# Fast numerical generation and encryption of computer-generated Fresnel holograms

TL;DR: Experimental results reveal that the proposed method can be applied to generate and encrypt holograms with a small number of computations, and can be decoded and reconstructed to the original 3D scene with good fidelity.

Abstract: In the past two decades, generation and encryption of holographic images have been identified as two important areas of investigation in digital holography. The integration of these two technologies has enabled images to be encrypted with more dimensions of freedom on top of simply employing the encryption keys. Despite the moderate success attained to date, and the rapid advancement of computing technology in recent years, the heavy computation load involved in these two processes remains a major bottleneck in the evolution of the digital holography technology. To alleviate this problem, we have proposed a fast and economical solution which is capable of generating, and at the same time encrypting, holograms with numerical means. In our method, the hologram formation mechanism is decomposed into a pair of one-dimensional (1D) processes. In the first stage, a given three-dimensional (3D) scene is partitioned into a stack of uniformed spaced horizontal planes and converted into a set of hologram sublines. Next, the sublines are expanded to a hologram by convolving it with a 1D reference signal. To encrypt the hologram, the reference signal is first convolved with a key function in the form of a maximum length sequence (also known as MLS, or M-sequence). The use of a MLS has two advantages. First, an MLS is spectrally flat so that it will not jeopardize the frequency spectrum of the hologram. Second, the autocorrelation function of an MLS is close to a train of Kronecker delta function. As a result, the encrypted hologram can be decoded by correlating it with the same key that is adopted in the encoding process. Experimental results reveal that the proposed method can be applied to generate and encrypt holograms with a small number of computations. In addition, the encrypted hologram can be decoded and reconstructed to the original 3D scene with good fidelity.

## Summary (2 min read)

### 1. Introduction

- Numerical generation and encryption of holographic data are two important topics in the realm of holography that have instigated numerous research works in the past two decades.
- Basically, heavy computation is involved in the hologram generation process, and the addition of numerical encryption will further lower the computation efficiency.
- Second, the authors employed the set of 1D maximum length sequence (M-sequence) [24] as the key signals.
- The decryption process can be achieved with numerical means with a small amount of arithmetic operations.
- The vast difference between the reconstructed form of an encrypted hologram that has been decodedwith, andwithout, the right encryption key is illustrated.

### A. Fast Generation of Fresnel Holograms

- Given a set of 3D, self-illuminating object points.
- Without loss of generality, the authors assumed that the source image has identical size and resolution as the hologram.
- Compared with the direct implementation in Eq. (1), the alternative hologram generation method based on Eqs. (4) and (5) is significantly faster as it only involves a pair of one-dimensional processes, i.e., the formation of the diffraction pattern Oðx; τÞ in each scan plane and the subsequent generation of the overall diffraction pattern Dðx; yÞ.
- The authors further noted that Oðx; τÞ can be derived with an economical hardware solution based on field programmable gate array (FPGA) at a throughput of over 100M pixels per second [25].

### B. Fast Encryption of Holograms

- The authors now propose to encrypt the hologram generated with Eq. (5) by convolving the conversion signal RðyÞ with a one-dimensional binary M-sequence signal.
- In fact, with the prime polynomial, the authors can generate the M-sequence.
- Subsequently, a new bit value is shifted to the output of the shift register.
- As the formation of Oðx; yÞ can be realized with the hardware solution, the computation loading on this part of the process is not taken into account.

### 3. Experimental Evaluation

- The authors shall evaluate their method with the standard image “Lenna” shown in Fig.
- The mean square error (MSE) is 3196, and the peak signal to noise ratio (PSNR) is 13:1dB, indicating an extremely low coding fidelity.
- Next the encrypted hologram is decrypted with the same encryption key S1ðyÞ, and the reconstructed image is shown in Fig. 4(c).
- The encryption method can be conducted with an encryption key of longer M-sequences.
- It can be seen that apart from a slight blurriness, together with minor ringing at the boundary and the junction between the upper and lower sections, the reconstructed images at each section are similar to the original one.

### 4. Conclusion

- The authors have proposed a method for fast numerical generation and encryption of a Fresnel hologram.
- This effectively merges the generation and encryption process into a single entity.
- First, the amount of complexity involved in the generation of the hologram is significantly smaller than that employing the direct table lookup technique.
- Experimental results reveal that the visual quality of a reconstructed image from a hologram that is generated and encrypted by their method is extremely poor, if not beyond recognition, if it is obtained without decrypting the hologram with the correct key.
- The latter can be integrated with existing image encryption schemes, such as [2–8] to foster stronger resistance of optical images towards illegal access and attacks.

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