# A Forward\(-\)Backward Splitting Equivalent Source Method Based on S\(-\)Difference
## 1 Introduction  
Near\(-\)field Acoustic Holography (NAH) technology [1] has been widely used for source identification and visualization of spatial sound fields. NAH technology has multiple solving methods [2; 3; 4], among which the Equivalent Source Method (ESM) [5] is widely used and researched in the NAH field due to its simple model, adaptability to any shaped sound source, and high reconstruction accuracy and computational efficiency. Based on the ESM theory [6], a relationship formula between equivalent source strength and measured sound pressure was established. By making the measured sound pressure on the hologram surface equal to the sound pressure generated by the equivalent source, the equivalent source strength can be solved. In practical engineering applications, the sound pressure matrix between the equivalent source surface and the measurement surface is always uncertain due to the limitation of the number of microphones and measurement conditions. To solve this underdetermined problem, [PERSON] et al. 's [7] regularization method is commonly used to find the optimal equivalent source strength. When using this method, the regularization parameter needs to be determined first, and the reconstruction result depends heavily on the accurate selection of the regularization parameter. The commonly used regularization parameter selection methods mainly include the L\(-\)curve method and Generalized Cross Validation (GCV) [8; 9]. In recent years, compressed sensing technology based on sparse regularization has been widely used in the field of image processing [10; 11]. [PERSON] et al. [12] solved the non\(-\)convex regularization function by analyzing the difference between the penalty function and the truncation function to solve the sparse recovery problem. [PERSON] et al. [13] combined sparsity and compressed sensing theory with near\(-\)field acoustic holography, and used the CVX convex optimizationtoolbox in MATLAB to solve the norm problem. [PERSON] et al. [14] proposed establishing a monopole and dipole point source model and developed a specific iterative solver to achieve sparsity by using \"norm\" for solution vectors. However, it was used for long-distance measurement and focused on high \(-\)frequency signals. [PERSON] et al. [15] proposed the Iterative Shrinking Threshold Algorithm (ISTA), which has a simple iterative format but a slow convergence rate. [PERSON] et al. [16] proposed the Wideband Acoustic Holography (WBH) algorithm, which first defined the residual function, and then used a two\(-\)step iterative method to solve the sparse solution of the equivalent source amplitude value, which has high reconstruction accuracy only in the mid\(-\)to\(-\)high\(-\)frequency range. [PERSON] et al. [17] proposed combining high\(-\)order beamforming with an improved fast iterative shrinkage threshold equivalent source method, which can identify smaller sound source main lobes. After that, [PERSON] [18] elaborated on the principles of five commonly used iterative algorithms, including ISTA, derived their formulas, and compared their respective advantages and applicable frequency ranges through numerical simulation analysis and experimental verification. Based on the above content, a forward-backward splitting (FBS) algorithm for the s\(-\)difference of equivalent source method near\(-\)field acoustic holography was proposed. The optimal solution formula of Tikhonov regularization was used as the input parameter for the FBS algorithm, and the step size was corrected in each iteration process. The s\(-\)difference truncation adjustment strategy was introduced to make it approach the optimal solution of the objective function. The FBS algorithm can expand the applicable frequency range of the sound field reconstruction and ensure its reconstruction accuracy and algorithm reliability.