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Ghost imaging with nonuniform thermal light fields Hu Li, Jianhong Shi,* and Guihua Zeng State Key Laboratory of Advanced Optical Communication Systems and Networks, Key Lab on Navigation and Location-based Service, Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China *Corresponding author: [email protected] Received May 6, 2013; revised August 6, 2013; accepted August 6, 2013; posted August 7, 2013 (Doc. ID 189949); published August 26, 2013 In practical thermal light ghost imaging, the nonuniform intensity distribution of the thermal light applied causes distortion of the retrieved image. We propose uniformly weighted arithmetics, including uniformly weighted ghost imaging and uniformly weighted differential ghost imaging (UWDGI), to improve the imaging quality of ghost imaging with nonuniform thermal light fields. Two kinds of nonuniform thermal light fields are discussed, including planar Gaussian distributed thermal light and near-field three-dimensional thermal light. The retrieved image using a uniformly weighted arithmetic has a lower distortion than that using the corresponding traditional ghost imaging arithmetic. In addition, the best imaging quality is achieved with the UWDGI arithmetic. © 2013 Optical Society of America OCIS codes: (100.2980) Image enhancement; (100.6890) Three-dimensional image processing; (110.6820) Thermal imaging. http://dx.doi.org/10.1364/JOSAA.30.001854
1. INTRODUCTION In classical thermal ghost imaging [1–6], an unknown object is illuminated by a series of independent random pseudothermal light patterns. Normally, the pseudothermal light patterns are generated by introducing a slowly rotating ground glass after a laser. For each pattern the light transmitted or reflected from the object is collected and measured by a single photoelectric detector. The unknown random light intensity pattern is split from the optical path with a beam splitter and measured by a planar array detector (CCD). In lensless thermal light ghost imaging [6–8], the distance between the CCD and the source is equal to that between the object and the source. However, in computational ghost imaging [9,10], the random light intensity pattern is known and is generated by manipulating a spatial light modulator, thereby removing the beam splitter and the reference CCD. The coupled records of the two detectors are combined to retrieve the image of the object by executing a specific correlation arithmetic. Except for compressive sensing ghost imaging [11,12], the other arithmetics are mainly a method of iteration. During iterative ghost imaging, the image gradually becomes legible as the iterations increase. In this paper, we discuss mainly the iteration arithmetics. The intensity distributions of speckle patterns projected onto the object in traditional ghost imaging (TGI) investigations are of a uniform or quasi-uniform distribution. However, in certain practical ghost imaging applications, the speckle patterns have a nonuniform intensity distribution. For instance, both the two-dimensional Gaussian distributed speckle field and the three-dimensional thermal light field are nonuniformly distributed. When discussing quantum sub-shot-noise imaging [13,14], Brida et al. pointed out that the nonuniform field will cause a local reduction of correlation. Also, the simulation results of ghost imaging demonstrate 1084-7529/13/091854-08$15.00/0
that the nonuniform thermal light field leads to distortion of the retrieved image. To our knowledge, ghost imaging with an obviously nonuniform thermal light field has not been discussed yet. In this paper, we propose two optimized ghost imaging arithmetics to eliminate the unfavorable influence caused by the nonuniform thermal light field. This paper is organized as follows. In Section 2, the preexisting ghost imaging arithmetics and the optimized ghost imaging arithmetics are introduced. In Section 3, with two kinds of nonuniform thermal light fields, the performances of ghost imaging using these four arithmetics are analyzed. In Section 4, the conclusion is presented.
2. OPTIMIZED GHOST IMAGING ARITHMETICS With the same optical setup, the imaging quality of ghost imaging mainly depends on its arithmetic. In this section, we introduce four kinds of thermal light ghost imaging arithmetic, including TGI, differential ghost imaging (DGI) [15,16], uniformly weighted ghost imaging (UWGI), and uniformly weighted differential ghost imaging (UWDGI). In this paper, we discuss four arithmetics based on lensless thermal light ghost imaging, a sketch of which is shown in Fig. 1. The distance between the source and the CCD equals the distance between the object and the source. The central point of the thermal light source is set as the origin of the coordinate system. For the ith iteration in thermal light ghost imaging, the intensity distribution of the reference speckle field measured by a CCD is recorded as an intensity matrix ⃗ρ; i. The intensity distribution of the signal speckle field projected onto the object is I s ⃗ρ; i. Then the total light intensity containing the unknown object’s information measured by a single photoelectric detector is recorded as Si, which has the form © 2013 Optical Society of America
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Fig. 1. Lensless thermal light ghost imaging system. BS, beam splitter; PD, photoelectric detector.
Z Si
I s ⃗ρ; iO ⃗ρd ⃗ρ;
(1)
where O ⃗ρ is the object function and ⃗ρ x; y; z is a space coordinate. The total light intensity collected by the reference CCD is recorded as Ri, which has the form Z Ri
I r ⃗ρ; id ⃗ρ;
(2)
where I r ⃗ρ; i is the intensity distribution of the reference speckle field. A. Traditional Ghost Imaging In TGI, the image is retrieved by calculating the intensity correlation or the intensity fluctuation correlation [15] between the signal beam and the reference beam. The initial TGI exploited the low-order intensity correlation; afterward higher-order correlation [17–21] was discussed for improving the performance of ghost imaging. Subsequently, further investigations [22,23] concluded that the best performance of TGI is achieved with the lowest-order intensity fluctuation correlation, in which the image is retrieved by TGI ⃗ρ hI r ⃗ρ − hI r ⃗ρiS − hSii;
for each iteration. According to Eq. (4), in DGI the ratio R∕hRi is introduced to eliminate the influence caused by the fluctuations of the total light intensity in the signal path and the reference path. C. Uniformly Weighted Ghost Imaging The first terms on the right-hand sides of Eqs. (3) and (4) are the same, without considering the weights of average intensity in different space points. Basically, the thermal light field used in a TGI experiment has a uniform or approximately uniform distribution, in which the weights for different space points are the same or similar. However, with a nonuniform thermal light field, the weights for different space points might be obviously different and therefore should be modified back into the imaging arithmetic. Introducing the reciprocal of the average light intensity for each space point as a weight into the TGI, we get the UWGI arithmetic. In UWGI, the image is retrieved by
I r ⃗ρ − 1 S − hSi : hI r ⃗ρi
UWGI ⃗ρ
(5)
(3)
P where hxi N1 N i1 xi and N is the total number of iterations. In a perfect world, I r has the same intensity patterns as I s , or they are mirror conjugates. The mutually correlated points in I r and I s fluctuate consistently over the same average intensity. Therefore, with increased iterations, the object part becomes brighter, and the background part becomes darker. B. Differential Ghost Imaging When the size of the object is relatively small in transmissiontype TGI or large in reflection-type TGI, which means that the weight of a single point intensity in Si is quite low, the background noise is large, and the imaging quality is therefore weak. To solve this problem, Ferri et al. proposed DGI. In DGI, the image is retrieved by hSiR DGI ⃗ρ I r ⃗ρ − hI r ⃗ρi S − : (4) hRi In addition, in the practical process of ghost imaging, the output power of the laser fluctuates and the speckle pattern changes for each iteration. Therefore, Si and Ri fluctuate
D. Uniformly Weighted Differential Ghost Imaging Similarly, we can introduce the reciprocal of the average light intensity for each space point as a weight into the DGI, resulting in the UWDGI arithmetic. In UWDGI, the image is retrieved by UWDGI ⃗ρ
I r ⃗ρ hSiR : −1 S− hI r ⃗ρi hRi
(6)
Clearly, UWDGI is provided with the advantages of both DGI and UWGI. It not only reduces the background noise but also makes up for the imaging distortion caused by the nonuniform field. E. Evaluations of the Ghost Imaging Arithmetics In the lensless thermal light ghost imaging configurations, we assume that the CCD response time is shorter than the correlation time of the speckle field, and the time interval of sampling is longer than the correlation time. These are the basic settings in typical ghost imaging experiments. In addition, we assume that the transverse correlation length is shorter than the pixel size, and the average light intensity distribution over
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the transverse plane is μ ⃗ρ . Specially, we assume that I r ⃗ρ I s ⃗ρ I ⃗ρ, and the light intensity I ⃗ρ has a negative exponential probability distribution for all positions ⃗ρ; the probability distribution function is
f I ⃗ρ
eI ⃗ρ ∕μ ⃗ρ μ ⃗ρ
0;
;
I ⃗ρ ≥ 0; I ⃗ρ < 0:
(7)
For simplification, we indicate I ⃗ρ , O ⃗ρ, and μ ⃗ρ by I i , Oi and μi , respectively. According to Eq. (7), we obtain that EI ni n!μni , DI i μ2i , EΔI i 0, EΔI i 2 μ2i , EΔI i 3 2μ3i , EΔI i 4 9μ4i , where Ex is the expectation value of x and Δx x − Ex. Only the light intensity fluctuations in correlated two spatial positions are coincident, while the light intensity fluctuations in any two uncorrelated spatial positions are statistically independent. Therefore the product of the light intensity fluctuations is expected to equal zero, i.e., EΔI i ΔI j μ2 for i j, while EΔI i ΔI j 0 for i ≠ j. We define a object position set Q and a complete position set Φ (Q ⊂ F), Oi ≠ 0 for any i ∈ Q and Oi 0P for any i ∈ F but not in Q. Consequently, we obtain S i∈Q I i × Oi P and R i∈Φ I i . Then for any a position coordinate i ∈ Φ, the corresponding second-order correlation value in the TGI is TGI ⃗ρi hΔI i ΔSi. Assume iterations N → ∞, TGI ⃗ρi μ2i Oi for i ∈ Q, while TGI ⃗ρi 0 for i ∈ F but not in Q. Therefore, for any a position i ∈ Q, the signal-to-noise ratio of imaging result at this position is hTGI ⃗ρi i ; SNR ⃗ρi p DTGI ⃗ρi
(8)
where DTGI ⃗ρi is the variance of TGI ⃗ρi . The variance of TGI ⃗ρi is calculated as DTGI ⃗ρi ETGI2 ⃗ρi − E 2 TGI ⃗ρi X X 1 8μ4i O2i 4μ3i Oi μm Om μ2i μ2m O2m N m≠i m≠i X (9) 2μ2i μm μn Om On ;
SNR ⃗ρi
p μ i Oi N r P P 2 2 P μ m Om 2 μ m μ n Om On 8μ2i O2i 4μi Oi μm Om m≠i
s N P ≈ μi Oi : 2 m≠n≠i μm μn Om On
m≠i
m≠n≠i
(10)
Clearly, Eq. (10) demonstrates that the nonuniform thermal light intensity distribution will result in the local dehomogenization of the signal-to-noise ratio (SNR). If the object is a binary amplitude object, the SNR in a spatial position with a higher average light intensity is larger than that with a lower average light intensity. The dehomogenization of the SNR will cause the distortion of the ghost image. In a similar way, the analysis results in DGI and TGI are the same. To reduce the distortion caused by the nonuniform thermal light field, we modify the TGI by introducing a weighting factor 1∕hI r ⃗ρi i 1∕μi as in the UWGI and UWDGI. Even in the case with a uniform thermal light field, the optimized arithmetics achieve the same performance as the traditional arithmetics. Specially, assume that the object is a binary amplitude object and the average light intensity is uniformly distributed; then Oi 1 for i ∈ Q, and otherwise Oi 0, μi μj for i; j ∈ Φ. In this case, the signal-to-noise ratio can be calculated as SNR ⃗ρi
r N ; 2P 2 − P 7
(11)
which depends only on the iterations and the total pixels of the object. Note that in TGI, TGI ⃗ρi ≈ 0 for i is not in Q; then SNR ⃗ρi ≈ CNR ⃗ρi , where CNR is the contrast-to-noise ratio defined in the work of Boyd’s group [19].
3. SIMULATIONS AND ANALYSIS Two cases of nonuniform thermal light fields used in the ghost imaging simulations are discussed in this section: the twodimensional thermal light field with Gaussian intensity distribution, and the three-dimensional thermal light field in the near field. The first field has a nonuniform intensity distribution in the lateral direction, while the second field has a nonuniform intensity distribution in the longitudinal direction.
m≠n≠i
where m; n ∈ Q; N is the number of iterations. The standard deviation sqrtDTGI ⃗ρi indicates the average noise intensity around the signal at the position ⃗ρi . Assume that the total number of illuminated pixels that indicate the object is P, which means that P is the total number of the elements in the set Q. Normally, in a typical experiment, P is a large number. Equation (9) demonstrates that the noise depends on the object function and the average light intensity distribution of the speckle field. Also, the term P 2μ2i m≠n≠i μm μn Om On has the greatest contribution to DTGI ⃗ρi . Substituting Eq. (9) into Eq. (8) and simplifying the expression, we obtain
A. Two-Dimensional Gaussian Thermal Light Field The light beam in practice, no matter whether a laser beam or a pseudothermal light beam, has Gaussian distribution characteristics in a transverse plane. In ghost imaging experiments, only part of the whole beam covers the object, named the effective beam. When the size of object is small compared with the whole beam, as is the situation in TGI, the intensity distribution of the effective beam is approximately uniform. However when the size of object is similar to that of the beam, then the intensity distribution of the effective beam is nonuniform. Here we exploit numerical simulation to compare the preexisting ghost imaging arithmetics with the optimized arithmetics in the second case. Based on the transmission-type ghost imaging simulation model [19], assume that the lateral coherence length of the
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pattern
object
TGI
DGI
UWGI
UWDGI
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Fig. 2. Simulation results with a two-dimensional Gaussian thermal light field whose total light intensity remains unchanged.
speckle patterns is less than the size of a pixel. The values of any two pixels are statistically uncorrelated, and any two patterns are independent. The average intensity of the generated patterns, one of which is shown in Fig. 2, conforms to a Gaussian distribution. The patterns contain 100 × 100 pixels; the peak of the Gaussian distribution is the center of the patterns, and the standard deviation of the Gaussian distribution is 50 pixels. The object, shown in Fig. 2, has a fabric structure and is illuminated by the random patterns. The bright pixels of the object stand for the transmission parts, and the dark pixels stand for the absorbed parts. For each iteration, the total light intensity remains unchanged. With 100,000 iterations, the images retrieved by TGI, DGI, UWGI, and UWDGI, according to Eqs. (3)–(6), respectively, are depicted in Fig. 2. Apparently, only the central part of the object is recovered well in TGI or DGI, while the whole image is recovered in UWGI or UWDGI. In addition, using differential arithmetics (DGI and UWDGI), the imaging for a weak absorption object is better. Furthermore, the situation in which the output power of laser has a random fluctuation is simulated. For each iteration, the total light intensity randomly fluctuates 5% over its average value. With the same object and iterations, the images retrieved by TGI, DGI, UWGI, and UWDGI are all depicted in Fig. 3. Compare Figs. 2 and 3: the differential arithmetics (DGI and UWDGI) can eliminate the influence caused by the fluctuation of the output power of laser. In addition, the imaging quality of UWDGI is the best of the four arithmetics, for its retrieved image has the lowest distortion. B. Three-Dimensional Thermal Light Field in the Near Field In the previous subsection, ghost imaging with a planar nonuniform thermal light field is discussed. Recently, the three-dimensional characteristics (or the spatial longitudinal coherence) of thermal light field [24–28] was investigated for its potential applications in three-dimension ghost imaging. According to classical diffraction theory, after a distance
of free-space diffraction, the complex amplitude at point x; y; z has the form ZZ Ex; y; z
Ex0 ; y0 ; z 0
eikr dx dy ; iλr 0 0
(12)
where Ex0 ; y0 ; z 0 is the initial complex amplitude at a component source point, λ is the central wavelength of the source, k 2π∕λ is the wavenumber, and r p x − x0 2 y − y0 2 z2 is the absolute distance between a component source point and the point x; y; z. From Eq. (12), it is observed that the intensity (the square of amplitude) decreases as the diffraction distance z increases. Therefore, the three-dimensional thermal light field is a kind of nonuniform field. Especially in the near field, the nonuniform characteristics of the three-dimensional thermal light field are obvious. Therefore, we focus on discussing TGI
DGI
UWGI
UWDGI
Fig. 3. Simulation results with a two-dimensional Gaussian thermal light field whose total light intensity has a 5% fluctuation over its average value.
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ghost imaging with near-field three-dimensional thermal light field in this subsection. Near-field three-dimensional ghost imaging has an optical setup that is similar to that of the lensless two-dimensional ghost imaging shown in Fig. 1. The difference is that in the three-dimensional case the reference light intensity distribution is three-dimensional and is obtained by scanning or is directly calculated. The four arithmetics are still working for the three-dimensional case. In the simulation, we calculate the reference threedimensional thermal light intensity distribution based on Eq. (12). The light source is located in the plane z 0. The size of the source is 5 mm, and its central point is the origin in the coordinate system. The wavelength of the light source is 633 nm. First, we analyze the three-dimensional ghost imaging with a single-point object (the simplest object) located at (0, 0, z 150 mm). For each iteration, the phase as well as the amplitude of every component source point is altered with a zeromean Gaussian random value. The complex amplitudes of any two different component source points are statistically independent. In the ith iteration, the reference light intensity distribution is calculated based on Eq. (12) and recorded as I r x; y; z; i, where the intervals of x, y, and z are (−0.2 mm, 0.2 mm), (−0.2 mm, 0.2 mm), and (100 mm, 200 mm), respectively. The corresponding Si and Ri can be calculated according to Eqs. (1) and (2), respectively. With 5000 iterations, substituting the calculated I r i, Si, and Ri into Eqs. (3)–(6) yields four image matrixes. However, the imaging results of the four arithmetics are all in the form of a three-dimensional matrix. Referring to the feature of ghost imaging that the correlation values at the points inside the object are far larger than those outside the object, we employ a threshold search method to visualize the image from a calculated three-dimensional matrix Ω. The basic steps of the threshold search method are given as follows.
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1. Choose a proper threshold as cc Ωmax , where * is the multiplication operator, cc is a cut coefficient, and Ωmax is the maximum of Ω ⃗ρ. ⃗ρ x; y; z, set Ω ⃗ρ 0, 2. For any point if Ω ⃗ρ < cc Ωmax . 3. For any x and y, search and retain the maximum Ωx; y; z; the other points are set to 0. Then the revised Ω describes the surface or shape of the three-dimensional object, symbolized as Γ. To analyze the performance of the four ghost imaging arithmetics with a single-point object, we set cc 0.12 and only execute step 1 and 2. The four stereograms obtained with the four arithmetics are shown in Fig. 4. For one of the arithmetics, the corresponding stereogram exhibits its threedimensional coherence zone under the condition cc 0.12. According to the ghost imaging arithmetic, the image of a complex object is approximately equal to the convolution of the object function and the image of a single point. Therefore, with the same cc, the more convergent the three dimension coherence zone is, the better the corresponding arithmetic is. Apparently, the three-dimensional coherence zones of DGI and UWDGI are more convergent than those of TGI and UWGI. Moreover, the three-dimensional zones of TGI and DGI contain discrete spreads, which are caused by the nonuniform intensity distribution of the field in the nearer distance, unlike that of UWGI and UWDGI. The discrete spreads are fake coherence zones; the phenomenon is more distinct in the stereogram of TGI in Fig. 4. Therefore, we conclude that the uniformly weighted arithmetics can eliminate the fake coherence zones, and the three-dimensional zone of UWDGI has the greatest convergence. Second, we analyze the three-dimensional ghost imaging with a complex object of three-dimensional structure. In the simulation, the optical setups are the same as that with a single-point object. The three-dimensional object is shown in Fig. 5, containing many columns of different heights. The light projected on the surfaces of the columns is reflected.
Fig. 4. Three-dimensional coherence zones obtained with the four arithmetics.
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Fig. 5. Three-dimensional object. The colors are used to distinguish the heights of different columns.
The surfaces of the columns are located at the distance interval of (100 mm, 120 mm); the difference between any two heights of columns is an integral multiple of 1 mm. With 50,000 iterations, another four three-dimensional matrixes representing four images are obtained. Then we use all steps of the threshold search method to visualize the object. Before we begin with step 1, we need to find the best threshold or the best cc for imaging. Traditionally, we apply visibility or the CNR to evaluate the performance of ghost imaging, without considering the brightness or distortion. If the average light intensity of the speckle fields projected on the object is nonuniform, we can only measure the SNR or CNR at a position ⃗ρi by repeating the experiments many times. However, the quality of the integral image is hard to evaluate by these parameters. In three-dimensional imaging, the acquisition of the shape of the three-dimensional object is quite important. Therefore, we define the shape distortion in the near-field threedimensional ghost imaging as Eq. (13), to evaluate the integral image quality: 2 TGI DGI UWGI UWDGI
1.8 1.6
distortion
1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
cc
Fig. 6. cc-distortion curves of the four arithmetics. Red-solid curve, TGI; blue-dashed curve, DGI; cyan-solid curve with circles, UWGI; green-dashed curve with rectangles, UWDGI.
P distortion
ρ jΓ ⃗ρ; cc
− O ⃗ρj ; ⃗ρj jO ρ
P
(13)
where O ⃗ρ is the object function and Γ ⃗ρ; cc is the retrieved image function with a variable cc. The smaller the distortion is, the better the imaging quality is. According to Eq. (13), the distortion is a function of cc. The cc distortion curves of the four ghost imaging arithmetics are plotted in Fig. 6. Based on the cc distortion curves, the best cc for different arithmetics can be found. In addition, Fig. 6 demonstrates that the smallest distortion of three-dimensional ghost imaging is achieved with the UWDGI, and even the UWGI performs better than the DGI. To intuitively show that the cc distortion curves reflect the varying quality trends for each arithmetic, the images achieved by the four arithmetics with cc 0.2, 0.35, 0.5, and 0.6 are depicted in Fig. 7. The four images in the first, second, third, and fourth line of Fig. 7 are obtained with TGI, DGI, UWGI, and UWDGI, respectively. Apparently, the varying quality trends in Fig. 7 agree with those in Fig. 6, which demonstrates that the distortion can reflect the quality of the near-field three-dimensional thermal light ghost imaging. Furthermore, the minimum distortion varies with the increase in iterations, as well. By simulation, the curves indicating the relations between the iterations and the minimum distortion of the four arithmetics are plotted in Fig. 8. The curves demonstrate that the TGI has the worst performance, the UWDGI has the best performance, and the imaging quality of UWGI surpasses that of DGI after a certain number of iterations. Eventually, the best cc or threshold for each arithmetic is determined. With 50,000 iterations, the best cc for TGI, DGI, UWGI, and UWDGI are 0.53, 0.24, 0.63 and 0.33, respectively. Substituting the best cc into the threshold search method, visible images of the object are obtained with the different arithmetics. The four visible images are shown in Fig. 9, which intuitively supports the theoretical analysis. All the simulation results demonstrate that the UWDGI has the best performance in ghost imaging with a nonuniform thermal light field. Note that in this paper we discussed thermal light threedimensional ghost imaging only at near distance, because the longitudinal resolution of this method is quite weak at long
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Fig. 7. Ghost images achieved by TGI, DGI, UWGI, and UWDGI with different values of cc.
1
The minimun distortion
0.8 TGI DGI UWGI UWDGI
0.6
0.4
0.2
Fig. 9. Final images obtained by the four arithmetics. 0
0
1
2
3 Iterations
4
5 4
x 10
Fig. 8. Iterations versus minimum distortion curves of the four arithmetics.
distance [24–28]. Recently, the Han group realized longdistance three-dimensional ghost imaging with a much better longitudinal resolution (about 60 cm longitudinal resolution with a distance of 1 km), by using time-resolved measurement [29].
4. CONCLUSION In conclusion, we proposed uniformly weighted arithmetics to improve the imaging quality of ghost imaging with a
nonuniform thermal light field. In the first section, we introduced four ghost imaging arithmetics, TGI, DGI, UWGI, and UWDGI. Then in the second section, we described two kinds of nonuniform thermal light fields, including the planar Gaussian distributed thermal light field and the three-dimensional thermal light field in the near field. In particular, we discussed the second kind of field in detail, because of the lack of investigations of three-dimensional ghost imaging. We introduced the shape distortion parameter to evaluate the quality of the integral image. With numerous simulations, we concluded that the uniformly weighted arithmetic is capable of eliminating the negative influence of the nonuniform field on the performance of ghost imaging, and the best imaging quality is achieved with the UWDGI. In addition, the longitudinal resolution in the simulation is less than 1 mm at a distance of
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about 120 mm. Our work is quite meaningful for the practical applications of ghost imaging.
ACKNOWLEDGMENT This work is supported by the National Natural Science Foundation of China (grants 60970109, 61170228).
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14. G. Brida, M. Genovese, A. Meda, and I. R. Berchera, “Experimental quantum imaging exploiting multimode spatial correlation of twin beams,” Phys. Rev. A 83, 033811 (2011). 15. W. Gong and S. Han, “A method to improve the visibility of ghost images obtained by thermal light,” Phys. Lett. A 374, 1005–1008 (2010). 16. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010). 17. Y. Bai and S. Han, “Ghost imaging with thermal light by thirdorder correlation,” Phys. Rev. A 76, 043828 (2007). 18. D. Z. Cao, J. Xiong, S. H. Zhang, L. F. Lin, L. Gao, and K. G. Wang, “Enhancing visibility and resolution in Nth-order intensity correlation of thermal light,” Appl. Phys. Lett. 92, 201102 (2008). 19. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-order thermal ghost imaging,” Opt. Lett. 34, 3343–3345 (2009). 20. Y. Zhou, J. Simon, J. B. Liu, and Y. H. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81, 043831 (2010). 21. X. H. Chen, I. N. Agafonov, K. H. Luo, Q. Liu, R. Xian, M. V. Chekhova, and L. A. Wu, “High-visibility, high-order lensless ghost imaging with thermal light,” Opt. Lett. 35, 1166–1168 (2010). 22. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Optimization of thermal ghost imaging: high-order correlations vs. background subtraction,” Opt. Express 18, 5562–5573 (2010). 23. H. Li, J. Shi, Z. Chen, and G. Zeng, “Detailed quality analysis of ideal high-order thermal ghost imaging,” J. Opt. Soc. Am. A 29, 2256–2262 (2012). 24. H. Liu and S. Han, “Spatial longitudinal coherence length of a thermal source and its influence on lensless ghost imaging,” Opt. Lett. 33, 824–826 (2008). 25. A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008). 26. D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009). 27. D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields. Part I. Theory and numerical investigation,” J. Opt. Soc. Am. A 28, 1896–1903 (2011). 28. D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields. Part II. Experimental investigation,” J. Opt. Soc. Am. A 28, 1904–1908 (2011). 29. W. Gong, C. Zhao, J. Jiao, E. Li, M. Chen, H. Wang, W. Xu, and S. Han, “Three-dimensional ghost imaging ladar,” arXiv:1301.5767 [quant-ph] (2013).
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Ghost imaging with nonuniform thermal light fields Hu Li, Jianhong Shi,* and Guihua Zeng State Key Laboratory of Advanced Optical Communication Systems and Networks, Key Lab on Navigation and Location-based Service, Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China *Corresponding author: [email protected] Received May 6, 2013; revised August 6, 2013; accepted August 6, 2013; posted August 7, 2013 (Doc. ID 189949); published August 26, 2013 In practical thermal light ghost imaging, the nonuniform intensity distribution of the thermal light applied causes distortion of the retrieved image. We propose uniformly weighted arithmetics, including uniformly weighted ghost imaging and uniformly weighted differential ghost imaging (UWDGI), to improve the imaging quality of ghost imaging with nonuniform thermal light fields. Two kinds of nonuniform thermal light fields are discussed, including planar Gaussian distributed thermal light and near-field three-dimensional thermal light. The retrieved image using a uniformly weighted arithmetic has a lower distortion than that using the corresponding traditional ghost imaging arithmetic. In addition, the best imaging quality is achieved with the UWDGI arithmetic. © 2013 Optical Society of America OCIS codes: (100.2980) Image enhancement; (100.6890) Three-dimensional image processing; (110.6820) Thermal imaging. http://dx.doi.org/10.1364/JOSAA.30.001854
1. INTRODUCTION In classical thermal ghost imaging [1–6], an unknown object is illuminated by a series of independent random pseudothermal light patterns. Normally, the pseudothermal light patterns are generated by introducing a slowly rotating ground glass after a laser. For each pattern the light transmitted or reflected from the object is collected and measured by a single photoelectric detector. The unknown random light intensity pattern is split from the optical path with a beam splitter and measured by a planar array detector (CCD). In lensless thermal light ghost imaging [6–8], the distance between the CCD and the source is equal to that between the object and the source. However, in computational ghost imaging [9,10], the random light intensity pattern is known and is generated by manipulating a spatial light modulator, thereby removing the beam splitter and the reference CCD. The coupled records of the two detectors are combined to retrieve the image of the object by executing a specific correlation arithmetic. Except for compressive sensing ghost imaging [11,12], the other arithmetics are mainly a method of iteration. During iterative ghost imaging, the image gradually becomes legible as the iterations increase. In this paper, we discuss mainly the iteration arithmetics. The intensity distributions of speckle patterns projected onto the object in traditional ghost imaging (TGI) investigations are of a uniform or quasi-uniform distribution. However, in certain practical ghost imaging applications, the speckle patterns have a nonuniform intensity distribution. For instance, both the two-dimensional Gaussian distributed speckle field and the three-dimensional thermal light field are nonuniformly distributed. When discussing quantum sub-shot-noise imaging [13,14], Brida et al. pointed out that the nonuniform field will cause a local reduction of correlation. Also, the simulation results of ghost imaging demonstrate 1084-7529/13/091854-08$15.00/0
that the nonuniform thermal light field leads to distortion of the retrieved image. To our knowledge, ghost imaging with an obviously nonuniform thermal light field has not been discussed yet. In this paper, we propose two optimized ghost imaging arithmetics to eliminate the unfavorable influence caused by the nonuniform thermal light field. This paper is organized as follows. In Section 2, the preexisting ghost imaging arithmetics and the optimized ghost imaging arithmetics are introduced. In Section 3, with two kinds of nonuniform thermal light fields, the performances of ghost imaging using these four arithmetics are analyzed. In Section 4, the conclusion is presented.
2. OPTIMIZED GHOST IMAGING ARITHMETICS With the same optical setup, the imaging quality of ghost imaging mainly depends on its arithmetic. In this section, we introduce four kinds of thermal light ghost imaging arithmetic, including TGI, differential ghost imaging (DGI) [15,16], uniformly weighted ghost imaging (UWGI), and uniformly weighted differential ghost imaging (UWDGI). In this paper, we discuss four arithmetics based on lensless thermal light ghost imaging, a sketch of which is shown in Fig. 1. The distance between the source and the CCD equals the distance between the object and the source. The central point of the thermal light source is set as the origin of the coordinate system. For the ith iteration in thermal light ghost imaging, the intensity distribution of the reference speckle field measured by a CCD is recorded as an intensity matrix ⃗ρ; i. The intensity distribution of the signal speckle field projected onto the object is I s ⃗ρ; i. Then the total light intensity containing the unknown object’s information measured by a single photoelectric detector is recorded as Si, which has the form © 2013 Optical Society of America
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Fig. 1. Lensless thermal light ghost imaging system. BS, beam splitter; PD, photoelectric detector.
Z Si
I s ⃗ρ; iO ⃗ρd ⃗ρ;
(1)
where O ⃗ρ is the object function and ⃗ρ x; y; z is a space coordinate. The total light intensity collected by the reference CCD is recorded as Ri, which has the form Z Ri
I r ⃗ρ; id ⃗ρ;
(2)
where I r ⃗ρ; i is the intensity distribution of the reference speckle field. A. Traditional Ghost Imaging In TGI, the image is retrieved by calculating the intensity correlation or the intensity fluctuation correlation [15] between the signal beam and the reference beam. The initial TGI exploited the low-order intensity correlation; afterward higher-order correlation [17–21] was discussed for improving the performance of ghost imaging. Subsequently, further investigations [22,23] concluded that the best performance of TGI is achieved with the lowest-order intensity fluctuation correlation, in which the image is retrieved by TGI ⃗ρ hI r ⃗ρ − hI r ⃗ρiS − hSii;
for each iteration. According to Eq. (4), in DGI the ratio R∕hRi is introduced to eliminate the influence caused by the fluctuations of the total light intensity in the signal path and the reference path. C. Uniformly Weighted Ghost Imaging The first terms on the right-hand sides of Eqs. (3) and (4) are the same, without considering the weights of average intensity in different space points. Basically, the thermal light field used in a TGI experiment has a uniform or approximately uniform distribution, in which the weights for different space points are the same or similar. However, with a nonuniform thermal light field, the weights for different space points might be obviously different and therefore should be modified back into the imaging arithmetic. Introducing the reciprocal of the average light intensity for each space point as a weight into the TGI, we get the UWGI arithmetic. In UWGI, the image is retrieved by
I r ⃗ρ − 1 S − hSi : hI r ⃗ρi
UWGI ⃗ρ
(5)
(3)
P where hxi N1 N i1 xi and N is the total number of iterations. In a perfect world, I r has the same intensity patterns as I s , or they are mirror conjugates. The mutually correlated points in I r and I s fluctuate consistently over the same average intensity. Therefore, with increased iterations, the object part becomes brighter, and the background part becomes darker. B. Differential Ghost Imaging When the size of the object is relatively small in transmissiontype TGI or large in reflection-type TGI, which means that the weight of a single point intensity in Si is quite low, the background noise is large, and the imaging quality is therefore weak. To solve this problem, Ferri et al. proposed DGI. In DGI, the image is retrieved by hSiR DGI ⃗ρ I r ⃗ρ − hI r ⃗ρi S − : (4) hRi In addition, in the practical process of ghost imaging, the output power of the laser fluctuates and the speckle pattern changes for each iteration. Therefore, Si and Ri fluctuate
D. Uniformly Weighted Differential Ghost Imaging Similarly, we can introduce the reciprocal of the average light intensity for each space point as a weight into the DGI, resulting in the UWDGI arithmetic. In UWDGI, the image is retrieved by UWDGI ⃗ρ
I r ⃗ρ hSiR : −1 S− hI r ⃗ρi hRi
(6)
Clearly, UWDGI is provided with the advantages of both DGI and UWGI. It not only reduces the background noise but also makes up for the imaging distortion caused by the nonuniform field. E. Evaluations of the Ghost Imaging Arithmetics In the lensless thermal light ghost imaging configurations, we assume that the CCD response time is shorter than the correlation time of the speckle field, and the time interval of sampling is longer than the correlation time. These are the basic settings in typical ghost imaging experiments. In addition, we assume that the transverse correlation length is shorter than the pixel size, and the average light intensity distribution over
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the transverse plane is μ ⃗ρ . Specially, we assume that I r ⃗ρ I s ⃗ρ I ⃗ρ, and the light intensity I ⃗ρ has a negative exponential probability distribution for all positions ⃗ρ; the probability distribution function is
f I ⃗ρ
eI ⃗ρ ∕μ ⃗ρ μ ⃗ρ
0;
;
I ⃗ρ ≥ 0; I ⃗ρ < 0:
(7)
For simplification, we indicate I ⃗ρ , O ⃗ρ, and μ ⃗ρ by I i , Oi and μi , respectively. According to Eq. (7), we obtain that EI ni n!μni , DI i μ2i , EΔI i 0, EΔI i 2 μ2i , EΔI i 3 2μ3i , EΔI i 4 9μ4i , where Ex is the expectation value of x and Δx x − Ex. Only the light intensity fluctuations in correlated two spatial positions are coincident, while the light intensity fluctuations in any two uncorrelated spatial positions are statistically independent. Therefore the product of the light intensity fluctuations is expected to equal zero, i.e., EΔI i ΔI j μ2 for i j, while EΔI i ΔI j 0 for i ≠ j. We define a object position set Q and a complete position set Φ (Q ⊂ F), Oi ≠ 0 for any i ∈ Q and Oi 0P for any i ∈ F but not in Q. Consequently, we obtain S i∈Q I i × Oi P and R i∈Φ I i . Then for any a position coordinate i ∈ Φ, the corresponding second-order correlation value in the TGI is TGI ⃗ρi hΔI i ΔSi. Assume iterations N → ∞, TGI ⃗ρi μ2i Oi for i ∈ Q, while TGI ⃗ρi 0 for i ∈ F but not in Q. Therefore, for any a position i ∈ Q, the signal-to-noise ratio of imaging result at this position is hTGI ⃗ρi i ; SNR ⃗ρi p DTGI ⃗ρi
(8)
where DTGI ⃗ρi is the variance of TGI ⃗ρi . The variance of TGI ⃗ρi is calculated as DTGI ⃗ρi ETGI2 ⃗ρi − E 2 TGI ⃗ρi X X 1 8μ4i O2i 4μ3i Oi μm Om μ2i μ2m O2m N m≠i m≠i X (9) 2μ2i μm μn Om On ;
SNR ⃗ρi
p μ i Oi N r P P 2 2 P μ m Om 2 μ m μ n Om On 8μ2i O2i 4μi Oi μm Om m≠i
s N P ≈ μi Oi : 2 m≠n≠i μm μn Om On
m≠i
m≠n≠i
(10)
Clearly, Eq. (10) demonstrates that the nonuniform thermal light intensity distribution will result in the local dehomogenization of the signal-to-noise ratio (SNR). If the object is a binary amplitude object, the SNR in a spatial position with a higher average light intensity is larger than that with a lower average light intensity. The dehomogenization of the SNR will cause the distortion of the ghost image. In a similar way, the analysis results in DGI and TGI are the same. To reduce the distortion caused by the nonuniform thermal light field, we modify the TGI by introducing a weighting factor 1∕hI r ⃗ρi i 1∕μi as in the UWGI and UWDGI. Even in the case with a uniform thermal light field, the optimized arithmetics achieve the same performance as the traditional arithmetics. Specially, assume that the object is a binary amplitude object and the average light intensity is uniformly distributed; then Oi 1 for i ∈ Q, and otherwise Oi 0, μi μj for i; j ∈ Φ. In this case, the signal-to-noise ratio can be calculated as SNR ⃗ρi
r N ; 2P 2 − P 7
(11)
which depends only on the iterations and the total pixels of the object. Note that in TGI, TGI ⃗ρi ≈ 0 for i is not in Q; then SNR ⃗ρi ≈ CNR ⃗ρi , where CNR is the contrast-to-noise ratio defined in the work of Boyd’s group [19].
3. SIMULATIONS AND ANALYSIS Two cases of nonuniform thermal light fields used in the ghost imaging simulations are discussed in this section: the twodimensional thermal light field with Gaussian intensity distribution, and the three-dimensional thermal light field in the near field. The first field has a nonuniform intensity distribution in the lateral direction, while the second field has a nonuniform intensity distribution in the longitudinal direction.
m≠n≠i
where m; n ∈ Q; N is the number of iterations. The standard deviation sqrtDTGI ⃗ρi indicates the average noise intensity around the signal at the position ⃗ρi . Assume that the total number of illuminated pixels that indicate the object is P, which means that P is the total number of the elements in the set Q. Normally, in a typical experiment, P is a large number. Equation (9) demonstrates that the noise depends on the object function and the average light intensity distribution of the speckle field. Also, the term P 2μ2i m≠n≠i μm μn Om On has the greatest contribution to DTGI ⃗ρi . Substituting Eq. (9) into Eq. (8) and simplifying the expression, we obtain
A. Two-Dimensional Gaussian Thermal Light Field The light beam in practice, no matter whether a laser beam or a pseudothermal light beam, has Gaussian distribution characteristics in a transverse plane. In ghost imaging experiments, only part of the whole beam covers the object, named the effective beam. When the size of object is small compared with the whole beam, as is the situation in TGI, the intensity distribution of the effective beam is approximately uniform. However when the size of object is similar to that of the beam, then the intensity distribution of the effective beam is nonuniform. Here we exploit numerical simulation to compare the preexisting ghost imaging arithmetics with the optimized arithmetics in the second case. Based on the transmission-type ghost imaging simulation model [19], assume that the lateral coherence length of the
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pattern
object
TGI
DGI
UWGI
UWDGI
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Fig. 2. Simulation results with a two-dimensional Gaussian thermal light field whose total light intensity remains unchanged.
speckle patterns is less than the size of a pixel. The values of any two pixels are statistically uncorrelated, and any two patterns are independent. The average intensity of the generated patterns, one of which is shown in Fig. 2, conforms to a Gaussian distribution. The patterns contain 100 × 100 pixels; the peak of the Gaussian distribution is the center of the patterns, and the standard deviation of the Gaussian distribution is 50 pixels. The object, shown in Fig. 2, has a fabric structure and is illuminated by the random patterns. The bright pixels of the object stand for the transmission parts, and the dark pixels stand for the absorbed parts. For each iteration, the total light intensity remains unchanged. With 100,000 iterations, the images retrieved by TGI, DGI, UWGI, and UWDGI, according to Eqs. (3)–(6), respectively, are depicted in Fig. 2. Apparently, only the central part of the object is recovered well in TGI or DGI, while the whole image is recovered in UWGI or UWDGI. In addition, using differential arithmetics (DGI and UWDGI), the imaging for a weak absorption object is better. Furthermore, the situation in which the output power of laser has a random fluctuation is simulated. For each iteration, the total light intensity randomly fluctuates 5% over its average value. With the same object and iterations, the images retrieved by TGI, DGI, UWGI, and UWDGI are all depicted in Fig. 3. Compare Figs. 2 and 3: the differential arithmetics (DGI and UWDGI) can eliminate the influence caused by the fluctuation of the output power of laser. In addition, the imaging quality of UWDGI is the best of the four arithmetics, for its retrieved image has the lowest distortion. B. Three-Dimensional Thermal Light Field in the Near Field In the previous subsection, ghost imaging with a planar nonuniform thermal light field is discussed. Recently, the three-dimensional characteristics (or the spatial longitudinal coherence) of thermal light field [24–28] was investigated for its potential applications in three-dimension ghost imaging. According to classical diffraction theory, after a distance
of free-space diffraction, the complex amplitude at point x; y; z has the form ZZ Ex; y; z
Ex0 ; y0 ; z 0
eikr dx dy ; iλr 0 0
(12)
where Ex0 ; y0 ; z 0 is the initial complex amplitude at a component source point, λ is the central wavelength of the source, k 2π∕λ is the wavenumber, and r p x − x0 2 y − y0 2 z2 is the absolute distance between a component source point and the point x; y; z. From Eq. (12), it is observed that the intensity (the square of amplitude) decreases as the diffraction distance z increases. Therefore, the three-dimensional thermal light field is a kind of nonuniform field. Especially in the near field, the nonuniform characteristics of the three-dimensional thermal light field are obvious. Therefore, we focus on discussing TGI
DGI
UWGI
UWDGI
Fig. 3. Simulation results with a two-dimensional Gaussian thermal light field whose total light intensity has a 5% fluctuation over its average value.
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ghost imaging with near-field three-dimensional thermal light field in this subsection. Near-field three-dimensional ghost imaging has an optical setup that is similar to that of the lensless two-dimensional ghost imaging shown in Fig. 1. The difference is that in the three-dimensional case the reference light intensity distribution is three-dimensional and is obtained by scanning or is directly calculated. The four arithmetics are still working for the three-dimensional case. In the simulation, we calculate the reference threedimensional thermal light intensity distribution based on Eq. (12). The light source is located in the plane z 0. The size of the source is 5 mm, and its central point is the origin in the coordinate system. The wavelength of the light source is 633 nm. First, we analyze the three-dimensional ghost imaging with a single-point object (the simplest object) located at (0, 0, z 150 mm). For each iteration, the phase as well as the amplitude of every component source point is altered with a zeromean Gaussian random value. The complex amplitudes of any two different component source points are statistically independent. In the ith iteration, the reference light intensity distribution is calculated based on Eq. (12) and recorded as I r x; y; z; i, where the intervals of x, y, and z are (−0.2 mm, 0.2 mm), (−0.2 mm, 0.2 mm), and (100 mm, 200 mm), respectively. The corresponding Si and Ri can be calculated according to Eqs. (1) and (2), respectively. With 5000 iterations, substituting the calculated I r i, Si, and Ri into Eqs. (3)–(6) yields four image matrixes. However, the imaging results of the four arithmetics are all in the form of a three-dimensional matrix. Referring to the feature of ghost imaging that the correlation values at the points inside the object are far larger than those outside the object, we employ a threshold search method to visualize the image from a calculated three-dimensional matrix Ω. The basic steps of the threshold search method are given as follows.
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1. Choose a proper threshold as cc Ωmax , where * is the multiplication operator, cc is a cut coefficient, and Ωmax is the maximum of Ω ⃗ρ. ⃗ρ x; y; z, set Ω ⃗ρ 0, 2. For any point if Ω ⃗ρ < cc Ωmax . 3. For any x and y, search and retain the maximum Ωx; y; z; the other points are set to 0. Then the revised Ω describes the surface or shape of the three-dimensional object, symbolized as Γ. To analyze the performance of the four ghost imaging arithmetics with a single-point object, we set cc 0.12 and only execute step 1 and 2. The four stereograms obtained with the four arithmetics are shown in Fig. 4. For one of the arithmetics, the corresponding stereogram exhibits its threedimensional coherence zone under the condition cc 0.12. According to the ghost imaging arithmetic, the image of a complex object is approximately equal to the convolution of the object function and the image of a single point. Therefore, with the same cc, the more convergent the three dimension coherence zone is, the better the corresponding arithmetic is. Apparently, the three-dimensional coherence zones of DGI and UWDGI are more convergent than those of TGI and UWGI. Moreover, the three-dimensional zones of TGI and DGI contain discrete spreads, which are caused by the nonuniform intensity distribution of the field in the nearer distance, unlike that of UWGI and UWDGI. The discrete spreads are fake coherence zones; the phenomenon is more distinct in the stereogram of TGI in Fig. 4. Therefore, we conclude that the uniformly weighted arithmetics can eliminate the fake coherence zones, and the three-dimensional zone of UWDGI has the greatest convergence. Second, we analyze the three-dimensional ghost imaging with a complex object of three-dimensional structure. In the simulation, the optical setups are the same as that with a single-point object. The three-dimensional object is shown in Fig. 5, containing many columns of different heights. The light projected on the surfaces of the columns is reflected.
Fig. 4. Three-dimensional coherence zones obtained with the four arithmetics.
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Fig. 5. Three-dimensional object. The colors are used to distinguish the heights of different columns.
The surfaces of the columns are located at the distance interval of (100 mm, 120 mm); the difference between any two heights of columns is an integral multiple of 1 mm. With 50,000 iterations, another four three-dimensional matrixes representing four images are obtained. Then we use all steps of the threshold search method to visualize the object. Before we begin with step 1, we need to find the best threshold or the best cc for imaging. Traditionally, we apply visibility or the CNR to evaluate the performance of ghost imaging, without considering the brightness or distortion. If the average light intensity of the speckle fields projected on the object is nonuniform, we can only measure the SNR or CNR at a position ⃗ρi by repeating the experiments many times. However, the quality of the integral image is hard to evaluate by these parameters. In three-dimensional imaging, the acquisition of the shape of the three-dimensional object is quite important. Therefore, we define the shape distortion in the near-field threedimensional ghost imaging as Eq. (13), to evaluate the integral image quality: 2 TGI DGI UWGI UWDGI
1.8 1.6
distortion
1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
cc
Fig. 6. cc-distortion curves of the four arithmetics. Red-solid curve, TGI; blue-dashed curve, DGI; cyan-solid curve with circles, UWGI; green-dashed curve with rectangles, UWDGI.
P distortion
ρ jΓ ⃗ρ; cc
− O ⃗ρj ; ⃗ρj jO ρ
P
(13)
where O ⃗ρ is the object function and Γ ⃗ρ; cc is the retrieved image function with a variable cc. The smaller the distortion is, the better the imaging quality is. According to Eq. (13), the distortion is a function of cc. The cc distortion curves of the four ghost imaging arithmetics are plotted in Fig. 6. Based on the cc distortion curves, the best cc for different arithmetics can be found. In addition, Fig. 6 demonstrates that the smallest distortion of three-dimensional ghost imaging is achieved with the UWDGI, and even the UWGI performs better than the DGI. To intuitively show that the cc distortion curves reflect the varying quality trends for each arithmetic, the images achieved by the four arithmetics with cc 0.2, 0.35, 0.5, and 0.6 are depicted in Fig. 7. The four images in the first, second, third, and fourth line of Fig. 7 are obtained with TGI, DGI, UWGI, and UWDGI, respectively. Apparently, the varying quality trends in Fig. 7 agree with those in Fig. 6, which demonstrates that the distortion can reflect the quality of the near-field three-dimensional thermal light ghost imaging. Furthermore, the minimum distortion varies with the increase in iterations, as well. By simulation, the curves indicating the relations between the iterations and the minimum distortion of the four arithmetics are plotted in Fig. 8. The curves demonstrate that the TGI has the worst performance, the UWDGI has the best performance, and the imaging quality of UWGI surpasses that of DGI after a certain number of iterations. Eventually, the best cc or threshold for each arithmetic is determined. With 50,000 iterations, the best cc for TGI, DGI, UWGI, and UWDGI are 0.53, 0.24, 0.63 and 0.33, respectively. Substituting the best cc into the threshold search method, visible images of the object are obtained with the different arithmetics. The four visible images are shown in Fig. 9, which intuitively supports the theoretical analysis. All the simulation results demonstrate that the UWDGI has the best performance in ghost imaging with a nonuniform thermal light field. Note that in this paper we discussed thermal light threedimensional ghost imaging only at near distance, because the longitudinal resolution of this method is quite weak at long
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Fig. 7. Ghost images achieved by TGI, DGI, UWGI, and UWDGI with different values of cc.
1
The minimun distortion
0.8 TGI DGI UWGI UWDGI
0.6
0.4
0.2
Fig. 9. Final images obtained by the four arithmetics. 0
0
1
2
3 Iterations
4
5 4
x 10
Fig. 8. Iterations versus minimum distortion curves of the four arithmetics.
distance [24–28]. Recently, the Han group realized longdistance three-dimensional ghost imaging with a much better longitudinal resolution (about 60 cm longitudinal resolution with a distance of 1 km), by using time-resolved measurement [29].
4. CONCLUSION In conclusion, we proposed uniformly weighted arithmetics to improve the imaging quality of ghost imaging with a
nonuniform thermal light field. In the first section, we introduced four ghost imaging arithmetics, TGI, DGI, UWGI, and UWDGI. Then in the second section, we described two kinds of nonuniform thermal light fields, including the planar Gaussian distributed thermal light field and the three-dimensional thermal light field in the near field. In particular, we discussed the second kind of field in detail, because of the lack of investigations of three-dimensional ghost imaging. We introduced the shape distortion parameter to evaluate the quality of the integral image. With numerous simulations, we concluded that the uniformly weighted arithmetic is capable of eliminating the negative influence of the nonuniform field on the performance of ghost imaging, and the best imaging quality is achieved with the UWDGI. In addition, the longitudinal resolution in the simulation is less than 1 mm at a distance of
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about 120 mm. Our work is quite meaningful for the practical applications of ghost imaging.
ACKNOWLEDGMENT This work is supported by the National Natural Science Foundation of China (grants 60970109, 61170228).
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