## 3 Noise Characteristics in the Reconstructed Images
RMSE is the most widely used image quality metric for IFTAs and it can be calculated as:  
\[RMSE=\sqrt{\frac{1}{MN}\sum_{m,n}Err(m,n)^{2}}, \tag{9}\]  
where \(Err(m,n)\) is the difference between the normalized grey values of the (\(m\)th, \(n\)th) pixels in the reconstructed and the target images. Since the RMSE is not able to consistently evaluate the image quality accurately, SSIM is proposed in this work as the image quality metric instead. The calculation of SSIM between images \(\mathbf{x}\) and \(\mathbf{y}\) can be expressed as [24; 29]:  
\[SIMM(x,y)=\left(\frac{2\mu_{x}\mu_{y}+C_{1}}{\mu_{x}^{2}+\mu_{y}^{2}+C_{1}} \right)\left(\frac{2\sigma_{x}\sigma_{y}+C_{2}}{\sigma_{x}^{2}+\sigma_{y}^{2}+ C_{2}}\right), \tag{10}\]  
where \(\mu_{x}\) and \(\mu_{y}\) are the means of the two image patterns, respectively; \(\sigma_{x}\) and \(\sigma_{y}\) the local standard deviations of the two image patterns, respectively; \(C_{1}\) and \(C_{2}\) are two small positive constants used to stabilize each term. In this work, the reconstructed and target images were divided into a matrix of small areas of \(11\times 11\) pixels and the local SSIM values between the corresponding areas were calculated individually. The overall SSIM between the reconstructed and target images are subsequently determined by averaging these local SSIM values across the whole area. The values for \(C_{1}\) and \(C_{2}\) are chosen to be 0.01 and 0.03, respectively. The value of SSIM can range from 0 to 1. A higher value indicates the reconstructed image has a better similarity with the target image and therefore a better quality. The relationship between the perceived image quality and the values of RMSE and SSIM is analyzed based on the LIVE Image Quality Assessment Database [29; 30; 31] released by Laboratory for Image and Video Engineering (LIVE) at University of Texas at Austin. This database provides the difference mean opinion scores (DMOS) of the subjective image quality ratings on 29 natural images with five types of distortions, including JPEG2000, JPEG, white Gaussian noise, Gaussian blur and Rayleigh fast-fading. The data was collected by performing a subjective experiment of image quality assessment according to the single-stimulus protocol. Participants in the experiment were asked to rate the quality of the images by 'Bad', 'Poor', 'Fair', 'Good' or 'Excellent'. These ratings were translated into scores of 20, 40, 60, 80 and 100, respectively. Since the primary source of the noise generated by the IFTAs is the white Gaussian noise, we only used the results of the images distorted by this type of noise in this study. Five different degrees of white Gaussian noise were evaluated in the database. In total, there are 145 stimuli in total, excluding the 29 original images. The original images are assumed to have a perfect image quality. The RMSE and SSIM of these 145 stimuli with respect to their corresponding original images were calculated. Their relationships with the perceived image quality are plotted in Figures 6 and 7. It can be seen that higher RMSE values are associated with poor image quality and higher SSIM values represent good image quality. SSIM values larger than 0.6 can always guarantee a good image quality while the corresponding boundary for RMSE is ~0.02. Linear relationships were observed in both figures. Linear fittings were applied to the results in these figures. Results show that SSIM has a more linear relationship with the perceived the image quality in the presence of white Gaussian noise than RMSE does. Since the images reconstructed from the phase-only holograms calculated by IFTAs are also distorted by the white Gaussian noise, SSIM is expected to be a reliable quality metric for the perceived image quality in IFTAs. In other words, we can use SSIM to evaluate the image reconstructed from phase-only holograms calculated by IFTAs.