## Fast full-color computational imaging with single-pixel detectors |

Optics Express, Vol. 21, Issue 20, pp. 23068-23074 (2013)

http://dx.doi.org/10.1364/OE.21.023068

Acrobat PDF (2327 KB)

### Abstract

Single-pixel detectors can be used as imaging devices by making use of structured illumination. These systems work by correlating a changing incident light field with signals measured on a photodiode to derive an image of an object. In this work we demonstrate a system that utilizes a digital light projector to illuminate a scene with approximately 1300 different light patterns every second and correlate these with the back scattered light measured by three spectrally-filtered single-pixel photodetectors to produce a full-color high-quality image in a few seconds of data acquisition. We utilize a differential light projection method to self normalize the measured signals, improving the reconstruction quality whilst making the system robust to external sources of noise. This technique can readily be extended for imaging applications at non-visible wavebands.

© 2013 Optical Society of America

## 1. Introduction

*computational ghost imaging*[9

9. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A **78**, 061802 (2008). [CrossRef]

10. M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Proc. Mag. **25**, 83–91 (2008). [CrossRef]

11. P. Sen, B. Chen, G. Garg, S. R. Marschner, M. Horowitz, M. Levoy, and H. P. A. Lensch, “Dual photography,” ACM Trans. Graph. **24**, 745–755 (2005). [CrossRef]

12. B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science **340**, 844–847 (2013). [CrossRef] [PubMed]

*sparseness*in natural images to significantly reduce the amount of information that is needed to reproduce it. Indeed, it is this feature of natural images that is at the heart of well-known lossy image compression algorithms, such as JPEG [13

13. G. K. Wallace, “The jpeg still picture compression standard,” Commun. ACM **34**, 30–44 (1991). [CrossRef]

## 2. Experimental setup

14. D. Preece, R. Bowman, A. Linnenberger, G. Gibson, S. Serati, and M. Padgett, “Increasing trap stiffness with position clamping in holographic optical tweezers,” Opt. Express **17**, 22718–22725 (2009). [CrossRef]

*queue*structures are used to pass data asynchronously between loops, and LabVIEW’s native parallelism allows separate loops to run on different processor cores for increased speed. Within each frame, the 22 bit-planes that are not used for synchronization are split into 11 pairs. Each pair consists of a pattern and its inverse. This allows us to make differential measurements, analogous to lock-in detection at 720Hz. Differential detection significantly reduces the influence of noise such as ambient light fluctuations on the measurement. Overall, the system is able to display and measure approximately 650 unique patterns per second, once differential measurement and synchronisation are taken into account.

## 3. Image reconstruction

*N*=

*x*×

*y*, where

*x*and

*y*is the number of pixels in

*x*and

*y*dimensions used for illumination. For each iteration,

*i*, a unique 2D intensity pattern

*I*(

_{i}*x*,

*y*) is projected onto the object and the corresponding reflected intensity (voltage signal), at spectral frequency

*μ*, is measured for each single-pixel photodetector,

*S*, thus we can write where

_{μi}*R*(

_{μ}*x*,

*y*) is the reflection function of the scene for a spectral frequency

*μ*and

*I′*(

_{i}*x*,

*y*) is the 2D pattern after propagation. In our case since the modulator is imaged onto the plane of the object,

*I′*(

_{i}*x*,

*y*) =

*I*(

_{i}*x*,

*y*). Provided that the entire scene is located within the detectors field of view, the signal measured is directly proportional to the projection of the scene reflectivity onto each illumination pattern, and can effectively be used as a weighting factor in the image reconstruction algorithm.

### 3.1. Iterative image reconstruction

15. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. **104**, 253603 (2010). [CrossRef] [PubMed]

16. B. Sun, S. S. Welsh, M. P. Edgar, J. H. Shapiro, and M. J. Padgett, “Normalized ghost imaging,” Opt. Express **20**, 16892–16901 (2012). [CrossRef]

*O*(

_{μ}*x*,

*y*) is the estimate of the scene and < ... > denotes an ensemble average. In this experiment the average differential signal tends to zero, as does each pixel in the average differential pattern, thus Eq. 2 can be re-written as for

*M*samples. The final result from our iterative reconstruction is shown in Fig. 2 for approximately 1 million measurements. The full-color image is obtained by combining the final three images reconstructed from each detector, corresponding to the red, green and blue color channels.

### 3.2. Inversion image reconstruction

*I*, and produce a measurement matrix,

_{i}**I**, containing all projected patterns, such that Similarly, the measured signals can written as a column vector such that the problem can be realised as a set of linear equations given by where

**O**

*is a set of unknowns, which once recovered can be reshaped to*

_{μ}*O*(

_{μ}*x*,

*y*), representing the estimate of the scene at spectral frequency

*μ*. In the case where

*M*≥

*N*, this problem can be solved by least-squares methods. However, as the image resolution increases, the size of

**I**makes performing this reconstruction computationally intensive, while using fewer measurements than the resolution size to address this issue results in an ill-conditioned problem and the image reconstruction quality rapidly decreases.

### 3.3. Compressive sensing

17. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett **95**, 131110 (2009). [CrossRef]

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

*ℓ*1

*magic toolbox*for the Matlab programming language (which is available at www.l1-magic.org). To employ this technique the system must be represented in an appropriate sparse basis, therefore we perform a 1D discrete cosine transform (DCT) on each reshaped pattern

*I*such that

_{i}**I**⇒

**I**

*. The ill-conditioned problem can then be expressed as where*

_{DCT}**O**

^{*}

*is a 1D vector containing the set of unknowns, for the scene at spectral frequency*

_{μ}*μ*, in DCT space. A solution for

**O**

^{*}

*is recovered when minimizing by using preconditioned conjugate gradients methods, where*

_{μ}*λ*is a regularisation parameter [19] and was assigned a value of 0.01 in this experiment for optimum performance. Performing an inverse DCT on

**O**

^{*}

*and with appropriate reshaping results in a solution for the scene,*

_{μ}*O*(

_{μ}*x*,

*y*).

## 4. Conclusions

## Acknowledgments

## References and links

1. | T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A |

2. | D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. |

3. | R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” coincidence imaging with a classical source,” Phys. Rev. Lett. |

4. | A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: Comparing entanglement and classical correlation,” Phys. Rev. Lett. |

5. | A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A |

6. | A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. |

7. | F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. |

8. | J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. |

9. | J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A |

10. | M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Proc. Mag. |

11. | P. Sen, B. Chen, G. Garg, S. R. Marschner, M. Horowitz, M. Levoy, and H. P. A. Lensch, “Dual photography,” ACM Trans. Graph. |

12. | B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science |

13. | G. K. Wallace, “The jpeg still picture compression standard,” Commun. ACM |

14. | D. Preece, R. Bowman, A. Linnenberger, G. Gibson, S. Serati, and M. Padgett, “Increasing trap stiffness with position clamping in holographic optical tweezers,” Opt. Express |

15. | F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. |

16. | B. Sun, S. S. Welsh, M. P. Edgar, J. H. Shapiro, and M. J. Padgett, “Normalized ghost imaging,” Opt. Express |

17. | O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett |

18. | D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory |

19. | K. Koh, S.-J. Kim, and S. P. Boyd, “An interior-point method for large-scale l1-regularized logistic regression.” J. Mach. Learn. Res. |

**OCIS Codes**

(110.1758) Imaging systems : Computational imaging

(110.4234) Imaging systems : Multispectral and hyperspectral imaging

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: July 22, 2013

Revised Manuscript: September 6, 2013

Manuscript Accepted: September 8, 2013

Published: September 23, 2013

**Citation**

Stephen S. Welsh, Matthew P. Edgar, Richard Bowman, Phillip Jonathan, Baoqing Sun, and Miles J. Padgett, "Fast full-color computational imaging with single-pixel detectors," Opt. Express **21**, 23068-23074 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23068

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### References

- T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A52, R3429–R3432 (1995). [CrossRef] [PubMed]
- D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett.74, 3600–3603 (1995). [CrossRef] [PubMed]
- R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” coincidence imaging with a classical source,” Phys. Rev. Lett.89, 113601 (2002). [CrossRef]
- A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: Comparing entanglement and classical correlation,” Phys. Rev. Lett.93, 093602 (2004). [CrossRef] [PubMed]
- A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A70, 013802 (2004). [CrossRef]
- A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett.94, 063601 (2005). [CrossRef] [PubMed]
- F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett.94, 183602 (2005).
- J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process.11, 949–993 (2012). [CrossRef]
- J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A78, 061802 (2008). [CrossRef]
- M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Proc. Mag.25, 83–91 (2008). [CrossRef]
- P. Sen, B. Chen, G. Garg, S. R. Marschner, M. Horowitz, M. Levoy, and H. P. A. Lensch, “Dual photography,” ACM Trans. Graph.24, 745–755 (2005). [CrossRef]
- B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science340, 844–847 (2013). [CrossRef] [PubMed]
- G. K. Wallace, “The jpeg still picture compression standard,” Commun. ACM34, 30–44 (1991). [CrossRef]
- D. Preece, R. Bowman, A. Linnenberger, G. Gibson, S. Serati, and M. Padgett, “Increasing trap stiffness with position clamping in holographic optical tweezers,” Opt. Express17, 22718–22725 (2009). [CrossRef]
- F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett.104, 253603 (2010). [CrossRef] [PubMed]
- B. Sun, S. S. Welsh, M. P. Edgar, J. H. Shapiro, and M. J. Padgett, “Normalized ghost imaging,” Opt. Express20, 16892–16901 (2012). [CrossRef]
- O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett95, 131110 (2009). [CrossRef]
- D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory52, 1289–1306 (2006). [CrossRef]
- K. Koh, S.-J. Kim, and S. P. Boyd, “An interior-point method for large-scale l1-regularized logistic regression.” J. Mach. Learn. Res.8, 1519–1555 (2007).

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