# Image Quality Evaluation of a Computer-Generated Phase Hologram

**DOI:**https://doi.org/10.1007/978-3-319-08234-9_277-1

## Synonyms

## Definition

Quality of reconstructed image from computer-generated phase hologram is evaluated objectively on its peak signal-to-noise ratio and brightness.

## Introduction

Hologram can record and reconstruct or playback an optical wavefront on the hologram plane. It uses interference between two waves, an object wave from the object to be recorded and a reference wave. The interference intensity pattern is recorded on a photosensitive material. Computer-generated hologram (CGH) simulates this optical phenomenon in a computer (Lohmann and Paris 1967). CGH is widely used to show not only for 2D images but also complex 3D images. Image quality of the reconstructed image from CGH is usually evaluated subjectively. For example, an observer compares two images and scores. Here shows basic research to evaluate reconstructed image quality of phase-type CGH objectively on its peak signal-to-noise ration and brightness (Yoshikawa and Yamaguchi 2015).

## Computer-Generated Hologram

*o*(

*x*

*,*

*y*) on the input image plane, and the result of

*O*(

*X*

*,*

*Y*) represents the complex amplitude of the object beam on the hologram plane. If the reference beam is collimated and its direction is perpendicular to the hologram, the complex amplitude of the reference beam

*R*(

*X*

*,*

*Y*) can be represented as the real-valued constant

*r*. The total complex amplitude on the hologram plane is the interference of the object and reference beam, represented as

*O*(

*X*

*,*

*Y*) +

*r*. The total intensity pattern

*C*} takes the real part of the complex number

*C*and

*C*∗ means the conjugate of

*C*. At the right most hand of the Eq. 1, the first term represents the object self-interference, and the second is the reference beam intensity. The third term is the interference of the object and the reference beams and contains holographic information.

### Calculation Without the Object Self-Interference

*r*ℜ{

*O*(

*X*

*,*

*Y*)} of Eq. 1. This idea is proposed at very early stage of CGH research (Waters 1966). The interference part can be written as:

*I*

_{max}and

*I*

_{min}are the maximum and the minimum values of

*I*

_{b}(

*X*

*,*

*Y*), respectively.

### Numerical Reconstruction of Phase Hologram

*t*(

*X*

*,*

*Y*) of the transmission phase CGH is assumed as:

Then *t*(*X**,**Y*) is inverse Fourier transformed to obtain the reconstructed image. In the case of a sine-wave phase grating, the maximum diffraction efficiency of 33.8% is obtained at Δ*φ* = 0.59. Therefore, this value is used unless denoted.

### Diffraction Efficiency

The diffraction efficiency (DE) is defined as the ratio of the intensities of the reconstructed image and the illumination light. It gives the brightness of the reconstructed image. In the numerical experiments, the reconstructed image intensity is obtained by summing up all intensities in the reconstructed image area as same size and position of the original image in the input image plane.

### Peak Signal-to-Noise Ratio

*W*and

*H*are horizontal and vertical pixel numbers of the image and

*J*and

*K*are intensities of the original and the reconstructed image.

## Numerical Experimental Results

*|R|*

^{2}

*/*

*|O|*

^{2}). The DE of the phase hologram becomes over ten times larger than that of the amplitude hologram (Yoshikawa 2015). Since the object self-interference (OSI) term of

*|O|*in Eq. 1 causes noise on the reconstructed image, it is known that larger beam ratio gives better PSNR. However, DE becomes smaller with larger beam ratio. The hologram calculated without OSI as shown in Eq. 3 gives PSNR of 25.0 dB with DE of 8.8%, which achieves both low noise and bright image simultaneously.

## Conclusion and Discussion

Image quality of phase CGH is evaluated objectively on the diffraction efficiency and the peak signal-to-noise ratio. For the transmission phase hologram, although it is obtained over ten times of diffraction efficiency against amplitude hologram (DE = 0.77%, PSNR = 38.9 dB), PSNR is not as good as that of the amplitude hologram. Since the evaluated hologram is very simple phase hologram, it is expected to evaluate other type of phase hologram.

## Cross-References

## References

- Lohmann, A.W., Paris, D.P.: Binary Fraunhofer holograms, generated by computer. Appl. Opt.
**6**(10), 1739–1748 (1967)CrossRefGoogle Scholar - Waters, J.P.: Holographic image synthesis utilizing theoretical methods. Appl. Phys. Lett.
**9**(11), 405407 (1966)CrossRefGoogle Scholar - Yoshikawa, H.: Image Quality Evaluation of a Computer-Generated Hologram, OSA topical meeting on Digital Holography and 3D Imaging. Shanghai, OSA (2015)Google Scholar
- Yoshikawa, H., Yamaguchi, T.: Image quality evaluation of a computer-generated phase hologram. In: 10th International Symposium on Display Holography, paper 4–4 (2015)Google Scholar