Ghost imaging (GI), as a nonlocal imaging method, can image an unknown object with a single-pixel detector at the object path^[1–6]. In the most recent two decades, GI has been receiving increasing interest and lots of achievements have been made, especially in the fields of remote sensing^[7–9], X-ray microscopy^[10,11], three-dimensional imaging^[12,13], and super-resolution imaging^[14–16]. The feasibility of GI has also been experimentally demonstrated^[7–20] from X-rays to microwave sources. However, there is still a long way to go for the practical application of GI, because some issues like imaging speed and moving target imaging without the prior knowledge of motion feature have not been solved and some physical mechanisms have not been clarified up to now. For example, for conventional imaging (CI), the imaging signal-to-noise ratio (SNR) is the same as the detection SNR. However, when the intensity of light illuminating the object plane is the same, the detection SNR of GI increases with the object’s transmission area because all the photons transmitted from the object illuminate the same single-pixel detector, but the imaging SNR of GI is reduced and is also related to the property of random coded patterns illuminating the object^[5,21–23], which is entirely different from CI. It is natural to ask what the quantitative relationship between the detection SNR and the imaging SNR of GI is. Although Erkmen and Shapiro have done some theoretical analysis on factors affecting the imaging SNR of GI, it was only applied to the random coded patterns with the Gaussian statistical property and the computation of the imaging SNR is relative complicated^[22]. Can we propose a new imaging SNR for GI that is used for any random patterns with different statistical properties and is easy to compute? For another example, the photon shot noise, which is the main factor affecting the imaging SNR of CI, will cause the detection signal’s intensity fluctuation, and thus it also affects the imaging SNR of GI due to the object’s information extraction that originates from the intensity fluctuation correlation of light fields for GI. It is natural to ask what conditions should be satisfied if GI is better than CI in photon shot noise cases. In this Letter, we propose an imaging SNR called SNRtran to evaluate the imaging quality of both GI and CI. Based on the deduced SNRtran, the influences of some parameters like the photon shot noise, the object’s transmission area, and the number density of photons illuminating the object plane on the imaging quality of GI are clarified, and the performance differences between CI and GI are also discussed by theoretical analysis and numerical simulations.

Figure 1(a) presents a typical schematic of computational GI. The light emitted from a pulsed laser uniformly illuminates a digital micro-mirror device (DMD) and a series of random coded patterns are prebuilt by modulating the mirrors of the DMD. Then the patterns reflected by the DMD are imaged onto an object by an optical imaging system with the focal length ft, and the photons transmitted through the object are imaged onto a bucket detector Dt by using another conventional imaging system with the focal length fr. In comparison with Fig. 1(a), a conventional imaging setup is shown in Fig. 1(b), where the DMD and the bucket detector Dt are replaced by a reflection mirror and a CCD camera, respectively.

In the framework of computational GI, the object’s image OGI can be reconstructed by computing the intensity correlation between the pattern’s intensity Irs(x,y) modulated by the DMD and the total intensity IBs recorded by the bucket detector Dt^[6], where s denotes the sth measurement and K is the total measurement number. In addition, 〈Irs(x,y)〉=1K∑s=1KIrs(x,y) represents the ensemble average of Irs(x,y), and ΔIrs(x,y)=Irs(x,y)−〈Irs(x,y)〉 is the intensity fluctuation of Irs(x,y).

When the numerical aperture of the lens with a focal length fr shown in Fig. 1(a) is large enough, and considering the photon shot noise of the detection system, the total intensity IBs can be expressed as where Inoises=∬dxdyIrs(x,y)O(x,y) and O(x,y) is the object’s transmission function. In addition, Inoises=Poisson(Isignals)−Isignals and Poisson(Isignals) represents the Poisson distribution with the mean value Isignals.

For the setup of GI displayed in Fig. 1(a), the bucket detector Dt collects all the photons transmitted through the object. If only considering the photon shot noise of the detection system, the detection signal-to-noise ratio (DSNR) of GI can be represented as where Io=〈Irs(x0,y0)〉 is the average number density of photons illuminating the object plane, the quantum efficiency of the detector is assumed to be 1, and Ao=∬dxdyO(x,y) is the transmission area of the object. In addition, 〈Isignals〉=Io∬dxdyO(x,y)=IoAo and std(Inoises)=1K∑s=1K(Inoises−〈Inoises〉)2=1K∑s=1K(δInoises)2 denotes the standard deviation of the noise vector Inoises. From Eq. (3), it is obviously observed that the value DSNRGI of GI depends on the number density of photons illuminating the object plane Io and the object’s transmission area Ao.

When the measurement number K is large enough or the patterns ΔIrs(x,y) conform to an orthogonal statistical distribution [namely the inner product of any two speckles ΔIrs(x,y) is zero and the mean value of ΔIrs(x,y) is also zero], Eq. (1) can be simplified as where μ(x,y)=〈ΔIrs(x,y)ΔIsignals〉〈Irs(x,y)〉〈Isignals〉 denotes the coherence function of the two signals and ⊗ denotes the convolution symbol. fs=δInoises/std(Inoises) is approximately a noise vector with the mean value 0 and the standard deviation value 1, and N(x,y)=1K∑s=1KIrs(x,y)Iofs. From Eq. (4), it is clearly seen that the first term of Eq. (4) corresponds to the object’s image and the spatial resolution of GI is determined by the function μ(x,y). The second term of Eq. (4) represents a random noise image, which will cause a degradation of GI quality. In addition, because the amplitude of the noise image is AoIo, the imaging SNR of GI will decrease as the object’s transmission area Ao is increased, and increase with the photon number density Io when only the photon shot noise of the detection system is considered, which is similar to the theoretical results of GI with a Gaussian speckle field^[22].

The imaging SNR of the object’s transmission region SNRGItran for GI can be defined as where δ(OGItran)=1NxNy∑xNx∑yNy[OGItran(x,y)−O¯GItran]2 and O¯GItran=1NxNy∑xNx∑yNyOGItran(x,y) are the standard deviation and mean value of the reconstructed object’s transmission region in the spatial domain, respectively.

Conventional imaging is based on the point-to-point information extraction mode, and thus the imaging SNR is also equal to the system’s DSNR. In comparison with GI, for the schematic shown in Fig. 1(b), if only the photon shot noise of the detection system is considered, then the imaging SNR of the object’s transmission region SNRCItran can be expressed as where Iphoton is the photon number received by the CCD camera at each pixel. In addition, Eq. (6) suggests that both DSNRCI and SNRCItran of CI increase with Iphoton.

To verify the analytical results, the parameters of the numerical simulation based on the schematic of Fig. 1 are set as follows: the wavelength of the laser is 532 nm and the transverse size of the laser beam illuminating the DMD is 10 mm by a conventional imaging system with the magnification 4×. The modulated area of the DMD is 64×64pixels and the speckle’s transverse size is set as 54.6μm. The speckles modulated by the DMD are Hadamard patterns and the measurement number K=4096, and thus the average number density of photons illuminating the DMD or the reflecting mirror is I0=Iphoton=2Io for the demonstration of the performance comparison between CI and GI. In addition, zt1=zt2=zr1=zr2=200mm, ft=fr=100mm, and the transmission apertures of both the lenses ft and fr are 25 mm. For the image reconstruction of GI, we have used the intensity fluctuation correlation reconstruction algorithm^[24]. When the object’s transmission area is Ao=20×20pixels2, Figs. 2(a) and 2(b) illustrate the relationship between the value DSNR/SNRtran and I0 for both CI and GI. The imaging results of both CI and GI are shown in Figs. 2(c)–2(f) when the number density of photons illuminating the DMD is I0=1,10,100, and 1000photons/pixels2, respectively. If I0 is fixed at 40photons/pixels2, the DSNR and SNRtran on the object’s transmission area Ao are shown in Fig. 3. From Figs. 2 and 3, it is obviously observed that for CI the values of both DSNRCI and SNRCItran increase with I0 and do not depend on Ao, whereas the value DSNRGI is proportional to IoAo for GI. What is more, the value of SNRGItran increases with Io but is reduced with the increase of Ao. Such simulation results displayed in Figs. 2 and 3 agree with the theoretical prediction described by Eqs. (3)–(6). In addition, the results shown in Figs. 3(b) and 3(f) also suggest that the imaging quality of CI will be better than that of GI when the object’s transmission area Ao is beyond a threshold value Aothreshold [for example, Aothreshold=2000pixels2 in Fig. 3(b)].

In order to further clarify the performance differences between CI and GI, Fig. 4 presents SNRGItran and SNRCItran for different I0 and Ao. From Fig. 4, we find that if Ao>Aothreshold=2000pixels2, the value SNRCItran will be always greater than that of GI and the influence of the photon number density I0 on SNRCItran and SNRGItran is displayed in Fig. 5 when Ao=2000pixels2 is fixed. It is clearly seen that if Ao=Aothreshold, both the imaging quality [see Figs. 5(c)–5(g)] and SNRtran [see Fig. 5(b)] of CI and GI are equivalent for the same I0. Furthermore, as shown in Fig. 6, for the diagonal value of SNRtran of GI [see Figs. 4 and 6(a)], the value of SNRGItran is also the same if AoIo maintains a constant value, which can be explained by Eq. (4) and means that imaging an object with a large transmission area needs a much higher DSNRGI compared with imaging an object with a small transmission area [see Eq. (4)]. Therefore, in order to obtain the same SNRGItran for two objects with the transmission areas A1 and A2, based on Eq. (3) and Eq. (4), the DSNRGI should satisfy DSNRA1/DSNRA2=A1/A2. In addition, although we have used Hadamard patterns to illuminate the object for GI, Eq. (4) is universal and the results described above can be applied to GI with any random pattern.

In conclusion, the defined SNRtran is valid to evaluate the imaging quality of both GI and CI for a transmission object. Both the analytical and simulated results have shown that for CI the value DSNRCI, which is the same as SNRCItran, increases with I0 and does not depend on Ao, whereas the DSNRGI is proportional to IoAo for GI. What is more, the SNRGItran will be enhanced as Io is increased but reduced with the increase of Ao. In addition, we can obtain the same SNRGItran when AoIo maintains a constant value, and SNRCItran will be larger than SNRGItran when Ao is beyond a threshold value. Such results are helpful for the solution selection of GI and CI in practical applications.