# A Forward\(-\)Backward Splitting Equivalent Source Method Based on S\(-\)Difference
## 5 Conclusions  
The near\(-\)field acoustic holography based on the equivalent source method is ill\(-\)posed, which requires the use of regularization methods to reduce the influence of input errors on the reconstruction accuracy. Due to the limitations of traditional regularization methods, a forward\(-\)backward splitting (FBS) algorithm based on s\(-\)differential approximation is proposed to approximate the solution, and a closed form solution of s\(-\)differential regularization is derived. Numerical simulations show that FBS is superior to Tikhonov regularization in terms of reconstruction accuracy and resolution. Additionally, compared to the ISTA and WBH regularization methods, FBS maintains higher reconstruction accuracy across the entire frequency range of 200-2500 Hz, and produces reconstructed sound pressure peaks that are closer to the theoretical values. The experimental verification of single and coherent sound sources shows that the sound field reconstruction within a wide frequency range of 200-2500 Hz is highly consistent with simulation results and  
Figure 12: Reconstructed sound pressure level diagram of 200 Hz and 2500 Hz coherent sound sources.  
accurately identifies the sound source location while maintaining high reconstruction accuracy, thus validating the effectiveness and stability of this method for reconstructing mid\(-\)to\(-\)low\(-\)frequency sound sources.  
Although the s\(-\)differential FBS algorithm proposed in this paper is able to accurately locate and identify the position of a single or coherent source in the near field, when the distance between the source plane and the holographic plane becomes large, the algorithm is limited in its ability to accurately reconstruct the radiated sound field at any position. However, when the distance between the source plane and the holographic plane becomes very large, the algorithm is limited in its ability to accurately reconstruct the radiated sound field at any location, and the algorithm should be further optimized for reconstructing the sound field of a long distance source to be better adapted to the holographic distance in subsequent work.  
To address the problem that the s\(-\)differential FBS algorithm needs to rethreshold the initial iteration when reconstructing the sound field in the low\(-\)frequency band, based on numerical simulation, the thresholds in the initial iteration are selected again. Based on numerical simulation and the two\(-\)step iterative shrinkage thresholding algorithm, the range of values is roughly determined, while in actual engineering applications, how to accurately select the initial threshold needs to be further explored.  
Conceptualization, J.M. and D.S.; methodology, J.M. and D.S.; software, J.L.; validation, Z.W. ; data curation, Z.W. ; writing--original draft preparation, J.L.; writing--review and editing, J.M.; funding acquisition, J.M. All authors have read and agreed to the published version of the manuscript.  
This research is supported by the Fundamental Research Funds for the Central Universities, CHD (300102211514, 300102210512).  
Not applicable.  
Data Availability Statement: Data are contained within the article.  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.