## Abstract

We propose an indirect method for generating a complex hologram and phase-only hologram from an amplitude hologram using the Hilbert transform. The Hilbert transform generates an imaginary part of complex amplitude from only an amplitude hologram, resulting in the reduction of the total computational complexity of complex and phase-only holograms. More importantly, the proposed method can reduce the hardware resources of dedicated hologram processors.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Digital holograms are generated using a computer [1–3], and their application includes holographic three-dimensional (3D) displays [4,5], projections [6,7], beam generation [8], and holographic optical elements [9]. Digital holograms are primarily categorized into amplitude holograms, phase-only holograms, and complex holograms [3]. A complex hologram includes the amplitude and phase part of a light wave, thus rendering the reconstructed light wave ideal. A complex hologram can be generally expressed as follows:

where $N$ is the total number of object points, and $a_j$ and $\theta _j$ are the amplitude and phase of $j$th object light on the hologram plane, respectively. However, a complex hologram cannot be displayed on a spatial light modulator (SLM) because current commercial SLMs modulate only amplitude or phase of light. Phase-only holograms are generated by taking only the phase part of complex holograms, and they can be displayed using a commercial SLM. A phase-only hologram can be expressed as follows: where ${\operatorname {Re}}\{{\cdot }\}$ and ${\operatorname {Im}}\{{\cdot }\}$ denote the operators adopting real and imaginary parts of a complex amplitude. To obtain phase-only holograms, we typically need to calculate Eq. (1) and then convert it to a phase-only hologram using Eq. (2). In contrast, an amplitude hologram is simply calculated as follows: Eqs. (1) and (2) include cosine and sine functions, whereas Eq. (3) for an amplitude hologram includes only a cosine function; therefore, the computational complexity of Eq. (3) will be lower than that required for complex and phase-only holograms.In this study, we propose an indirect method for generating a complex hologram and phase-only hologram from an amplitude hologram using the Hilbert transform [10]. The Hilbert transform requires low computational complexity, and generates a complex hologram from only an amplitude hologram. The total computational complexity of the proposed method will be small compared with that of complex and phase-only holograms. More importantly, as will be discussed later, the proposed method can reduce the hardware resources of dedicated hologram computers [3].

## 2. Proposed method

The Hilbert transform is widely used in signal processing and communication theory. In optical measurement, quantitative phase imaging [11] and optical coherence tomography [12] are both able to employ the transform. However, to our knowledge, the transform has to date not been employed to accelerate hologram generation or to reduce the hardware resources needed for dedicated hologram computers. The Hilbert transform [10] is defined as follows:

As an example, we illustrate the proposed method in the case of complex hologram generation using an amplitude hologram generated from point cloud data using the Hilbert transform. Using Eq. (3), we calculated an amplitude hologram from a point cloud as follows:

We then generated a complex hologram from an amplitude hologram using Eq. (7) and the Hilbert transform via the following steps.

- (1) We expanded the resolution of $u_a$ with $N_x \times N_y$ pixels to $2 N_x \times N_y$ by horizontal zero padding, and then extracted the first row.
- (2) The Hilbert transform converted the $j$th row of the amplitude hologram into that of a complex hologram; that is, we calculated $u_c(x_h, j)=\mathbb {H} \left [{u_a(x_h,j)} \right ]$.
- (3) We updated $j$ as $j\leftarrow j+1.$
- (4) We repeated the steps 2 and 3 up to and including the final row.
- (5) We obtained the final complex hologram by cropping the result of the Hilbert transform to $N_x \times N_y$ pixels.

## 3. Results and discussion

Figures 2 and 3 show numerically reconstructed images from complex holograms generated using Eq. (1), amplitude holograms using Eq. (3), and complex holograms obtained from amplitude holograms using the proposed method. The insets are the enlarged images of the reconstructed object lights. The calculation conditions used were as follows: the wavelength was 633nm, the pixel pitch of the hologram was 10$\mu$m, the distance between the holograms and point cloud was 0.8m, and the resolution of the holograms was $2,048 \times 2,048$ pixels. The centroid of object points was placed at (-15.4mm, 0.0mm, 0.8mm), and the size of the point cloud was approximately 2cm $\times$ 2cm $\times$ 2 cm.

In Figs. 2 and 3, the reconstructed images from the complex holograms of Eq. (1) and the proposed method include only object light without conjugate lights. In contrast, the reconstructed images of the amplitude holograms include conjugate lights in principle.

We evaluated the image quality using the peak signal-to-noise ratio (PSNR) and structural similarity (SSIM). The reference image for PSNR and SSIM was the reconstructed images generated by Eq. (1). In Fig. 2, the PSNR and SSIM of the amplitude hologram are 28.5 dB and 0.594, whereas those of the proposed method are 47.7 dB and 0.998, respectively. In Fig. 3, the PSNR and SSIM of the amplitude hologram are 24.9 dB and 0.463, whereas those of the proposed method are 39.4 dB and 0.998, respectively. The proposed method was able to obtain almost the same image quality as the complex holograms of Eq. (1).

We measured the calculation times of Eq. (1) and the proposed method. This was done using a computer with an AMD Ryzen5 3600X CPU with 64GB memory, and Visual C++ 2015 as a compiler. We used eight CPU threads. In Fig. 2, the calculation times for Eq. (1) and the proposed method were 24 and 15 s, respectively. The number of object points was 11,646. The calculation time of the Hilbert transform was only 83 ms. The proposed method calculated approximately 1.6 times faster than Eq. 1. In Fig. 3, the calculation times of Eq. (1) and the proposed method were 93 and 59 s, respectively. The number of object points was 44,647. The proposed method calculated approximately 1.6 times faster than Eq. (1). All the calculations in this study were performed by using our wave optics library [14].

Figure 4 shows the optical reconstructions of a phase-only hologram obtained using Eq. (1) and the proposed method. A phase-modulated type SLM was used in this study (HoloEye PLUTO-2). Similar to the numerical reconstructions, the image quality for Eq. (1) and the proposed method was essentially almost the same.

The proposed method indicated not only has a calculation speed advantage but also has efficient implementation with dedicated computers. We developed dedicated hologram computers, referred to as “HOlographic ReconstructioN (HORN)” [15,16]. A HORN computer for phase-only holograms includes a large number of “complex amplitude units” (CAUs), which include cosine and sine circuits and the two product-sum circuits. Accordingly, by operating a large number of CAUs in parallel, the HORN computer can calculate Eq. (1) rapidly. The proposed method anticipates increasing the number of CAUs because it can generate phase-only holograms from amplitude holograms; the CAUs of which require only a cosine circuit and one product-sum circuit. Current HORN computers have been implemented on field-programmable gate arrays (FPGAs); current FPGAs have a logic block, which is used for user logic circuits, and have embedded CPUs. Since the computational complexity of the Hilbert transform is low, the embedded CPU will simultaneously perform the Hilbert transform while the logic block performs hologram calculations. This will result in further efficient generation of phase-only holograms compared with current HORN computers.

## 4. Conclusion

In this paper, we proposed the Hilbert transform-based hologram generation. The proposed method was able to calculate complex holograms and phase-only holograms from amplitude holograms. The total computational complexity of the proposed method was reduced compared with the direct calculation of Eqs. (1) and (2). Moreover, we discussed the efficient implementation of phase-only holograms into dedicated hologram processors. In future work, we will implement the proposed method for FPGAs.

## Funding

Japan Society for the Promotion of Science (19H01097, 19H04132).

## Disclosures

The authors declare no conflicts of interest.

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