1. Introduction

Incoherent digital holography (IDH) uses the self-interference principle to create holograms of objects under spatial incoherent light, such as fluorescence, light emitting diodes (LEDs) and sunlight, to reconstruct the three-dimensional volume of objects without the need for a scanning device [1,2]. In the IDH, the Michelson interferometer or spatial light modulator (SLM) is used as a wavefront division device and thus, self-interference incoherent digital holographic (SIDH) and Fresnel incoherent correlation holographic (FINCH) systems are proposed. Moreover, both of these methods have been applied in biofluorescence imaging [3–5], nanoparticle tracking [6,7], phase difference imaging [8], temperature measurements [9] and 3D natural light holographic [10,11] cameras. Notably, FINCH has lateral superresolution capability and a simple structure [12,13], which promotes the commercialization of FINCH-based 3D super resolution imaging devices [14]. In this paper, we select the FINCH imaging system to verify the effectiveness of our proposed method.

The FINCH system uses an SLM to load the designed lens phase by pixel multiplexing method [15] or polarization multiplexing [16] and splits the light emitted from a point source into modulated and unmodulated beams with different curvatures. These two beams interfere with each other in the camera plane and create a coaxial point hologram. According to the linear space invariance property of the incoherent imaging system, the object hologram of the whole 3D object can be regarded as the incoherent intensity superposition of all the point holograms. The reconstructed field of a coaxial hologram contains the image, twin image, and zeroth-order terms, which are overlapped and usually separated by a phase-shifting strategy. In the FINCH system, an SLM is also used as a phase shift device. SLM must be triggered synchronously with the camera first, and then the camera is used to capture at least 3 different phase-shift holograms at multiple exposures. If there is any movement or deformation of the object during this process, the hologram computationally synthesized by the phase-shift algorithm will be incorrect, which is why phase-shift FINCH cannot achieve dynamic imaging. Recently, there has been increasing interest in the single-shot FINCH method. For example, an off-axis configuration is used to introduce the linear carrier-frequency phase to the recorded self-interference hologram, so the undesired terms can be filtered out in the Fourier domain [17–20]. However, this approach sacrifices the resolution of the reconstructed image and the space-bandwidth product of the camera. Parallel phase shifting techniques can extract four sub-holograms with different phase shifts from an image [21–23], but the resolution of sub-holograms is reduced by 2 × 2 pixels compared with the original image recorded by the polarization camera. By loading a grating phase on the spatial light modulator or using a prepared checkerboard grating, the two beams that interfere with each other in the camera plane are divided into four copies so that the image acquired subsequently contains four sub-holograms with different phase shifts [24–26]. This method does not reduce the resolution of the hologram. Nevertheless, to avoid overlapping between the sub-holograms, each sub-hologram cannot exceed a certain size, meaning that the imaging field of view is only a quarter of the image sensor. The deep learning-based phase-shifting method can predict multiple holograms with different phase shifts from an experimentally captured hologram [27,28]; however, a large amount of labeled data is inevitably needed to train the network, and the method has the problem of insufficient generalizability, so it is not a general method for reconstructing the 3D information of arbitrary objects. In the FINCH system, the image of the object at different depths can also be reconstructed by a cross-correlation between the object hologram and the point spread holograms (PSHs), but this approach suffers from reconstruction background noise and requires recording the PSHs of the optical system in advance [29].

Realizing single-shot FINCH imaging without sacrificing the space-bandwidth product remains a challenge. Here, we propose a dual-channel Fresnel incoherent compressive holography (DC-FINCH) method. With this method, we introduce a dual-channel simultaneous phase-shifting setup [30] into the FINCH imaging system for the first time. As a result, we simultaneously obtain a pair of holograms with a phase shift of δ captured by the two cameras. Moreover, arbitrarily changing the value of δ requires only a simple adjustment of the system. In addition, different reconstruction algorithms are flexibly used depending on the characteristics of the imaged objects. For simple and sparse samples, the twin image is eliminated by the two-step phase shift algorithm [31], and the zeroth-order term can be directly filtered by a Gaussian high-pass filter in the Fourier domain. Finally, the back propagation (BP) algorithm is used to quickly reconstruct the images of the object at different depths. Obviously, as the number of object points increases, the DC bias in the recorded hologram increases. Therefore, for complex samples with many object points, the low signal-to-noise ratio (SNR) of the reconstructed image is mainly due to the bias term noise. In this case, we first physically removed the bias term by subtracting the two holograms and then adopted the compressive sensing (CS) algorithm with total variation (TV) regularization to eliminate the twin-image effect [32,33]. Finally, we achieve artefact-free reconstruction of complex objects, although this requires additional computing time. We successfully implemented video recording of holograms of 3D moving objects and numerically adjusted the optical focus of each frame to recover high-fidelity 3D sectioning images. The results show that the proposed method can alleviate the inherent contradiction between imaging speed, the space-bandwidth product, and the signal-to-noise ratio to some extent. Furthermore, our method is also suitable for other incoherent digital holographic systems where wavefront division is achieved by a birefringent crystal lens [13], a liquid crystal GRIN lens [34], geometric phase lens [35], triangular interferometer [36], Michelson interferometer [37] or Conoscopic holography [38]. In summary, the DC-FINCH system has the advantages of simple structure, wide applicability, strong flexibility and the potential to ensure measurement consistency of the forward physics imaging model [39] and implement multiplexed reconstruction [40] in the field of incoherent light.

2. Methodology

2.1 DC-FINCH system principle

Figure 1 shows a schematic of the DC-FINCH system, which combines a dual-channel simultaneous phase-shifting setup with the FINCH system. According to the nature of incoherent light, the object hologram captured by the camera can be regarded as an incoherent intensity superposition of all point holograms. To simplify the analysis without loss of generality, we analyze the complex amplitude of the diffraction light field of the point source located at $({x_0},{y_0},{z_0})$ by scalar diffraction theory. The point source illuminated by spatially incoherent light (or a self-luminous point source) is located ${z_0}$ away from the refractive lens L. L has a focal length ${f_0}$, and it is used to collect and collimate the light emitted from the object point. The distance between the SLM and L is d, and the complex amplitude of the light field on the front plane of the SLM can be expressed as follows:

 

Fig. 1. Schematic of DC-FINCH. (a) Optical setup. L - refractive lens; Pi - polarizers; SLM - spatial light modulator; BS - non-polarized beam splitter; HWP - half-wave plate. (b) Phase shifting holograms and Jones matrix of polarization elements in two channels. Cam – camera.

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Assume that the active axis of the SLM matches the y-axis (horizontal direction) and that the polarizer P1 is oriented to an angle of $\alpha$ with respect to the y-axis. In this case, for the object light, the polarization component along the y-axis is modulated by the quadratic phase $\varphi ({\bar{{\boldsymbol r}}_{{\boldsymbol s}}}) ={-} i\pi ({x_s}^2 + {y_s}^2)/\lambda {f_\textrm{a}}$ loaded on the SLM, while the polarization component along the x-axis (vertical direction) is not modulated. The light field $U({{{\bar{{\boldsymbol r}}}_{{\boldsymbol s}}}} )$ on the back plane of the SLM can be regarded as a vector superposition of linearly polarized spherical waves with different curvatures in the x-axis direction and y-axis direction, which can be expressed in Jones matrix form:

The dual-channel simultaneous phase shift setup included a half-wave plate (HWP), a non-polarized beam splitter (BS), two polarizers and two identical cameras. The angle between the fast axis of the HWP and the y-axis is 22.5°, and polarizers P2 and P3 are oriented at angles of 0° and 90°, respectively, with respect to the orientation of the y-axis. According to the Jones matrix of the typical optical elements, we described the polarization modulation process of the dual-channel simultaneous phase shift setup in the Jones matrix form. Only the changes in the polarization state and phase delay of the light field caused by optical elements are considered, and the spatial phase changes in the light field caused by the diffraction propagation process are not considered; thus, the coordinates of the light field expression are ignored. Therefore, the light field on the plane of camera 1 and camera 2 can be expressed in Jones matrix form as follows:

We obtain holograms ${I_{CAM1}}$ and ${I_{CAM2}}$ from the two cameras at the same time, and there is a phase shift of π between this pair of holograms. Their intensity can be expressed as follows:

Several variants of the dual-channel simultaneous phase shift setup exist. According to the requirements of imaging experiments, one can be selected to obtain a pair of holograms with a specific phase shift. For example, both the configurations in Fig. 2(a) and Fig. 1 can simultaneously yield a pair of holograms with π phase shifts, but the former has a simpler and more compact structure. The configuration in Fig. 2(b) can simultaneously yield a pair of holograms with π/2 phase shifts. In the configuration shown in Fig. 2(c), the fast axis of the quarter wave plate (QWP) is oriented to an angle of 45° with respect to the y-axis to transform orthogonal linearly polarized light into left-handed circularly polarized light and right-handed circularly polarized light. By utilizing the geometric phase effect of circularly polarized light, holograms with arbitrary phase shifts in the range of 0-2π can be obtained by rotating the polarization axes of polarizers P1 and P2. The derivations of the interference formulas in the Jones matrix form for Fig. 2(b) and (c) can be found in Supplement 1, S1 and S2..

 

Fig. 2. Alternative configurations for DC-FINCH. P1, P2 - polarizers; BS - nonpolarized beam splitter; PBS - polarized beam splitter; HWP - half-wave plate; QWP - quarter wave plate.

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2.2 Compressive reconstruction algorithm model

From the analysis in section 2.1, the object light $U({{{\bar{{\boldsymbol r}}}_{{\boldsymbol s}}}} )$ is divided into a pair of orthogonal linearly polarized light with a spherical phase difference $\varphi ({\bar{{\boldsymbol r}}_{{\boldsymbol s}}})$ by SLM. Then, these two beams propagate through a polarizer to the camera plane with a propagation distance of ${z_h}$. At this time, the projections of two such beams with different curvatures along the polarizer axis will interfere with each other and create a hologram. According to previous reports, the intensity distribution of the self-interference hologram of the FINCH system is similar to that of the Fresnel zone plate, which encodes the 3D position of the object point. Combined with Eq. (2) above, the intensity distribution of the recorded point source hologram, which is the point spread function (PSF) of the FINCH system, can be described by:

2.3 Two-step phase-shifting filter reconstruction algorithm model

For reconstructing simple and sparse objects, the two-step phase-shifting filtering reconstruction algorithm is more attractive owing to the shorter reconstructing time. An alternative configuration for the DC-FINCH system, as shown in Fig. 2(b) can easily achieved just by adding a quarter wave plate in one of the channels of the optical setup shown in Fig. 1. As a result, we can obtain holograms ${H_0}$ and ${H_{\pi /2}}$ with phase shifts of 0 and π/2, respectively, in the new experimental system and switch a reconstruction algorithm as well. In this case, the complex-valued hologram ${H_T}$ without the twin image term is a superposition according to the following:

Generally, in conventional 2-step phase-shifting interferometry (PSI) [42–44], the bias term $(1 + i){C_2}$ in Eq. (11) is filtered out by a Gaussian high-pass filter window in the Fourier domain:

2. Results and discussion of the proof-of-principle experiment

3.1 Optical experimental setup

To validate the proposed method, a DC-FINCH experimental system is built, as shown in Fig. 3. We used digital micromirror devices (DMD, Fldiscovery F4300, 1920 × 1080 pixels, 10.8 µm pixel pitch) to display a static or dynamic sense and treat this as object 1. Two transmission charts with the letters “SCNU” pattern and the Chinese character “guang” pattern are treated as objects 2 and 3, respectively. Objects 1, 2 and 3 are located at positions in front of the refractive lens L at distances of 150 mm, 140 mm, and 127 mm, respectively. The DMD and the two transmission charts are illuminated by two identical spatial incoherent light-emitting diodes (LEDs). The angle between the polarization axis of the polarizer P₁ and the active axis of the phase-only SLM (Hamamatsu, X15213-16, 1280 × 1024 pixels, 12.5 µm pixel pitch) is 45°. We use the SLM to display a diffractive lens phase with a focal length of 300 mm and to divide the diffracted wavefront of the objects collected by L into two beams with different curvatures by means of polarization multiplexing. The dual-channel simultaneous phase shift setup consists of an achromatic HWP, a non-polarized beam splitter BS₂, two identical cameras (Sony IMX304, TRI120S-MC), and polarizers P₂ and P₃. The angle between the fast axis of the HWP and the active axis of the SLM is 22.5°, while the polarizers P₂ and P₃ are oriented to angles of 0° and 90°, respectively, with respect to the active axis of the SLM. In this case, we can obtain a pair of holograms with a phase shift of π from the two cameras in a single exposure.

 

Fig. 3. Experimental arrangement of DC-FINCH. LEDi - light-emitting diodes; BSi - non-polarized beam splitters; L, L_A1, L_B1, L_A2, and L_B2 - lenses; Pi - polarizers; TCi - transmission charts; DMD - digital micromirror devices; HWP - half-wave plate; SLM - phase-only spatial light modulator.

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3.2 2D imaging results for a single sample

We implement 2D imaging of a single sample via the proposed method. With the experimental setup presented in Fig. 3, we first used only DMD to display four static patterns of different complexities in the time sequence, and then four pairs of holograms with π phase shifts captured by camera 1 and camera 2 were acquired sequentially in measurements 1, as shown in Fig. 4(a1)-(a8). Subtraction between each pair of holograms is performed to obtain a real-valued synthetic hologram ${H_{cs}}$ without the bias term noise [Fig. 4(b1)-(b4)]. The diffraction patterns become visible in the synthetic hologram when the DC bias is removed, and in this case, the reconstructed image quality deteriorates only by the twin image effect [ Fig. 5(d1)-(d4)]. Figures 4(c1)-(c8) show four pairs of holograms with a phase shift of π/2 obtained in measurements 2 by simply adjusting the experimental system to the configuration shown in Fig. 2(b). Subsequently, four complex-valued synthetic holograms ${H_T}$ without the twin image term are calculated by Eq. (11), whose amplitude and phase profiles are presented in Fig. 4(d1)-(d4) and Fig. 4(e1)-(e4), respectively. Although the twin image term has been removed from the synthetic hologram ${H_T}$, the amplitude and phase images still have strong nonlinear noise contributed by the bias term. Gaussian high-pass filtering is used to remove the bias term of the hologram ${H_T}$ to obtain a new hologram ${H_F}$. The nonlinear noise in the amplitude profile of ${H_F}$ is greatly reduced [Fig. 4(f1)-(f4)], and the contrast of the phase profile of ${H_F}$ is enhanced [Fig. 4(g1)-(g4)], so the signal-to-noise ratio (SNR) of the reconstructed image is expected to be greatly improved.

 

Fig. 4. Phase-shifting holograms acquired from the measurements of two configurations of the DC-FINCH system and the corresponding computational synthetic holograms. (a1-a8) Four pairs of object holograms with 0 and π phase shifts and their computational synthetic real-valued hologram (b1-b4) in measurements 1. (c1-c8) Four pairs of object holograms with 0 and π/2 phase shifts and amplitude (d1-d4) and phase (e1-e4) images of their computational synthetic complex-valued hologram. The amplitude (f1-f4) and phase (g1-g4) images of the computational synthetic complex-valued hologram filtered out by a Gaussian high-pass filter.

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Fig. 5. Experimental results of direct imaging and DC-FINCH imaging of 2D objects by different algorithms. The first column is a regular wide-field image (a1-a4); the second and third columns are reconstructed images by the BP algorithm of synthetic holograms ${H_T}$ without (b1-b4) and with (c1-c4) Gaussian high-pass filters; and the fourth and fifth columns are reconstructed images generated by the BP algorithm (b1-b4) and CS algorithm (c1-c4), respectively, of synthetic holograms ${H_{\textrm{cs}}}$. The intensity profiles are shown on the right side of each figure for comparison. The PSNR values are presented in the lower left corner.

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We numerically reconstruct the computational synthetic holograms in Fig. 4 by different algorithms, and the results are shown in Fig. 5. Figure 5(a1)-(a4) presents the direct imaging results of the 2D objects through loading the corresponding imaging lens phase on the SLM, which were used as ground truths for calculating the peak signal-to-noise ratio (PSNR) of the reconstructed images. The PSNR is a standard metric used to evaluate the quality of the reconstructed images, which is defined as:

3.3 3D imaging results for multiple samples

We recorded the phase-shifting holograms of a 3D object consisting of different patterns at different depth sections, which are a reflective DMD with a pattern of a flying bird, a transmission chart with the letters “SCNU” and another transmission chart with the Chinese character “guang”. Since the reconstruction of the FINCH hologram involves numerically adjusting the optical focus, we have the flexibility to refocus the image at any desired depth plane. Figure 6(a) shows a series of reconstructions of real-valued synthetic holograms ${H_{cs}}$ generated by the CS algorithm, where three reconstructions with the best focus at each section of the object patterns are presented in Fig. 6(d1)-(d3), respectively. Obviously, when the image is focused on the pattern of the bird, the patterns of the letters and Chinese character at other depths are blurred, and vice versa. For comparison, we numerally reconstructed synthetic holograms ${H_{cs}}$ and ${H_F}$ via the BP algorithm, as shown in Fig. 6(e1)-(f3). The process of adjusting the focus of reconstructed images by different algorithms is provided in Visualization 1.

 

Fig. 6. 3D experimental results of DC-FINCH. (a) Series of reconstructions of the real-valued synthetic hologram ${H_{\textrm{cs}}}$ shown in (b) generated by the CS algorithm. Images of the 3D object reconstructed with the best focus at the axial position in section 1, section 2, and section 3 by (d1-d3) the CS algorithm and (e1-e3) the BP algorithm from the real-valued synthetic hologram and (f1-f3) the BP algorithm from the complex-valued synthetic hologram ${H_F}$ shown in (c) and (d).

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It is worth noting that, compared with the traditional Fresnel back propagation algorithm, our first proposed strategy in IDH, which combines the advantages of experimental system and reconstruction algorithm, not only removes the twin-image effect but also suppresses the artefacts introduced by out-of-focus light, thereby achieves high-dimensional and image fidelity reconstructions. The three-dimensional reconstruction of axially overlapping objects is still a challenge in the fields of both coherent and incoherent holography. The two transmission charts are placed axially overlapping to perform 3D imaging and the 4-step phase-shifting strategy is used to compare with our proposed method. As is shown in Fig. 7, the crosstalk noise introduced by out-of-focus light from the axially overlapping object can be suppressed as well by our single-shot imaging method. Moreover, the proposed method is applicable to the grayscale objects as well (see Supplement 1 for more details)

 

Fig. 7. 3D reconstruction results of axially overlapping objects. (a-d) the captured holograms with 0, π/2, π, 3π/2 phase shifts. (e) The phase image of the computational synthetic complex-valued hologram by 4-step phase-shifting strategy and its BP reconstruction result at 140 mm (g) and 197 mm (h). (f) The computational synthetic real-valued hologram ${H_{\textrm{cs}}}$ and its CS reconstruction result at 140 mm (i) and 197 mm (j).

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3.4 Video recording and 3D reconstruction

To further validate the 3D dynamic imaging ability of the proposed method, we use the DMD to display a dynamic scene at a speed consistent with the acquisition frame rate of the camera and implement a video recording of the self-interference holograms by the DC-FINCH system. The captured hologram is 2800 × 2800 pixels, and the camera used in our experimental system can record dynamic 3D objects at a frame rate of 8 fps. The first 6 frames of the synthetic hologram video used for reconstruction are presented in Fig. 8(a). Obviously, each frame encodes the sample’s 3D position information, and we achieve the best focus on the flying bird, the letter “SCNU” and the Chinese character “guang” at 100 mm, 137 mm and 196 mm reconstructed depth positions, respectively, where the CS algorithm was used for artefact-free reconstruction [Fig. 8(b)-(d)]. See Visualization 2 for more details. We view this imaging process as high-speed 3D videography, which involves 3D information about the object as well as 1D temporal information. The relationship between the reconstruction distance ${z_r}$ and the actual distance of the object ${z_0}$ is investigated and estimated in Supplement 1. In addition, by using a high-acquisition frame rate camera or reducing the number of pixels in the recorded hologram, the imaging speed can be improved. For example, when the camera used in the experimental system in this paper records holograms of 1000 × 1000 pixels, the imaging speed can be enhanced to 32 frames/second so that the experimental system can implement video recording and reconstruction of the dynamic scene displayed on the DMD at the same frame rate, as is provided in Visualization 3.

 

Fig. 8. Experimental results of the video recordings. (a) The first 6 frames of real-valued synthetic hologram video. The last three rows are reconstructed videos (6 frames) generated by the CS algorithm at reconstruction distances of 100 mm (b), 137 mm (c), and 196 mm (d).

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3. Conclusion

In summary, we first introduced a dual-channel simultaneous phase-shift interferometry setup into the FINCH imaging system and successfully acquired a pair of phase-shifting holograms in a single shot. Then, according to the computational real-valued and complex-valued holograms, we use the CS algorithm and the two-step phase-shift filtering algorithm to reconstruct the images of 2D, 3D and dynamic objects for comparison. The experimental results show that both algorithms can realize high-fidelity reconstruction for simple and sparse objects; however, for complex objects, increasing the undesired nonlinear bias term noise will deteriorate the quality of the reconstructed image when the two-step phase-shift filtering algorithm is used. Our solution is to physically remove the bias term noise by subtracting the two holograms collected by the DC-FINCH system and then eliminate the twin-term effect by the CS algorithm. As a result, this strategy can not only obtain an artefact-free reconstructed image of a 2D object but also suppress the crosstalk of out-of-focus plane information to a certain extent when the imaging target is a 3D object. In other words, we need to flexibly select different reconstruction algorithms depending on the complexity of the object. The proposed method enables video recording of dynamic 3D scene holograms via the numerical refocusing of images at an arbitrary desired depth plane while ensuring high temporal resolution and image fidelity. However, the proposed method requires more time in the process of iterative computation, and we will continue to improve the CS algorithm by adding new physical constraints to increase the convergence rate in future work. In addition, the proposed method has great advantages in the 3D tracing task of sparse particle probes, and we expect this technique to be applied to fast 3D fluorescence imaging.