# Encyclopedia of Computer Graphics and Games

Living Edition
| Editors: Newton Lee

# Image Quality Evaluation of a Computer-Generated Phase Hologram

• Hiroshi Yoshikawa
Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-08234-9_277-1

## 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

The Fourier hologram can be calculated with the Fourier transform of an original image. Figure 1a shows the image location in the input image plane for the Fourier transform, and Fig. 1b is a synthesized CGH. Figure 1c shows a numerically reconstructed image from the CGH. As one can see from Fig. 1c, the reconstructed image includes the desired image appeared as same position of the original image and the conjugate image that appears as the point symmetry to the center. The direct light (or non-diffracted light) is eliminated numerically in the figure, but it usually appears at the center and should be taken into account to evaluate image quality. The original image should be placed off-center not to overlap with the direct light and the conjugate image. Therefore, the original image is located center in vertical and right most side in horizontal. For the hologram calculation, the pixel value other than the original 2D image is set to zero. The random phase is multiplied to each pixel to make the reconstructed image diffusing and bright. Then 2D Fourier transform is applied to the transmittance distribution of 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
$${\displaystyle \begin{array}{l}I\left(X,Y\right)={\left|O\Big(X,Y\Big)+r\right|}^2\\ {}\quad ={\left|O\left(X,Y\right)\right|}^2+{r}^2+ rO\left(X,Y\right)+{rO}^{\ast}\left(X,Y\right)\\ {}\kern3em ={\left|O\left(X,Y\right)\right|}^2+{r}^2+2 r\mathit{\Re}\left\{O\left(X,Y\right)\right\},\end{array}}$$
(1)
is a real physical light distribution on the hologram, where ℜ{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

In the CGH, it is quite easy to use only the interference term 2rℜ{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:
$${I}_b\left(X,Y\right)=\Re \left\{O\left(X,Y\right)\right\}.$$
(2)
The normalization defined in Eq. 3 is applied to make final fringe intensity positive,
$${I}_n\left(X,Y\right)=\frac{I_b\left(X,Y\right)-{I}_{\mathrm{min}}}{I_{\mathrm{max}}-{I}_{\mathrm{min}}},$$
(3)
where Imax and Imin are the maximum and the minimum values of Ib(X,Y), respectively.

### Numerical Reconstruction of Phase Hologram

For reconstruction simulation, the complex amplitude transmittance t(X,Y) of the transmission phase CGH is assumed as:
$$t\left(X,Y\right)=\exp \left[-i\Delta \phi {I}_n\left(X,Y\right)\right].$$
(4)

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 Δφ = 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

The peak signal-to-noise ratio (PSNR) is defined as the ratio of the maximum signal power and the noise power. The reconstructed image is extracted from numerically reconstructed image plane such as Fig. 1c and normalized to 8-bit grayscale image that has same mean intensity of the original image, and then the PSNR is calculated as:
$${\displaystyle \begin{array}{ll}& \mathrm{PSNR}\\ {}& =10{\log}_{10}\frac{255^2 WH}{\sum_{i=0}^{W-1}{\sum}_{j=0}^{H-1}{\left[J\left(i,j\right)-K\Big(i,j\Big)\right]}^2}\left[\mathrm{dB}\right],\end{array}}$$
(5)
where 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

Figure 2 shows DE (solid line) and PSNR (dashed line) of the rigorous calculation against the beam ratio (BR, defined as |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.

## References

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