Abstract

A fiber bundle is widely used for endoscopic imaging due to its direct image delivery capability. However, there exists an inevitable pixelation artifact, which limits spatial resolution to the diameter of individual fibers. In this Letter, we present a method that can eliminate this artifact and achieve diffraction-limited spatial resolution. We exploited the binary control of a digital micromirror device to measure a transmission matrix of a fiber bundle and to subsequently control mode mixing among individual fibers. In doing so, we achieved a 22 kHz scanning rate of a diffraction-limited focused spot and obtained fluorescence endoscope imaging (58μm×58μm) with near video-rate (10.3 Hz) acquisition. Our study lays a foundation for developing an ultrathin and high-resolution microendoscope.

© 2014 Optical Society of America

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2013 (4)

2012 (6)

D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, Opt. Express 20, 1733 (2012).
[CrossRef]

I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, Opt. Express 20, 10583 (2012).
[CrossRef]

T. Cizmar and K. Dholakia, Nat. Commun. 3, 1027 (2012).
[CrossRef]

S. Bianchi and R. Di Leonardo, Lab Chip 12, 635 (2012).
[CrossRef]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, Nature 491, 232 (2012).
[CrossRef]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, Phys. Rev. Lett. 109, 203901 (2012).
[CrossRef]

2011 (4)

D. R. Rivera, C. M. Brown, D. G. Ouzounov, I. Pavlova, D. Kobat, W. W. Webb, and C. Xu, Proc. Natl. Acad. Sci. U.S.A. 108, 17598 (2011).
[CrossRef]

W. M. Lee and S. H. Yun, Opt. Lett. 36, 4608 (2011).
[CrossRef]

R. P. J. Barretto, T. H. Ko, J. C. Jung, T. J. Wang, G. Capps, A. C. Waters, Y. Ziv, A. Attardo, L. Recht, and M. J. Schnitzer, Nat. Med. 17, 223 (2011).
[CrossRef]

D. Akbulut, T. J. Huisman, E. G. van Putten, W. L. Vos, and A. P. Mosk, Opt. Express 19, 4017 (2011).
[CrossRef]

2010 (3)

P. Kim, E. Chung, H. Yamashita, K. E. Hung, A. Mizoguchi, R. Kucherlapati, D. Fukumura, R. K. Jain, and S. H. Yun, Nat. Methods 7, 303 (2010).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 1 (2010).
[CrossRef]

2005 (1)

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, Nat. Methods 2, 941 (2005).
[CrossRef]

2004 (1)

T. Matsunaga, Y. Kawakami, K. Namba, and M. Fujii, Cancer 101, 2164 (2004).
[CrossRef]

2001 (1)

F. Koenig, J. Knittel, and H. Stepp, Science 292, 1401 (2001).
[CrossRef]

2000 (1)

1997 (1)

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, Science 276, 2037 (1997).
[CrossRef]

1954 (1)

H. H. Hopkins and N. S. Kapany, Nature 173, 39 (1954).
[CrossRef]

Akbulut, D.

Andresen, E. R.

Attardo, A.

R. P. J. Barretto, T. H. Ko, J. C. Jung, T. J. Wang, G. Capps, A. C. Waters, Y. Ziv, A. Attardo, L. Recht, and M. J. Schnitzer, Nat. Med. 17, 223 (2011).
[CrossRef]

Barretto, R. P. J.

R. P. J. Barretto, T. H. Ko, J. C. Jung, T. J. Wang, G. Capps, A. C. Waters, Y. Ziv, A. Attardo, L. Recht, and M. J. Schnitzer, Nat. Med. 17, 223 (2011).
[CrossRef]

Bertolotti, J.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, Nature 491, 232 (2012).
[CrossRef]

Bianchi, S.

S. Bianchi and R. Di Leonardo, Lab Chip 12, 635 (2012).
[CrossRef]

Blum, C.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, Nature 491, 232 (2012).
[CrossRef]

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 1 (2010).
[CrossRef]

Boppart, S. A.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, Science 276, 2037 (1997).
[CrossRef]

Bouma, B. E.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, Science 276, 2037 (1997).
[CrossRef]

Bouwmans, G.

Brezinski, M. E.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, Science 276, 2037 (1997).
[CrossRef]

Brown, C. M.

D. R. Rivera, C. M. Brown, D. G. Ouzounov, I. Pavlova, D. Kobat, W. W. Webb, and C. Xu, Proc. Natl. Acad. Sci. U.S.A. 108, 17598 (2011).
[CrossRef]

Capps, G.

R. P. J. Barretto, T. H. Ko, J. C. Jung, T. J. Wang, G. Capps, A. C. Waters, Y. Ziv, A. Attardo, L. Recht, and M. J. Schnitzer, Nat. Med. 17, 223 (2011).
[CrossRef]

Caravaca-Aguirre, A. M.

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Cheung, E. L. M.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, Nat. Methods 2, 941 (2005).
[CrossRef]

Choi, W.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, Phys. Rev. Lett. 109, 203901 (2012).
[CrossRef]

Choi, Y.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, Phys. Rev. Lett. 109, 203901 (2012).
[CrossRef]

Chudoba, C.

Chung, E.

P. Kim, E. Chung, H. Yamashita, K. E. Hung, A. Mizoguchi, R. Kucherlapati, D. Fukumura, R. K. Jain, and S. H. Yun, Nat. Methods 7, 303 (2010).
[CrossRef]

Cizmar, T.

T. Cizmar and K. Dholakia, Nat. Commun. 3, 1027 (2012).
[CrossRef]

Cocker, E. D.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, Nat. Methods 2, 941 (2005).
[CrossRef]

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, Phys. Rev. Lett. 109, 203901 (2012).
[CrossRef]

Dholakia, K.

T. Cizmar and K. Dholakia, Nat. Commun. 3, 1027 (2012).
[CrossRef]

Di Leonardo, R.

S. Bianchi and R. Di Leonardo, Lab Chip 12, 635 (2012).
[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, Phys. Rev. Lett. 109, 203901 (2012).
[CrossRef]

Farahi, S.

Fink, M.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 1 (2010).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Flusberg, B. A.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, Nat. Methods 2, 941 (2005).
[CrossRef]

Fujii, M.

T. Matsunaga, Y. Kawakami, K. Namba, and M. Fujii, Cancer 101, 2164 (2004).
[CrossRef]

Fujimoto, J. G.

X. D. Li, C. Chudoba, T. Ko, C. Pitris, and J. G. Fujimoto, Opt. Lett. 25, 1520 (2000).
[CrossRef]

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, Science 276, 2037 (1997).
[CrossRef]

Fukumura, D.

P. Kim, E. Chung, H. Yamashita, K. E. Hung, A. Mizoguchi, R. Kucherlapati, D. Fukumura, R. K. Jain, and S. H. Yun, Nat. Methods 7, 303 (2010).
[CrossRef]

Gigan, S.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 1 (2010).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Gu, R. Y.

Hopkins, H. H.

H. H. Hopkins and N. S. Kapany, Nature 173, 39 (1954).
[CrossRef]

Huisman, T. J.

Hung, K. E.

P. Kim, E. Chung, H. Yamashita, K. E. Hung, A. Mizoguchi, R. Kucherlapati, D. Fukumura, R. K. Jain, and S. H. Yun, Nat. Methods 7, 303 (2010).
[CrossRef]

Jain, R. K.

P. Kim, E. Chung, H. Yamashita, K. E. Hung, A. Mizoguchi, R. Kucherlapati, D. Fukumura, R. K. Jain, and S. H. Yun, Nat. Methods 7, 303 (2010).
[CrossRef]

Jung, J. C.

R. P. J. Barretto, T. H. Ko, J. C. Jung, T. J. Wang, G. Capps, A. C. Waters, Y. Ziv, A. Attardo, L. Recht, and M. J. Schnitzer, Nat. Med. 17, 223 (2011).
[CrossRef]

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, Nat. Methods 2, 941 (2005).
[CrossRef]

Kahn, J. M.

Kapany, N. S.

H. H. Hopkins and N. S. Kapany, Nature 173, 39 (1954).
[CrossRef]

Kawakami, Y.

T. Matsunaga, Y. Kawakami, K. Namba, and M. Fujii, Cancer 101, 2164 (2004).
[CrossRef]

Kim, M.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, Phys. Rev. Lett. 109, 203901 (2012).
[CrossRef]

Kim, P.

P. Kim, E. Chung, H. Yamashita, K. E. Hung, A. Mizoguchi, R. Kucherlapati, D. Fukumura, R. K. Jain, and S. H. Yun, Nat. Methods 7, 303 (2010).
[CrossRef]

Knittel, J.

F. Koenig, J. Knittel, and H. Stepp, Science 292, 1401 (2001).
[CrossRef]

Ko, T.

Ko, T. H.

R. P. J. Barretto, T. H. Ko, J. C. Jung, T. J. Wang, G. Capps, A. C. Waters, Y. Ziv, A. Attardo, L. Recht, and M. J. Schnitzer, Nat. Med. 17, 223 (2011).
[CrossRef]

Kobat, D.

D. R. Rivera, C. M. Brown, D. G. Ouzounov, I. Pavlova, D. Kobat, W. W. Webb, and C. Xu, Proc. Natl. Acad. Sci. U.S.A. 108, 17598 (2011).
[CrossRef]

Koenig, F.

F. Koenig, J. Knittel, and H. Stepp, Science 292, 1401 (2001).
[CrossRef]

Kucherlapati, R.

P. Kim, E. Chung, H. Yamashita, K. E. Hung, A. Mizoguchi, R. Kucherlapati, D. Fukumura, R. K. Jain, and S. H. Yun, Nat. Methods 7, 303 (2010).
[CrossRef]

Lagendijk, A.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, Nature 491, 232 (2012).
[CrossRef]

Lee, K. J.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, Phys. Rev. Lett. 109, 203901 (2012).
[CrossRef]

Lee, W. M.

Lerosey, G.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 1 (2010).
[CrossRef]

Li, X. D.

Mahalati, R. N.

Matsunaga, T.

T. Matsunaga, Y. Kawakami, K. Namba, and M. Fujii, Cancer 101, 2164 (2004).
[CrossRef]

Mizoguchi, A.

P. Kim, E. Chung, H. Yamashita, K. E. Hung, A. Mizoguchi, R. Kucherlapati, D. Fukumura, R. K. Jain, and S. H. Yun, Nat. Methods 7, 303 (2010).
[CrossRef]

Monneret, S.

Moser, C.

Mosk, A. P.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, Nature 491, 232 (2012).
[CrossRef]

D. Akbulut, T. J. Huisman, E. G. van Putten, W. L. Vos, and A. P. Mosk, Opt. Express 19, 4017 (2011).
[CrossRef]

Namba, K.

T. Matsunaga, Y. Kawakami, K. Namba, and M. Fujii, Cancer 101, 2164 (2004).
[CrossRef]

Niv, E.

Ouzounov, D. G.

D. R. Rivera, C. M. Brown, D. G. Ouzounov, I. Pavlova, D. Kobat, W. W. Webb, and C. Xu, Proc. Natl. Acad. Sci. U.S.A. 108, 17598 (2011).
[CrossRef]

Papadopoulos, I. N.

Pavlova, I.

D. R. Rivera, C. M. Brown, D. G. Ouzounov, I. Pavlova, D. Kobat, W. W. Webb, and C. Xu, Proc. Natl. Acad. Sci. U.S.A. 108, 17598 (2011).
[CrossRef]

Piestun, R.

Pitris, C.

X. D. Li, C. Chudoba, T. Ko, C. Pitris, and J. G. Fujimoto, Opt. Lett. 25, 1520 (2000).
[CrossRef]

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, Science 276, 2037 (1997).
[CrossRef]

Piyawattanametha, W.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, Nat. Methods 2, 941 (2005).
[CrossRef]

Popoff, S.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 1 (2010).
[CrossRef]

Popoff, S. M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Psaltis, D.

Recht, L.

R. P. J. Barretto, T. H. Ko, J. C. Jung, T. J. Wang, G. Capps, A. C. Waters, Y. Ziv, A. Attardo, L. Recht, and M. J. Schnitzer, Nat. Med. 17, 223 (2011).
[CrossRef]

Rigneault, H.

Rivera, D. R.

D. R. Rivera, C. M. Brown, D. G. Ouzounov, I. Pavlova, D. Kobat, W. W. Webb, and C. Xu, Proc. Natl. Acad. Sci. U.S.A. 108, 17598 (2011).
[CrossRef]

Schnitzer, M. J.

R. P. J. Barretto, T. H. Ko, J. C. Jung, T. J. Wang, G. Capps, A. C. Waters, Y. Ziv, A. Attardo, L. Recht, and M. J. Schnitzer, Nat. Med. 17, 223 (2011).
[CrossRef]

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, Nat. Methods 2, 941 (2005).
[CrossRef]

Southern, J. F.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, Science 276, 2037 (1997).
[CrossRef]

Stepp, H.

F. Koenig, J. Knittel, and H. Stepp, Science 292, 1401 (2001).
[CrossRef]

Tearney, G. J.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, Science 276, 2037 (1997).
[CrossRef]

van Putten, E. G.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, Nature 491, 232 (2012).
[CrossRef]

D. Akbulut, T. J. Huisman, E. G. van Putten, W. L. Vos, and A. P. Mosk, Opt. Express 19, 4017 (2011).
[CrossRef]

Vos, W. L.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, Nature 491, 232 (2012).
[CrossRef]

D. Akbulut, T. J. Huisman, E. G. van Putten, W. L. Vos, and A. P. Mosk, Opt. Express 19, 4017 (2011).
[CrossRef]

Wang, T. J.

R. P. J. Barretto, T. H. Ko, J. C. Jung, T. J. Wang, G. Capps, A. C. Waters, Y. Ziv, A. Attardo, L. Recht, and M. J. Schnitzer, Nat. Med. 17, 223 (2011).
[CrossRef]

Waters, A. C.

R. P. J. Barretto, T. H. Ko, J. C. Jung, T. J. Wang, G. Capps, A. C. Waters, Y. Ziv, A. Attardo, L. Recht, and M. J. Schnitzer, Nat. Med. 17, 223 (2011).
[CrossRef]

Webb, W. W.

D. R. Rivera, C. M. Brown, D. G. Ouzounov, I. Pavlova, D. Kobat, W. W. Webb, and C. Xu, Proc. Natl. Acad. Sci. U.S.A. 108, 17598 (2011).
[CrossRef]

Xu, C.

D. R. Rivera, C. M. Brown, D. G. Ouzounov, I. Pavlova, D. Kobat, W. W. Webb, and C. Xu, Proc. Natl. Acad. Sci. U.S.A. 108, 17598 (2011).
[CrossRef]

Yamashita, H.

P. Kim, E. Chung, H. Yamashita, K. E. Hung, A. Mizoguchi, R. Kucherlapati, D. Fukumura, R. K. Jain, and S. H. Yun, Nat. Methods 7, 303 (2010).
[CrossRef]

Yang, T. D.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, Phys. Rev. Lett. 109, 203901 (2012).
[CrossRef]

Yoon, C.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, Phys. Rev. Lett. 109, 203901 (2012).
[CrossRef]

Yun, S. H.

W. M. Lee and S. H. Yun, Opt. Lett. 36, 4608 (2011).
[CrossRef]

P. Kim, E. Chung, H. Yamashita, K. E. Hung, A. Mizoguchi, R. Kucherlapati, D. Fukumura, R. K. Jain, and S. H. Yun, Nat. Methods 7, 303 (2010).
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Ziv, Y.

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Supplementary Material (2)

» Media 1: MOV (1283 KB)     
» Media 2: AVI (2350 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

Schematic experimental setup for recording a transmission matrix and fluorescence endomicroscopy. The output beam from a He–Ne laser was divided into sample and reference beams using a beam splitter, BS1. Another beam splitter, BS2, recombines the two beams. OL1 and OL2, objective lenses (PLN 40×, Olympus); DS, dichroic beam splitters (T647lpxr, Chroma); TL1, TL2, and TL3, tube lenses (LA1979-A, Thorlabs); LF, long wavelength filter (FELH0650, Thorlabs); DG, diffraction grating (830 Grooves); D, aperture; PMT, photomultiplier tube (H5784-20, Hamamatsu); camera (M3, IDT, 500 fps); IP, input plane of a fiber bundle; SP, sample plane. The x, y, and z are the spatial coordinates in the SP. The magnification from the DMD to the IP is 1/111 and that from the SP to the camera is 55.6.

Fig. 2.
Fig. 2.

Transmission matrix recording and diffraction-limited focusing. (a) Representative maps of the binary speckle basis, i.e., the binary sequences S in the main text, displayed on the DMD. White (black) pixels indicate that mirrors are in an on (off) state. (b) Intensity maps at the SP corresponding to the individual binary speckle basis in Fig. 2(a). The x and y are the orthogonal coordinates at the SP, and p is the index of the speckle basis. Scale bar, 10 μm. (c) A binary input pattern on the DMD identified from the measured transmission matrix for focusing the laser beam at the SP. (d) Output intensity image at the SP when the binary pattern in Fig. 2(c) was displayed on the DMD. The inset figure is the 3.33× enlarged and 10× oversaturated display of the focus. Scale bar, 5 μm.

Fig. 3.
Fig. 3.

Point-spread function of the focusing through a fiber bundle. A focused spot was scanned along the x-z plane and its intensity map was recorded by the camera. (a) Intensity map of the focus at the x-z plane. The z is the optical axis, and x is one of the transverse coordinates in the SP. Scale bar, 10 μm. (b) and (c) Line profiles along the lateral and axial directions of (a), respectively, showing their respective FWHMs to be 1.07 and 8.99 μm.

Fig. 4.
Fig. 4.

Pixelation-free endoscopic imaging through fiber bundle. (a) and (c) Conventional transmission images of 2 μm fluorescence beads (Skyblue, Spherotech) and a cancer cell line (SNU-1074), respectively, recorded with an LED illuminated through a fiber bundle. (b) and (d) Endoscopic fluorescence images of the same samples as in (a) and (c), respectively, recorded by the proposed method (Media 1 and Media 2). Scale bar, 20 μm.

Equations (2)

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Oip=jtijSjp.
tij=pOipSpj1.

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