Optical scanning holography (OSH), one of the hologram acquisition methods, was invented by Poon [1] and has a long history. The dual output heterodyne detection scheme of OSH makes it possible to record the complex hologram of an object without twin-image noise [2, 3]. The incoherent mode of OSH can record the hologram of a diffusely reflected object without speckle noise [4, 5]. OSH scans the object line by line and instantaneously records a hologram of each scaned point using a single-pixel optical sensor. This means that OSH does not require a digital camera to record the hologram, unlike a conventional digital hologram [6]. Conventional digital holography records the hologram of an object using image sensor, which has a limited number of pixels. This means the size of the recorded hologram is limited by the pixel number of the image sensor. On the other hand, OSH scans an object and acquires the hologram using a single-pixel optical sensor, which means that the size of the hologram is virtually unlimited. Several numerical reconstruction algorithms have been proposed for digital holography [7–11]. In this paper, we propose a method that reconstructs the OSH with segmentation. In section II, we review OSH, which records the complex hologram of an object scene by encoding the 3D distribution of the object scene with a Fresnel zone plate (FZP). Conventional images are formed using defined points on the imaging plane, which enables the image processing by use of simple divide-and-stitch work. However, holographic information is distributed throughout the recorded hologram, which makes the simple divide-and-stitch approach fail. In section III, we propose a divide-and-recombine technique for numerical processing of the hologram recorded by OSH without losing holographic information. First, we segment the hologram which contains holographic information of the object scene. Here we derive the size of the segmented holograms required so as not to lose holographic information. After that, we reconstruct the segmented holograms simultaneously. Finally, we stitch the reconstructions of the segmented holograms while preserving holographic information. In section IV, we record the hologram of a die experimentally and perform the proposed reconstruction technique. This shows that the proposed technique corresponds to reconstruction of the hologram without loss.

Since the details of OSH were published previously [12], in this section we review OSH only briefly, for the sake of completenession. Figure 1 shows a typical OSH setup that captures the hologram of an object’s scene. The OSH comprises a scanning-beam-generation unit, a scanning unit, a photo-detection unit, and an electronic-processing unit.

Figure 1. Setup for optical scanning holography (OSH). M, mirrors; BS, beam splitters; BE, beam expander; AOM, acousto-optic modulators; PD, photo-detector; L, Lens.

The scanning-beam-generation unit consists of a Mach-Zender interferometer, acousto-optic modulators (AOM1,2), beam expanders (BE1,2), and a lens (L1). AOM1,2 are modulated in time with frequencies Ω and Ω + ΔΩ. The scanning beam is generated through a beam splitter (BS2). The frequency of the scanning beam is ΔΩ, which is the frequency difference between the modulated frequencies Ω and Ω + ΔΩ. The upper path, with a lens (L1) and a beam expander (BE1) generates a spherical wave. The lower path, with a beam expander (BE2) generates a plane wave. The spherical and plane waves interfere at the beam splitter (BS2). The spatial distribution of the scanning beam becomes a FZP. which varies in time. The scanning beam, called the time-dependent Fresnel zone plate (TD-FZP), is given by

(1)$$I_{\text{FZP}}=\frac{j}{\lambda z}\text{ exp}\left[\left(\frac{-\text{π}}{{\text{NA}}^2{z}^2}+\frac{j\pi }{\lambda z}\right)\left(x^2+y^2\right)\right],$$

where numerical aperture (NA) is defined as the inverse sine of the half-cone angle subtended by the spherical wave generated through BE1 and L1, and λ is the wavelength of the laser beam. The scanning unit scans the object, I_O (x, y;z) by Galvo scan mirrors. Photo-detection unit consists of photodetectors (PD1,2) and a lens (L2). In the photo-detection unit, a photodetector (PD1) detects the beating signal generated by the interference between the spherical and plane waves. The beating signal goes to a lock-in-amplifier (LIA) as reference signal. A photodetector (PD2) detects the light reflected by the object through the lens (L2), and the detected light goes to the LIA as the object signal. In the electronic processing unit, the LIA generates in-phase (i_I (t)) and quadrature-phase (i_Q (t))signals which are stored in a digital computer after analog to digital conversion (ADC). The stored signals are rearranged according to the scan position. The in-phase and quadrature-phase signals become the real and imaginary parts of a complex hologram. Finally, the complex hologram is synthesized by rearranging the real and imaginary parts of the complex hologram in the following manner:

(2)$$\begin{array}{l}\text{H}(\text{x},\text{y})={\text{i}}_{\text{I}}(\text{x},\text{y})-{\text{ji}}_{\text{Q}}(\text{x},\text{y})\\ ={\displaystyle \int I_O(x,y,z)\otimes \frac{j}{\lambda z}\exp \left[\left(\frac{-\pi }{{\text{NA}}^2{z}^2}+\frac{j\pi }{\lambda z}\right)\left(x^2+y^2\right)\right]},\end{array}$$

where the symbol represents the two-dimentional convolution.

As reviewed in section II, the hologram recoreded using OSH is the encoded pattern between the 3D object and the FZP. This means that information of each point of the 3D object is spread on the hologram plane with the extent of the FZP. Here the extent of an the FZP on the object is determined by the NA of the FZP, and the distance between the focal point of the FZP and the object. According to the scanning of OSH, FZP beam scans the object in a row by row manner with a X-Y scanner.. The collected light at this scan position, which goes to the photodetector (PD2), contains the encoded pattern between the FZP and the object’s intensity distribution, where the radius of the FZP at the object’s depth location is given by

(3)$$R_{\text{fzp}}=z\times \tan \left(\theta \right),$$

where z is the distance between the focal point of the spherical wave that generates the FZP and the object’s depth location, and θ is the half-angle of the spherical wave. This shows that the size of the hologram recorded by OSH must be larger than the size of the object, to contain the complete holographic information supported by the FZP. The minimum size of the hologram that records the complete holographic information of the object of L_x × L_y is (L_x + R_fzp) × (L_y + R_fzp) as shown in Fig. 2. Here we note that the size of the hologram recorded by OSH could be as large as the scanning area, and that the numerical reconstruction time drastically increase according to the increasing size of the hologram. We propose a fractional reconstruction technique, in which we segment the hologram that is recorded by OSH and reconstructs the segments simultaneously. The fractional reconstruction technique is composed of segmentation and reconstruction stages. In the segmentation stage, we segment the hologram. Here the segmented sub-hologram should be larger than the object’s sub-area. The minimum size of the sub-hologram that supports the object’s sub-area is depited in Fig. 3. Figure 4 shows a flow chart, with its first three blocks illustrating procedures discussed so far. The inputs i_I (x, y) and i_Q (x, y) are added in a complex manner to make a complex hologram. After that, the complex hologram is segmented into sub-holograms according to Fig. 3. Finally, we end with reconstructing the sub-holograms using convolution in the frequency domain. In the following we discuss the reconstruction of sub-holograms, and the arrangement of the sub-holograms to synthesizing the reconstructed image. First, we reconstruct the segmented sub-holograms simultaneously, and recombine the reconstructions of the sub-holograms as shown in Fig. 5. The last two blocks of the flow chart shown in Fig. 4 summarize the procedures.

Figure 2. Relationship between the size of the hologram and that of the object’s scene.

Figure 3. Segmentation stage. (n, m) is the row and column of the sub-hologram, and of the corresponding sub-area of the object.

Figure 4. Flow chart for the proposied reconstruction technique.

Figure 5. Reconstruction and recombination stage.

We record the complex hologram of a die experimentally. In the experiments, we use a HeNe laser that generates 630-nm light and drives the AOM 1,2 at 40 MHz and 40.01 Mhz respectively. We set the NA of the scanning beam to 0.02 and the die as the target object. The size of the object is 3 mm × 3 mm. the depth location of the object is 15cm, the size of the scan area is 3 mm × 3 mm and the pixel number of the hologram is 2000 × 2000 pixels. The recorded hologram is shown in Fig. 6. We can see the die with a fringe pattern, which contains the depth information of the object. The size of the FZP is 400 × 400, which is given by Eq. (3). In the segmentation stage, we set the size of the segmented hologram to be 4 times as large as that of FZP in each dimension; that is, 800 × 800. In the reconstruction stage, we reconstruct the sub-holograms then and recombine those reconstructed holograms according to Fig. 5. Figure 7(a) shows the reconstruction of the whole hologram, and Fig. 7(b) is the image of the recombined reconstructions of the sub-holograms. Both holograms are reconstructing at a depth location of 15 cm. We can see that the die is focused at the corresponding depth location. We calculate the root mean square error (RMSE) and Peak signal-to-noise ratio (PSNR) to check the correspondence between the reconstructed images from the proposed and conventional methods. The RMSE and PSNR are given by

Figure 6. Recorded complex hologram: (a) real part of the complex hologram and (b) imaginary part of the complex hologram.

Figure 7. Reconstruction of the hologram: (a) with a conventional method and (b) with the proposed technique.

(4)$$\text{RMSE}=\sqrt{\frac{{\sum }_{x=1}^{nx}{\sum }_{y=1}^{ny}{(x_1(x,y)-x_2(x,y))}^2}{n}},$$

(5)$$\text{PSNR}=20{\log }_{10}\left(\frac{\max (x_1(x,y))}{\text{RMSE}}\right),$$

where x₁ (x, y) is the image reconstructed according to the proposed technique, x₂ (x, y) is the reconstructed image of the whole hologram, n is the total number of samples in the hologram, and max (x₁ (x, y)) is the maximum value of the reconstructed image according to the proposed technique. The RMSE value is calculated to be 0.0135, and the PSNR value is 35.6945 dB. This shows that the reconstructed images correspond to each other. Compared to the conventional technique, the main advantage of the proposed technique is to reconstruct the hologram by parts before finishing the recording of the hologram. The experiment requires 30 seconds to record the hologram of the object using a Galva scan. This delays the reconstruction of the hologram by 30 seconds, using the conventional method. However, using the proposed technique, we need to wait only for the recording of the first segmentation of the hologram. In the experiment, the reconstruction time for the segmentation is 0.03 seconds. This means that we can finish reconstructing the entire hologram just 0.03 seconds after finishing the scanning.

We propose a technique that reconstructs a hologram recorded by OSH. In OSH, the size of the hologram increases as the scan area increases. This means that the size of the hologram recorded by OSH could be as large as the scan area and the reconstruction time increases according to the size of the hologram. We propose a reconstruction technique in which we segment a large hologram into sub-holograms, and reconstruct the sub-holograms simultaneously. The experimental results show that the image reconstructed using the proposed method matches that reconstructed using a conventional method.

This work was partly supported by an Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No. 2020-0-00981, Development of Digital Holographic Metrology Technology for Phase Retrieval; 50%) and by an Institute of Civil Military Technology Cooperation grant funded by the Defense Acquisition Program Administration and Ministry of Trade, Industry and Energy of the Korean government (No. 18-CM-DP-24, Development of digital HOE for immersive exhibition applications; 50%).