch, “The thermal response of laser irradiated tissue,” IEEE J. Quantum Electron. 20(12), 1471–1481 (1984).
    [CrossRef]
  22. A. L. McKenzie, “Physics of thermal processes in laser-tissue interaction,” Phys. Med. Biol. 35(9), 1175–1210 (1990).
    [CrossRef] [PubMed]
  23. S. W. Allison, G. T. Gillies, D. W. Magnuson, and T. S. Pagano, “Pulsed laser damage to optical fibers,” Appl. Opt. 24(19), 3140–3144 (1985).
    [CrossRef] [PubMed]
  24. M. B. Shemirani, W. Mao, R. A. Panicker, and J. M. Kahn, “Principal modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling,” J. Lightwave Technol. 27(10), 1248–1261 (2009).
    [CrossRef]

2013

2012

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[CrossRef] [PubMed]

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip 12(3), 635–639 (2012).
[CrossRef] [PubMed]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[CrossRef] [PubMed]

2009

2006

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

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

2005

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[CrossRef] [PubMed]

1990

A. L. McKenzie, “Physics of thermal processes in laser-tissue interaction,” Phys. Med. Biol. 35(9), 1175–1210 (1990).
[CrossRef] [PubMed]

1985

S. W. Allison, G. T. Gillies, D. W. Magnuson, and T. S. Pagano, “Pulsed laser damage to optical fibers,” Appl. Opt. 24(19), 3140–3144 (1985).
[CrossRef] [PubMed]

B. Fischer and S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett. 46(2), 113 (1985).
[CrossRef]

1984

A. J. Welch, “The thermal response of laser irradiated tissue,” IEEE J. Quantum Electron. 20(12), 1471–1481 (1984).
[CrossRef]

1976

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28(2), 88 (1976).
[CrossRef]

Allison, S. W.

Bianchi, S.

Candes, E. J.

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

Caravaca-Aguirre, A. M.

Cheung, E. L. M.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[CrossRef] [PubMed]

Choi, W.

Y. Choi, C. Yoon, M. Kim, J. Yang, and W. Choi, “Disorder-mediated enhancement of fiber numerical aperture,” Opt. Lett. 38(13), 2253–2255 (2013).
[CrossRef] [PubMed]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[CrossRef] [PubMed]

Choi, Y.

Y. Choi, C. Yoon, M. Kim, J. Yang, and W. Choi, “Disorder-mediated enhancement of fiber numerical aperture,” Opt. Lett. 38(13), 2253–2255 (2013).
[CrossRef] [PubMed]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[CrossRef] [PubMed]

Cižmár, T.

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[CrossRef] [PubMed]

Cocker, E. D.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[CrossRef] [PubMed]

Conkey, D. B.

Dasari, R. R.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[CrossRef] [PubMed]

Dholakia, K.

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[CrossRef] [PubMed]

Di Fabrizio, E.

Di Leonardo, R.

Donoho, D. L.

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

Fang-Yen, C.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[CrossRef] [PubMed]

Farahi, S.

Ferrara, L.

Fischer, B.

B. Fischer and S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett. 46(2), 113 (1985).
[CrossRef]

Flusberg, B. A.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[CrossRef] [PubMed]

Gillies, G. T.

Gu, R. Y.

Jung, J. C.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[CrossRef] [PubMed]

Kahn, J. M.

Kim, M.

Y. Choi, C. Yoon, M. Kim, J. Yang, and W. Choi, “Disorder-mediated enhancement of fiber numerical aperture,” Opt. Lett. 38(13), 2253–2255 (2013).
[CrossRef] [PubMed]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[CrossRef] [PubMed]

Lee, K. J.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[CrossRef] [PubMed]

Liberale, C.

Magnuson, D. W.

Mahalati, R. N.

Mao, W.

McKenzie, A. L.

A. L. McKenzie, “Physics of thermal processes in laser-tissue interaction,” Phys. Med. Biol. 35(9), 1175–1210 (1990).
[CrossRef] [PubMed]

Moser, C.

Niv, E.

Pagano, T. S.

Panicker, R. A.

Papadopoulos, I. N.

Piestun, R.

Piyawattanametha, W.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[CrossRef] [PubMed]

Psaltis, D.

Rajamanickam, V. P.

Schnitzer, M. J.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[CrossRef] [PubMed]

Shemirani, M. B.

Sternklar, S.

B. Fischer and S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett. 46(2), 113 (1985).
[CrossRef]

Tao, T.

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

Welch, A. J.

A. J. Welch, “The thermal response of laser irradiated tissue,” IEEE J. Quantum Electron. 20(12), 1471–1481 (1984).
[CrossRef]

Yang, J.

Yang, T. D.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[CrossRef] [PubMed]

Yariv, A.

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28(2), 88 (1976).
[CrossRef]

Yoon, C.

Y. Choi, C. Yoon, M. Kim, J. Yang, and W. Choi, “Disorder-mediated enhancement of fiber numerical aperture,” Opt. Lett. 38(13), 2253–2255 (2013).
[CrossRef] [PubMed]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[CrossRef] [PubMed]

Ziegler, D.

Appl. Opt.

Appl. Phys. Lett.

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28(2), 88 (1976).
[CrossRef]

B. Fischer and S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett. 46(2), 113 (1985).
[CrossRef]

IEEE J. Quantum Electron.

A. J. Welch, “The thermal response of laser irradiated tissue,” IEEE J. Quantum Electron. 20(12), 1471–1481 (1984).
[CrossRef]

IEEE Trans. Inf. Theory

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

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

J. Lightwave Technol.

Lab Chip

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip 12(3), 635–639 (2012).
[CrossRef] [PubMed]

Nat. Commun.

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[CrossRef] [PubMed]

Nat. Methods

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

A. L. McKenzie, “Physics of thermal processes in laser-tissue interaction,” Phys. Med. Biol. 35(9), 1175–1210 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[CrossRef] [PubMed]

Other

J. A. Buck, Fundamentals of Optical Fibers, 2nd ed. (John Wiley, 2004).

J. Mertz, Introduction to Optical Microscopy (Roberts and Company, 2010).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2004).

G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (John Wiley, 2010), Chap. 3.

A. J. Welch, M. J. C. Van Gemert, W. M. Star, and T. Optics, Optical-Thermal Response of Laser-Irradiated Tissue, 2nd ed. (Springer, 2011), Chap. 3.

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Figures (6)

Fig. 1
Fig. 1

Experimental setup for imaging through multi-mode fiber (MMF) [3]. (a) In calibration, a camera is placed at the MMF output to record intensity patterns appearing there. (b) In imaging, an object is placed at the MMF output and illuminated by a sequence of intensity patterns, and the reflected power values coupled back into the MMF are recorded.

Fig. 2
Fig. 2

Noise amplification for OBR when sampling using random (dashed) and spot (solid) intensity sampling patterns as a function of the number of LP modes per polarization under constraints on (a) maximum irradiance and (b) total power. Noise amplification values are obtained from Eqs. (27)(29) and Eqs. (32)(34). Intensity patterns are observed 5 μm from the MMF fiber tip. The MMF has a 50-μm core diameter and a varying NA. The number of sampling patterns is 16N (corresponding to 2 × oversampling in the x- and y-dimensions for spots). Intensity patterns are sampled on a 416 × 416 grid.

Fig. 3
Fig. 3

Images reconstructed using OBR: (a) original algorithm and (b) new algorithm using an estimate of the object spatial frequency spectrum. Also shown are: (c) the object reflectance and (d) the object providing the estimated spatial frequency spectrum, which is unrelated to the actual object. In (a) and (b), values of the image error RMSD σ( w ˜ i ) within the MMF core are given as a percent of the maximum image intensity. Intensity patterns and object are at 5 μm from the MMF fiber tip. The MMF has 50-μm core diameter and 0.45 NA. The number of sampling patterns is 16N (corresponding to 2 × oversampling in the x- and y-dimensions for spots). Intensity patterns are sampled on a 256 × 256 grid.

Fig. 4
Fig. 4

Images reconstructed using OBR: (a) using original method of Eqs. (42) and (43), and (b) using reduced-noise method of Eqs. (44) and (45). The ratio of the noise constraint to the maximum noise constraint is α/ α max =10dB . The shot noise-limited SNR is σ noise / p ¯ =22 dB , where p ¯ is the average power. Values of the image error RMSD σ( w ˜ i ) within the MMF core are given as a percent of the maximum image intensity. The intensity patterns and object are at 5 μm from MMF fiber tip. The MMF has 50-μm core diameter and 0.45 NA. The number of sampling patterns is 16N (corresponding to 2 × oversampling in the x- and y-dimensions for spots). The intensity patterns are sampled on a 256 × 256 grid.

Fig. 5
Fig. 5

Errors of images reconstructed using OBR (dashed lines) and localized reconstruction (solid lines) as a function of power noise levels under constraints on (a) maximum irradiance and (b) total power. The intensity patterns and object are at 5 μm from MMF fiber tip. The MMF has 50-μm core diameter and 0.45 NA. The number of sampling patterns is 16N (corresponding to 2 × oversampling in the x- and y-dimensions for spots). Intensity patterns are sampled on a 256 × 256 grid.

Fig. 6
Fig. 6

Images reconstructed with OBR (a-c) and localized reconstruction (d-f) for various levels of shot noise. σ shot / p ¯ is −25 dB for Figs. (a) and (d); −15 dB for Figs. (b) and (e); and −10 dB for Figs. (c) and (f). The sampling patterns are constrained by total power. Values of the image error RMSD σ( w ˜ i ) within the MMF core are given as a percent of the maximum image intensity. The intensity patterns and object are at 5 μm from MMF fiber tip. The MMF has 50-μm core diameter and 0.45 NA. The number of sampling patterns is 16N (corresponding to 2 × oversampling in the x- and y-dimensions for spots). The intensity patterns are sampled on a 256 × 256 grid.

Equations (45)

Equations on this page are rendered with MathJax. Learn more.

E lm (ρ,ϕ)= c 1 J 1 ( κ l,m ρ)sin(lϕ) (core) E lm (ρ,ϕ)= c 2 K l ( γ l,m ρ)sin(lϕ) (cladding)
E P1f ( x,y )= l=0,m=0,q=0 l max , m max , q max c lmq E lmq ( x,y ),
E P2f ( x,y )= l=0,m=0,q=0 l max , m max , q max c lmq P z [ E lmq ( x,y ) ],
E P2b ( x,y )=r( x,y ) E P2f ( x,y ).
p= l=0,m=0,q=0 l max , m max , q max | E lmq ( x,y ) P z [ E P2b ( x,y ) ]dxdy | 2 .
p= c | E P2f ( x,y )r( x,y ) | 2 dxdy
p= | c ˜ E P2f ( x,y ) | 2 | r( x,y ) | 2 dxdy
p= I ˜ P2f ( x,y )R( x,y )dxdy,
p =  I ˜ r,
w=V D ˜ 1 U T p,
D ˜ i,j 1 ={ 1/ D i,j 0 for  i=j4N otherwise .
w=V D ˜ 1 U T ( p+n )
w=V V T r+ V T D ˜ 1 U T n
w ˜ V T D ˜ 1 U T n,
n= σ thermal n ^ (thermal)
n= σ shot n ^ p  (shot)
n= σ intensity n ^ p (intensity),
p=UD V T r
n= σ intensity n ^ p
w ˜ =V D ˜ 1 U T n
n ˜ U T n,
E[ i=1 L w ˜ i 2 ]=E[ i=1 4N ( D ii 1 n ˜ i ) 2 ]= i=1 4N D ii 2 E[ n ˜ i 2 ] 1 M E[ i=1 M n ˜ i 2 ] i=1 4N D ii 2
E[ i=1 M n ˜ i 2 ]=E[ i=1 M n i 2 ]= i=1 M E[ n i 2 ]= i=1 M σ intensity 2 μ ( p i ) 2 + σ intensity 2 σ 2 ( p i ).
μ( p i )=E[ i i T r ]=E[ r 1 ] j=1 L I ˜ ij = 1 2 j=1 L I ˜ ij
σ 2 ( p i )= 1 M σ 2 ( r 1 ) i=1 4N D ii 2 = 1 12M i=1 4N D ii 2 ,
E[ i=1 L w ˜ i 2 ]= 1 M ( i=1 4N D ii 2 )[ i=1 M ( 1 2 j=1 L I ˜ ij ) 2 ] σ intensity 2 .
σ( w ˜ i )= 1 L i=1 4N D ii 2 1 M i=1 M ( 1 2 j=1 L I ˜ ij ) 2 σ intensity (intensity noise)
σ( w ˜ i )= 1 L i=1 4N D ii 2 1 M i=1 M ( 1 2 j=1 L I ˜ ij ) σ shot (shot noise)
σ( w ˜ i )= 1 L i=1 4N D ii 2 σ thermal (thermal noise),
w=T[ s( I ˜ r+n ) ]
w ˜ =T( sn ),
σ( w ˜ i )= 1 2 σ intensity (intensity noise)
σ( w ˜ i )= 1 2M i=1 M ( j=1 P I ˜ ij ) 1 σ shot (shot noise)
σ( w ˜ i )= 1 M i=1 M ( j=1 P I ˜ ij ) 2 σ thermal (thermal noise),
argmin w p I ˜ w 2  s.t. V 2 T w=0,
argmin w wr 2
argmin w p I ˜ w 2  s.t V 2 T w=0= argmin w UD V T rUD V T w 2 s.t. V 2 T w=0 = argmin w D V T ( rw ) 2 s.t. V 2 T w=0 = argmin w V 1 T ( wr ) 2 s.t. V 2 T w=0
argmin w wr 2 s.t. p= I ˜ w= argmin w V T ( wr ) 2 s.t. V 1 T r= V 1 T w = argmin w ( V 1 T ( wr ) 2 +  V 2 T ( wr ) 2 ) = argmin w V 1 T ( wr ) 2 s.t. V 2 T w=0
Image [ w( x,y )r( x,y ) ] 2 dxdy
Image [ r( x,y )( q( x,y )δ( x,y ) ) ] 2 dxdy
Image [ r ˜ ( x,y )( q ˜ ( x,y ) δ ˜ ( x,y ) ) ] 2 dxdy,
w= V ˜ D ˜ 1 U T p,
v ˜ i T = argmin v ˜ i T ( r T F DFT )( v ˜ i T V 1 F DFT δ T F DFT ) 2 ,
v ˜ i T  <α d T  i,
v ˜ i T = argmin v ˜ i T  ( r T F DFT )( v ˜ i T V 1 F DFT δ T F DFT ) 2 s.t v ˜ i T  <α d T .

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