## Adaptive compressive ghost imaging based on wavelet trees and sparse representation |

Optics Express, Vol. 22, Issue 6, pp. 7133-7144 (2014)

http://dx.doi.org/10.1364/OE.22.007133

Acrobat PDF (2197 KB)

### Abstract

Compressed sensing is a theory which can reconstruct an image almost perfectly with only a few measurements by finding its sparsest representation. However, the computation time consumed for large images may be a few hours or more. In this work, we both theoretically and experimentally demonstrate a method that combines the advantages of both adaptive computational ghost imaging and compressed sensing, which we call adaptive compressive ghost imaging, whereby both the reconstruction time and measurements required for any image size can be significantly reduced. The technique can be used to improve the performance of all computational ghost imaging protocols, especially when measuring ultra-weak or noisy signals, and can be extended to imaging applications at any wavelength.

© 2014 Optical Society of America

## 1. Introduction

1. 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]

4. B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A **77**(4), 043809 (2008). [CrossRef]

14. E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory **52**(2), 489–509 (2006). [CrossRef]

17. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. **95**(13), 131110 (2009). [CrossRef]

18. W. L. Gong and S. S. Han, “Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints,” Phys. Lett. A **376**(17), 1519–1522 (2012). [CrossRef]

20. A. Averbuch, S. Dekel, and S. Deutsch, “Adaptive compressed image sensing using dictionaries,” SIAM J. Imaging Sci. **5**(1), 57–89 (2012). [CrossRef]

1. 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]

4. B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A **77**(4), 043809 (2008). [CrossRef]

21. M. F. Li, Y. R. Zhang, X. F. Liu, X. R. Yao, K. H. Luo, H. Fan, and L. A. Wu, “A double-threshold technique for fast time-correspondence imaging,” Appl. Phys. Lett. **103**, 211119 (2013). [CrossRef]

22. V. Studer, J. Bobin, M. Chahid, H. Moussavi, E. J. Candès, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” in Proceedings of the National Academy of Sciences, (2012), 109(26), E1679–E1687. [CrossRef]

## 2. Adaptive compressive ghost imaging model and results

*q*×

*q*pixels and then convert it from the space domain to the wavelet domain. The wavelet decomposition procedure [23

23. S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. **11**(7), 674–693 (1989). [CrossRef]

20. A. Averbuch, S. Dekel, and S. Deutsch, “Adaptive compressed image sensing using dictionaries,” SIAM J. Imaging Sci. **5**(1), 57–89 (2012). [CrossRef]

23. S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. **11**(7), 674–693 (1989). [CrossRef]

*q*×

*q*pixels image and total

*L*-level wavelet transform, where

*q*is the

*k*th power of two, we first produce

*m*(

_{j}*m*<

_{j}*q*

^{2}) full screen scattered speckles of 2

^{j}^{−1}× 2

^{j}^{−1}pixel size (

*j*=

*L*) by using sparse random matrices. After the

^{j−1}× 2

^{j−1}pixel size (

*j*=

*L*− 1). The number of frames

*m*is

_{j}*b*% (5 ≤

_{j}*b*≤ 30) of the number of elements in the union. Each frame corresponds to a measurement value recorded by the point detector. Once all the measurements on the finer scale have been done, a finer area image can be retrieved by a CS reconstruction algorithm. We still perform another one-level wavelet decomposition on this finer image, and find the large coefficients against a new threshold in the three child-quadrants that represent the information of sharp edges. This entire process is repeated until finally the speckles reach a size of 1 × 1 pixel (

_{j}*j*= 1). After that, the wavelet transform of this image is created, which is then converted back to a real space image using the inverse transform. A flowchart of the whole process is shown in Fig. 2.

*x*by point-scanning with an SLM, we measure the scalar product of the signal with truly random speckles: where

*y*is the measurement vector of dimension

*M*, Φ is an

*M*×

*N*sensing matrix,

*e*is the noise vector of dimension

*M*, and

*x′*is the sparse representation coefficient. When

*M*<

*N*, the problem is ill-conditioned [15

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

7. M. Fornasier and H. Rauhut, “Iterative thresholding algorithms,” Appl. Comput. Harmon. Anal. **25**(2), 187–208 (2008). [CrossRef]

8. N. B. Karahanoglu and H. Erdogan, “A* orthogonal matching pursuit: best-first search for compressed sensing signal recovery,” Digit. Sig. Process. **22**(4), 555–568 (2012). [CrossRef]

29. S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. **20**(1), 33–61 (1998). [CrossRef]

*l*

_{1}– norm serves as a measure of sparsity [16], and by minimizing it we can obtain the optimal solution of such a problem: where

*ε*is the allowed error, and

*l*norm is defined as

_{p}*l*

_{1}minimization problem is the so-called basis pursuit [29

29. S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. **20**(1), 33–61 (1998). [CrossRef]

*I*and has good generality as well as noise robustness, which is very suitable for our imaging system. The solution can be approximately sparse. Given that, we choose TVAL3 as the compressive reconstruction algorithm instead of other standard algorithms to address the optimization problem of Eq. (2) and to smooth the noisy image. Here, we perform total variation on the union of interest at each level to acquire a considerable reduction ratio. In addition, we use sparse speckles to obtain better image quality.

*c*. When the coefficients are sorted in descending order, we find that only the first

_{j}*b*% coefficients are relatively large. The

_{j}*a*th coefficient of this sequence is the threshold needed to be set, here

*a*=

*c**

_{j}*b*%. According to the data structure of the wavelet subtree, we put all the locations of selected wavelet coefficients in the

_{j}*j*th decomposition together to form a union set. The next procedure is to measure these areas in the real image. However, unlike the former detection approach, our strategy regards these regions as a whole, and generates sparse speckles using a much smaller number of measurements than the total number of pixels in these regions, then calculate the inner product of the pattern matrix and the target to simulate the total light intensity measured in the

*j*th decomposition. An area image can be obtained by applying CS instead of SLM point-scanning. Once all the sampling steps have been completed, a wavelet transform [Fig. 3(g)] of the image will be created. The real image is recovered by performing the inverse wavelet transform, as shown in Fig. 3(h).

*T*represents the original image consisting of

_{o}*s*×

*t*pixels, and

*T̃*the retrieved image. Naturally, the larger the PSNR value, the better the quality of the image recovered. The PSNR of Fig. 3(h) is 28.055 dB. The whole three-level wavelet decomposition corresponds to 64 × 64, 128 × 128 and 256 × 256 pixel square regions, respectively, so the total signal dimension is 86,016. In this case, after setting appropriate thresholds 81.000 (

*j*= 2,

*b*= 24.697%) and 60.221 (

_{j}*j*= 1,

*b*= 16.787%), there are (4, 096 + 4, 496 + 12, 224) = 20, 816 patterns that need to be scanned in CCGI. However, our method can perform 60%, 90% and 90% compressive sampling on each respective square region, which decreases the sampling rate by 3.847%, while

_{j}*b*also can be changed by choosing suitable thresholds. We can see that the total acquisition rate is roughly proportional to the area of detail or the number of nonzero high frequency wavelet coefficients.

_{j}*μ*m

^{2}. The frame modulation frequency can reach 32,552 Hz [31

31. Texas Instruments, “DLP discovery 4100 chipset data sheet (Rev. A),” (2013), "http://www.ti.com/lit/er/dlpu008a/dlpu008a.pdf""”.

*μ*s, with a dead time of 200

*μ*s, to ensure that the light intensity transmitted did not exceed the saturation limitation of the PMT. The PMT is active only when the rising edges of the pattern triggered signals arrive. Here we also used a total 3-level wavelet transform and performed 45.117% and 22.559% finer measurements, respectively, for a series of largest wavelet coefficients on the 2 × 2 and 1 × 1 scale by setting different thresholds. For this case we had

*m*

_{3}= 2, 464 for the first 64 × 64 coarse image,

*m*

_{2}= 7, 392 for the second stage, and

*m*

_{1}= 14, 784 for the third step, so the total number of measurements needed was 24,640, approximately 28.646% of 86,016, and 37.598% of the number of original image pixels 65,536. Of course, we could have fixed the number of finer measurements for each scale beforehand, but in this way some image quality would have been sacrificed. Besides, the diameter

*D*of the imaging lens we used was 25.4 mm, with a focal length

*f*of 100 mm, and central wavelength

*λ*of 550 nm. As is known, the minimum angle resolution

*d*is the diameter. Considering the magnification

*β*and micro-mirror size

*ρ*, we obtain the following inequality:

*ρ*. Since the DMD actually replaces the function of a traditional charge-coupled device (CCD) detector, the image should match the size of the screen to obtain optimal results. For objects that are very small, it would therefore be necessary to magnify the image first by a microscope. The reconstructed image of the test chart is given in Fig. 5(b).

*a priori*to be compressive.

## 3. Discussion

### 3.1. Analysis of results

32. J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data,” IEEE J. Sel. Top. Signal Processing **4**(2), 288–297 (2010). [CrossRef]

### 3.2. Potential applications

## 4. Summary

## Acknowledgments

## References and links

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

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

3. | D. Zhang, Y. H. Zhai, L. A. Wu, and X. H. Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. |

4. | B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A |

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

6. | Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A |

7. | M. Fornasier and H. Rauhut, “Iterative thresholding algorithms,” Appl. Comput. Harmon. Anal. |

8. | N. B. Karahanoglu and H. Erdogan, “A* orthogonal matching pursuit: best-first search for compressed sensing signal recovery,” Digit. Sig. Process. |

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

10. | W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. |

11. | P. Sen and S. Darabi, “Compressive dual photography,” Computer Graphics Forum |

12. | S. Li, X. R. Yao, W. K. Yu, L. A. Wu, and G. J. Zhai, “High-speed secure key distribution over an optical network based on computational correlation imaging,” Opt. Lett. |

13. | W. K. Yu, S. Li, X. R. Yao, X. F. Liu, L. A. Wu, and G. J. Zhai, “Protocol based on compressed sensing for high-speed authentication and cryptographic key distribution over a multiparty optical network,” Appl. Opt. |

14. | E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory |

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

16. | E. J. Candès, “Compressive sampling,” in |

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

18. | W. L. Gong and S. S. Han, “Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints,” Phys. Lett. A |

19. | M. Aßmann and M. Bayer, “Compressive adaptive computational ghost imaging,” Sci. Rep. |

20. | A. Averbuch, S. Dekel, and S. Deutsch, “Adaptive compressed image sensing using dictionaries,” SIAM J. Imaging Sci. |

21. | M. F. Li, Y. R. Zhang, X. F. Liu, X. R. Yao, K. H. Luo, H. Fan, and L. A. Wu, “A double-threshold technique for fast time-correspondence imaging,” Appl. Phys. Lett. |

22. | V. Studer, J. Bobin, M. Chahid, H. Moussavi, E. J. Candès, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” in Proceedings of the National Academy of Sciences, (2012), 109(26), E1679–E1687. [CrossRef] |

23. | S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. |

24. | S. Mallat, |

25. | J. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Signal Proces. |

26. | A. Said and W. Pearlman, “A new, fast, and efficient image codec based on set partitioning in hierarchical trees,” IEEE Trans. Circ. Syst. Video Technol. |

27. | S. G. Chang, B. Yu, and M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. Image Process. |

28. | J. Haupt, R. Nowak, and R. Castro, “Adaptive sensing for sparse signal recovery,” in Proceedings of the 2009 IEEE Digital Signal Processing Workshop and 5th IEEE Signal Processing Education Workshop, (Marco Island, FL, Jan., 2009), 702–707. |

29. | S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. |

30. | C. B. Li, “An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing,” Master Thesis, Rice University, (2010). |

31. | Texas Instruments, “DLP discovery 4100 chipset data sheet (Rev. A),” (2013), "http://www.ti.com/lit/er/dlpu008a/dlpu008a.pdf""”. |

32. | J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data,” IEEE J. Sel. Top. Signal Processing |

33. | R. Berinde and P. Indyk, “Sequential sparse matching pursuit,” in Proc. 47th Annu. Allerton Conf. Commun. Control Comput., (2009), 36–43. |

**OCIS Codes**

(110.2990) Imaging systems : Image formation theory

(200.4740) Optics in computing : Optical processing

(110.1085) Imaging systems : Adaptive imaging

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: December 27, 2013

Revised Manuscript: February 25, 2014

Manuscript Accepted: March 6, 2014

Published: March 19, 2014

**Citation**

Wen-Kai Yu, Ming-Fei Li, Xu-Ri Yao, Xue-Feng Liu, Ling-An Wu, and Guang-Jie Zhai, "Adaptive compressive ghost imaging based on wavelet trees and sparse representation," Opt. Express **22**, 7133-7144 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-7133

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### References

- D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995). [CrossRef] [PubMed]
- A. Gatti, E. Brambilla, M. Bache, L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004). [CrossRef] [PubMed]
- D. Zhang, Y. H. Zhai, L. A. Wu, X. H. Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. 30(18), 2354–2356 (2005). [CrossRef] [PubMed]
- B. I. Erkmen, J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77(4), 043809 (2008). [CrossRef]
- J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008). [CrossRef]
- Y. Bromberg, O. Katz, Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009). [CrossRef]
- M. Fornasier, H. Rauhut, “Iterative thresholding algorithms,” Appl. Comput. Harmon. Anal. 25(2), 187–208 (2008). [CrossRef]
- N. B. Karahanoglu, H. Erdogan, “A* orthogonal matching pursuit: best-first search for compressed sensing signal recovery,” Digit. Sig. Process. 22(4), 555–568 (2012). [CrossRef]
- M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Proc. Mag. 25(2), 83–91 (2008). [CrossRef]
- W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008). [CrossRef]
- P. Sen, S. Darabi, “Compressive dual photography,” Computer Graphics Forum 28(2), 609–618 (2009). [CrossRef]
- S. Li, X. R. Yao, W. K. Yu, L. A. Wu, G. J. Zhai, “High-speed secure key distribution over an optical network based on computational correlation imaging,” Opt. Lett. 38(12), 2144–2146 (2013). [CrossRef] [PubMed]
- W. K. Yu, S. Li, X. R. Yao, X. F. Liu, L. A. Wu, G. J. Zhai, “Protocol based on compressed sensing for high-speed authentication and cryptographic key distribution over a multiparty optical network,” Appl. Opt. 52(33), 7882–7888 (2013). [CrossRef]
- E. J. Candès, J. Romberg, T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006). [CrossRef]
- D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). [CrossRef]
- E. J. Candès, “Compressive sampling,” in Proc. Int. Cong. Math, (European Mathematical Society, Madrid, Spain, 2006), 3, pp. 1433–1452.
- O. Katz, Y. Bromberg, Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009). [CrossRef]
- W. L. Gong, S. S. Han, “Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints,” Phys. Lett. A 376(17), 1519–1522 (2012). [CrossRef]
- M. Aßmann, M. Bayer, “Compressive adaptive computational ghost imaging,” Sci. Rep. 3, 1545 (2013).
- A. Averbuch, S. Dekel, S. Deutsch, “Adaptive compressed image sensing using dictionaries,” SIAM J. Imaging Sci. 5(1), 57–89 (2012). [CrossRef]
- M. F. Li, Y. R. Zhang, X. F. Liu, X. R. Yao, K. H. Luo, H. Fan, L. A. Wu, “A double-threshold technique for fast time-correspondence imaging,” Appl. Phys. Lett. 103, 211119 (2013). [CrossRef]
- V. Studer, J. Bobin, M. Chahid, H. Moussavi, E. J. Candès, M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” in Proceedings of the National Academy of Sciences, (2012), 109(26), E1679–E1687. [CrossRef]
- S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. 11(7), 674–693 (1989). [CrossRef]
- S. Mallat, A wavelet tour of signal processing, the sparse way (Elsevier, 2009), pp. 340–346.
- J. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Signal Proces. 41(12), 3445–3462 (1993). [CrossRef]
- A. Said, W. Pearlman, “A new, fast, and efficient image codec based on set partitioning in hierarchical trees,” IEEE Trans. Circ. Syst. Video Technol. 6(3), 243–250 (1996). [CrossRef]
- S. G. Chang, B. Yu, M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. Image Process. 9(9), 1532–1546 (2000). [CrossRef]
- J. Haupt, R. Nowak, R. Castro, “Adaptive sensing for sparse signal recovery,” in Proceedings of the 2009 IEEE Digital Signal Processing Workshop and 5th IEEE Signal Processing Education Workshop, (Marco Island, FL, Jan., 2009), 702–707.
- S. S. Chen, D. L. Donoho, M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20(1), 33–61 (1998). [CrossRef]
- C. B. Li, “An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing,” Master Thesis, Rice University, (2010).
- Texas Instruments, “DLP discovery 4100 chipset data sheet (Rev. A),” (2013), "http://www.ti.com/lit/er/dlpu008a/dlpu008a.pdf"" ”.
- J. Yang, Y. Zhang, W. Yin, “A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data,” IEEE J. Sel. Top. Signal Processing 4(2), 288–297 (2010). [CrossRef]
- R. Berinde, P. Indyk, “Sequential sparse matching pursuit,” in Proc. 47th Annu. Allerton Conf. Commun. Control Comput., (2009), 36–43.

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