## 4 The Verification of CGH
### Generation of Modified Off-Axis Reference Beam CGH  
This section details the simulation of Fourier-transform hologram recording and reconstruction using a modified off-axis reference beam CGH based on the Burch method. The workflow includes:  
1. Loading a simulated object light image (_ef\({}_{1}\)_ = 0.0-1.0). 2. Performing a Fast Fourier Transform (FFT) to obtain the spectrum. 3. Interfering the spectrum with a tilted reference beam. 4. Encoding the hologram via the Burch method to generate the CGH. 5. Applying an inverse Fourier transform to the CGH to reconstruct the optical field and produce the final image. (1) Modified off-axis reference beam CGH via the Burch method  
Figures 11-13 illustrate the workflow of modified off-axis reference beam holography (Burch method), including the modified off-axis reference beam CGH encoding flowchart by using the Burch method (Figure 11), Burch method CGHs under varying Gaussian noise coefficients (_ef\({}_{1}\)_ = 0.0-1.0) (Figure 12), and Burch method reconstructed images under varying Gaussian noise coefficients (_ef\({}_{1}\)_ = 0.0 1.0) (Figure 13). Figure 11: The modified off-axis reference beam CGH encoding flowchart by using the Burch method. Figure 12: Burch method CGHs under varying Gaussian noise coefficients (_ef\({}_{1}\)_ = 0.0–1.0). The corresponding PSNR and SSIM values of Burch-method-reconstructed images under varying Gaussian noise coefficients (_ef_1 = 0.0-1.0) are shown in Figure 14.  
The noise coefficient demonstrates nonlinear impacts on imaging quality (Figure 14). PSNR sharply decreases from 27.71 dB to 14.08 dB (49.2% reduction) as the noise coefficient increases from 0.1 to 1.0. SSIM shows stronger noise robustness, declining moderately from 0.9904 to 0.9042 (8.7% reduction). Key observations: In a low-noise regime (0.1-0.5), PSNR rapidly declines by 28.3% (27.71\(\rightarrow\)19.86 dB), and SSIM remains near-lossless (0.9904\(\rightarrow\)0.979, 1.1% reduction). For the critical threshold (noise coefficient \(\approx\) 0.5), PSNR falls below 20 dB (visually perceptible degradation), and SSIM retains high fidelity (0.9623). In a high-noise regime (\(\geq\)0.7), there is accelerated degradation of PSNR \(<\) 16.3 dB and SSIM \(<\) 0.95; severe detail loss occurs. Analysis reveals a signal-to-noise \"saturation\" effect. While high-noise PSNR reduction (29.1%) matches low-noise trends, the perceptual quality collapses abruptly. For noise-sensitive systems, maintaining a noise coefficient \(\leq\)0.3 ensures visually lossless performance (PSNR \(>\) 22.97 dB, SSIM \(>\) 0.97). (2) Lee-encoded continuous grayscale CGHs:  
In Figures 15-17, the flowchart of Lee-encoded continuous grayscale CGH encoding, Lee-encoded continuous grayscale CGHs under varying Gaussian noise coefficients (_ef_1 = 0.0-1.0), and Lee-encoded continuous grayscale reconstructed images under varying Gaussian noise coefficients (_ef_1 = 0.0-1.0) are displayed. [PERSON]'s method decomposes complex functions into four orthogonal non-negative components (Equations (12) and  
Figure 14: PSNR and SSIM values of Burch-method-reconstructed images under varying Gaussian noise coefficients (_ef_1 = 0.0-1.0). Figure 13: Burch method reconstructed images under varying Gaussian noise coefficients (_ef_1 = 0.0–1.0). (13)). By applying delayed sampling, each sampling unit is divided into four subgrids corresponding to distinct basis vectors. Noise is distributed across multiple components during decomposition, with partial energy canceled via orthogonality, thereby reducing noise sensitivity in individual components. Grayscale encoding records component amplitudes through multi-level transmittance, smoothing noise-induced fluctuations and avoiding abrupt distortions inherent to binary encoding. PSNR and SSIM values of Lee-encoded continuous grayscale reconstructed images under varying Gaussian noise coefficients (_ef_1 = 0.0-1.0) are shown in Figure 18.  
Figure 16: Lee-encoded continuous grayscale CGHs under varying Gaussian noise coefficients (_ef_1 = 0.0–1.0).