- Email: [email protected]
Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints
Physics Letters A 376 (2012) 1519–1522
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints Wenlin Gong ∗ , 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 Sciences, PO Box 800-211, Shanghai 201800, PR China
a r t i c l e
i n f o
Article history: Received 25 October 2011 Received in revised form 28 February 2012 Accepted 12 March 2012 Available online 14 March 2012 Communicated by P.R. Holland
a b s t r a c t Ghost imaging via sparsity constraints (GISC) can nonlocally realize super-resolution imaging. Factors influencing the quality of lensless super-resolution GISC are investigated and the experimental results show that, the quality of GISC is enhanced as the object’s sparse ratio in the representation basis or the spatial transverse coherence lengths on the object plane are decreased. The differences between ghost imaging (GI) and GISC are also discussed. © 2012 Elsevier B.V. All rights reserved.
Keywords: Ghost imaging Super-resolution Image reconstruction Compressive sensing
1. Introduction Based on the quantum or classical correlation of fluctuating light fields, ghost imaging (GI) can nonlocally image an unknown object by measuring the intensity correlation function between two light fields [1–18]. Lots of practical applications are benefited from its particular scheme advantages [4–9]. However, one of the main drawbacks of GI is the long acquisition times required for reconstructing images with a good signal-to-noise ratio (SNR) [18]. Also, the speckle’s transverse size on the object plane in the test path limits the spatial resolution of GI [9–11]. Recently, employing the sparsity of the object and a new sparse reconstruction theory called compressive sensing (CS, also called compressive sampling), a new imaging approach called ghost imaging via sparsity constraints (GISC) was shown to be able to reconstruct pseudo-thermal GI with high quality from far fewer measurements than that the method of the intensity correlation measurement uses [19,20], and even realize super-resolution imaging [21]. Compared with GI, GISC has much higher efficiency in the process of image extraction [19,20] and the recovered image’s spatial resolution can beat the diffraction limit of GI [21]. However, because CS theory is based on the imaging object’s sparsity in the representation basis and random measurement, GISC,
*
Corresponding author. Fax: +86 21 69918537. E-mail address: [email protected] (W. Gong).
0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.03.027
which has utilized the CS theory, will also depend on the coherence of measurement matrix, the object’s sparsity and the type of measurement matrixes [22–25]. In order to clarify the factors influencing the quality of super-resolution GISC with pseudo-thermal source, the effects of both the object’s sparsity and the spatial transverse coherence length of the light field located at the object plane on GISC are investigated. 2. Image reconstruction methods Fig. 1 presents the setup of standard lensless GI and GISC with pseudo-thermal light. The pseudo-thermal source, which is obtained by modulating a laser beam (the wavelength λ = 650 nm) with a rotating ground glass disk, is divided by a beam splitter (BS) into a test and a reference paths. The transverse size of the light on the disk D can be modulated by a diaphragm. In the test path, the object is at a distance of z from the source plane, followed immediately by a bucket detector D t . In the reference path, the light propagates directly to a charge-coupled device (CCD) camera D r . Denote the s-th speckle intensity at pixel (i , j ) recorded by the CCD camera D r as I rs (i , j ), where the indices i = 1, . . . , N x and j = 1, . . . , N y represent the horizontal and vertical pixel coordinates, respectively. s = 1, . . . , N s is the sampling frame index and N s is the total measurement number. The s-th intensity recorded by the detector D t is denoted by I ts . The indices r and t represent, respectively, the reference path and the test path. Then thermal-light GI can be reconstructed by measuring the correlation between I ts and
1520
W. Gong, S. Han / Physics Letters A 376 (2012) 1519–1522
Fig. 1. Standard schematic of lensless GI and GISC with pseudo-thermal light.
I rs (i , j ) according to the image reconstruction method discussed in Refs. [16,17], namely the GI linear image reconstruction algorithm:
R (i , j ) = G
(2,2)
Ns 1 r (i , j ) = I s (i , j ) I ts − γ I svir , Ns
(1)
s =1
where G (2,2) (i , j ) is the second-order correlation function. Moreover,
I r (i , j ) I rs (i , j ) = s ,
(2)
I svir
I svir =
t
Is =
1
Ny Nx
Nx N y 1 Ns
I rs (i , j );
γ=
i =1 j =1
Ns
I ts ;
s =1
I svir =
Ns 1
Ns
I ts I svir
I svir ,
,
(3)
(4)
s =1
where I svir denotes the space average value of the s-th speckle intensity recorded by the CCD camera D r , which is similar to the reference bucket signal S 2 in Ref. [17] when the test detector is also a bucket detector. And · · · denotes the function’s ensemble average value. To recast GISC, we re-formulate it in the CS framework. Each of the speckle intensity I rs (i , j ) is reshaped as a row vector Φ (1 × N, N = N x × N y ). After N s measurements, we can build a random projection matrix A (N s × N). At the same time, the intensities I ts recorded by the test detector D t are arranged as a column vector Y (N s × 1). If we denote the unknown object image as an N-dimensional column vector X (N × 1), then we have
Y = AX.
(5)
CS theory maintains that if X is not sparse, then X can be represented as X = Ψ · α such that α is sparse (namely there are only K non-zero entries in the column vector α , K N and Ψ denotes the transform operator to the sparse basis. For example, some objects are sparse in real-space domain (such as a transmission double-slit), and some objects are sparse in Fourier space (such as a pure-phase double-slit)). Furthermore, if the projection matrix A satisfies the restricted isometry property (RIP) (we emphasize that for ghost imaging with thermal light, the intensity fluctuation of light field obeys to Gaussian statistical distribution and the projection matrix essentially satisfies RIP), then GISC can be reconstructed by solving the following convex optimization program even if the measurement number N s ≈ O ( K ) N [25,26]:
T GISC = Ψ · α ;
which minimizes:
1 2
Y − A · Ψ · α 22 + τ α 1 , (6)
Fig. 2. The effect of the object’s sparsity on GI and GISC with D = 2.0 mm and z = 500 mm. (1) The cross-section curve of the normalized correlation function of intensity distribution on the reference detection plane (the FWHM of g (2) (xr , xr = 0) is x = 162 μm). Images shown in (2) and (3) are the results reconstructed by GI method (averaged 2000 observations) and GISC method (with 1000 measure1 2 ments), respectively. (a) A transmission double-slit, β = 15 ; (b) a four-slit, β = 15 ;
3 4 ; (d) an eight-slit, β = 15 . The slit width a = 60 μm, slit height (c) a six-slit, β = 15 h = 500 μm and center-to-center separation d = 120 μm for above all objects.
where T GISC is the object’s image recovered by GISC reconstruction algorithm, τ is a nonnegativeparameter, V 2 denotes the Euclidean norm of V , and V 1 = i |υi | is the 1 -norm of V . 3. Experimental results and discussion For the object of the dimension length N and K non-zero entries in the representation basis, the object’s sparse ratio (β ) in the representation basis can be expressed as
β=
K N
(7)
,
where the smaller the value β is, the sparser the image is in the representation basis. In GI schemes, the spatial transverse coherence length of light field (namely the speckle’s transverse size) on the object plane can be obtained by measuring the normalized correlation function of light intensity distribution on the reference detection plane D r [16], namely
g (2) xr , xr = 0 =
I 1 (xr ) I 2 (xr = 0) , I 1 (xr ) I 2 (xr = 0)
(8)
where the normalized correlation function’s full-width at halfmax (FWHM) x is equivalent to the spatial transverse coherence length of light field, which determines the diffraction limit of GI system [9–11]. To demonstrate the effects of the object’s sparse ratio β and the spatial transverse coherence length of light field x on GI and GISC, the experimental reconstruction results for some transmission apertures are summarized in Figs. 2 and 3, respectively. For GI, we have utilized the image reconstruction method discussed in Refs. [16,17], while for the GISC reconstruction, we have utilized the gradient projection for sparse reconstruction (GPSR) algorithm [26] and the objects are represented in the real-space basis because the imaging targets we propose are sparse in real-space domain. Figs. 2(2) and 2(3), respectively, present the results obtained by GI and GISC reconstructions in different β . The different spatial transverse coherence lengths x on the object plane, as shown in Fig. 3 (left column), can be achieved by modulating the transverse sizes of the light on the disk D since x ≈ λDz [10]. The images recovered by GI and GISC methods in different x are displayed in Fig. 3(a)–(d) (right column).
W. Gong, S. Han / Physics Letters A 376 (2012) 1519–1522
1521
Fig. 4. The performance between the reconstruction fidelity (MSE) and the sparse ratio β or spatial transverse coherence lengths x for GI and GISC reconstruction results. (a) Relationship between β and MSE based on the results obtained in Fig. 2; (b) Relationship between x and MSE based on the results obtained in Fig. 3.
Fig. 3. The reconstruction results of GI and GISC in different D with z = 850 mm. Left column: the cross-section curves of the normalized correlation function of intensity distribution on the reference detection plane; Right column: recovered images by GI (averaged 2000 observations, upper left) and GISC (with 1000 measurements, upper right) methods and their cross-section curves. Black solid curves are the cross-section of the original object (with the slit width a = 30 μm, and center-to-center separation d = 60, 90, 120, 150, 180 μm), blue dotted curves and red dash–dot curves display the results reconstructed by GI and GISC, respectively. (a) D = 1.0 mm (namely the FWHM of g (2) (xr , xr = 0) is x ≈ 530 μm); (b) D = 2.0 mm (x ≈ 268 μm); (c) D = 4.0 mm (x ≈ 138 μm); and (d) D = 8.0 mm (x ≈ 70 μm). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
In order to evaluate quantitatively the quality of images reconstructed by GI and GISC methods in different β or x, the reconstruction fidelity is estimated by calculating the mean-squared error (MSE) of the reconstructions with respect to the original object T 0 [19]:
MSE =
1 N
2
T GISC/GI (i , j ) − T 0 (i , j ) ,
(9)
sparse property of the images [21]. In addition, the MSE of the image reconstructed by GISC increases with the transverse coherence lengths x because the coherence of measurement is enhanced (Figs. 3(a)–(d) and 4(b)). Based on the above experimental results, both GI and GISC closely depend on the sparsity of images [18–21]. For GI, as the increase of the spatial transverse coherence lengths x, the uncertainties of momentum will be decreased (namely the modes of electromagnetic filed will become much purer), thus the reconstructed image’s spatial resolution will reduce as the GI linear image reconstruction algorithm is used [14]. Different from GI, employing the sparsity of images and nonlinear sparse reconstruction algorithm, super-resolved imaging can be achieved and the quality of images recovered by GISC method doesn’t only depends on the sparsity of the imaging objects, but also depends on the type of measurement matrixes and the coincidence degree of the equality described by Eq. (5) [21–23]. Compared with the reconstruction approaches based on additional a priori information of optical system, super-resolution can also be obtained but the improvement degree is limited in practice because of the influence of detection noise [27–29]. However, GISC is robust to noise according to CS theory [21–25] and can be a universal super-resolution imaging method. In practice, GISC will be very useful to the applications such as microscopy, astronomy and so on. 4. Conclusion In conclusion, we have analyzed the effect of the object’s sparsity and spatial transverse coherence property of thermal light field on GI and GISC. The experimental results have demonstrated that the quality of the images recovered by GISC is reduced as the increase of the sparse ratio in the representation basis or the transverse coherence lengths x, which is similar to the results obtained by GI. We also show that the reconstruction results of GISC are always better than that obtained by GI method. Acknowledgements
i, j
where smaller MSE means the quality of recovered image is better. Calculated from the results shown in Figs. 2 and 3, the dependence of the value of MSE on β or x are depicted in Figs. 4(a) and 4(b), respectively. Apparently, the MSE of the result recovered by GI increases with the value β or x, and the spatial resolution, as shown in Figs. 2(1) and 3(a)–(d), is determined by the spatial transverse coherence lengths x, which is in accordance with the results presented in Refs. [9–11]. Therefore, when the speckle’s transverse size on the object plane x is larger than the object’s center-to-center separation, the image of the object cannot be reconstructed by GI. However, super-resolution imaging can be achieved by GISC method and the reconstruction fidelity is enhanced as the decrease of the value β due to the utilization of the
The work was supported by the Hi-Tech Research and Development Program of China under Grant Project No. 2011AA120101, No. 2011AA120102, and Shanghai Natural Science Foundation under Grant Project No. 09JC1415000. References [1] T.B. Pittman, Y.H. Shih, D.V. Strekalov, A.V. Sergienko, Phys. Rev. A 52 (1995) R3429. [2] A. Gatti, E. Brambilla, M. Bache, L.A. Lugiato, Phys. Rev. Lett. 93 (2004) 093602. [3] R.S. Bennink, S.J. Bentley, R.W. Boyd, J.C. Howell, Phys. Rev. Lett. 92 (2004) 033601. [4] J. Cheng, S. Han, Phys. Rev. Lett. 92 (2004) 093903. [5] W. Gong, P. Zhang, X. Shen, S. Han, Appl. Phys. Lett. 95 (2009) 071110. [6] W. Gong, S. Han, Opt. Lett. 36 (2011) 394.
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[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
W. Gong, S. Han / Physics Letters A 376 (2012) 1519–1522
N. Tian, Q. Guo, A. Wang, D. Xu, L. Fu, Opt. Lett. 36 (2011) 3302. R.E. Meyers, K.S. Deacon, Y. Shih, Appl. Phys. Lett. 98 (2011) 111115. P. Zhang, W. Gong, X. Shen, S. Han, Opt. Lett. 34 (2009) 1222. W. Gong, S. Han, J. Opt. Soc. Am. B 27 (2010) 675. F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, L.A. Lugiato, Phys. Rev. Lett. 94 (2005) 183602. D. Zhang, Y.-H. Zhai, L.-A. Wu, X.-H. Chen, Opt. Lett. 30 (2005) 2354. Y.-H. Zhai, X.-H. Chen, D. Zhang, L.-A. Wu, Phys. Rev. A 72 (2005) 043805. G. Scarcelli, V. Berardi, Y. Shih, Phys. Rev. Lett. 96 (2006) 063602. Y. Bromberg, O. Katz, Y. Silbergerg, Phys. Rev. A 79 (2009) 053840. W. Gong, S. Han, Phys. Lett. A 374 (2010) 1005. F. Ferri, D. Magatti, L.A. Lugiato, A. Gatti, Phys. Rev. Lett. 104 (2010) 253603. A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, L.A. Lugiato, J. Mod. Opt. 53 (2006) 739.
[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
O. Katz, Y. Bromberg, Y. Silberberg, Appl. Phys. Lett. 95 (2009) 131110. J. Liu, J. Zhu, C. Lu, S. Huang, Opt. Lett. 35 (2010) 1206. W. Gong, S. Han, arXiv:0911.4750v3 [quant-ph], 2009. E.J. Candès, in: International Congress of Mathematicians, vol. III, Eur. Math. Soc., Zürich, 2006, pp. 1433–1452, and references therein. D.L. Donoho, Y. Tsaig, IEEE Trans. Inform. Theory 54 (2006) 4789. E.J. Candès, C. R. Acad. Sci. Paris, Ser. I 346 (2008) 589. E.J. Candès, M.B. Wakin, IEEE Signal Process. Mag. 25 (2008) 21. M.A.T. Figueiredo, R.D. Nowak, S.J. Wright, IEEE J. Sel. Top. Sig. Proc. 1 (2007) 586. J.L. Harris, J. Opt. Soc. Am. 54 (1964) 931. S.G. Mallat, IEEE Trans. Pattern Anal. Machine Intell. 11 (1989) 674. B.R. Hunt, International Journal of Imaging Systems and Technology 6 (1995) 297.
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints Wenlin Gong ∗ , 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 Sciences, PO Box 800-211, Shanghai 201800, PR China
a r t i c l e
i n f o
Article history: Received 25 October 2011 Received in revised form 28 February 2012 Accepted 12 March 2012 Available online 14 March 2012 Communicated by P.R. Holland
a b s t r a c t Ghost imaging via sparsity constraints (GISC) can nonlocally realize super-resolution imaging. Factors influencing the quality of lensless super-resolution GISC are investigated and the experimental results show that, the quality of GISC is enhanced as the object’s sparse ratio in the representation basis or the spatial transverse coherence lengths on the object plane are decreased. The differences between ghost imaging (GI) and GISC are also discussed. © 2012 Elsevier B.V. All rights reserved.
Keywords: Ghost imaging Super-resolution Image reconstruction Compressive sensing
1. Introduction Based on the quantum or classical correlation of fluctuating light fields, ghost imaging (GI) can nonlocally image an unknown object by measuring the intensity correlation function between two light fields [1–18]. Lots of practical applications are benefited from its particular scheme advantages [4–9]. However, one of the main drawbacks of GI is the long acquisition times required for reconstructing images with a good signal-to-noise ratio (SNR) [18]. Also, the speckle’s transverse size on the object plane in the test path limits the spatial resolution of GI [9–11]. Recently, employing the sparsity of the object and a new sparse reconstruction theory called compressive sensing (CS, also called compressive sampling), a new imaging approach called ghost imaging via sparsity constraints (GISC) was shown to be able to reconstruct pseudo-thermal GI with high quality from far fewer measurements than that the method of the intensity correlation measurement uses [19,20], and even realize super-resolution imaging [21]. Compared with GI, GISC has much higher efficiency in the process of image extraction [19,20] and the recovered image’s spatial resolution can beat the diffraction limit of GI [21]. However, because CS theory is based on the imaging object’s sparsity in the representation basis and random measurement, GISC,
*
Corresponding author. Fax: +86 21 69918537. E-mail address: [email protected] (W. Gong).
0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.03.027
which has utilized the CS theory, will also depend on the coherence of measurement matrix, the object’s sparsity and the type of measurement matrixes [22–25]. In order to clarify the factors influencing the quality of super-resolution GISC with pseudo-thermal source, the effects of both the object’s sparsity and the spatial transverse coherence length of the light field located at the object plane on GISC are investigated. 2. Image reconstruction methods Fig. 1 presents the setup of standard lensless GI and GISC with pseudo-thermal light. The pseudo-thermal source, which is obtained by modulating a laser beam (the wavelength λ = 650 nm) with a rotating ground glass disk, is divided by a beam splitter (BS) into a test and a reference paths. The transverse size of the light on the disk D can be modulated by a diaphragm. In the test path, the object is at a distance of z from the source plane, followed immediately by a bucket detector D t . In the reference path, the light propagates directly to a charge-coupled device (CCD) camera D r . Denote the s-th speckle intensity at pixel (i , j ) recorded by the CCD camera D r as I rs (i , j ), where the indices i = 1, . . . , N x and j = 1, . . . , N y represent the horizontal and vertical pixel coordinates, respectively. s = 1, . . . , N s is the sampling frame index and N s is the total measurement number. The s-th intensity recorded by the detector D t is denoted by I ts . The indices r and t represent, respectively, the reference path and the test path. Then thermal-light GI can be reconstructed by measuring the correlation between I ts and
1520
W. Gong, S. Han / Physics Letters A 376 (2012) 1519–1522
Fig. 1. Standard schematic of lensless GI and GISC with pseudo-thermal light.
I rs (i , j ) according to the image reconstruction method discussed in Refs. [16,17], namely the GI linear image reconstruction algorithm:
R (i , j ) = G
(2,2)
Ns 1 r (i , j ) = I s (i , j ) I ts − γ I svir , Ns
(1)
s =1
where G (2,2) (i , j ) is the second-order correlation function. Moreover,
I r (i , j ) I rs (i , j ) = s ,
(2)
I svir
I svir =
t
Is =
1
Ny Nx
Nx N y 1 Ns
I rs (i , j );
γ=
i =1 j =1
Ns
I ts ;
s =1
I svir =
Ns 1
Ns
I ts I svir
I svir ,
,
(3)
(4)
s =1
where I svir denotes the space average value of the s-th speckle intensity recorded by the CCD camera D r , which is similar to the reference bucket signal S 2 in Ref. [17] when the test detector is also a bucket detector. And · · · denotes the function’s ensemble average value. To recast GISC, we re-formulate it in the CS framework. Each of the speckle intensity I rs (i , j ) is reshaped as a row vector Φ (1 × N, N = N x × N y ). After N s measurements, we can build a random projection matrix A (N s × N). At the same time, the intensities I ts recorded by the test detector D t are arranged as a column vector Y (N s × 1). If we denote the unknown object image as an N-dimensional column vector X (N × 1), then we have
Y = AX.
(5)
CS theory maintains that if X is not sparse, then X can be represented as X = Ψ · α such that α is sparse (namely there are only K non-zero entries in the column vector α , K N and Ψ denotes the transform operator to the sparse basis. For example, some objects are sparse in real-space domain (such as a transmission double-slit), and some objects are sparse in Fourier space (such as a pure-phase double-slit)). Furthermore, if the projection matrix A satisfies the restricted isometry property (RIP) (we emphasize that for ghost imaging with thermal light, the intensity fluctuation of light field obeys to Gaussian statistical distribution and the projection matrix essentially satisfies RIP), then GISC can be reconstructed by solving the following convex optimization program even if the measurement number N s ≈ O ( K ) N [25,26]:
T GISC = Ψ · α ;
which minimizes:
1 2
Y − A · Ψ · α 22 + τ α 1 , (6)
Fig. 2. The effect of the object’s sparsity on GI and GISC with D = 2.0 mm and z = 500 mm. (1) The cross-section curve of the normalized correlation function of intensity distribution on the reference detection plane (the FWHM of g (2) (xr , xr = 0) is x = 162 μm). Images shown in (2) and (3) are the results reconstructed by GI method (averaged 2000 observations) and GISC method (with 1000 measure1 2 ments), respectively. (a) A transmission double-slit, β = 15 ; (b) a four-slit, β = 15 ;
3 4 ; (d) an eight-slit, β = 15 . The slit width a = 60 μm, slit height (c) a six-slit, β = 15 h = 500 μm and center-to-center separation d = 120 μm for above all objects.
where T GISC is the object’s image recovered by GISC reconstruction algorithm, τ is a nonnegativeparameter, V 2 denotes the Euclidean norm of V , and V 1 = i |υi | is the 1 -norm of V . 3. Experimental results and discussion For the object of the dimension length N and K non-zero entries in the representation basis, the object’s sparse ratio (β ) in the representation basis can be expressed as
β=
K N
(7)
,
where the smaller the value β is, the sparser the image is in the representation basis. In GI schemes, the spatial transverse coherence length of light field (namely the speckle’s transverse size) on the object plane can be obtained by measuring the normalized correlation function of light intensity distribution on the reference detection plane D r [16], namely
g (2) xr , xr = 0 =
I 1 (xr ) I 2 (xr = 0) , I 1 (xr ) I 2 (xr = 0)
(8)
where the normalized correlation function’s full-width at halfmax (FWHM) x is equivalent to the spatial transverse coherence length of light field, which determines the diffraction limit of GI system [9–11]. To demonstrate the effects of the object’s sparse ratio β and the spatial transverse coherence length of light field x on GI and GISC, the experimental reconstruction results for some transmission apertures are summarized in Figs. 2 and 3, respectively. For GI, we have utilized the image reconstruction method discussed in Refs. [16,17], while for the GISC reconstruction, we have utilized the gradient projection for sparse reconstruction (GPSR) algorithm [26] and the objects are represented in the real-space basis because the imaging targets we propose are sparse in real-space domain. Figs. 2(2) and 2(3), respectively, present the results obtained by GI and GISC reconstructions in different β . The different spatial transverse coherence lengths x on the object plane, as shown in Fig. 3 (left column), can be achieved by modulating the transverse sizes of the light on the disk D since x ≈ λDz [10]. The images recovered by GI and GISC methods in different x are displayed in Fig. 3(a)–(d) (right column).
W. Gong, S. Han / Physics Letters A 376 (2012) 1519–1522
1521
Fig. 4. The performance between the reconstruction fidelity (MSE) and the sparse ratio β or spatial transverse coherence lengths x for GI and GISC reconstruction results. (a) Relationship between β and MSE based on the results obtained in Fig. 2; (b) Relationship between x and MSE based on the results obtained in Fig. 3.
Fig. 3. The reconstruction results of GI and GISC in different D with z = 850 mm. Left column: the cross-section curves of the normalized correlation function of intensity distribution on the reference detection plane; Right column: recovered images by GI (averaged 2000 observations, upper left) and GISC (with 1000 measurements, upper right) methods and their cross-section curves. Black solid curves are the cross-section of the original object (with the slit width a = 30 μm, and center-to-center separation d = 60, 90, 120, 150, 180 μm), blue dotted curves and red dash–dot curves display the results reconstructed by GI and GISC, respectively. (a) D = 1.0 mm (namely the FWHM of g (2) (xr , xr = 0) is x ≈ 530 μm); (b) D = 2.0 mm (x ≈ 268 μm); (c) D = 4.0 mm (x ≈ 138 μm); and (d) D = 8.0 mm (x ≈ 70 μm). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
In order to evaluate quantitatively the quality of images reconstructed by GI and GISC methods in different β or x, the reconstruction fidelity is estimated by calculating the mean-squared error (MSE) of the reconstructions with respect to the original object T 0 [19]:
MSE =
1 N
2
T GISC/GI (i , j ) − T 0 (i , j ) ,
(9)
sparse property of the images [21]. In addition, the MSE of the image reconstructed by GISC increases with the transverse coherence lengths x because the coherence of measurement is enhanced (Figs. 3(a)–(d) and 4(b)). Based on the above experimental results, both GI and GISC closely depend on the sparsity of images [18–21]. For GI, as the increase of the spatial transverse coherence lengths x, the uncertainties of momentum will be decreased (namely the modes of electromagnetic filed will become much purer), thus the reconstructed image’s spatial resolution will reduce as the GI linear image reconstruction algorithm is used [14]. Different from GI, employing the sparsity of images and nonlinear sparse reconstruction algorithm, super-resolved imaging can be achieved and the quality of images recovered by GISC method doesn’t only depends on the sparsity of the imaging objects, but also depends on the type of measurement matrixes and the coincidence degree of the equality described by Eq. (5) [21–23]. Compared with the reconstruction approaches based on additional a priori information of optical system, super-resolution can also be obtained but the improvement degree is limited in practice because of the influence of detection noise [27–29]. However, GISC is robust to noise according to CS theory [21–25] and can be a universal super-resolution imaging method. In practice, GISC will be very useful to the applications such as microscopy, astronomy and so on. 4. Conclusion In conclusion, we have analyzed the effect of the object’s sparsity and spatial transverse coherence property of thermal light field on GI and GISC. The experimental results have demonstrated that the quality of the images recovered by GISC is reduced as the increase of the sparse ratio in the representation basis or the transverse coherence lengths x, which is similar to the results obtained by GI. We also show that the reconstruction results of GISC are always better than that obtained by GI method. Acknowledgements
i, j
where smaller MSE means the quality of recovered image is better. Calculated from the results shown in Figs. 2 and 3, the dependence of the value of MSE on β or x are depicted in Figs. 4(a) and 4(b), respectively. Apparently, the MSE of the result recovered by GI increases with the value β or x, and the spatial resolution, as shown in Figs. 2(1) and 3(a)–(d), is determined by the spatial transverse coherence lengths x, which is in accordance with the results presented in Refs. [9–11]. Therefore, when the speckle’s transverse size on the object plane x is larger than the object’s center-to-center separation, the image of the object cannot be reconstructed by GI. However, super-resolution imaging can be achieved by GISC method and the reconstruction fidelity is enhanced as the decrease of the value β due to the utilization of the
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