Morphology separation in ghost imaging via sparsity constraint Xuyang Xu, Enrong Li,∗ Hong Yu, Wenlin Gong, and Shensheng Han Key Laboratory for Quantum Optics and Center for Cold Atom Physics of CAS, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, Shanghai 201800, China ∗ [email protected]
Abstract: The separation of morphology components in ghost imaging via sparsity constraint is investigated by adapting the morphology component analysis technique based on the fact that different morphology components can be sparsely expressed in different basis. The successful separation of reconstructed image plays an important role in the ability to identify it, analyze it, enhance it and more. This approach is first studied with numerical simulations and then verified with both table-top and outdoor experimental data. Results show that it can not only separate different morphology components but also improve the quality of the reconstructed image. © 2014 Optical Society of America OCIS codes: (110.1758) Computational imaging; (270.1670) Coherent optical effects; (100.4994) Pattern recognition, image transforms.
References and links 1. M. D’Angelo and Y. Shih, “Quantum imaging,” Laser Phys. Lett. 2, 567–596 (2005). 2. J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in x-ray diffraction,” Phys. Rev. Lett. 92, 093903 (2004). 3. J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. 11, 949–993 (2012). 4. R. S. Bennink, S. J. Bentley, and R. W. Boyd, “’Two-photon’ coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002). 5. D. Zhang, Y. H. Zhai, L. A. Wu, and X. H. Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. 30, 2354–2356 (2005). 6. D. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005). 7. A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006). 8. P. Zerom, K. W. C. Chan, J. C. Howell, and R. W. Boyd, “Entangled-photon compressive ghost imaging,” Phys. Rev. A 84, 061804 (2011). 9. Y. Bai, H. Liu, and S. Han, “Transmission area and correlated imaging,” Opt. Express 15, 6062–6068 (2007). 10. P. Zhang, W. Gong, X. Shen, S. Han, and R. Shu, “Homodyne detection in ghost imaging with thermal light,” Phys. Rev. A 80, 033827 (2009). 11. M. Bache, E. Brambilla, A. Gatti, and L. Lugiato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004). 12. P. Zhang, W. Gong, X. Shen, D. Huang, and S. Han, “Improving resolution by the second-order correlation of light fields,” Opt. Lett. 34, 1222–1224 (2009). 13. Y. Bai and S. Han, “Ghost imaging with thermal light by third-order correlation,” Phys. Rev. A 76, 043828 (2007). 14. D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, “Enhancing visibility and resolution in Nth-order intensity correlation of thermal light,” Appl. Phys. Lett. 92, 201102 (2008). 15. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009). 16. C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
#209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14375
17. W. Gong and S. Han, “Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints,” Phys. Lett. A 376, 1519–1522 (2012). 18. H. Wang, S. Han, and M. I. Kolobov, “Quantum limits of super-resolution of optical sparse objects via sparsity constraint,” Opt. Express 20, 23235–23252 (2012). 19. H. Wang and S. Han, “Coherent ghost imaging based on sparsity constraint without phase-sensitive detection,” Europhys. Lett. 98, 24003 (2012). 20. R. G. Baraniuk, “Compressive sensing [lecture notes],” IEEE Signal Process. Mag. 24, 118–121 (2007). 21. J. Du, W. Gong, and S. Han, “The influence of sparsity property of images on ghost imaging with thermal light,” Opt. Lett. 37, 1067–1069 (2012). 22. D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomic d ecomposition,” IEEE Trans. Inf. Theory 47, 2845–2862 (2001). 23. X. Qu, X. Cao, D. Guo, C. Hu, and Z. Chen, “Combined sparsifying transforms for compressed sensing MRI,” Electron. Lett. 46, 121–123 (2010). 24. M. J. Fadili, J. L. Starck, J. Bobin, and Y. Moudden, “Image decomposition and separation using sparse representations: An overview,” Proc. IEEE 98, 983–994 (2010). 25. S. Mallat, A Wavelet Tour of Signal Processing (Access Online via Elsevier, 1999). 26. E. Candes, L. Demanet, D. Donoho, and L. Ying, “Fast discrete curvelet transforms,” Multiscale Model. Simul. 5, 861–899 (2006). 27. N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE Trans. Comput. 100, 90–93 (1974). 28. J. Bobin, J. L. Starck, J. M. Fadili, Y. Moudden, and D. L. Donoho, “Morphological component analysis: An adaptive thresholding strategy,” IEEE Trans. Image Process. 16, 2675–2681 (2007). 29. M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sign. Proces. 1, 586–597 (2007). 30. S. Chen and D. Donoho, “Basis pursuit,” in “Signals, Systems and Computers, 1994. 1994 Conference Record of the Twenty-Eighth Asilomar Conference on,” (IEEE, 1994), vol. 1, pp. 41–44. 31. J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007). 32. D. L. Donoho, A. Maleki, and A. Montanari, “Message-passing algorithms for compressed sensing,” Proc. Natl. Acad. Sci. 106, 18914–18919 (2009). 33. D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc. 90, 1200–1224 (1995). 34. W. Gong, C. Zhao, J. Jiao, E. Li, M. Chen, H. Wang, W. Xu, and S. Han, “Three-dimensional ghost imaging ladar,” arXiv preprint arXiv:1301.5767 (2013).
1.
Introduction
Ghost imaging (GI) is obtained by correlating the light field detected by a high resolution detector directly with the light intensity collected by a bucket detector which detects the intensity transmitted by or reflected from the target [1–9]. To improve the the sampling efficiency and quality of image reconstruction, different approaches have been proposed, such as homodyne detection GI [10, 11], high order correlation GI [12–14] and so on. Recently, researchers have found that by applying compressed sensing techniques [15], the usage of sparsity prior of images in GI proves an efficient way to improve the sampling efficiency and reconstruction in GI, which can be termed as ghost imaging via sparsity constraint (GISC) [16–19]. In GISC, one generally needs a proper transformation in which the image can be sparsely represented to improve the quality of the reconstructed image using the sparse reconstruction methods [20, 21]. Since an image generally contains different morphology components, a combined dictionary [22] can be used to get a sparser representation and a better reconstruction than using a single dictionary [23]. For example, an image may contain the point-part and curve-part [24] that can be respectively sparsely expressed in wavelet [25] and curvelet [26] or the texture-part and cartoon-part which are sparse in discrete cosine basis [27] and wavelet respectively. In this work, we realize the separation of different morphology components using morphology component analysis (MCA) [28] in GISC for the first time. Using the proposed method, we can not only get the components of interest, but also improve the image quality. In the rest of the paper, we will first describe the experimental setup and the sampling model of GISC in section 2, followed by the simulation and experimental results in section 3. Conclu-
#209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14376
sions are given in section 4. 2.
The experimental setup and sampling model
The experimental setup of GISC is shown in Fig. 1. A laser beam passes through a rotating ground glass to create a pseudothermal light and is then divided into two paths by a beam splitter (BS). In the reference path, the intensity distribution is recorded by a charge-coupled device (CCD) directly and in the object path the light goes through the object and is then focused onto a one-pixel detector by a lens.
Fig. 1. The experimental setup of ghost imaging via sparsity constraint(GISC).
The image of the pseudothermal light field recorded by the CCD has m × n pixels and is reshaped into a vector (1 × N, N = m × n) as a row of the measurement matrix denoted by A . Repeating this process K times, K N, a K × N-matrix A can be obtained. At the same time, we can get a column vector Y of length K from the bucket detector. The unknown object can be denoted by a column vector X of length N and represented as X = Ψ θ by a combined dictionary Ψ which consists of dictionaries Ψ 1 and Ψ 2 (here we only consider the combination of two dictionaries without loss of generality). Then one may have Y = AX + ε = A Ψ θ + ε ,
(1)
with ε the additive noise. The reconstruction of X can be regarded as finding X = Ψ θ that Y − A X 22 + τ θ 1 . Considering the combined dictionary, the problem can be minimizes 12 Y expressed as: 1 Y − A X 22 + τ (θ 1 1 + θ 2 1 ). X = Ψ 1 θ 1 + Ψ 2 θ 2 ; which minimizes: Y 2
(2)
In the above, vv2 denotes the l2 Euclid norm of vector v , vv1 the l1 Euclid norm and τ a nonnegative parameter which should be chosen according to the sparsity of transform coefficients: the sparser the image is, the larger it should be. To solve the convex optimization problem, we employ the gradient projection for sparse reconstruction (GPSR) approach [29], although there are also other algorithms [30–32]. Once we get the representation coefficients of the combined dictionary, we can then start the MCA approach with these coefficients. The largest value of all the representation coefficients #209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14377
is chosen as our initial threshold and the stopping threshold is estimated by the noise of reconstructed image [33]. In each iteration, we first compute the residual of the current estimation of the target image and representation coefficients using the selected dictionary; then we may have the estimation of morphological parts by reconstructing from the corresponding coefficients after being hard thresheld using the threshold that, starting from the initial one, decreases linearly until it reaches the stopping one. 3. 3.1.
Results and discussion Simulation result and discussion
We first test the effect of MCA in GISC by simulation. The object which consists of some “points” and “lines” is shown in Fig. 2(a) with a size of 64 × 64 pixels. The number of sampling is set to be 1400. first A is generated as a random gaussian matrix. Then Y is calculated according to Eq. (1) and gaussian noise of different levels (16dB, 19dB, 22dB, 25dB) is then added. The wavelet and curvelet dictionary is used to form our combined dictionary in the reconstruction. We use the discrete orthogonal wavelet transform with the Mallat algorithm. Daubechies 2 wavelet is chosen as the mother function of the transform which is also known as Haar transform. The decomposition level of Haar transform is set to 2. The curvelet used in this work is based on the wrapping of specially selected fourier samples. The coarsest level of the curvelet is 3 and the number of angles at the second coarsest level is 16. As shown in Fig. 2, it can be seen that the “point” and “line” components are successfully separated with the proposed approach and the quality of the images obtained from the combination of two components is better than the one from GISC without the MCA approach. The main reason of the improvement is that the stopping threshold is estimated by the noise level of reconstructed image and thus the hard thresholding will remove much of the noise component in the image without much affecting the components of interest since the noise is generally non-sparse in the representation basis while the morphology components are.
(a)
(b)
(c)
(d)
(e)
Fig. 2. The object and simulation results. (a) the object of 64 × 64 pixels used in the simulation; (b-e) simulation results with SNR=16dB,19dB,22dB,25dB respectively. The first row shows the GPSR reconstruction results, the third row the separated “point” component, the fourth row the separated “line” component and the second row the combination of the two separated components.
The relative mean squared error (RMSE) is evaluated to characterize the quality of reconm,n 2 2 structed image: RMSE = ∑m,n i, j=1 (Ii j − Ii j ) /(∑i, j=1 Ii j ), where Ii j , Ii j are the pixel values of reconstructed image and the original image, respectively, with i, j the pixel indices. From Fig.
#209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14378
(a)
(b)
Fig. 3. The RMSE values of the reconstructed images using MCA in GISC and GISC without MCA. (a)The RMSE values of results from data with different SNR. (b)The RMSE values of the results reconstructed from different numbers of sampling.
3(a) we can see that the RMSE values of the image reconstructed using the MCA technique in GISC is smaller than that reconstructed without applying MCA in different noise level. Figure 3(b) shows the RMSE values of the reconstructed images with different numbers of sampling and it can be seen that the image quality is also improved when MCA is applied. 3.2.
Experimental result and discussion
(a)
(b)
(c)
(d)
(e)
Fig. 4. The experimental results of ghost imaging via sparse constraint. (a) the target object; (b-f) experimental results with K = 800, 1200, 1600, 2000 respectively. The first row shows the GPSR reconstructed results, the third row the “point” component, the forth row the “line” component and the second row the combination of the two separated components.
To experimentally demonstrate the separation of different components of the object image in GISC, we carried out experiments using the object as shown in Fig. 4(a), which is the image reconstructed from 4000 samplings using the MCA technique in GISC. It will be taken as the reference image because it has better quality than that reconstructed with GISC without using MCA according to the simulation result shown in Fig. 3(b). The size of each “point” is about 232.5μ m × 232.5μ m and the width of each “line” is about 93μ m. The field-of-view has a size of 64 × 64 pixels. We choose a proper size of diaphragm and the distance between ground glass and object to make sure that the average full width at half maximum (FWHM) of the speckle is approximately the size of one pixel. From the results corresponding to different
#209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14379
numbers of sampling K ranging from 800 to 2000 as shown in Figs. 4(b)–4(f), it’s obvious that the separation of “point” component and “line” component is successful. We can see that the quality of “point” and “line” component increases with sampling numbers from the results shown in the third and forth rows of Fig. 4.
Fig. 5. The RMSE with different number of samplings in the experiment. The blue line is the RMSE of GISC and the red one is the RMSE of sum of the separated components when MCA is used in GISC.
It is also confirmed that the quality of image combined from the two separated components via MCA in GISC is better than the reconstructed image without MCA as shown in Fig. 5, which is consistent with the simulation results shown in Fig. 3(b). The table-top experiment shows that it can improve RMSE of reconstructed image by almost 50%.
(a)
(b)
(c)
(d)
(e)
Fig. 6. The results of a natural scene. (a) the original scene captured with a telescope; (b) the result of GISC; (c) the “point” component; (d) the “line” component; (e) the combination of the two components. The first to third row of (b-e) is just three of these different slices and last row is the sum of all slices.
We then applied this approach to GISC of a natural scene as shown in Fig. 6(a). The distance between the target and the GI system is about 1km and the GI experiment is described in detail in [34]. In each measurement of the experiment, the bucket detector gets a pulse signal with a finite time duration due to the 3D structure of the target. Using the signal data from the recorded #209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14380
pulses with a certain time delay, one may reconstruct a “slice” of the 3D target and then get the whole image of the target shown in Fig. 6(b) by combing all the slices. A combined waveletcurvelet dictionary is used for the slice reconstruction and morphology separation. We may see that the “point” (“point” means the support of patch is the same in two direction) component shown in Fig. 6(c) are successfully separated from the “line” components in Fig. 6(d). Although a reference image is not available, if we investigate the results shown in row 1, 2, 3 of Figs. 6(b) and 6(e), we can see that the noise is reduced while few image details are sacrificed. The last row in Fig. 6 is the combination of all different slices, visually the image shown in Fig. 6(e) combined from the separated components appear more smooth than that shown in Fig. 6(b) and the details also are maintained. This shows that the proposed approach may improve the reconstruction performance when applied to real world applications. 4.
Conclusion
In summary the morphology component separation in GISC has been studied by numerical simulations and verified by both table-top and outdoor experiments. The results show that it can not only extract different morphology components, but also improve the quality of reconstructed image by the combination of these different components in GISC. The successful separation of reconstructed image can help us to analyze the target, enhance it and identify it in future applications. Acknowledgments The work was supported by the Hi-Tech Research and Development Program of China under Grant Projects No.2013AA122901 and No.2013AA122902
#209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14381
Abstract: The separation of morphology components in ghost imaging via sparsity constraint is investigated by adapting the morphology component analysis technique based on the fact that different morphology components can be sparsely expressed in different basis. The successful separation of reconstructed image plays an important role in the ability to identify it, analyze it, enhance it and more. This approach is first studied with numerical simulations and then verified with both table-top and outdoor experimental data. Results show that it can not only separate different morphology components but also improve the quality of the reconstructed image. © 2014 Optical Society of America OCIS codes: (110.1758) Computational imaging; (270.1670) Coherent optical effects; (100.4994) Pattern recognition, image transforms.
References and links 1. M. D’Angelo and Y. Shih, “Quantum imaging,” Laser Phys. Lett. 2, 567–596 (2005). 2. J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in x-ray diffraction,” Phys. Rev. Lett. 92, 093903 (2004). 3. J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. 11, 949–993 (2012). 4. R. S. Bennink, S. J. Bentley, and R. W. Boyd, “’Two-photon’ coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002). 5. D. Zhang, Y. H. Zhai, L. A. Wu, and X. H. Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. 30, 2354–2356 (2005). 6. D. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005). 7. A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006). 8. P. Zerom, K. W. C. Chan, J. C. Howell, and R. W. Boyd, “Entangled-photon compressive ghost imaging,” Phys. Rev. A 84, 061804 (2011). 9. Y. Bai, H. Liu, and S. Han, “Transmission area and correlated imaging,” Opt. Express 15, 6062–6068 (2007). 10. P. Zhang, W. Gong, X. Shen, S. Han, and R. Shu, “Homodyne detection in ghost imaging with thermal light,” Phys. Rev. A 80, 033827 (2009). 11. M. Bache, E. Brambilla, A. Gatti, and L. Lugiato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004). 12. P. Zhang, W. Gong, X. Shen, D. Huang, and S. Han, “Improving resolution by the second-order correlation of light fields,” Opt. Lett. 34, 1222–1224 (2009). 13. Y. Bai and S. Han, “Ghost imaging with thermal light by third-order correlation,” Phys. Rev. A 76, 043828 (2007). 14. D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, “Enhancing visibility and resolution in Nth-order intensity correlation of thermal light,” Appl. Phys. Lett. 92, 201102 (2008). 15. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009). 16. C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
#209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14375
17. W. Gong and S. Han, “Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints,” Phys. Lett. A 376, 1519–1522 (2012). 18. H. Wang, S. Han, and M. I. Kolobov, “Quantum limits of super-resolution of optical sparse objects via sparsity constraint,” Opt. Express 20, 23235–23252 (2012). 19. H. Wang and S. Han, “Coherent ghost imaging based on sparsity constraint without phase-sensitive detection,” Europhys. Lett. 98, 24003 (2012). 20. R. G. Baraniuk, “Compressive sensing [lecture notes],” IEEE Signal Process. Mag. 24, 118–121 (2007). 21. J. Du, W. Gong, and S. Han, “The influence of sparsity property of images on ghost imaging with thermal light,” Opt. Lett. 37, 1067–1069 (2012). 22. D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomic d ecomposition,” IEEE Trans. Inf. Theory 47, 2845–2862 (2001). 23. X. Qu, X. Cao, D. Guo, C. Hu, and Z. Chen, “Combined sparsifying transforms for compressed sensing MRI,” Electron. Lett. 46, 121–123 (2010). 24. M. J. Fadili, J. L. Starck, J. Bobin, and Y. Moudden, “Image decomposition and separation using sparse representations: An overview,” Proc. IEEE 98, 983–994 (2010). 25. S. Mallat, A Wavelet Tour of Signal Processing (Access Online via Elsevier, 1999). 26. E. Candes, L. Demanet, D. Donoho, and L. Ying, “Fast discrete curvelet transforms,” Multiscale Model. Simul. 5, 861–899 (2006). 27. N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE Trans. Comput. 100, 90–93 (1974). 28. J. Bobin, J. L. Starck, J. M. Fadili, Y. Moudden, and D. L. Donoho, “Morphological component analysis: An adaptive thresholding strategy,” IEEE Trans. Image Process. 16, 2675–2681 (2007). 29. M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sign. Proces. 1, 586–597 (2007). 30. S. Chen and D. Donoho, “Basis pursuit,” in “Signals, Systems and Computers, 1994. 1994 Conference Record of the Twenty-Eighth Asilomar Conference on,” (IEEE, 1994), vol. 1, pp. 41–44. 31. J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007). 32. D. L. Donoho, A. Maleki, and A. Montanari, “Message-passing algorithms for compressed sensing,” Proc. Natl. Acad. Sci. 106, 18914–18919 (2009). 33. D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc. 90, 1200–1224 (1995). 34. W. Gong, C. Zhao, J. Jiao, E. Li, M. Chen, H. Wang, W. Xu, and S. Han, “Three-dimensional ghost imaging ladar,” arXiv preprint arXiv:1301.5767 (2013).
1.
Introduction
Ghost imaging (GI) is obtained by correlating the light field detected by a high resolution detector directly with the light intensity collected by a bucket detector which detects the intensity transmitted by or reflected from the target [1–9]. To improve the the sampling efficiency and quality of image reconstruction, different approaches have been proposed, such as homodyne detection GI [10, 11], high order correlation GI [12–14] and so on. Recently, researchers have found that by applying compressed sensing techniques [15], the usage of sparsity prior of images in GI proves an efficient way to improve the sampling efficiency and reconstruction in GI, which can be termed as ghost imaging via sparsity constraint (GISC) [16–19]. In GISC, one generally needs a proper transformation in which the image can be sparsely represented to improve the quality of the reconstructed image using the sparse reconstruction methods [20, 21]. Since an image generally contains different morphology components, a combined dictionary [22] can be used to get a sparser representation and a better reconstruction than using a single dictionary [23]. For example, an image may contain the point-part and curve-part [24] that can be respectively sparsely expressed in wavelet [25] and curvelet [26] or the texture-part and cartoon-part which are sparse in discrete cosine basis [27] and wavelet respectively. In this work, we realize the separation of different morphology components using morphology component analysis (MCA) [28] in GISC for the first time. Using the proposed method, we can not only get the components of interest, but also improve the image quality. In the rest of the paper, we will first describe the experimental setup and the sampling model of GISC in section 2, followed by the simulation and experimental results in section 3. Conclu-
#209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14376
sions are given in section 4. 2.
The experimental setup and sampling model
The experimental setup of GISC is shown in Fig. 1. A laser beam passes through a rotating ground glass to create a pseudothermal light and is then divided into two paths by a beam splitter (BS). In the reference path, the intensity distribution is recorded by a charge-coupled device (CCD) directly and in the object path the light goes through the object and is then focused onto a one-pixel detector by a lens.
Fig. 1. The experimental setup of ghost imaging via sparsity constraint(GISC).
The image of the pseudothermal light field recorded by the CCD has m × n pixels and is reshaped into a vector (1 × N, N = m × n) as a row of the measurement matrix denoted by A . Repeating this process K times, K N, a K × N-matrix A can be obtained. At the same time, we can get a column vector Y of length K from the bucket detector. The unknown object can be denoted by a column vector X of length N and represented as X = Ψ θ by a combined dictionary Ψ which consists of dictionaries Ψ 1 and Ψ 2 (here we only consider the combination of two dictionaries without loss of generality). Then one may have Y = AX + ε = A Ψ θ + ε ,
(1)
with ε the additive noise. The reconstruction of X can be regarded as finding X = Ψ θ that Y − A X 22 + τ θ 1 . Considering the combined dictionary, the problem can be minimizes 12 Y expressed as: 1 Y − A X 22 + τ (θ 1 1 + θ 2 1 ). X = Ψ 1 θ 1 + Ψ 2 θ 2 ; which minimizes: Y 2
(2)
In the above, vv2 denotes the l2 Euclid norm of vector v , vv1 the l1 Euclid norm and τ a nonnegative parameter which should be chosen according to the sparsity of transform coefficients: the sparser the image is, the larger it should be. To solve the convex optimization problem, we employ the gradient projection for sparse reconstruction (GPSR) approach [29], although there are also other algorithms [30–32]. Once we get the representation coefficients of the combined dictionary, we can then start the MCA approach with these coefficients. The largest value of all the representation coefficients #209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14377
is chosen as our initial threshold and the stopping threshold is estimated by the noise of reconstructed image [33]. In each iteration, we first compute the residual of the current estimation of the target image and representation coefficients using the selected dictionary; then we may have the estimation of morphological parts by reconstructing from the corresponding coefficients after being hard thresheld using the threshold that, starting from the initial one, decreases linearly until it reaches the stopping one. 3. 3.1.
Results and discussion Simulation result and discussion
We first test the effect of MCA in GISC by simulation. The object which consists of some “points” and “lines” is shown in Fig. 2(a) with a size of 64 × 64 pixels. The number of sampling is set to be 1400. first A is generated as a random gaussian matrix. Then Y is calculated according to Eq. (1) and gaussian noise of different levels (16dB, 19dB, 22dB, 25dB) is then added. The wavelet and curvelet dictionary is used to form our combined dictionary in the reconstruction. We use the discrete orthogonal wavelet transform with the Mallat algorithm. Daubechies 2 wavelet is chosen as the mother function of the transform which is also known as Haar transform. The decomposition level of Haar transform is set to 2. The curvelet used in this work is based on the wrapping of specially selected fourier samples. The coarsest level of the curvelet is 3 and the number of angles at the second coarsest level is 16. As shown in Fig. 2, it can be seen that the “point” and “line” components are successfully separated with the proposed approach and the quality of the images obtained from the combination of two components is better than the one from GISC without the MCA approach. The main reason of the improvement is that the stopping threshold is estimated by the noise level of reconstructed image and thus the hard thresholding will remove much of the noise component in the image without much affecting the components of interest since the noise is generally non-sparse in the representation basis while the morphology components are.
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Fig. 2. The object and simulation results. (a) the object of 64 × 64 pixels used in the simulation; (b-e) simulation results with SNR=16dB,19dB,22dB,25dB respectively. The first row shows the GPSR reconstruction results, the third row the separated “point” component, the fourth row the separated “line” component and the second row the combination of the two separated components.
The relative mean squared error (RMSE) is evaluated to characterize the quality of reconm,n 2 2 structed image: RMSE = ∑m,n i, j=1 (Ii j − Ii j ) /(∑i, j=1 Ii j ), where Ii j , Ii j are the pixel values of reconstructed image and the original image, respectively, with i, j the pixel indices. From Fig.
#209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14378
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Fig. 3. The RMSE values of the reconstructed images using MCA in GISC and GISC without MCA. (a)The RMSE values of results from data with different SNR. (b)The RMSE values of the results reconstructed from different numbers of sampling.
3(a) we can see that the RMSE values of the image reconstructed using the MCA technique in GISC is smaller than that reconstructed without applying MCA in different noise level. Figure 3(b) shows the RMSE values of the reconstructed images with different numbers of sampling and it can be seen that the image quality is also improved when MCA is applied. 3.2.
Experimental result and discussion
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Fig. 4. The experimental results of ghost imaging via sparse constraint. (a) the target object; (b-f) experimental results with K = 800, 1200, 1600, 2000 respectively. The first row shows the GPSR reconstructed results, the third row the “point” component, the forth row the “line” component and the second row the combination of the two separated components.
To experimentally demonstrate the separation of different components of the object image in GISC, we carried out experiments using the object as shown in Fig. 4(a), which is the image reconstructed from 4000 samplings using the MCA technique in GISC. It will be taken as the reference image because it has better quality than that reconstructed with GISC without using MCA according to the simulation result shown in Fig. 3(b). The size of each “point” is about 232.5μ m × 232.5μ m and the width of each “line” is about 93μ m. The field-of-view has a size of 64 × 64 pixels. We choose a proper size of diaphragm and the distance between ground glass and object to make sure that the average full width at half maximum (FWHM) of the speckle is approximately the size of one pixel. From the results corresponding to different
#209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14379
numbers of sampling K ranging from 800 to 2000 as shown in Figs. 4(b)–4(f), it’s obvious that the separation of “point” component and “line” component is successful. We can see that the quality of “point” and “line” component increases with sampling numbers from the results shown in the third and forth rows of Fig. 4.
Fig. 5. The RMSE with different number of samplings in the experiment. The blue line is the RMSE of GISC and the red one is the RMSE of sum of the separated components when MCA is used in GISC.
It is also confirmed that the quality of image combined from the two separated components via MCA in GISC is better than the reconstructed image without MCA as shown in Fig. 5, which is consistent with the simulation results shown in Fig. 3(b). The table-top experiment shows that it can improve RMSE of reconstructed image by almost 50%.
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Fig. 6. The results of a natural scene. (a) the original scene captured with a telescope; (b) the result of GISC; (c) the “point” component; (d) the “line” component; (e) the combination of the two components. The first to third row of (b-e) is just three of these different slices and last row is the sum of all slices.
We then applied this approach to GISC of a natural scene as shown in Fig. 6(a). The distance between the target and the GI system is about 1km and the GI experiment is described in detail in [34]. In each measurement of the experiment, the bucket detector gets a pulse signal with a finite time duration due to the 3D structure of the target. Using the signal data from the recorded #209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14380
pulses with a certain time delay, one may reconstruct a “slice” of the 3D target and then get the whole image of the target shown in Fig. 6(b) by combing all the slices. A combined waveletcurvelet dictionary is used for the slice reconstruction and morphology separation. We may see that the “point” (“point” means the support of patch is the same in two direction) component shown in Fig. 6(c) are successfully separated from the “line” components in Fig. 6(d). Although a reference image is not available, if we investigate the results shown in row 1, 2, 3 of Figs. 6(b) and 6(e), we can see that the noise is reduced while few image details are sacrificed. The last row in Fig. 6 is the combination of all different slices, visually the image shown in Fig. 6(e) combined from the separated components appear more smooth than that shown in Fig. 6(b) and the details also are maintained. This shows that the proposed approach may improve the reconstruction performance when applied to real world applications. 4.
Conclusion
In summary the morphology component separation in GISC has been studied by numerical simulations and verified by both table-top and outdoor experiments. The results show that it can not only extract different morphology components, but also improve the quality of reconstructed image by the combination of these different components in GISC. The successful separation of reconstructed image can help us to analyze the target, enhance it and identify it in future applications. Acknowledgments The work was supported by the Hi-Tech Research and Development Program of China under Grant Projects No.2013AA122901 and No.2013AA122902
#209794 - $15.00 USD (C) 2014 OSA
Received 8 Apr 2014; revised 26 May 2014; accepted 29 May 2014; published 4 Jun 2014 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.014375 | OPTICS EXPRESS 14381
