## Optimization of thermal ghost imaging: high-order correlations vs. background subtraction

Optics Express, Vol. 18, Issue 6, pp. 5562-5573 (2010)

http://dx.doi.org/10.1364/OE.18.005562

Acrobat PDF (473 KB)

### 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

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]

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]

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]

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. 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]

24. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. **95**, 131110 (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]

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

## 2. Conventional thermal ghost imaging

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. 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]

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]

*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 ⟨

*I*(

^{n}*x⃗*)⟩ =

*n*!⟨

*I*(

*x⃗*)⟩

*=*

^{n}*n*!

*μ*and ⟨

^{n}*I*(

*x⃗*)

*I*(

*x⃗*′)⟩ = ⟨

*I*(

*x⃗*)⟩⟨

*I*(

*x⃗*′)⟩ =

*μ*

^{2}for

*x⃗*≠

*x⃗*′.

*I*and the reference signal

_{o}*I*(

_{r}*x⃗*):

*X*for the

*N*measurements {

*X*

^{(s)}},

*s*= 1,⋯ ,

*N*. Here

*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).

^{2}

*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 Δ

^{2}

*G*(

*x⃗*

_{in}) and Δ

^{2}

*G*(

*x⃗*

_{out}) are generally not the same. It follows from Eq. (1) that ⟨

*G*(

*x⃗*)⟩ = ⟨

*I*(

_{o}I_{r}*x⃗*)⟩ and

**34**, 3343–3345 (2009). [CrossRef] [PubMed]

*x⃗*) is the gamma function, the CNR of

*G*(

*x⃗*) is found to be [22

**34**, 3343–3345 (2009). [CrossRef] [PubMed]

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]

*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.

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]

**96**, 063602 (2006). [CrossRef] [PubMed]

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]

*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]

*G*′(

*x⃗*) performs much better than

*G*(

*x⃗*) for the relevant limit

*T*≫ 1.

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]

*M*[

*I*(

_{r}*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

*g*′

_{1,1}(

*x⃗*) performs approximately as well as

*G*′(

*x⃗*).

*M*[

*I*]

_{o}*M*[

*I*(

_{r}*x⃗*)] in Eq. (9) as a constant in the limit of large

*N*, that is, taking

*M*[

*I*] → ⟨

_{o}*I*⟩ and

_{o}*M*[

*I*(

_{r}*x⃗*)] → ⟨

*I*(

_{r}*x⃗*)⟩. Such a procedure leads to the the prediction that the contrast-to-noise ratio scales as

## 3. High-order thermal ghost imaging

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]

**34**, 3343–3345 (2009). [CrossRef] [PubMed]

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]

*M*[

*I*(

^{m}_{o}I^{n}_{r}*x⃗*)],

*M*[

*I*] and

^{m}_{o}*M*[

*I*(

^{n}_{r}*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.

*M*[

*I*]

^{m}_{o}^{-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

*C*= ⟨

_{m}*I*⟩ and

^{m}_{o}*B*= ⟨

_{n}*I*(

^{n}_{r}*x⃗*)⟩, and we would like to compare the qualities of the ghost images calculated by

*g*(

_{mn}*x⃗*) is equivalent to the unnormalized ghost image

*G*(

_{mn}*x⃗*) =

*M*[

*I*(

^{m}_{o}I^{n}_{r}*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

**34**, 3343–3345 (2009). [CrossRef] [PubMed]

*g*(

_{mn}*x⃗*) that scales as CNR(

*g*

_{opt}) ∝ √

*N*/

*T*, with the optimal exponents

*m*~ √

*T*and

*n*= 1 or 2.

*g*′ (

_{mn}*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

*g*(

_{mn}*x*) are

*m*≈ 20 and

*n*= 2, and CNR(

*g*

_{20,2}) ≈ 20CNR(

*g*

_{1,1}).

*g*′(

_{mn}*xmn*), it should be realized that, for large

*N*,

*M*[

*I*(

_{o}^{m}I^{n}_{r}*xmn*)] and

*M*[

*I*(

^{n}_{r}*xmn*)] behave as normal random variables with means

*g*′ (

_{mn}*x⃗*) is a ratio of two normal random variables, and the form of the probability density function of

*g*′(

_{mn}*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]

*M*[

*I*(

^{n}_{r}*x⃗*)] ≠ 0. In particular, when

*B*≳ 3

_{n}*β*it has been shown that the function

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

*ρ*(

_{mn}*x⃗*) is the covariance of

*M*[

*I*(

^{m}_{o}I^{n}_{r}*x⃗*)] and

*M*[

*I*(

^{n}_{r}*x⃗*)]. Moreover, if

*B*≫

_{n}*β*, the probability density function of

_{n}*g*′(

_{mn}*x⃗*) tends to be a normal distribution.

*g*′ (

_{mn}*x⃗*) = ⟨

*g*′ (

_{mn}*x⃗*)⟩ in the denominator of

*t*(

_{mn}*x⃗*). The probability density function for

*g*′ (

_{mn}*x⃗*) is then found to be

*g*′(

_{mn}*x⃗*) is given by

*g*′ (

_{mn}*x⃗*) is given by

*T*≫

*m*and

*n*, it can be shown that

*T*≫

*m, n*is found to be

*g*′) scales as √

_{mn}*N*/

*T*, which is the same as the scaling law for CNR(

*g*

_{opt}) and CNR(

*G*′). Figure 4 shows the theoretically predicted CNR(

*g*′) calculated using Eq. (29). Numerical results using

_{mn}*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.

*T*limit of Δ

*(*

_{mn}*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.

*m*, one can take a very large value of

*m*in order to achieve high visibility, which is given by [22

**34**, 3343–3345 (2009). [CrossRef] [PubMed]

*m*and a small

*n*are preferable in high-order thermal ghost imaging.

## 4. Effects of shot noise

*known*intensity

*I*with

_{a}*a*=

*o,r*are

*e*is the electron charge and

*n*

_{Δ}is the average number of photoelectrons ejected by a detector element, which is defined by

*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.

*n*moment of

^{th}*i*is given by

_{a}*ϕ*(

_{n}*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]

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

*S*(

*n,k*) is the Stirling number of the second kind. It is then seen that the high-order correlations of

*I*and

_{o}*I*(

_{r}*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]

*I*}

_{o}*(*

_{m}I^{n}_{r}*x⃗*)⟩ instead of ⟨{

*I*}

_{o}*{*

_{m}*I*(

_{r}*x⃗*)}

*n*⟩.

*N*limit. The mean of the ghost image is given by

*O*(

*x⃗*) is the binary object function. The corresponding CNR is

*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]

*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 ⟨{

*I*}

_{o}*{*

_{m}*I*(

_{r}*x⃗*)}

*⟩ and ⟨*

_{n}*I*(

^{m}_{o}I^{n}_{r}*x⃗*)⟩ is less than 5% for

*m*≤ 50,

*n*≤ 4 and

*T*≥ 100 with

*n*

_{Δ}200.

^{2}and the pixel size of the CCD camera (reference detector) is about 10

*μ*m

^{2}. Therefore, the energy of light on a CCD pixel is (assuming the CCD has the same quantum efficiency as the bucket detector) about 1nJ/(5mm

^{2}) × (10

*μ*m

^{2}). The wavelength of the laser is 780 nm. Therefore, the number of photoelectrons per pixel per pulse is

## 5. Conclusions

*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

*g*′(

_{mn}*x⃗*) scales as √

*N/T*, which is independent of the exponents of the high-order correlation.

*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]. 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

## 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 |

2. | A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost Imaging with Thermal Light: Comparing Entanglement and Classical Correlation,” Phys. Rev. Lett. |

3. | A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A |

4. | A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, “Two-Photon Imaging with Thermal Light,” Phys. Rev. Lett. |

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

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. |

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. |

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

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 |

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

11. | R. Meyers, K. S. Deacon, and Y. H. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A |

12. | J. Cheng and S. Han, “Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction,” Phys. Rev. Lett. |

13. | G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. |

14. | L. Basano and P. Ottonello, “Experiment in lensless ghost imaging with thermal light,” Appl. Phys. Lett. |

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. |

16. | L. Basano and P. Ottonello, “Use of an intensity threshold to improve the visibility of ghost images produced by incoherent light,” Appl. Opt. |

17. | Y. Bai and S. Han, “Ghost imaging with thermal light by third-order correlation,” Phys. Rev. A. |

18. | L.-H. Ou and L.-M. Kuang, “Ghost imaging with third-order correlated thermal light,” J. Phys. B: At. Mol. Opt. Phys. |

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. |

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 |

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 |

22. | K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-Order Thermal Ghost Imaging,” Opt. Lett. |

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. |

25. | L. Basano and P. Ottonello, “A conceptual experiment on single-beam coincidence detection with pseudothermal light,” Opt. Express |

26. | D. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A |

27. | Y. Cai and F. Wang, “Lensless imaging with partially coherent light,” Opt. Lett. |

28. | B. I. Erkmen and J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A |

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

30. | D. V. Hinkley, “On the Ratio of Two Correlated Normal Random Variables,” Biometrika |

31. | A. Cedilnik, K. Košmelj, and A. Blejec, “Ratio of Two Random Variables: A Note on the Existence of its Moments,” Metodološki zvezki |

32. | R. C. Geary, “The Frequency Distribution of the Quotient of Two Normal Variates,” J. Roy. Statistical Society |

33. | K. N. Boyadzhiev, “Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals,” Abstract and Applied Analysis |

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

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- D. Cao, J. Xiong, and K. Wang, "Geometrical optics in correlated imaging systems," Phys. Rev. A 71, 013801 (2005). [CrossRef]
- Y. Cai and F. Wang, "Lensless imaging with partially coherent light," Opt. Lett. 32, 205-207 (2007). [CrossRef] [PubMed]
- B. I. Erkmen and J. H. Shapiro, "Signal-to-noise ratio of Gaussian-state ghost imaging," Phys. Rev. A 79, 023833 (2009). [CrossRef]
- 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]
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- S. Roman, The Umbral Calculus (Academic Press, New York, 1984), pp. 63-67 and 82-87.

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