## 5 Conclusions  
In this study, we conducted a comprehensive evaluation of three demodulation algorithms, namely FSD, FSD-HSO, and FSD-PSO, using synthetic interferograms. Our aim was to gain a deeper understanding of the capabilities and limitations of these techniques in the context of optical interferometry. Through the use of synthetic interferograms, we were able to control various parameters, such as the amplitude, position, and width of Gaussian components within the interferograms. These numerical experiments allowed us to assess the performance of the demodulation methods across different scenarios, including those with positive and negative Gaussians and varying fringe frequencies. This study yielded several important findings and insights:  
1. Method Robustness: The standard FSD technique exhibited satisfactory performance, particularly with low-frequency interferograms, where it successfully recovered the phase in most cases. However, challenges arose in scenarios involving central peaks and multiple Gaussians, as observed in Figures 2a and 2d. 2. FSD-HSO Performance: FSD-HSO demonstrated improved performance compared to the standard FSD method. It produced better results with fewer artifacts. Nevertheless, some regions in the interferograms still exhibited noticeable artifacts, impacting the quality of the demodulated phase. 3. Superiority of FSD-PSO: FSD-PSO emerged as the standout performer among the three algorithms. It consistently delivered more accurate and faithful results, with reduced artifacts and a closer resemblance to the original phase. 4. Frequency Sensitivity: Our experiments with varying Gaussian widths and amplitudes provided insights into the frequency sensitivity of the methods. Both FSD and FSD-HSO exhibited limitations as fringe frequencies increased, leading to distortions in Gaussian peaks. FSD-PSO demonstrated a more stable response, although it also exhibited some sensitivity to higher frequencies. To quantitatively evaluate the algorithms' performance, we introduced kurtosis and RMS parameters. These parameters offered insights into the recovery of Gaussian function widths and divergence from the original phase as fringe frequencies increased. Our study highlights the strengths of FSD-PSO as a superior demodulation technique. One important consideration about the use of optimizers with the FSD technique is that adds computational complexity and also increases the demodulation time. It is essential to acknowledge other limitations and potential drawbacks:  
Figure 7: RMS vs. Gaussian width \(\left(F_{xx},\ F_{yy}\right)\). Computational Resources: FSD-PSO, being computationally intensive, may require substantial computational resources for real-time applications. This could be a limiting factor in scenarios where real-time results are essential. Parameter Tuning: The optimal performance of FSD-PSO may rely on fine-tuning numerous algorithm parameters. Achieving these optimal settings can be a non-trivial task and may require expertise. Sensitivity to Extremely High Frequencies: While FSD-PSO demonstrates greater stability in high-frequency scenarios compared to its counterparts, it may still exhibit sensitivity to extremely high fringe frequencies, necessitating careful consideration of application-specific requirements. In conclusion, our research contributes to a better understanding of the demodulation capabilities of FSD and its optimized variants FSD-HSO and FSD-PSO. FSD-PSO, in particular, stands out as a promising approach for accurate phase recovery from interferograms. However, researchers and practitioners should be mindful of the computational demands and parameter-tuning requirements associated with this technique. Further research is warranted to explore its practical applicability in real-world optical interferometry scenarios and address the identified limitations.