1206 OPTICS LETTERS / Vol. 35, No. 8 / April 15, 2010

High-quality quantum-imaging algorithm andexperiment based on compressive sensing

Liu Jiying,1,* Zhu Jubo,1,2 Lu Chuan,2 and Huang Shisheng1

1Department of Mathematics and Systems Science, College of Science, National University of Defense Technology,Changsha 410073, China

2Department of Physics, College of Science, National University of Defense Technology, Changsha 410073, China*Corresponding author: gene0572@163.com

Received December 2, 2009; revised February 5, 2010; accepted February 23, 2010;posted March 3, 2010 (Doc. ID 120610); published April 15, 2010

Quantum imaging (QI) has some unique advantages, such as nonlocal imaging and enhanced space reso-lution. However, the quality of the reconstructed images and the time of data acquisition leave much to bedesired. Based on the framework of compressive sensing, we propose an optimization criterion for high-quality QI whereby total variation restriction is specifically utilized for noise suppression. The correspond-ing reported algorithm uses a combination of a greedy strategy and the interactive reweight least-squaresmethod. The simulation and the actual imaging experiment both show a significant improvement of the pro-posed algorithm the over previous imaging method. © 2010 Optical Society of America

OCIS codes: 270.4180, 100.6640.

Since the first experiment was completed by Pittmanet al. [1] in 1995, the theoretic and experimental in-vestigation of quantum imaging (QI) has achievedgreat progress. In 2000, Boto et al. [2] pointed outthat the entangled N photon could break through thelimitation of Rayleigh diffraction and improve thespatial resolution by a factor of N. Recently, Gatti etal. [3], Zhai et al. [4], Xiong et al. [5], Bai and Han [6],and Cai and Zhu [7] reported QI experiments that in-volved the substitution of chaotic thermal light forentangled photons. The principle on which thermallight QI is based is the second-order spatial correla-tion of thermal radiation. This principle was demon-strated by Hanbury-Brown and Twiss [8] in 1956. QIhas at least two unique features: (1) generating theimage of the unknown object in a nonlocal manner [9]and (2) improving the spatial resolution of the imag-ing system.

However, chaotic thermal light QI suffers from twoinherent drawbacks. The maximal visibility modula-tion of thermal light QI is limited to 33% [9]. In ad-dition, it takes a significant amount of time to accu-mulate sufficient data for high signal-to-noise ratio(SNR) image reconstruction [10,11]. Thus, it is natu-ral to ask whether high-visibility and high-SNRquantum images can be attained with fewer mea-surements.

Compressive sensing (CS) has provided the math-ematical basis [12–15]. Employing the sparsity of im-ages, high-quality images could be reconstructedeven when far fewer data are measured. In thepresent study, the process of data generation is math-ematically reformatted within the framework of CS.Furthermore, to smooth the noisy image, total varia-tion (TV) [16] restriction is added to the criterion ofoptimization. A corresponding algorithm is proposedfor solving the optimization problem, and this algo-rithm combines the greedy strategy [17] with the in-teractive reweight least-squares is method [18]. Fi-nally, the performance of our algorithm was validated

by a chaotic pseudothermal light QI experiment.

0146-9592/10/081206-3/$15.00 ©

A classic chaotic thermal light QI schematic isshown in Fig. 1. The chaotic thermal light is dividedinto two paths by a splitter. One of them is the testpath, where an object mask is illuminated and thetransmitted light is converged by a lens and recordedby a single-pixel bucket detector. The other path isthe reference path, and the intensity of the light fieldis recorded by a 2D CCD array. The image can begenerated by calculating the mutual correlation be-tween the single-pixel detector and the 2D CCD ar-ray. QI is also known as ghost imaging [19,20].

Denote the measurements of the single pixel detec-tor as aj, j=1, . . . ,J, where J is the amount of mea-surements. The data collected by the CCD array isMj, j=1, . . . ,J. The conventional QI formula is basedon the correlation value of the aj and the Mj, i.e.,

Ic = �j

�aj · Mj� � ��j

aj · �j

Mj� , �1�

where Ic is the n�n-dimensional image of object andn is the size of the CCD array.

To recast QI based on CS theory, we must first re-formulate it in the CS framework. The Mj can be re-shaped into a (row) vector mT

j. These mTj can then be

rearranged into a projection matrix M row by row. Atthe same time, the measurements of the single pixeldetector can be rearranged into a (column) vector a.

Fig. 1. (Color online) Schematic of chaotic thermal light

QI.

2010 Optical Society of America

April 15, 2010 / Vol. 35, No. 8 / OPTICS LETTERS 1207

If we denote the unknown image as a N-dimensionalvector x, where N=n2, then the data acquisition canbe described as

a = M · x. �2�

CS theory maintains that if the image in questionmeets the sparsity requirement, i.e., there exists arepresentation matrix D such that x=D ·� and K�N, where K is the number of nonzero entries in co-efficient �. Furthermore, if the projection matrix Mhas the restricted isometry property [13], then high-quality image reconstruction can be achieved via theoptimization in Eq. (3),

�s = arg min��a − M · D · ��22 + ����1, xs = D · �s,

�3�

where xs is the reconstructed image and � controlsthe strength of sparse restriction.

Candes et al. [13] have proved that some randommatrices will obey the restricted isometry propertywith a probability of almost one. In QI, the character-istic of chaotic thermal light determines the random-ness of the projection M. It is important to emphasizethat xs from Eq. (3) is an estimate of the intensity ofthe 2D optical field behind the object, which is notsimilar to the second-order spatial correlation fromEq. (1). Consequently, the visibility modulation is nolonger limited to the maximum of 33%.

We analyzed the performance of Eq. (3) under dif-ferent noise levels. A semireal numerical experimentwas designed to this end. The projection matrix Mwas sampled from a real system, and J � 2000. Thena dual-slit image was generated numerically [see Fig.2(a)] to serve as the object mask. The measurementsof the single pixel detector were formed by using Eq.(2) and subsequently contaminated by white Gauss-ian noise with different SNRs. The discrete cosinetransform was chosen as matrix D.

Figure 2 shows the images reconstructed via Eqs.(1) and (3) when SNR=15 dB and 40 dB. The curvewith triangles in Fig. 3 shows the rms error (RMSE)of the reconstruction due to the different SNRs.These results indicate that, with the increase of SNR,the reconstruction quality improvement of the con-

Fig. 2. (Color online) Results of semireal experiments. (a)Numerical dual-slit image; (b), (c), (d) reconstructed imagesvia conventional method, sparse recovery algorithm with-out and with TV restriction, respectively, when SNR=15 dB; (e), (f), (g) corresponding results when SNR

=40 dB.

ventional method is limited; however, the spare re-covery from Eq. (3) showed a significant improve-ment. In addition, when the SNR is below 15 dB, theresults of the spare recovery are completely obscure.

To suppress the influence of noise, TV restrictionwas added to the process of sparse recovery. Themodified optimization is formulated in Eq. (4),

�sg = arg min��a − M · D · ��22 + ����1

+ ��G · D · ��1, xsg = D · �sg, �4�

where xsg is the reconstructed image with TV restric-tion, � controls the strength of TV restriction, and Gis the 2D difference operator [18].

Algorithm 1, proposed for solving the optimizationproblem in Eq. (4), is a compromise between the abil-ity to obtain an accurate solution and retainingefficiency. In this algorithm, the outer iteration(while...end) realizes the greedy strategy, and it en-sures the sparsity of coefficient �. The inner iteration(for...end) is the process of reweighted least squares,and it contains a TV restriction.

Figure 2 also shows the images of sparse recoverywith TV. The curve with circles in Fig. 3 representsthe RMSE error of reconstruction due to differentSNRs. A comparison reveals that the results of Eqs.(3) and (4) are similar when SNR is high, whereasthe results of Eq. (4) are much better than those ofEq. (3) when SNR is low. These data indicate that theoptimization in Eq. (4) and algorithm 1 can signifi-cantly suppress the influence of noise.

Algorithm 1

Initialize: tol ,� ,� ,S ,� ,I for certain values,and k=0,�k=0,�=�

While ��k−�k−1�2 tolrk=a−M ·D ·�k

pk=DT ·MT ·rk

�add= � : p�� is one of the S largest entries in �p�=�� ��add /��

T T −1 T

Fig. 3. (Color online) RMSE curves of sparse recoverywithout and with TV.

�k,1= ��MD�� MD�+�G� G�� �MD�� a

1208 OPTICS LETTERS / Vol. 35, No. 8 / April 15, 2010

D� and G� are submatrixes of D and G,whose columns are indexed by �

For i=1 to IWD=diag�fD�M ·D� ·�k,i−a��;WG=diag�fG�G� ·D� ·�k,i��

where: fD�x�= �x−1, if x�

�−1, if x�� ;fG�x�= �x−1, if x�

0,if x�� �k,i= ��MD��TWFMD�+�G�

TWRG��−1�MD��TWFaEnd

�k= ��k�� :�k��= ��k,I�� , if ��

0,if �� EndOutput: xk=D ·�k.

A chaotic pseudothermal light QI experiment wascarried out to validate the performance of the pro-posed algorithm. A snapshot of our experimental sys-tem is shown in Fig. 4. A light beam (produced by Fig.4, 1–6) passed through a rotating ground glass (Fig.4, 7) to generate chaotic light. In Fig. 4 8 is a 1:1beam splitter; 9 and 11 are two CCD arrays. The firstis a counterpart of the CCD in Fig. 1, and the secondserves as a single pixel detector by summing its 2Dmeasurements to a single value. An object mask (10)was placed between the splitter (8) and the CCD (11).The number of measurements was 2000.

Figure 5(a) is a picture of the object mask. Figures5(b) and 5(c) are the corresponding image reconstruc-tions obtained via the conventional method and algo-rithm 1, respectively. The latter result is a clear im-provement over the former in many aspects, such asvisibility and the background noise of the images. Itis estimated that it requires at least 10 times thenumber of measurements for the conventionalmethod to achieve the imaging quality seen in Fig.5(c). The result of optimization in Eq. (3) will be ob-scured by the noise (the SNR is about 15 dB).

Inherent inefficiencies have limited the applicationand extension of QI. The present Letter proposes ahigh-quality image reconstruction algorithm withinthe framework of compressive sensing. In addition toemploying the sparsity of images as part of the usualCS reconstruction process, the algorithm emphasizes

Fig. 4. (Color online) Snapshot of our QI experimentsystem.

the suppression of the influence of noise contained inactual measured data. This is achieved by adding theTV minimization to the original CS optimizationfunction.

It is well known that computational efficiency is in-tractable in sparse recovery. Although we have em-phasized this problem in our algorithm, the con-sumption of memory and time still substantiallyexceeds that of the conventional imaging method,particularly when the images are large. Future re-search will focus on the further alleviation of thesecomputational burdens.

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Fig. 5. (Color online) Reconstructed images from the cha-otic pseudothermal light QI experiment. (a) Object mask;(b), (c) images calculated by using the conventional methodand algorithm 1.