## Preconditioning for multiplexed imaging with spatially coded PSFs |

Optics Express, Vol. 19, Issue 13, pp. 12540-12550 (2011)

http://dx.doi.org/10.1364/OE.19.012540

Acrobat PDF (1158 KB)

### Abstract

We propose a preconditioning method to improve the convergence of iterative reconstruction algorithms in multiplexed imaging based on convolution-based compressive sensing with spatially coded point spread functions (PSFs). The system matrix is converted to improve the condition number with a preconditioner matrix. The preconditioner matrix is calculated by Tikhonov regularization in the frequency domain. The method was demonstrated with simulations and an experiment involving a range detection system with a grating based on the multiplexed imaging framework. The results of the demonstrations showed improved reconstruction fidelity by using the proposed preconditioning method.

© 2011 OSA

## 1. Introduction

2. D. J. Brady, *Optical imaging and spectroscopy* (Wiley-OSA, 2009). [CrossRef]

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

5. E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Sig. Process. Mag. **25**, 21–30 (2008). [CrossRef]

6. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express **17**, 13040–13049 (2009). [CrossRef] [PubMed]

10. A. Ashok and M. A. Neifeld, “Compressive light field imaging,” Proc. SPIE **7690**, 76900Q (2010). [CrossRef]

11. J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci. **2**, 1098–1128 (2009). [CrossRef]

13. Y. Rivenson, A. Stern, and B. Javidi, “Single exposure super-resolution compressive imaging by double phase encoding,” Opt. Express **18**, 15094–15103 (2010). [CrossRef] [PubMed]

6. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express **17**, 13040–13049 (2009). [CrossRef] [PubMed]

14. J. Hahn, S. Lim, K. Choi, R. Horisaki, and D. J. Brady, “Video-rate compressive holographic microscopic tomography,” Opt. Express **19**, 7289–7298 (2011). [CrossRef] [PubMed]

15. R. Horisaki and J. Tanida, “Multi-channel data acquisition using multiplexed imaging with spatial encoding,” Opt. Express **18**, 23041–23053 (2010). [CrossRef] [PubMed]

15. R. Horisaki and J. Tanida, “Multi-channel data acquisition using multiplexed imaging with spatial encoding,” Opt. Express **18**, 23041–23053 (2010). [CrossRef] [PubMed]

16. N. Nguyen, P. Milanfar, S. Member, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Proc. **10**, 573–583 (2001). [CrossRef]

17. K. Choi, R. Horisaki, J. Hahn, S. Lim, D. L. Marks, T. J. Schulz, and D. J. Brady, “Compressive holography of diffuse objects,” Appl. Opt. **49**, H1–H10 (2010). [CrossRef] [PubMed]

18. A. Ashok and M. A. Neifeld, “Pseudorandom phase masks for superresolution imaging from subpixel shifting,” Appl. Opt. **46**, 2256–2268 (2007). [CrossRef] [PubMed]

19. A. Mahalanobis, M. Neifeld, V. K. Bhagavatula, T. Haberfelde, and D. Brady, “Off-axis sparse aperture imaging using phase optimization techniques for application in wide-area imaging systems,” Appl. Opt. **48**, 5212–5224 (2009). [CrossRef] [PubMed]

16. N. Nguyen, P. Milanfar, S. Member, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Proc. **10**, 573–583 (2001). [CrossRef]

17. K. Choi, R. Horisaki, J. Hahn, S. Lim, D. L. Marks, T. J. Schulz, and D. J. Brady, “Compressive holography of diffuse objects,” Appl. Opt. **49**, H1–H10 (2010). [CrossRef] [PubMed]

## 2. The proposed preconditioning method

15. R. Horisaki and J. Tanida, “Multi-channel data acquisition using multiplexed imaging with spatial encoding,” Opt. Express **18**, 23041–23053 (2010). [CrossRef] [PubMed]

*𝒢*,

*𝒫*, and

*ℱ*are the captured data, the spatial PSFs, and the object data, respectively, and

*x*and

*c*are indexes of the spatial coordinate and the channels.

**∈ ℝ**

*g*^{Nx×1},

**Φ**∈ ℝ

^{Nx×(Nx×Nc)}, and

**∈ ℝ**

*f*^{(Nx×Nc)×1}are the vectorized captured data, the system matrix, and the vectorized object data, respectively;

*N*and

_{x}*N*are the numbers of detectors and channels of the object; ℝ

_{c}

^{a}^{×}

*is an*

^{b}*a*×

*b*matrix of real numbers;

*C**∈ ℝ*

_{c}^{Nx×Nx}is a circulant matrix indicating the PSF for the

*c*-th channel in the object;

*D**∈ ℂ*

_{c}^{Nx×Nx},

**0**∈ ℝ

^{Nx×Nx}, and

**∈ ℂ**

*F*^{Nx×Nx}are a diagonal matrix indicating the optical transfer function (OTF) for the

*c*-th channel in the object, a zero matrix, and a Fourier transform matrix, respectively; and ℂ

^{a×b}is an

*a*×

*b*matrix of complex numbers.

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

**,**

*P**α*, and ℛ(·) are a preconditioner matrix, a regularization parameter, and a regularizer, respectively; and ||·||

_{ℓ}_{2}denotes

*ℓ*

_{2}norm. Here, Tikhonov inversion of the system matrix

**Φ**is chosen as the preconditioner matrix

**to reduce the condition number of**

*P***Φ**[16

16. N. Nguyen, P. Milanfar, S. Member, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Proc. **10**, 573–583 (2001). [CrossRef]

**is close to an identity matrix.**

*P*Φ**∈ ℝ**

*P*^{(Nx×Nc)×Nx}is given by where

*i*-th row and column in

*d**∈ ℂ*

_{i}^{1×Nc}and

*D*_{0})

*(*

_{i}

*D*_{1})

*··· (*

_{i}

*D*_{Nc−1})

*] and*

_{i}*denotes the element in the*

_{i}*i*-th row and column in a diagonal matrix;

*λ*and

**∈ ℝ**

*I′*^{Nc×Nc}are a regularization parameter and an identity matrix, respectively; and

*T*and

*H*show the transpose and Hermitian conjugate transpose matrices, respectively. A small

*λ*realizes quick convergence, but on the other hand, it boosts noise on the captured data.

## 3. Demonstrations

20. J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Proc. **16**, 2992–3004 (2007). [CrossRef]

*α*in Eq. (5) and

*λ*in Eq. (7) were chosen experimentally.

### 3.1. Simulations

**in the simulations are shown in Fig. 1. The size of the objects was 256 ×256 ×4. The object shown in Fig. 1(a) consisted of multiple rotated Shepp-Logan phantoms, which were sparse in the two-dimensional total variation (TV) domain [21**

*f*21. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D **60**, 259–268 (1992). [CrossRef]

*House*image in the Spectral image database [22

22. “Spectral image database,” http://spectral.joensuu.fi/multispectral/spectralimages.php.

**of Fig. 1 with the PSFs in Fig. 2 is shown in Fig. 3. The size of the captured data was 256 ×256. The measurement signal-to-noise ratio (SNR) was 60 dB.**

*g*### 3.2. Experiment

*μ*m × 6.45

*μ*m, respectively. The grating modulated the PSFs differently at each object distance, as shown in Fig. 7. Two planar objects, the printed characters “a” and “b”, were located at different distances from the imaging optics. The objects were passively illuminated with incoherent interior lights.

## 4. Conclusions

## References and links

1. | A. Kak and M. Slaney, |

2. | D. J. Brady, |

3. | D. L. Donoho, “Compressed sensing,” IEEE Trans. Info. Theory |

4. | R. Baraniuk, “Compressive sensing,” IEEE Sig. Process. Mag. |

5. | E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Sig. Process. Mag. |

6. | D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express |

7. | M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express |

8. | R. Horisaki, K. Choi, J. Hahn, J. Tanida, and D. J. Brady, “Generalized sampling using a compound-eye imaging system for multi-dimensional object acquisition,” Opt. Express |

9. | M. Shankar, N. P. Pitsianis, and D. J. Brady, “Compressive video sensors using multichannel imagers,” Appl. Opt. |

10. | A. Ashok and M. A. Neifeld, “Compressive light field imaging,” Proc. SPIE |

11. | J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci. |

12. | R. F. Marcia and R. M. Willett, “Compressive coded aperture superresolution image reconstruction,” in “IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2008),” (2008), pp. 833–836. |

13. | Y. Rivenson, A. Stern, and B. Javidi, “Single exposure super-resolution compressive imaging by double phase encoding,” Opt. Express |

14. | J. Hahn, S. Lim, K. Choi, R. Horisaki, and D. J. Brady, “Video-rate compressive holographic microscopic tomography,” Opt. Express |

15. | R. Horisaki and J. Tanida, “Multi-channel data acquisition using multiplexed imaging with spatial encoding,” Opt. Express |

16. | N. Nguyen, P. Milanfar, S. Member, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Proc. |

17. | K. Choi, R. Horisaki, J. Hahn, S. Lim, D. L. Marks, T. J. Schulz, and D. J. Brady, “Compressive holography of diffuse objects,” Appl. Opt. |

18. | A. Ashok and M. A. Neifeld, “Pseudorandom phase masks for superresolution imaging from subpixel shifting,” Appl. Opt. |

19. | A. Mahalanobis, M. Neifeld, V. K. Bhagavatula, T. Haberfelde, and D. Brady, “Off-axis sparse aperture imaging using phase optimization techniques for application in wide-area imaging systems,” Appl. Opt. |

20. | J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Proc. |

21. | L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D |

22. | “Spectral image database,” http://spectral.joensuu.fi/multispectral/spectralimages.php. |

**OCIS Codes**

(110.1758) Imaging systems : Computational imaging

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: April 7, 2011

Revised Manuscript: June 1, 2011

Manuscript Accepted: June 2, 2011

Published: June 14, 2011

**Citation**

Ryoichi Horisaki and Jun Tanida, "Preconditioning for multiplexed imaging with spatially coded PSFs," Opt. Express **19**, 12540-12550 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-12540

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### References

- A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).
- D. J. Brady, Optical imaging and spectroscopy (Wiley-OSA, 2009). [CrossRef]
- D. L. Donoho, “Compressed sensing,” IEEE Trans. Info. Theory 52, 1289–1306 (2006). [CrossRef]
- R. Baraniuk, “Compressive sensing,” IEEE Sig. Process. Mag. 24, 118–121 (2007). [CrossRef]
- E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Sig. Process. Mag. 25, 21–30 (2008). [CrossRef]
- D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009). [CrossRef] [PubMed]
- M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express 15, 14013–14027 (2007). [CrossRef] [PubMed]
- R. Horisaki, K. Choi, J. Hahn, J. Tanida, and D. J. Brady, “Generalized sampling using a compound-eye imaging system for multi-dimensional object acquisition,” Opt. Express 18, 19367–19378 (2010). [CrossRef] [PubMed]
- M. Shankar, N. P. Pitsianis, and D. J. Brady, “Compressive video sensors using multichannel imagers,” Appl. Opt. 49, B9–B17 (2010). [CrossRef] [PubMed]
- A. Ashok and M. A. Neifeld, “Compressive light field imaging,” Proc. SPIE 7690, 76900Q (2010). [CrossRef]
- J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci. 2, 1098–1128 (2009). [CrossRef]
- R. F. Marcia and R. M. Willett, “Compressive coded aperture superresolution image reconstruction,” in “IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2008),” (2008), pp. 833–836.
- Y. Rivenson, A. Stern, and B. Javidi, “Single exposure super-resolution compressive imaging by double phase encoding,” Opt. Express 18, 15094–15103 (2010). [CrossRef] [PubMed]
- J. Hahn, S. Lim, K. Choi, R. Horisaki, and D. J. Brady, “Video-rate compressive holographic microscopic tomography,” Opt. Express 19, 7289–7298 (2011). [CrossRef] [PubMed]
- R. Horisaki and J. Tanida, “Multi-channel data acquisition using multiplexed imaging with spatial encoding,” Opt. Express 18, 23041–23053 (2010). [CrossRef] [PubMed]
- N. Nguyen, P. Milanfar, S. Member, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Proc. 10, 573–583 (2001). [CrossRef]
- K. Choi, R. Horisaki, J. Hahn, S. Lim, D. L. Marks, T. J. Schulz, and D. J. Brady, “Compressive holography of diffuse objects,” Appl. Opt. 49, H1–H10 (2010). [CrossRef] [PubMed]
- A. Ashok and M. A. Neifeld, “Pseudorandom phase masks for superresolution imaging from subpixel shifting,” Appl. Opt. 46, 2256–2268 (2007). [CrossRef] [PubMed]
- A. Mahalanobis, M. Neifeld, V. K. Bhagavatula, T. Haberfelde, and D. Brady, “Off-axis sparse aperture imaging using phase optimization techniques for application in wide-area imaging systems,” Appl. Opt. 48, 5212–5224 (2009). [CrossRef] [PubMed]
- J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Proc. 16, 2992–3004 (2007). [CrossRef]
- L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992). [CrossRef]
- “Spectral image database,” http://spectral.joensuu.fi/multispectral/spectralimages.php .

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