Abstract

In this paper we present the design and implementation of a Compressive Sensing Microscopy (CSM) imaging system, which uses the Compressive Sensing (CS) method to realize optical-sectioning imaging. The theoretical aspect of the proposed system is investigated using the mathematical model of the CS method and an experimental prototype is constructed to verify the CSM design. Compared to conventional optical-sectioning microscopes (such as Laser Scanning Confocal Microscopes (LSCMs) or Programmable Array Microscopes (PAMs)), the CSM system realizes optical-sectioning imaging using a single-pixel photo detector and without any mechanical scanning process. The complete information of the imaging scene is reconstructed from the CS measurements numerically.

©2010 Optical Society of America

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References

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  1. J. A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005).
    [Crossref] [PubMed]
  2. M. Liang, R. L. Stehr, and A. W. Krause, “Confocal pattern period in multiple-aperture confocal imaging systems with coherent illumination,” Opt. Lett. 22(11), 751–753 (1997).
    [Crossref] [PubMed]
  3. Q. S. Hanley, P. J. Verveer, M. J. Gemkow, D. Arndt-Jovin, and T. M. Jovin, “An optical sectioning programmable array microscope implemented with a digital micromirror device,” J. Microsc. 196(3), 317–331 (1999).
    [Crossref] [PubMed]
  4. Q. S. Hanley, D. Verveer, and T. M. Jovin, “Optical-sectioning fluorescence spectroscopy in a programmable array microscope,” Appl. Spectrosc. 52(6), 783–792 (1998).
    [Crossref]
  5. E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
    [Crossref]
  6. E. Candès, “Compressive sampling,” Proc. Int. Congress of Mathematics 3, 1433–1452, Madrid, Spain, (2006).
  7. M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
    [Crossref] [PubMed]
  8. R. G. Baraniuk, “Compressive Sensing,” IEEE Signal Process. Mag. 24(4), 118–121 (2007).
    [Crossref]
  9. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, K. F. Ting Sun, Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
    [Crossref]
  10. Y. Wu, P. Ye, Z. Wang, G. R. Arce, and D. W. Prather, “A Single-Pixel Optical Sectioning Programmable Array Microscope (SP-PAM),” Proc. SPIE 7596, 75960D (2010).
    [Crossref]
  11. P. Ye, J. L. Paredes, Y. Wu, C. Chen, G. R. Arce, and D. W. Prather, “Compressive confocal microscopy: 3D reconstruction algorithms,” Proc. SPIE 7210, 72100G (2009).
    [Crossref]
  12. S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “A method for large-scale l1- regularized least squares,” IEEE J. STSP 1(4), 606–617 (2007).
  13. D. Dudley, W. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14 (2003).
    [Crossref]
  14. J. W. Goodman, Introduction to Fourier Optics, 3rd Edition (Roberts & Company Publishers, 2004), Chap. 6.

2010 (1)

Y. Wu, P. Ye, Z. Wang, G. R. Arce, and D. W. Prather, “A Single-Pixel Optical Sectioning Programmable Array Microscope (SP-PAM),” Proc. SPIE 7596, 75960D (2010).
[Crossref]

2009 (1)

P. Ye, J. L. Paredes, Y. Wu, C. Chen, G. R. Arce, and D. W. Prather, “Compressive confocal microscopy: 3D reconstruction algorithms,” Proc. SPIE 7210, 72100G (2009).
[Crossref]

2008 (2)

E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, K. F. Ting Sun, Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

2007 (3)

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

R. G. Baraniuk, “Compressive Sensing,” IEEE Signal Process. Mag. 24(4), 118–121 (2007).
[Crossref]

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “A method for large-scale l1- regularized least squares,” IEEE J. STSP 1(4), 606–617 (2007).

2005 (1)

J. A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005).
[Crossref] [PubMed]

2003 (1)

D. Dudley, W. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14 (2003).
[Crossref]

1999 (1)

Q. S. Hanley, P. J. Verveer, M. J. Gemkow, D. Arndt-Jovin, and T. M. Jovin, “An optical sectioning programmable array microscope implemented with a digital micromirror device,” J. Microsc. 196(3), 317–331 (1999).
[Crossref] [PubMed]

1998 (1)

1997 (1)

Arce, G. R.

Y. Wu, P. Ye, Z. Wang, G. R. Arce, and D. W. Prather, “A Single-Pixel Optical Sectioning Programmable Array Microscope (SP-PAM),” Proc. SPIE 7596, 75960D (2010).
[Crossref]

P. Ye, J. L. Paredes, Y. Wu, C. Chen, G. R. Arce, and D. W. Prather, “Compressive confocal microscopy: 3D reconstruction algorithms,” Proc. SPIE 7210, 72100G (2009).
[Crossref]

Arndt-Jovin, D.

Q. S. Hanley, P. J. Verveer, M. J. Gemkow, D. Arndt-Jovin, and T. M. Jovin, “An optical sectioning programmable array microscope implemented with a digital micromirror device,” J. Microsc. 196(3), 317–331 (1999).
[Crossref] [PubMed]

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, K. F. Ting Sun, Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

R. G. Baraniuk, “Compressive Sensing,” IEEE Signal Process. Mag. 24(4), 118–121 (2007).
[Crossref]

Boyd, S.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “A method for large-scale l1- regularized least squares,” IEEE J. STSP 1(4), 606–617 (2007).

Candès, E.

E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

Chen, C.

P. Ye, J. L. Paredes, Y. Wu, C. Chen, G. R. Arce, and D. W. Prather, “Compressive confocal microscopy: 3D reconstruction algorithms,” Proc. SPIE 7210, 72100G (2009).
[Crossref]

Conchello, J. A.

J. A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005).
[Crossref] [PubMed]

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, K. F. Ting Sun, Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Donoho, D.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, K. F. Ting Sun, Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Dudley, D.

D. Dudley, W. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14 (2003).
[Crossref]

Duncan, W.

D. Dudley, W. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14 (2003).
[Crossref]

Gemkow, M. J.

Q. S. Hanley, P. J. Verveer, M. J. Gemkow, D. Arndt-Jovin, and T. M. Jovin, “An optical sectioning programmable array microscope implemented with a digital micromirror device,” J. Microsc. 196(3), 317–331 (1999).
[Crossref] [PubMed]

Gorinevsky, D.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “A method for large-scale l1- regularized least squares,” IEEE J. STSP 1(4), 606–617 (2007).

Hanley, Q. S.

Q. S. Hanley, P. J. Verveer, M. J. Gemkow, D. Arndt-Jovin, and T. M. Jovin, “An optical sectioning programmable array microscope implemented with a digital micromirror device,” J. Microsc. 196(3), 317–331 (1999).
[Crossref] [PubMed]

Q. S. Hanley, D. Verveer, and T. M. Jovin, “Optical-sectioning fluorescence spectroscopy in a programmable array microscope,” Appl. Spectrosc. 52(6), 783–792 (1998).
[Crossref]

Jovin, T. M.

Q. S. Hanley, P. J. Verveer, M. J. Gemkow, D. Arndt-Jovin, and T. M. Jovin, “An optical sectioning programmable array microscope implemented with a digital micromirror device,” J. Microsc. 196(3), 317–331 (1999).
[Crossref] [PubMed]

Q. S. Hanley, D. Verveer, and T. M. Jovin, “Optical-sectioning fluorescence spectroscopy in a programmable array microscope,” Appl. Spectrosc. 52(6), 783–792 (1998).
[Crossref]

Kelly,

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, K. F. Ting Sun, Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Kim, S. J.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “A method for large-scale l1- regularized least squares,” IEEE J. STSP 1(4), 606–617 (2007).

Koh, K.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “A method for large-scale l1- regularized least squares,” IEEE J. STSP 1(4), 606–617 (2007).

Krause, A. W.

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, K. F. Ting Sun, Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Liang, M.

Lichtman, J. W.

J. A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005).
[Crossref] [PubMed]

Lustig, M.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “A method for large-scale l1- regularized least squares,” IEEE J. STSP 1(4), 606–617 (2007).

Paredes, J. L.

P. Ye, J. L. Paredes, Y. Wu, C. Chen, G. R. Arce, and D. W. Prather, “Compressive confocal microscopy: 3D reconstruction algorithms,” Proc. SPIE 7210, 72100G (2009).
[Crossref]

Pauly, J. M.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Prather, D. W.

Y. Wu, P. Ye, Z. Wang, G. R. Arce, and D. W. Prather, “A Single-Pixel Optical Sectioning Programmable Array Microscope (SP-PAM),” Proc. SPIE 7596, 75960D (2010).
[Crossref]

P. Ye, J. L. Paredes, Y. Wu, C. Chen, G. R. Arce, and D. W. Prather, “Compressive confocal microscopy: 3D reconstruction algorithms,” Proc. SPIE 7210, 72100G (2009).
[Crossref]

Slaughter, J.

D. Dudley, W. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14 (2003).
[Crossref]

Stehr, R. L.

Takhar, D.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, K. F. Ting Sun, Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Ting Sun, K. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, K. F. Ting Sun, Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Verveer, D.

Verveer, P. J.

Q. S. Hanley, P. J. Verveer, M. J. Gemkow, D. Arndt-Jovin, and T. M. Jovin, “An optical sectioning programmable array microscope implemented with a digital micromirror device,” J. Microsc. 196(3), 317–331 (1999).
[Crossref] [PubMed]

Wakin, M.

E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

Wang, Z.

Y. Wu, P. Ye, Z. Wang, G. R. Arce, and D. W. Prather, “A Single-Pixel Optical Sectioning Programmable Array Microscope (SP-PAM),” Proc. SPIE 7596, 75960D (2010).
[Crossref]

Wu, Y.

Y. Wu, P. Ye, Z. Wang, G. R. Arce, and D. W. Prather, “A Single-Pixel Optical Sectioning Programmable Array Microscope (SP-PAM),” Proc. SPIE 7596, 75960D (2010).
[Crossref]

P. Ye, J. L. Paredes, Y. Wu, C. Chen, G. R. Arce, and D. W. Prather, “Compressive confocal microscopy: 3D reconstruction algorithms,” Proc. SPIE 7210, 72100G (2009).
[Crossref]

Ye, P.

Y. Wu, P. Ye, Z. Wang, G. R. Arce, and D. W. Prather, “A Single-Pixel Optical Sectioning Programmable Array Microscope (SP-PAM),” Proc. SPIE 7596, 75960D (2010).
[Crossref]

P. Ye, J. L. Paredes, Y. Wu, C. Chen, G. R. Arce, and D. W. Prather, “Compressive confocal microscopy: 3D reconstruction algorithms,” Proc. SPIE 7210, 72100G (2009).
[Crossref]

Appl. Spectrosc. (1)

IEEE J. STSP (1)

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “A method for large-scale l1- regularized least squares,” IEEE J. STSP 1(4), 606–617 (2007).

IEEE Signal Process. Mag. (3)

E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

R. G. Baraniuk, “Compressive Sensing,” IEEE Signal Process. Mag. 24(4), 118–121 (2007).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, K. F. Ting Sun, Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

J. Microsc. (1)

Q. S. Hanley, P. J. Verveer, M. J. Gemkow, D. Arndt-Jovin, and T. M. Jovin, “An optical sectioning programmable array microscope implemented with a digital micromirror device,” J. Microsc. 196(3), 317–331 (1999).
[Crossref] [PubMed]

Magn. Reson. Med. (1)

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Nat. Methods (1)

J. A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005).
[Crossref] [PubMed]

Opt. Lett. (1)

Proc. SPIE (3)

Y. Wu, P. Ye, Z. Wang, G. R. Arce, and D. W. Prather, “A Single-Pixel Optical Sectioning Programmable Array Microscope (SP-PAM),” Proc. SPIE 7596, 75960D (2010).
[Crossref]

P. Ye, J. L. Paredes, Y. Wu, C. Chen, G. R. Arce, and D. W. Prather, “Compressive confocal microscopy: 3D reconstruction algorithms,” Proc. SPIE 7210, 72100G (2009).
[Crossref]

D. Dudley, W. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14 (2003).
[Crossref]

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 3rd Edition (Roberts & Company Publishers, 2004), Chap. 6.

E. Candès, “Compressive sampling,” Proc. Int. Congress of Mathematics 3, 1433–1452, Madrid, Spain, (2006).

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Figures (9)

Fig. 1
Fig. 1 Schematic drawing of the CSM system.

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Fig. 2
Fig. 2 Example CS measurement patterns (64 × 64). (a) Random pattern (30% “on” pixels). (b) ODHE pattern [inverse Hadamard transform of a sampling impulse in the Hadamard space location (6,7)]. (c) MSBHE pattern (BS = 32). (d) MSBHE pattern (BS = 16).

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Fig. 3
Fig. 3 Simulated image-reconstruction results. (a) A thin specimen, which is placed in the focal-plane of the objective lens (the “in-focus” plane). (b) A thick specimen, which contains the “in-focus” plane and one “out-of-focus” plane ( + 1.5-μm away from the focal-plane in the depth-direction). (c) A thick specimen, which contains the “in-focus” plane and two “out-of-focus” planes ( + 1.5-μm and + 3.0-μm away from the focal-plane in the depth-direction respectively). Images in column (d) are reconstructed using 128 × 128 MSBHE patterns, with a BS value of 64. Images in column (e) are reconstructed using 128 × 128 MSBHE patterns, with a BS value of 32. Images in column (f) are reconstructed using 128 × 128 MSBHE patterns, with a BS value of 16. A 2-D median-filter with a window of 3 × 3 is used to remove noise-spikes in the reconstructed images.

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Fig. 4
Fig. 4 PSNR values of the reconstructed images shown in Fig. 3.

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Fig. 5
Fig. 5 Detector response of the CSM system when an infinitely thin and uniform specimen is used as the imaging target. The specimen is placed in different optical planes of the objective lens. The x-axis in this figure indicates the position of different optical planes with respect to the focal-plane of the objective lens. The y-axis indicates the normalized detector response calculated using Eq. (6).

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Fig. 6
Fig. 6 Mechanical construction of the CSM system.

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Fig. 7
Fig. 7 (a) CCD image of the imaging scene captured with white light illumination. (b) CCD image of the imaging scene captured with 532-nm laser and a set of fluorescent filters.

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Fig. 8
Fig. 8 Experimental results obtained from the experimental CSM system. (a) and (b) are reconstructed using BS = 16 MSBHE patterns 16. (c) and (d) are reconstructed using BS = 32 MSBHE patterns. (e) and (f) are reconstructed using conventional Hadamard patterns. Top-row images are 10-μm away from the bottom-row images in the depth-direction.

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Fig. 9
Fig. 9 Optical-sections obtained with the experimental CSM system. The adjacent optical sections are 1-μm away from each other in the depth-direction.

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Equations (6)

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

y k = < ϕ k , x > + v ,    k = 1 , 2 , ... ,    K   or   Y = Φ x + v = Φ φ θ + v ,
min x ∈ R N ‖ Φ φ θ − Y ‖ 2 2 + λ ‖ θ ‖ 1 ,
P k ( u , v ) = { | h e x ( u , v ) | 2 ⊗ | u k g ( u , v ) | 2 (incoherent illumination) | h e x ( u , v ) ⊗ u k g ( u , v ) | 2 (coherent illumination)   .
E k ( u , v ) = P k ( u , v ) • O ( u , v ) = { [ | h e x ( u , v ) | 2 ⊗ | u k g ( u , v ) | 2 ] • O ( u , v )   (incoherent illumination) [ | h e x ( u , v ) ⊗ u k g ( u , v ) | 2 ] • O ( u , v )   (coherent illumination),  
I k ( ξ , η ) = | h e m ( ξ , η ) | 2 ⊗ | E k g ( ξ , η ) | 2 ,
y k ( ξ , η ) = < ϕ k ( ξ , η ) , I k ( ξ , η ) > + v .

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