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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 6 — Mar. 15, 2010
  • pp: 5562–5573
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Optimization of thermal ghost imaging: high-order correlations vs. background subtraction

Kam Wai C. Chan, Malcolm N. O’Sullivan, and Robert W. Boyd  »View Author Affiliations


Optics Express, Vol. 18, Issue 6, pp. 5562-5573 (2010)
http://dx.doi.org/10.1364/OE.18.005562


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Abstract

We compare the performance of high-order thermal ghost imaging with that of conventional (that is, lowest-order) thermal ghost imaging for different data processing methods. Particular attention is given to high-order thermal ghost imaging with background normalization and conventional ghost imaging with background subtraction. The contrast-to-noise ratio (CNR) of the ghost image is used as the figure of merit for the comparison. We find analytically that the CNR of the normalized high-order ghost image is inversely proportional to the square root of the number of transmitting pixels of the object. This scaling law is independent of the exponents used in calculating the high-order correlation and is the same as that of conventional ghost imaging with background subtraction. We find that no data processing procedure performs better than lowest-order ghost imaging with background subtraction. Our results are found to be able to explain the observations of a recent experiment [Chen et al., arXiv:0902.3713v3 [quant-ph]].

© 2010 Optical Society of America

1. Introduction

Ghost imaging is a novel imaging technique that acquires the image of an object with the processes of object interrogation and image formation performed separately [1

1. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995). [CrossRef] [PubMed]

, 2

2. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost Imaging with Thermal Light: Comparing Entanglement and Classical Correlation,” Phys. Rev. Lett. 93, 093602 (2004); [CrossRef] [PubMed]

, 3

3. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004). [CrossRef]

, 4

4. A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, “Two-Photon Imaging with Thermal Light,” Phys. Rev. Lett. 94, 063601 (2005). [CrossRef] [PubMed]

]. In a typical ghost imaging setup, two mutually correlated beams of light are used: an object field that illuminates the object but is not spatially resolved by its detector and a reference field that does not interact with the object but is spatially resolved by its detector. The ghost image is then obtained by measuring the intensity cross-correlation function of the object and reference fields.

There are two main categories of light sources used for ghost imaging: entangled photons produced in parametric down-conversion [1

1. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995). [CrossRef] [PubMed]

] and spatially incoherent pseudo-thermal light produced by a laser beam with a rotating diffuser [4

4. A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, “Two-Photon Imaging with Thermal Light,” Phys. Rev. Lett. 94, 063601 (2005). [CrossRef] [PubMed]

] or a spatial light modulator [5

5. Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009). [CrossRef]

]. The setup that uses pseudo-thermal light has attracted a lot of attention recently due to its potential practical applications as well as the interpretation of its underlying physical process [6

6. G. Scarcelli, V. Berardi, and Y. Shih “Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations?” Phys. Rev. Lett. 96, 063602 (2006). [CrossRef] [PubMed]

, 7

7. A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations?” Phys. Rev. Lett. 98, 039301 (2007). [CrossRef] [PubMed]

, 8

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

, 9

9. L.-G. Wang, S. Qamar, S.-Y. Zhu, and M. S. Zubairy, “Hanbury Brown-Twiss effect and thermal light ghost imaging: A unified approach,” Phys. Rev. A 79, 033835 (2009). [CrossRef]

, 10

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

, 11

11. R. Meyers, K. S. Deacon, and Y. H. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801(R) (2008). [CrossRef]

]. While there are similarities and differences in the ghost images produced by the two different light sources, the most notable difference between the two is that a thermal ghost image always lies on a noisy background whereas the ghost image obtained using entangled photons can achieve 100% visibility.

In the following, we first calculate the CNR of the conventional thermal ghost imaging. The image quality of the high-order thermal ghost imaging is then studied in detail in Section 3. Throughout this paper, the analyses of the ghost image quality are made using classical arguments [22

22. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-Order Thermal Ghost Imaging,” Opt. Lett. 34, 3343–3345 (2009). [CrossRef] [PubMed]

, 25

25. L. Basano and P. Ottonello, “A conceptual experiment on single-beam coincidence detection with pseudothermal light,” Opt. Express 15, 12386–12394 (2007). [CrossRef] [PubMed]

]. We will consider the effects of shot noise using a semi-classical approach in Section 4. The results are summarized in Section 5.

2. Conventional thermal ghost imaging

A typical thermal ghost imaging setup is shown in Fig. 1(a). Here we consider the lensless ghost imaging configuration [6

6. G. Scarcelli, V. Berardi, and Y. Shih “Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations?” Phys. Rev. Lett. 96, 063602 (2006). [CrossRef] [PubMed]

, 12

12. J. Cheng and S. Han, “Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction,” Phys. Rev. Lett. 92, 093903 (2004). [CrossRef] [PubMed]

, 13

13. G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006). [CrossRef]

, 14

14. L. Basano and P. Ottonello, “Experiment in lensless ghost imaging with thermal light,” Appl. Phys. Lett. 89, 091109 (2006) [CrossRef]

, 26

26. D. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005). [CrossRef]

, 27

27. Y. Cai and F. Wang, “Lensless imaging with partially coherent light,” Opt. Lett. 32, 205–207 (2007). [CrossRef] [PubMed]

]. The spatial patterns of the light beams are generated by a rotating ground glass plate to mimic a thermal light source. To simplify the analysis, the object is taken to be a binary object, i.e., the object mask has either zero or unity transmission. The average size of the speckles is taken to be approximately the same as the size of the pixels of the reference detector. As a result, we can perform a spatial discretization of the intensity of light falling onto the object surface, and the light falling onto a given pixel can be assumed to be statistically independent of that falling onto other pixels (see Fig. 1(b)).

We assume that the photodetector response time is short compared to the correlation time of the speckle field, as is the case in many typical situations. In addition, we assume that the illuminating beam has an essentially uniform intensity distribution over the transverse extent of the object. Specifically, the average intensity of the light beam is taken to have a negative exponential probability distribution of mean ⟨I(x⃗)⟩ = μ for all the positions x⃗, where I(x⃗) is the speckle intensity in the object plane and ⟨⋯⟩ denotes ensemble average. We assume that the light source is sufficiently intense that shot noise in the photocurrent leaving each detector element is negligibly small. In Section 4 we examine the conditions under which this assumption is valid. Together with the assumption on the size of the speckles made above, we can conclude that ⟨In(x⃗)⟩ =n!⟨I(x⃗)⟩n =n!μn and ⟨I(x⃗)I(x⃗′)⟩ = ⟨I(x⃗)⟩⟨I(x⃗′)⟩ = μ 2 for x⃗x⃗′.

The conventional thermal ghost image is obtained by calculating the cross-correlation of the object signal Io and the reference signal Ir(x⃗):

Fig. 1. (a) Schematic of the lenless thermal ghost imaging setup. (b) The simplified object model for analysis. The object plane (left) and the reference detector plane (right) are discretized by pixels of finite sizes. BS: beam-splitter.
G(x)=M[IoIr(x)],
(1)

where

M[X]1Ns=1NX(s)
(2)

is defined to be the sample average of the quantity X for the N measurements {X (s)}, s = 1,⋯ ,N. Here

Io=1Δbeamaread2yO(y)I(y)y𝓣I(y)
(3)
Ir(x)=1Δareaofpixelatxd2yI(y)I(x)
(4)

and

are the object and reference signals respectively, and Δ is the area of a pixel. The pixel index y⃗ in Eq. (3) is summed over the transmitting regions 𝓣 of the binary object mask O(y⃗), with the number of transmitting pixels denoted by T according to the assumption made previously. In the general situation in which the speckle size can be larger than the size of a pixel of the reference detector, T is given by T = (transmitting area of object)/(average speckle size).

We define the contrast-to-noise ratio (CNR) to be

CNR(G)G(xin)G(xout)12[Δ2G(xin)+Δ2G(xout)],
(5)

where Δ2 G(x⃗) = ⟨G(x⃗)2⟩ - ⟨G(x⃗)⟩2. Here x⃗ in and x⃗ out represent the pixel positions inside and outside the transmitting regions of the object. The variances Δ2 G(x⃗ in) and Δ2 G(x⃗ out) are generally not the same. It follows from Eq. (1) that ⟨G(x⃗)⟩ = ⟨IoIr(x⃗)⟩ and

Δ2G(x)=1N[Io2Ir2(x)IoIr(x)2].
(6)

Using the result [22

22. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-Order Thermal Ghost Imaging,” Opt. Lett. 34, 3343–3345 (2009). [CrossRef] [PubMed]

]

IomIrn(x)={Γ(T+m+n)n!Γ(T+n)μm+n,forx=xin,Γ(T+m)n!Γ(T)μm+n,forx=xout,
(7)

where Γ(x⃗) is the gamma function, the CNR of G(x⃗) is found to be [22

22. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-Order Thermal Ghost Imaging,” Opt. Lett. 34, 3343–3345 (2009). [CrossRef] [PubMed]

, 25

25. L. Basano and P. Ottonello, “A conceptual experiment on single-beam coincidence detection with pseudothermal light,” Opt. Express 15, 12386–12394 (2007). [CrossRef] [PubMed]

]

CNR(G)=NT2+5T+11/2.
(8)

Note that the numerical coefficients of T in the denominator of Eq. (8) depend on the specifics of the model used for the correlations of the speckle intensities. Of more general applicability is the scaling law CNR(G) ∝= √N/T for T ≫ 1 that characterizes the quality of the ghost image for this data processing method.

An alternative approach for forming the ghost image is to calculate the correlation of the background-subtracted object and reference signals [2

2. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost Imaging with Thermal Light: Comparing Entanglement and Classical Correlation,” Phys. Rev. Lett. 93, 093602 (2004); [CrossRef] [PubMed]

, 3

3. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004). [CrossRef]

, 6

6. G. Scarcelli, V. Berardi, and Y. Shih “Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations?” Phys. Rev. Lett. 96, 063602 (2006). [CrossRef] [PubMed]

, 11

11. R. Meyers, K. S. Deacon, and Y. H. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801(R) (2008). [CrossRef]

, 12

12. J. Cheng and S. Han, “Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction,” Phys. Rev. Lett. 92, 093903 (2004). [CrossRef] [PubMed]

, 13

13. G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006). [CrossRef]

, 14

14. L. Basano and P. Ottonello, “Experiment in lensless ghost imaging with thermal light,” Appl. Phys. Lett. 89, 091109 (2006) [CrossRef]

, 15

15. F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light,” Phys. Rev. Lett. 94, 183602 (2005). [CrossRef] [PubMed]

]. Specifically, one calculates the ghost image as

G(x)=M[(IoM[Io])(Ir(x)M[Ir(x)])]
=M[IoIr(x)]M[Io]M[Ir(x)].
(9)

It follows from Eq. (7) that the contrast-to-noise ratio of this method is given by

CNR(G)=N1T+7/23/N.
(10)

The scaling law of this procedure is thus given by CNR(G′) ∝ = √N/T [28

28. B. I. Erkmen and J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009). [CrossRef]

]. We thus see that G′(x⃗) performs much better than G(x⃗) for the relevant limit T ≫ 1.

Still another procedure for determining the ghost image is to calculate the normalized correlation function [29

29. 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). [CrossRef] [PubMed]

]

g1,1(x)=M[IoIr(x)]M[Io]M[Ir(x)].
(11)

By dividing the correlation function by M[Ir(x⃗)], artifacts associated with the nonuniform response of the CCD detector are largely removed from the final ghost image. We will show below in Section 3 that

CNR(g1,1)N2T+3.
(12)

Thus g1,1 (x⃗) performs approximately as well as G′(x⃗).

It is remarked that at times certain workers have advocated taking the background term M[Io]M[Ir(x⃗)] in Eq. (9) as a constant in the limit of large N, that is, taking M[Io] → ⟨Io⟩ and M[Ir(x⃗)] → ⟨Ir(x⃗)⟩. Such a procedure leads to the the prediction that the contrast-to-noise ratio scales as

CNR(G)NT+7/2.
(13)

However, we note that this procedure is somewhat unphysical, as there is no operational way to determine an ensemble average without performing an infinite number of measurements.

3. High-order thermal ghost imaging

The procedure of high-order ghost image entails calculating the correlation of powers of the signal and reference intensities rather than the the correlation of the intensities themselves. Different configurations for performing high-order thermal ghost imaging have been described [17

17. Y. Bai and S. Han, “Ghost imaging with thermal light by third-order correlation,” Phys. Rev. A. 76,043828 (2007). [CrossRef]

, 18

18. L.-H. Ou and L.-M. Kuang, “Ghost imaging with third-order correlated thermal light,” J. Phys. B: At. Mol. Opt. Phys. 40, 1833–1844 (2007). [CrossRef]

, 19

19. D.-Z. Cao, J. Xiong, S.-H. Zhang, L.-F. Lin, L. Gao, and K. Wang, “Enhancing visibility and resolution in Nth-order intensity correlation of thermal light,” Appl. Phys. Lett. 92, 201102 (2008). [CrossRef]

, 20

20. I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and A. N. Penin, “High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light,” Phys. Rev. A 77, 053801 (2008). [CrossRef]

, 21

21. Q. Liu, X.-H. Chen, K.-H. Luo, W. Wu, and L.-A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A 79, 053844 (2009). [CrossRef]

, 22

22. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-Order Thermal Ghost Imaging,” Opt. Lett. 34, 3343–3345 (2009). [CrossRef] [PubMed]

, 23

23. X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, and L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv:0902.3713v3 [quant-ph].

]. The simplest procedure is to use the same setup as in Fig. 1(a), where there is only a single reference detector, and to calculate the normalized high-order intensity correlation [19

19. D.-Z. Cao, J. Xiong, S.-H. Zhang, L.-F. Lin, L. Gao, and K. Wang, “Enhancing visibility and resolution in Nth-order intensity correlation of thermal light,” Appl. Phys. Lett. 92, 201102 (2008). [CrossRef]

, 20

20. I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and A. N. Penin, “High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light,” Phys. Rev. A 77, 053801 (2008). [CrossRef]

, 23

23. X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, and L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv:0902.3713v3 [quant-ph].

]

gmn(full)(x)=M[IomIrn(x)]M[Iom]M[Irn(x)].
(14)

Note that the exponents m and n in the denominator are inside the averages, so that gmn (full) (x⃗) is unity when Io and Ir(x⃗) are uncorrelated. As mentioned in the last section, the normalization over M[Irm (x⃗)] also has the advantage of eliminating artifacts in the reference detector that appear as a multiplicative function of x⃗ on the reference signal.

Unlike as in the case of Eq. (9), the use of sample averages in Eq. (14) imposes difficulties on the analysis of the properties of the ghost image, as the quantities M[ImoInr (x⃗)],M[Imo] and M[Inr(x⃗)] are also random variables when the number of measurements N is finite. It might be tempting to treat the denominator as a constant to simplify the analysis. However, this assumption is inappropriate to describe the noise properties of the normalized ghost image, as we demonstrate below.

To simplify the subsequent analysis, we note that M[Imo]-1 is a multiplicative factor that is independent of the position x⃗. Therefore, it does not affect the scaling law for the quality of the ghost image and for the present can be safely taken as a constant. We define Cm = ⟨Imo⟩ and Bn = ⟨Inr(x⃗)⟩, and we would like to compare the qualities of the ghost images calculated by

gmn(x)=1CmBnM[IomIrn(x)],
(15)

and by

gmn(x)=1CmM[IomIrn(x)]M[Irn(x)].
(16)

It should be noted that gmn(x⃗) is equivalent to the unnormalized ghost image Gmn(x⃗) = M[ImoInr(x⃗)]. In particular, g 1,1(x⃗) ∝= G(x⃗) with G(x⃗) = G 1,1(x⃗) given in Eq. (1). Therefore, according to Ref. [22

22. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-Order Thermal Ghost Imaging,” Opt. Lett. 34, 3343–3345 (2009). [CrossRef] [PubMed]

], there is an optimal contrast-to-noise ratio for the high-order ghost image gmn(x⃗) that scales as CNR(g opt) ∝ √N/T, with the optimal exponents m ~ √T and n = 1 or 2.

Before presenting the analysis for gmn′ (x⃗), we show the results of numerical simulations using Eqs. (15) and (16). In the simulations, we have taken the illuminating beam to be spatially incoherent, i.e., I(x⃗) = ∣E(x⃗)∣2 with ⟨E *(x⃗)E(x⃗)⟩ = μδ(x⃗ - x⃗′), where the electric field E(x⃗) follows circular complex Gaussian statistics. The object mask used is depicted in Fig. 2(a), and the number of transmitting pixels of the object is T = 410. The number of samplings used is N = 150000. For the value of T used in the simulation, the optimal exponents for the correlation gmn(x) are m ≈ 20 and n = 2, and CNR(g 20,2) ≈ 20CNR(g 1,1).

From Figs. 2(c) and 2(d), it is seen that the conventional ghost image g 1,1 (x⃗) is barely visible, while the optimal high-order ghost image g 20,2(x⃗) has very good quality. It is noted that the averaged speckle pattern M[Ir(x⃗)] (Fig. 2(b)) is not uniform. More importantly, it is correlated with the ghost image as can be seen by comparing Figs. 2(b) and 2(c). On the contrary, when Eq. (16) is used to calculate the ghost image, Figs. 2(e) and 2(f) show that the quality of the normalized conventional ghost image is the same as that of the high-order ghost image. It is also noted that while the visibility of the high-order ghost image is much larger than that of the conventional ghost image (see the scale bars in Fig. 2), the signal fluctuations of the high-order ghost image are also larger than that of the conventional ghost image. This is shown explicitly in Fig. 3 in which the probability density functions of the ghost image signals inside and outside the transmitting regions are plotted.

Fig. 2. Numerical simulations. (a) The object mask (50 × 50 pixels). (b) The averaged speckle pattern M[Ir(x⃗)]. (c) The conventional ghost image g 1,1 (x⃗). (d) The optimal high-order ghost image g opt(x⃗) = g 20,2(x⃗). (e) The normalized conventional ghost image g1,1(x⃗). (f) The normalized high-order ghost image g20,2(x⃗). The number of transmitting pixels is T = 410 and the number of samplings is N = 150000. The latter is chosen to be large to make the ghost image in (c) visible.

To understand the properties of gmn′(xmn), it should be realized that, for large N, M[IomInr(xmn)] and M[Inr(xmn)] behave as normal random variables with means

Amn(x)=IomIrn(x),Bn=Irn(x),
(17)

and variances

αmn2(x)=Δ2[IomIrn(x)]N,βn2=Δ2[Irn(x)]N,
(18)

by virtue of the central limit theorem. This implies that gmn′ (x⃗) is a ratio of two normal random variables, and the form of the probability density function of gmn′(x⃗) is given in Ref. [30

30. D. V. Hinkley, “On the Ratio of Two Correlated Normal Random Variables,” Biometrika 56, 635–639 (1969). [CrossRef]

]. It is noted that in general the moments of the ratio of two normal random variables do not exist, especially when the denominator can vanish [31

31. A. Cedilnik, K. Košmelj, and A. Blejec, “Ratio of Two Random Variables: A Note on the Existence of its Moments,” Metodološki zvezki 3, 1–7 (2006).

]. However, in our case, M[Inr(x⃗)] ≠ 0. In particular, when Bn ≳ 3βn it has been shown that the function

tmn(x)=BnCmgmn(x)Amn(x)βn2Cm2gmn2(x)2ρmn(x)Cmgmn(x)+αmn2(x)
(19)

is normally distributed with zero mean and unity variance [32

32. R. C. Geary, “The Frequency Distribution of the Quotient of Two Normal Variates,” J. Roy. Statistical Society 93, 442–446 (1930). [CrossRef]

], i.e.,

P(tmn(x))=12πexp[tmn2(x)2].
(20)
Fig. 3. The probability density functions (pdf) of gmn and gmn′ inside and outside the transmitting regions of the object with Pi(gmn) ≡P(gmn(x⃗i)), i = in or out. (a) The pdf for the conventional ghost image Pi(g 1,1). (b) The pdf for the optimal high-order ghost image Pi(g 20,2). (c) The pdf for the normalized conventional ghost image Pi(g1,1). (d) The pdf for the normalized high-order ghost image Pi(g20,2). The number of transmitting pixels is T =410 and the number of samplings is N = 150000. These give ⟨g 1,1 (x⃗ in)⟩ = ⟨g 1,1′ (x⃗ in)⟩ = 1.0024 and ⟨g 20,2(x⃗)⟩ = ⟨g20,2(x⃗ in)) = 1.0998. The data points are accumulated by repeating the simulations for 500 times. The solid curves (overlapping with the numerical data) are the theoretical results using Eq. (20).

Here ρmn(x⃗) is the covariance of M[ImoInr(x⃗)] and M[Inr(x⃗)]. Moreover, if Bnβn, the probability density function of gmn′(x⃗) tends to be a normal distribution.

Using Eq. (7), we find that

Amn(x)={Γ(T+m+n)n!Γ(T+n)μm+n,forx=xin,Γ(T+m)n!Γ(T)μm+n,forx=xout,
(21)
Bn=n!μn,Cm=Γ(T+m)Γ(T)μm,
(22)
αmn2(x)=A2m2n(x)Amn2(x)N,βn2=B2nBn2N,
(23)

and

ρmn(x)=Am2n(x)Amn(x)BnN.
(24)

The theoretical probability density functions of the ghost images are plotted in Fig. 3(b) using Eq. (20) and an excellent agreement between the theory and the simulation results is seen.

An approximation to Eq. (20) can be made by putting gmn′ (x⃗) = ⟨gmn′ (x⃗)⟩ in the denominator of tmn (x⃗). The probability density function for gmn′ (x⃗) is then found to be

P(gmn(x))12πΔmn(x)exp[[gmn(x)gmn(x)]22Δmn2(x)].
(25)

The mean of gmn′(x⃗) is given by

gmn(x)=Amn(x)CmBn(x)={Γ(T)Γ(T+m+n)Γ(T+m)Γ(T+n),forx=xin,1,forx=xout,
(26)

and the variance of gmn′ (x⃗) is given by

Δmn2(xin)=(2n)!N(n!)2[Γ(T+2m+2n)Γ(T)2Γ(T+m)2Γ(T+2n)+Γ(T+m+n)2Γ(T)2Γ(T+m)2Γ(T+n)2
2Γ(T+m+n)Γ(T+m+2n)Γ(T)2Γ(T+m)2Γ(T+n)Γ(T+2n)],
(27)

and

Δmn2(xout)=(2n)!N(n!)2[Γ(T+2m)Γ(T)Γ(T+m)21].
(28)

Using these results, we calculate the CNR as

CNR(gmn)=gmn(xin)gmn(xout)12[Δmn2(xin)+Δmn2(xout)].
(29)

For example, CNR(g1,1)=N/(2T+3)andCNR(g1,2)=N/(3T/2+6).

When Tm and n, it can be shown that

g'mn(xin)~1+mnT,gmn(xout)=1,
(30)

and

Δmn2(xin)Δmn2(xout)~m2(2n)!NT(n!)2.
(31)

Then the CNR with Tm, n is found to be

CNR(gmn)~NT[n(n!)(2n)!].
(32)

Therefore, CNR(gmn′) scales as √N/T, which is the same as the scaling law for CNR(g opt) and CNR(G′). Figure 4 shows the theoretically predicted CNR(gmn′) calculated using Eq. (29). Numerical results using N = 20000 are also shown and agree with the analytical results. The deviation of the numerical data from the theory for g 5,5′ and g 20,2′ is due to the finite number of samplings used in the simulations and is found to improve when a larger N is used.

From the large T limit of Δmn(x⃗), we see that the noise of the high-order ghost image increases linearly with the object signal order m and exponentially with the reference signal order n. Moreover, the CNR is independent of m. Finally, Eq. (32) demonstrates that the optimal value for n is 1 or 2.

Fig. 4. Plots of the contrast-to-noise ratio CNR(gmn′) (normalized by √N) against the number of transmitting pixels T for various values of m and n. The theoretical curves are plotted using Eq. (29). The data points are from simulations using N = 20000. Also shown are CNR(G) from Eq. (8), CNR(G′) from Eq. (10) and CNR(g opt) from Ref. [22]. We note that under all circumstances the function G′ produces the best image, with procedures g1,1 and g2,2 performing nearly as well.

Since the CNR is not quite dependent on m, one can take a very large value of m in order to achieve high visibility, which is given by [22

22. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-Order Thermal Ghost Imaging,” Opt. Lett. 34, 3343–3345 (2009). [CrossRef] [PubMed]

]

Vgmn(xin)gmn(xout)gmn(xin)+gmn(xout)tanh(mn2T).
(33)

The results explain the experimental observations [23

23. X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, and L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv:0902.3713v3 [quant-ph].

] that a large m and a small n are preferable in high-order thermal ghost imaging.

4. Effects of shot noise

In Sections 2 and 3, the contrast-to-noise ratios are calculated under the assumption that the light beams are sufficiently intense so that the noise properties of the signals are dominated by the statistics of the speckle fields. Depending on the illumination power on the photodetectors, other noise sources, such as shot noise and dark current, contribute to a certain extent to the signal statistics. The intent of the present section is to study the effect of detection noise on the quality of the ghost image and to determine the conditions under which these additional noise sources are negligible so that the results of Sections 2 and 3 are valid. For definiteness, we assume that shot noise is the only noise source that is present, and that sources of technical noise have been rendered negligibly small.

Based on the assumptions made to the bucket and reference detectors in Section 2, the pho-tocurrents output from the detectors during the reception of an optical pulse of known intensity Ia with a = o,r are

ia=enΔIaμ,
(34)

where e is the electron charge and n Δ is the average number of photoelectrons ejected by a detector element, which is defined by

nΔ=ημΔτhv,
(35)

where η is the quantum efficiency of the detector, Δ is the area of the pixel, τ is the integration time of detection process, ν is the frequency of the light, and h is the Planck constant. We have assumed that the quantum efficiencies of the bucket and reference detectors are the same for convenience.

According to the semiclassical theory of photodetection, the photocurrent is governed by Poisson statistics. The nth moment of ia is given by

ianshotnoise=enϕn(nΔμIa)en(nΔμ)n{Ia}n,
(36)

where ϕn(z) are the exponential polynomials [33

33. K. N. Boyadzhiev, “Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals,” Abstract and Applied Analysis 2009, 168672 (2009). [CrossRef]

, 34

34. S. Roman, The Umbral Calculus (Academic Press, New York, 1984), pp. 63–67 and 82–87.

]. The ensemble averages of the object and reference signals over the intensity of the speckle fields in the calculations of the mean and variance of the ghost image are thus replaced by

IomIrn(x){Io}m{Ir(x)}n
=(μnΔ)m+nϕm(nΔμIo)ϕn(nΔμIr(x)),
(37)

with m,n ≥ 0. Equation (37) can also be written as

(μnΔ)m+nϕm(nΔμIo)ϕn(nΔμIr(x))=(μnΔ)m+ni=0mj=0nSmiSnjIoiIrj(x)μi+jnΔi+j,
(38)

where S(n,k) is the Stirling number of the second kind. It is then seen that the high-order correlations of Io and Ir(x⃗) also involve the contributions from lower order correlations. Note that, for the case of single beam ghost imaging [5

5. Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009). [CrossRef]

], the Poisson shot noise appears only in the bucket detector. In this case, the cross-correlation in the calculation of the ghost image is ⟨{Io}mInr(x⃗)⟩ instead of ⟨{Io}m{Ir(x⃗)}n⟩.

As an illustration, we calculate the CNR with shot noise for the conventional ghost image with background subtraction [cf Eq. (9)] in the large N limit. The mean of the ghost image is given by

G(sn)'(x)=(μnΔ)2ϕ1(nΔμIonΔμIo)ϕ1(nΔμIr(x)nΔμIr(x))
(39)

and the variance is

Δ2G(sn)(x)=1N{(μnΔ)4ϕ2(nΔμIonΔμIo)ϕ2(nΔμIr(x)nΔμIr(x))
[G(sn)(x)]2}.
(40)

Using ϕ 1(z) = z, ϕ 2(z) = z 2 +z and Eq. (21), the signal-to-noise ratio is found to be

G(sn)(x)ΔG(sn)(x)=NO(x)[T+7O(x)]+O(x)[4nΔ1+nΔ2],
(41)

where O(x⃗) is the binary object function. The corresponding CNR is

CNR(G(sn))=N(T+7/2)+(4nΔ1+nΔ2)/2.
(42)

Equation (41) gives the correct dependence on the intensity μ when compared to the full quantum approach [28

28. B. I. Erkmen and J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009). [CrossRef]

]. The form of CNR(G (sn)′) is the same as that in Eq. (13) except for an additional second term in the denominator. When T ≫ 1 or n Δ > 10, the contribution from shot noise to the CNR becomes insignificant. For high-order thermal ghost imaging, the reference signal order n is kept to be small. It is checked numerically that the difference between ⟨{Io}m{Ir(x⃗)}n⟩ and ⟨ImoInr(x⃗)⟩ is less than 5% for m ≤ 50, n ≤ 4 and T ≥ 100 with n Δ 200.

In our laboratory, we perform pseudo-thermal ghost imaging experiment using a pulsed laser with power ~ 100 mW, pulse duration ~ 100 fs and repetition rate ~ 15 Hz. After scattered by the diffuser (ground glass plate), the energy of the light registered by the bucket detector is measured to be about 1 nJ. The object transmission area is about 5 mm2 and the pixel size of the CCD camera (reference detector) is about 10 μm2. Therefore, the energy of light on a CCD pixel is (assuming the CCD has the same quantum efficiency as the bucket detector) about 1nJ/(5mm2) × (10μm2). The wavelength of the laser is 780 nm. Therefore, the number of photoelectrons per pixel per pulse is

nΔ=ημΔτhc/λ=109/(5×106)×(10×1012)(6.63×1034)×(3×108)/(780×109)~7800
(43)

With more than a thousand photons per pixel, the shot noise is thus negligible.

5. Conclusions

We have analyzed the performances of several common methods of processing the data of thermal ghost imaging. It is found that the contrast-to-noise ratio (CNR) of conventional thermal ghost imaging using simply the averages of the product of the object and reference signal (G(x⃗)) scales as √N/T, whereas that utilizing the correlation with background subtraction (G′(x⃗)) or background normalization g 1,1′(x⃗) scales as √N/T, in which N is the number of measurements and T is the number of transmitting pixels of the object. We have also shown analytically that the CNR of the normalized high-order ghost image gmn′(x⃗) scales as √N/T, which is independent of the exponents of the high-order correlation.

We note that, by choosing a large object signal order m and a small reference signal order n in the high-order ghost imaging, both the visibility and the contrast-to-noise ratio can be increased substantially at the same time. The results are consistent with the experimental observations reported in Ref. [23

23. X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, and L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv:0902.3713v3 [quant-ph].

]. Finally, a semiclassical photodetection analysis is performed to justify the approach and the applicable regime of the classical method used in the calculations in this paper.

Acknowledgments

This research is supported by Rochester Optical Manufacturing Company and the US Army Research Office under a Multidisciplinary University Research Initiative (MURI) grant.

References and links

1.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995). [CrossRef] [PubMed]

2.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost Imaging with Thermal Light: Comparing Entanglement and Classical Correlation,” Phys. Rev. Lett. 93, 093602 (2004); [CrossRef] [PubMed]

3.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004). [CrossRef]

4.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, “Two-Photon Imaging with Thermal Light,” Phys. Rev. Lett. 94, 063601 (2005). [CrossRef] [PubMed]

5.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009). [CrossRef]

6.

G. Scarcelli, V. Berardi, and Y. Shih “Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations?” Phys. Rev. Lett. 96, 063602 (2006). [CrossRef] [PubMed]

7.

A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations?” Phys. Rev. Lett. 98, 039301 (2007). [CrossRef] [PubMed]

8.

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

9.

L.-G. Wang, S. Qamar, S.-Y. Zhu, and M. S. Zubairy, “Hanbury Brown-Twiss effect and thermal light ghost imaging: A unified approach,” Phys. Rev. A 79, 033835 (2009). [CrossRef]

10.

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

11.

R. Meyers, K. S. Deacon, and Y. H. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801(R) (2008). [CrossRef]

12.

J. Cheng and S. Han, “Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction,” Phys. Rev. Lett. 92, 093903 (2004). [CrossRef] [PubMed]

13.

G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006). [CrossRef]

14.

L. Basano and P. Ottonello, “Experiment in lensless ghost imaging with thermal light,” Appl. Phys. Lett. 89, 091109 (2006) [CrossRef]

15.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light,” Phys. Rev. Lett. 94, 183602 (2005). [CrossRef] [PubMed]

16.

L. Basano and P. Ottonello, “Use of an intensity threshold to improve the visibility of ghost images produced by incoherent light,” Appl. Opt. 46, 6291–6296 (2007). [CrossRef] [PubMed]

17.

Y. Bai and S. Han, “Ghost imaging with thermal light by third-order correlation,” Phys. Rev. A. 76,043828 (2007). [CrossRef]

18.

L.-H. Ou and L.-M. Kuang, “Ghost imaging with third-order correlated thermal light,” J. Phys. B: At. Mol. Opt. Phys. 40, 1833–1844 (2007). [CrossRef]

19.

D.-Z. Cao, J. Xiong, S.-H. Zhang, L.-F. Lin, L. Gao, and K. Wang, “Enhancing visibility and resolution in Nth-order intensity correlation of thermal light,” Appl. Phys. Lett. 92, 201102 (2008). [CrossRef]

20.

I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and A. N. Penin, “High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light,” Phys. Rev. A 77, 053801 (2008). [CrossRef]

21.

Q. Liu, X.-H. Chen, K.-H. Luo, W. Wu, and L.-A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A 79, 053844 (2009). [CrossRef]

22.

K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-Order Thermal Ghost Imaging,” Opt. Lett. 34, 3343–3345 (2009). [CrossRef] [PubMed]

23.

X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, and L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv:0902.3713v3 [quant-ph].

24.

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

25.

L. Basano and P. Ottonello, “A conceptual experiment on single-beam coincidence detection with pseudothermal light,” Opt. Express 15, 12386–12394 (2007). [CrossRef] [PubMed]

26.

D. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005). [CrossRef]

27.

Y. Cai and F. Wang, “Lensless imaging with partially coherent light,” Opt. Lett. 32, 205–207 (2007). [CrossRef] [PubMed]

28.

B. I. Erkmen and J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009). [CrossRef]

29.

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). [CrossRef] [PubMed]

30.

D. V. Hinkley, “On the Ratio of Two Correlated Normal Random Variables,” Biometrika 56, 635–639 (1969). [CrossRef]

31.

A. Cedilnik, K. Košmelj, and A. Blejec, “Ratio of Two Random Variables: A Note on the Existence of its Moments,” Metodološki zvezki 3, 1–7 (2006).

32.

R. C. Geary, “The Frequency Distribution of the Quotient of Two Normal Variates,” J. Roy. Statistical Society 93, 442–446 (1930). [CrossRef]

33.

K. N. Boyadzhiev, “Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals,” Abstract and Applied Analysis 2009, 168672 (2009). [CrossRef]

34.

S. Roman, The Umbral Calculus (Academic Press, New York, 1984), pp. 63–67 and 82–87.

OCIS Codes
(030.4280) Coherence and statistical optics : Noise in imaging systems
(110.1650) Imaging systems : Coherence imaging

ToC Category:
Imaging Systems

History
Original Manuscript: January 4, 2010
Revised Manuscript: January 26, 2010
Manuscript Accepted: January 27, 2010
Published: March 3, 2010

Citation
Kam Wai C. Chan, Malcolm N. O'Sullivan, and Robert W. Boyd, "Optimization of thermal ghost imaging: high-order correlations vs. background subtraction," Opt. Express 18, 5562-5573 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-5562


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References

  1. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995). [CrossRef] [PubMed]
  2. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost Imaging with Thermal Light: Comparing Entanglement and Classical Correlation," Phys. Rev. Lett. 93, 093602 (2004); [CrossRef] [PubMed]
  3. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Correlated imaging, quantum and classical," Phys. Rev. A 70, 013802 (2004). [CrossRef]
  4. A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, "Two-Photon Imaging with Thermal Light," Phys. Rev. Lett. 94, 063601 (2005). [CrossRef] [PubMed]
  5. Y. Bromberg, O. Katz, and Y. Silberberg, "Ghost imaging with a single detector," Phys. Rev. A 79, 053840 (2009). [CrossRef]
  6. G. Scarcelli, V. Berardi, and Y. Shih "Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations?" Phys. Rev. Lett. 96, 063602 (2006). [CrossRef] [PubMed]
  7. A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, "Comment on Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations?" Phys. Rev. Lett. 98, 039301 (2007). [CrossRef] [PubMed]
  8. B. I. Erkmen and J. H. Shapiro, "Unified theory of ghost imaging with Gaussian-state light," Phys. Rev. A 77, 043809 (2008). [CrossRef]
  9. L.-G. Wang, S. Qamar, S.-Y. Zhu, and M. S. Zubairy, "Hanbury Brown-Twiss effect and thermal light ghost imaging: A unified approach," Phys. Rev. A 79, 033835 (2009). [CrossRef]
  10. J. H. Shapiro, "Computational ghost imaging," Phys. Rev. A 78, 061802 (2008). [CrossRef]
  11. R. Meyers, K. S. Deacon, and Y. H. Shih, "Ghost-imaging experiment by measuring reflected photons," Phys. Rev. A 77, 041801 (2008). [CrossRef]
  12. J. Cheng and S. Han, "Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction," Phys. Rev. Lett. 92, 093903 (2004). [CrossRef] [PubMed]
  13. G. Scarcelli, V. Berardi, and Y. Shih, "Phase-conjugate mirror via two-photon thermal light imaging," Appl. Phys. Lett. 88, 061106 (2006). [CrossRef]
  14. L. Basano and P. Ottonello, "Experiment in lensless ghost imaging with thermal light," Appl. Phys. Lett. 89, 091109 (2006) [CrossRef]
  15. F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light," Phys. Rev. Lett. 94, 183602 (2005). [CrossRef] [PubMed]
  16. L. Basano and P. Ottonello, "Use of an intensity threshold to improve the visibility of ghost images produced by incoherent light," Appl. Opt. 46, 6291-6296 (2007). [CrossRef] [PubMed]
  17. Y. Bai and S. Han, "Ghost imaging with thermal light by third-order correlation," Phys. Rev. A. 76, 043828 (2007). [CrossRef]
  18. L.-H. Ou and L.-M. Kuang, "Ghost imaging with third-order correlated thermal light," J. Phys. B: At. Mol. Opt. Phys. 40, 1833-1844 (2007). [CrossRef]
  19. D.-Z. Cao, J. Xiong, S.-H. Zhang, L.-F. Lin, L. Gao, and K. Wang, "Enhancing visibility and resolution in Nth order intensity correlation of thermal light," Appl. Phys. Lett. 92, 201102 (2008). [CrossRef]
  20. I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and A. N. Penin, "High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light," Phys. Rev. A 77, 053801 (2008). [CrossRef]
  21. Q. Liu, X.-H. Chen, K.-H. Luo, W. Wu, and L.-A. Wu, "Role of multiphoton bunching in high-order ghost imaging with thermal light sources," Phys. Rev. A 79, 053844 (2009). [CrossRef]
  22. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, "High-Order Thermal Ghost Imaging," Opt. Lett. 34, 3343-3345 (2009). [CrossRef] [PubMed]
  23. X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, L.-A. Wu, "Arbitrary-order lensless ghost imaging with thermal light," arXiv:0902.3713v3 [quant-ph].
  24. O. Katz, Y. Bromberg, and Y. Silberberg, "Compressive ghost imaging," Appl. Phys. Lett. 95, 131110 (2009). [CrossRef]
  25. L. Basano and P. Ottonello, "A conceptual experiment on single-beam coincidence detection with pseudothermal light," Opt. Express 15, 12386-12394 (2007). [CrossRef] [PubMed]
  26. D. Cao, J. Xiong, and K. Wang, "Geometrical optics in correlated imaging systems," Phys. Rev. A 71, 013801 (2005). [CrossRef]
  27. Y. Cai and F. Wang, "Lensless imaging with partially coherent light," Opt. Lett. 32, 205-207 (2007). [CrossRef] [PubMed]
  28. B. I. Erkmen and J. H. Shapiro, "Signal-to-noise ratio of Gaussian-state ghost imaging," Phys. Rev. A 79, 023833 (2009). [CrossRef]
  29. 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). [CrossRef] [PubMed]
  30. D. V. Hinkley, "On the Ratio of Two Correlated Normal Random Variables," Biometrika 56, 635-639 (1969). [CrossRef]
  31. A. Cedilnik, K. Košmelj, and A. Blejec, "Ratio of Two Random Variables: A Note on the Existence of its Moments," Metodološki zvezki 3, 1-7 (2006).
  32. R. C. Geary, "The Frequency Distribution of the Quotient of Two Normal Variates," J. Roy. Statistical Society 93, 442-446 (1930). [CrossRef]
  33. K. N. Boyadzhiev, "Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals," Abstract and Applied Analysis 2009, 168672 (2009). [CrossRef]
  34. S. Roman, The Umbral Calculus (Academic Press, New York, 1984), pp. 63-67 and 82-87.

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