Correlated imaging, which is a computational imaging method developed in 1995, claims that the image of an unknown object can be obtained using a single-pixel detector^[1–8]. Similar to optical imaging and diffraction techniques in traditional information acquisition systems, there are two main imaging modes for correlated imaging: one is ghost imaging (GI)^[3,9], and the other is Fourier-transform ghost diffraction (FGD)^[2,9,10]. For GI, all the photons transmitted/reflected from the object are usually collected by a bucket detector, which is actually an incoherent imaging method^[3–5], whereas for FGD, the target is illuminated by an incoherent source and a single point-like detector is adopted to detect the object’s information, but the technique is, in essence, corresponding to coherent imaging^[2,9,10]. Up to now, many theoretical and experimental results have demonstrated that correlated imaging has some important applications in remote sensing^[11,12], three-dimensional imaging^[13–15], and biomedical imaging^[16–18]. Before 2011, all the investigations on correlated imaging were focused on static targets^{[1–10,16,19–21]}. However, different from conventional imaging, the acquisition of high-quality correlated imaging usually needs a mass of random measurements, which will cause image blurring when there is relative motion between the target and the illumination system during the sampling process^[22,23]. With the development of GI and FGD techniques, a series of works were recently focused on GI for moving targets^[23–32]. In 2012, Zhang investigated the effect of lateral motion on the spatial resolution of GI; he found that high-quality GI for a moving target with a constant speed can be obtained by correcting the speckle patterns recorded by a charge-coupled device (CCD) camera in the reference path^[23]. Later, Zhang et al. showed that the spatial resolution degradation caused by the target’s random lateral shaking could be overcome by FGD, without correcting the speckle patterns recorded by the reference detector^[24]. Subsequently, based on the priori information of motion estimation and image correction, some motion deblurring methods on GI for axially moving targets and laterally moving targets were reported^[26–32]. However, the data processing of deblurring for these methods is complicated, and the deblurring ability is also restricted by the accuracy of motion estimation. In this paper, a motion deblurring imaging system based on FGD with pseudo-thermal light is proposed without using the priori information of motion estimation and image correction, even if the target has both lateral and axial random relative motion at the same time. Based on the principle of FGD, the condition of motion deblurring for the proposed optical system is deduced and its physical explanation is clarified. The validity of the proposed method is also demonstrated by experiments.

Figure 1 presents the schematic of a motion deblurring imaging system based on the Fourier-transform ghost diffraction technique for a target with lateral and axial translation motion. By passing a laser beam through a slowly rotating ground glass disk (RGGD), the pseudo-thermal light source S is obtained and then is divided into a test and a reference path by a beam splitter (BS). In the test path, the light propagates through a target and then is collected onto a CCD camera Dt by a lens L2 with the focal length f2. In the reference path, the light goes through another lens with the focal length f1 and then to a CCD camera Dr that is fixed on the focal plane of lens L1. Moreover, the target is placed on a motion platform, and the platform is driven by two stepping motors so it can move simultaneously in lateral and axial directions.

For correlated imaging with thermal light, the second-order correlation function of the intensity fluctuation recorded by two detectors is^[2–4]$$\begin{gathered} ΔG(2,2)(xr,xt)=|∫dx1∫dx2G(1,1)(x1,x2) \\ ×hr*(x1,xr)ht(x2,xt)|2, \end{gathered}$$(1)where G(1,1)(x1,x2) is the first-order correlation function of the light field at the source plane, ht(x2,xt) is the impulse response function in the test path, and hr*(x1,xr) denotes the phase conjugate of the impulse response function in the reference path.

For the schematic shown in Fig. 1, when the transmission apertures of both lenses L1 and L2 are large enough and under the condition of paraxial approximation, the impulse response function of the reference path is hr(x1,xr)∝exp{jπλf1(1−zf1)xr2−2jπλf1x1xr},(2)where z is the distance between the RGGD plane and the lens L1 plane, and λ is the wavelength of the illumination source.

Because of the lateral and axial motion of the target in the process of sampling, the distance between the source plane and the target plane is time-dependent; it can be expressed as z1(t). Simultaneously, the deviation of the target’s center position from the optical axis is also time-dependent. If the target’s center position deviates ξ from the optical axis at time t, then the target’s transmission function will be changed from T(x) to T[x−ξ(t)]. Thus, the impulse response function of the test path can be expressed as $$\begin{gathered} ht(x2,xt)∝∫dxexp[jπλz1(t)(x2−x)2]T[x−ξ(t)] \\ ×exp[jπλf2(1−z2(t)f2)xt2−2jπλf2xxt], \end{gathered}$$(3)where z2(t) is the distance between the target plane and the lens L2 plane.

If the light field generated by the RGGD is fully spatially incoherent and the intensity distribution is assumed to be uniform as a constant intensity I0, then G(1,1)(x1,x2)=I0δ(x1−x2),(4)where δ(x) is the Dirac delta function. Substituting Eqs. (2)–(4) into Eq. (1), the second-order correlation function of intensity fluctuation can be simplified as $$\begin{gathered} ΔG(2,2)(xr,xt)∝|∫dx2∫dxexp[2jπλ(xrf1−xz1(t))x2] \\ ×exp[jπλz1(t)(x22+x2)−2jπλf2xxt] \\ ×T[x−ξ(t)]|2. \end{gathered}$$(5)

If the diameter of the laser spot illuminating the RGGD plane is D and z1(t)≪D2λ (namely, the target should be located in the near field of the illumination source^[21]), after some calculation, we can obtain $$\begin{gathered} ΔG(2,2)(xr,xt)∝|∫dxT(x−ξ(t))exp[−2jπλ(xtf2−xrf1)x]|2 \\ =|F˜[T(x)]2πλ(xtf2−xrf1)exp[−2jπλ(xtf2−xrf1)ξ(t)]|2 \\ =|F˜[T(x)]2πλ(xtf2−xrf1)|2, \end{gathered}$$(6)where F˜ denotes the Fourier transform of a function. If we set xt=0, then Eq. (6) can be rewritten as ΔG(2,2)(xr,xt=0)∝|F˜[T(x)]−2πλf1xr|2.(7)

From Eq. (7), it is observed that the result of FGD does not depend on z1(t) and ξ(t), which means that both the lateral and axial motion between the target plane and the source plane has no effect on the quality of FGD. Therefore, a high-resolution image of the target can always be obtained by the phase-retrieval method from the FGD patterns, even if the amplitude and mode of the target’s motion are random.

According to Klyshko’s picture^[4,33], the physical essence of motion deblurring imaging system based on FGD technique shown in Fig. 1 can be explained as follows: by the reversibility of the light field, as displayed in Fig. 2, a coherent point light source, emitting the light from the detector Dt, first goes through lens L2 and then is collimated to a parallel light beam. Next, the parallel light successively transmits through the target and source S. Because source S acts as a phase-conjugated reflection mirror^[3], the light field reflected by source S is transformed into the far field by lens L1 and the target’s FGD patterns will appear on the CCD camera Dr. What is more, it is clearly seen that the imaging process of Fig. 2 corresponds to the case that the target is illuminated by a parallel coherent light and is detected by the detector in the Fourier plane. Therefore, when the target’s motion range in the lateral direction is smaller than the effective transmission aperture of the lens L2 and the target is moving simultaneously along the axial direction, high-quality FGD can always be obtained even if the lateral and axial motion modes are random. In addition, it is emphasized that source S also acts as a low-pass filter; thus, the imaging resolution of the present system is determined by the transverse size of the speckle pattern on the target plane, which is the same as the case of GI with a bucket detector^[9].

To verify the validity of the proposed motion deblurring imaging system based on FGD technique and analytical results, the experimental parameters of the setup in Fig. 1 are set as follows: the wavelength of the laser is λ=532nm, z=400mm, z1+z2=400mm, f1=f2=200mm, and the effective transmission apertures of both lenses L1 and L2 are 10.0 mm. In addition, the transverse size of the BS is 25 mm, the transverse pixel size of the CCD camera is 6.9 µm, and the sampling frequency is 150 frame/s (fps). The target used for demonstration is a three-slit (slit width a=200μm, slit height h=1600μm, and center-to-center separation d=800μm), and the measurement number used for image reconstruction is 10,000. Moreover, the target does a back-and-forth motion along the lateral and axial directions, and the motion ranges are both restricted. When the target has only lateral motion and the transverse size of the source S is set as D=3.0mm, Fig. 3 presents the experimental results of the motion deblurring method via FGD in different lateral motion amplitudes. The FGD patterns are shown in Figs. 3(a)–3(d) where the maximum lateral motion amplitudes deviating from the optical axis are set as 0, 0.5, 1, and 2 mm, respectively. Based on phase-retrieval Fienup’s iterative algorithm^[34,35], the target’s image in the spatial domain, as displayed in the upper right corners of Figs. 3(a)–3(d), can be retrieved from the FGD patterns. For comparison, when the transverse size of the laser beam on the RGGD plane is D=6.0mm, the corresponding results of conventional Fourier diffraction, which is achieved using the test path and removing the RGGD in Fig. 1, are shown in Figs. 3(e)–3(h). It is clearly seen that as the maximum lateral motion amplitude deviating from the optical axis is increased and the effective transmission aperture of the lens L2 is larger than the target’s lateral motion amplitude, high-quality FGD can always be achieved and it is the same as the static case [namely, Fig. 3(a)], which accords with the theoretical analysis above. However, because the target is illuminated by a divergent light source, the imaging resolution will be dramatically decayed with the increase of the lateral motion amplitude for the conventional Fourier diffraction technique, which is attributed to the shift of the Fourier patterns caused by the target’s lateral motion. Therefore, in comparison with conventional Fourier diffraction techniques, even if a divergent light source illuminates the object and the light source is incoherent, the ability of motion deblurring for the proposed method is also valid, which can dramatically reduce the requirement of the source and is helpful to imaging in the waveband without a coherent source.

When the target’s center position is located in the optical axis, the target has only axial motion; Fig. 4 gives the experimental results in different axial motion amplitudes. The other experimental parameters are the same as Fig. 3. When the target goes back and forth in the sampling process, the axial motion ranges around z1=160mm are ±10, ±20, and ±40mm; the target’s FGD patterns and corresponding reconstructed image in the spatial domain are displayed in Figs. 4(a)–4(c). As predicted by Eq. (7), the target’s motion in the axial direction also has no effect on the quality of FGD. Similar to Fig. 3, the corresponding results of conventional Fourier diffraction are shown in Figs. 4(d)–4(f). In comparison with the case of lateral motion, the influence of the target’s motion in axial direction to the patterns obtained by conventional Fourier diffraction is relatively weaker because the optical system’s phase difference in the axial direction is very small in the present experiments [For conventional Fourier diffraction with a divergent light source, the quadratic wave distribution of the light field caused by the source’s phase difference on the target plane is exp[jπλz1(t)x22] and usually the distance z1(t) is much larger than x22 in the condition of paraxial approximation].

To further verify the capability of the proposed deblurring imaging system, the target’s maximum lateral motion amplitude is 2.0 mm and its axial motion range around z1=160mm is ±40mm; Figs. 5(a) and 5(b) show the reconstruction results obtained by the proposed FGD system and conventional Fourier diffraction. It is obvious that even if the target has relative motion in both lateral and axial directions, there is no influence on the quality of FGD and the imaging resolution is also the same as the static case, which is consistent with the analytical result described by Eq. (7) and the phenomenological explanation in Fig. 2. In addition, although we only give the experimental demonstration of translational motion deblurring for imaging a pure amplitude target in the present work, the proposed method is also valid to imaging a complex-valued target, which is superior to the method in Ref. [25], where only the target’s amplitude information can be obtained.

In conclusion, on the basis of the property of FGD, we have proposed a motion deblurring imaging method with pseudo-thermal light which can resist axial and lateral motion blurring. Even if a divergent light source illuminates the target and the target is positioned in the near-field region of the source, high-resolution imaging can always be reconstructed by a phase-retrieval algorithm from the FGD patterns, which cannot be achieved by conventional Fourier diffraction. What is more, in comparison with the previous GI deblurring approaches, this method does not use the priori information of motion estimation and need not correct the speckle’s distribution recorded by the reference detector. This technique is simple and robust, which provides a useful approach to overcome the image blurring caused by random high-frequency shaking, and is helpful to photography and optical surface detection.