## Normalized ghost imaging |

Optics Express, Vol. 20, Issue 15, pp. 16892-16901 (2012)

http://dx.doi.org/10.1364/OE.20.016892

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### Abstract

We present an experimental comparison between different iterative ghost imaging algorithms. Our experimental setup utilizes a spatial light modulator for generating known random light fields to illuminate a partially-transmissive object. We adapt the weighting factor used in the traditional ghost imaging algorithm to account for changes in the efficiency of the generated light field. We show that our normalized weighting algorithm can match the performance of differential ghost imaging.

© 2012 OSA

## 1. Introduction

9. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. **104**, 253603 (2010). [CrossRef] [PubMed]

## 2. Experimental setup

*λ*= 632.8nm through a polarizing beam splitter and a half-wave plate, before illuminating the SLM window. The speckle field is generated by modulation of the SLM and the returning light field is then magnified by a simple telescope system consisting of 150mm and 450mm biconvex lenses. The object is located at the focus plane of the 450mm lens, which is also the image plane of the SLM window. A 50 : 50 beam splitter is placed before the object in order to split the speckle field into two beams; the object beam (

*I*(

*x*)) and the reference beam (

_{S}*I*(

*x*)). The object beam illuminates the object and is then collected by a bucket detector, thus providing an computational GI setup. The additional reference beam for monitoring the light differentiates our system from previous experimental computational GI configurations. Since we are generating a computer hologram that is then sent to the SLM to create the speckle field, we can therefore predict the light field at the reference arm, negating the demand for a CCD camera, and requiring only a second bucket detector. It should be noted that for TGI based on our computational GI setup, only the object bucket detector is needed. The additional bucket detector in the reference arm is only required for NGI and DGI. Light intensities detected by the object and reference bucket detectors are indicated by

_{R}*S*and

*R*respectively, and the speckle field is described by

*I*(

*x*,

*y*). As we use a 50 : 50 beam splitter, it is understood that

*I*(

*x*,

*y*) = 2

*I*(

*x*,

_{S}*y*) = 2

_{S}*I*(

*x*,

_{R}*y*).

_{R}## 3. Iterative ghost imaging algorithms

_{2}, is reconstructed by correlating the speckle field intensity measured at

*S*and

*R*, then adding together each successive frame with a suitable weighting factor. The transmitted light power detected after the object can be expressed as where the laser area is

*A*and

_{l}*T*(

*x*,

_{S}*y*) is the (intensity) object transmission function, while the background reference is expressed as

_{S}### 3.1. Traditional Ghost Imaging

*O*(

*x*,

*y*) is retrieved from the correlation between

*S*and

*I*(

*x*,

*y*). We define for each iteration,

*i*, the contribution to the reconstruction to be [7

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

*M*iterations. We obtain the final reconstruction by averaging over all iterations such that

*O*(

*x*,

*y*) = 〈

*O*(

_{i}*x*,

*y*)〉. It is easy to understand the reconstruction as being derived from the weighted sum of the speckle field for each measurement. Therefore

*S*is the weight for the speckle field for each measurement. One drawback of using this algorithm is that the reconstruction is heavily weighted to the size of the signal

*S*and is thus susceptible to fluctuations in the generated light field. These fluctuations can arise from either changes to the laser power or the efficiency of the SLM in computational GI.

### 3.2. Differential ghost imaging

9. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. **104**, 253603 (2010). [CrossRef] [PubMed]

*et al*, utilizes a second bucket detector to extract a reference signal which is used in the reconstruction to weight the speckle field based on the average transmission signal relative to the average reference signal. Similarly, each contribution to the reconstruction can be expressed as Thus we obtain the final result by summing for all iterations. We observe the second term in brackets on the right hand side of Eq. (3) and Eq. (4) are both identical however the first term in brackets of Eq. (4) is now weighted according to the average value of

*S*, which is normalized to the average value of

*R*. As demonstrated in [9

9. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. **104**, 253603 (2010). [CrossRef] [PubMed]

### 3.3. Normalized ghost imaging

#### 3.3.1. Normalized ghost imaging with two detectors

*S*measured by the bucket detector results in a greater weight for that particular speckle field, therefore external noise sources can still affect the overall reconstruction. There exists another iterative algorithm which instead normalizes each individual measurement

*S*, as well as the running average, according to the reference signal

*R*, resulting in an arguably more intuitive approach for dealing with time varying noise sources. We call this approach normailized GI (NGI). The algorithm used to describe each contribution to the reconstruction in NGI is given by where we have assumed

#### 3.3.2. Normalized ghost imaging with a single detector

*S*for an arbitrary number of previous iterations. Calculating

*R*negates the requirement for an additional detector, whilst improving the performance of the reconstruction compared to TGI, thus single-detector NGI (SNGI) is identical to the TGI experimental setup, with only a modified algorithm.

### 3.4. Signal-to-noise ratio analysis

*O*(

*x*,

*y*) can be expressed as for which it can be shown that 〈

*O*(

*x*,

*y*)〉 = 0, thus the second term on the right hand side (RHS) of in Eq. (10) may be omitted. Again, under the assumptions of uniform illumination and perfect resolution, the noise of DGI can be expressed as where

*δS*and

*δR*are the zero-mean deviation of

*S*and

*R*, thus the noise of NGI is shown to be

*N*=

_{s}*A*is the number of speckles in the field. The SNR contribution for NGI is found to be identical to that of the DGI algorithm derived in [9

_{l}/A_{s}**104**, 253603 (2010). [CrossRef] [PubMed]

*et al*, the difference is always greater than 1 and dependent only upon the variation in the object transmission function.

## 4. Experiment results

*k⃗*of each simulated plane wave is Gaussian distributed. Figure 2 shows a typical example of the speckle patterns generated on the SLM and the exponentially distributed intensity for many patterns, implying that the speckle hologram has complex-Gaussian statistics, thereby a good approximation for real speckle fields [10]. A binary transmissive object, 5mm × 5mm in size, is located after a 3× magnification telescope in the image plane of the SLM. Since we know both the object and the random speckle field projected to the SLM, we are able to simulate the expected results for comparison with our experiment. Experimental and simulated reconstruction results after 10000 iterations are shown in Fig. 3. The simulated reconstruction is produced assuming no external noise sources. The partially transmissive object used is indicated in the bottom right of Fig. 3. It is clear that the DGI and NGI algorithms provide very similar results, as predicted from the theory, and both show improved background subtraction compared to TGI.

*R*. Thus we can negate the requirement for the reference detector and return the system to a true single element camera, which we call single-detector NGI (SNGI). The two major factors that dominate the value of

*R*are from the different speckle patterns displayed on the SLM and fluctuations of the incident laser power. We can computationally predict changes to the value of

*R*due to the speckle pattern, whereas fluctuations of the laser power can be simulated by using a rolling average for a particular series of

*S*measurements. The bottom row in Fig. 3 shows the experimental results for reconstructing the object using the SNGI algorithm. We observe similar results compared with DGI and NGI algorithms indicating an improved performance compared with the TGI algorithm for single element camera.

**104**, 253603 (2010). [CrossRef] [PubMed]

*T*= 1), we measured the SNR’s for the final object reconstruction obtained after 5000 random speckle iterations. The beam size used was 10 × 10mm and the speckle size at the plane of the object was found to be

*δ*∼ 90

_{s}*μ*m, providing around

*N*∼ 12500 speckles. The experimental results and theoretical predictions for the SNR’s of each iterative algorithm are shown in Fig. 4. Note that the

_{s}*y*-axis has been normalized to the number of iterations. We observe close quantitative agreement between the theory and the measurements. The results indicate that for low transmissive objects, all algorithms reconstruct with similar SNR, while for more transmissive objects the DGI and NGI algorithms become more efficient in comparison to TGI due to the differential nature of the reconstruction. Furthermore, we observe that when using a single detector, SNGI is a more efficient algorithm for reconstructing objects of all transmissions compared to TGI. We observe that for increasing transmissive objects SNGI becomes less efficient than NGI, for which the reason is the subject of ongoing research. Similar to [9

**104**, 253603 (2010). [CrossRef] [PubMed]

## 5. Normalization in matrix inverse algorithms

### 5.1. Introduction to matrix inverse algorithms and compressive sensing

*M*speckle patterns, each containing

*N*pixels can be represented by a

*M*×

*N*matrix. If the object is also represented as an

*N*element column vector, then the vector containing the measured signals is a

*M*element vector. This relationship is expressed as

*M*×

*N*matrix is square, such that its inverse can be calculated and the object vector determined. However when

*M*<

*N*and or

*N*is large, the system is ill-conditioned and calculating the inverse of the matrix is not straightforward. Problems of this type are wide spread in physics and techniques for solving them have been developed. Within our system the appeal is to reconstruct the image of

*N*pixels from

*M*measurements where

*M*<

*N*. That this is possible is based on the fact that natural images are sparse and the reconstruction can be obtained by solving a convex optimization problem [11], which is a generalization of a linear least squares problem. In contrast to iterative methods, compressive GI (CGI) needs to take all measurements, represented here, in some compressible basis (in this case a discrete cosine transform which has been applied to each row of the

*M*×

*N*matrix). Solving the convex optimization problem requires minimizing the

*ℓ*1 norm [12

12. D. L. Donoho and Y. Tsaig, “Fast solution of l_{1}-norm minimization problems when the solution may be sparse,” IEEE Trans. Inf. Theory **54**, 4789–4812 (2008). [CrossRef]

### 5.2. Normalized compressive ghost imaging

*S*′ ≡

*S/R*, we can apply the CGI technique [13

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

## 6. Conclusion

## Acknowledgments

## References and links

1. | R. S. Bennink, S. J. Bentley, and R. W. Boyd, “‘Two-photon’ coincidence imaging with a classical source,” Phys. Rev. Lett. |

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

3. | 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. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. |

5. | 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. |

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

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

8. | M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Processing Magazine |

9. | F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. |

10. | J. Goodman, |

11. | S. Boyd and L. Vandenberghe, |

12. | D. L. Donoho and Y. Tsaig, “Fast solution of l |

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

**OCIS Codes**

(030.4280) Coherence and statistical optics : Noise in imaging systems

(030.6140) Coherence and statistical optics : Speckle

(110.1650) Imaging systems : Coherence imaging

(200.1130) Optics in computing : Algebraic optical processing

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: March 23, 2012

Revised Manuscript: May 14, 2012

Manuscript Accepted: June 27, 2012

Published: July 11, 2012

**Citation**

Baoqing Sun, Stephen S. Welsh, Matthew P. Edgar, Jeffrey H. Shapiro, and Miles J. Padgett, "Normalized ghost imaging," Opt. Express **20**, 16892-16901 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16892

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### References

- R. S. Bennink, S. J. Bentley, and R. W. Boyd, “‘Two-photon’ coincidence imaging with a classical source,” Phys. Rev. Lett.89, 113601 (2002). [CrossRef] [PubMed]
- 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]
- A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A70, 013802 (2004). [CrossRef]
- A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett.94, 063601 (2005). [CrossRef] [PubMed]
- 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).
- J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A78, 061802 (2008). [CrossRef]
- Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A79, 053840 (2009). [CrossRef]
- M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Processing Magazine25, 83–91 (2008). [CrossRef]
- F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett.104, 253603 (2010). [CrossRef] [PubMed]
- J. Goodman, Statistical Optics (Wiley, 2000).
- S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).
- D. L. Donoho and Y. Tsaig, “Fast solution of l1-norm minimization problems when the solution may be sparse,” IEEE Trans. Inf. Theory54, 4789–4812 (2008). [CrossRef]
- O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett.95(13), 131110 (2009). [CrossRef]

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