# 3D Acoustic Mapping in Automotive Wind Tunnel: Algorithm and Problem Analysis on Simulated Data
## 3 Volumetric Acoustic Mapping Based on IRLS
### IRLS Solution Strategy
All of them lead to a particular cost function to minimize with respect to the regularization parameter. These cost functions share the property of having a unique global minimum in most application conditions, thus giving a reliable estimation of the amount of regularization needed. The cost function chosen here returns the value with the maximum probability of occurrence, i.e., the maximum _a posteriori_ estimation of \(\eta^{2}\):  
\[\mathbf{J}_{MAP}(\eta^{2})=\sum_{k=1}^{M}\ln(s_{k}^{2}+\eta^{2})+(M-2)\ln\left( \frac{1}{M}\sum_{k=1}^{M}\frac{|y_{k}|^{2}}{s_{k}^{2}+\eta^{2}}\right) \tag{8}\]  
\[\eta^{2}_{MAP}=\arg\min_{\eta^{2}}\ \mathbf{J}_{MAP}(\eta^{2}). \tag{9}\]  
The solutions based on energy minimization described above suffer similar issues to those of beamformers. In fact, the \(L_{2}\)-norm minimization mechanism spreads the energy of single source into several potential sources [17]. In fact, even a single point source is mapped into a \"main lobe\" and other artifacts, thus leading to bad spatial and poor dynamics. For these reasons, they are not well suited for three-dimensional acoustic mapping. In addition, it has been proven to suffer of severe underestimation of source strength [16] especially when dealing with volumetric imaging. In fact, it so happens that the size and shape of the real sources do not change, while only the ROI increases its size; therefore, its discretization requires more potential sources to keep the same spatial resolution. This consideration suggests the hypothesis of _sparsity_ of the source field, which means that it can be approximated by few non-zeros elements in a given representation. In fact, the basis adopted in the direct operator is crucial when sparse representation is sought, as the hypothesis can be fulfilled in some representation and not in others [21]. In this paper, only monopoles are used to build the matrix \(\mathbf{G}\), but other elementary sources can be adopted (dipoles, plane waves, spherical harmonics, etc.). It is known that a priori assumptions such as sparsity and choice of basis are mandatory to achieve satisfactory results in source quantification. Sparsity can be enforced by minimizing the \(L_{p}\)-norm of solution jointly with the fitting error, thus having the following problem:  
\[\mathbf{\hat{q}}(\eta^{2},p)=\arg\min_{\mathbf{q}}\left(\|\mathbf{G}\mathbf{q }-\mathbf{p}\|_{2}^{2}+\eta^{2}\|\mathbf{q}\|_{p}^{p}\right) \tag{10}\]  
with \(0\leq p<2\). True sparse representation of the source field is achieved when \(p=0\). Unfortunately, the exact solution of this problem is known to be NP-hard. A method to get a sparse solution is the Iteratively Re-weighted Least Squares [25], and it is the method adopted in this work. This approach is exploited in several acoustic imaging techniques, for example, Generalized Inverse Beamforming (GIBF) [17], Equivalent Source Method (ESM) [16], or Iterated Bayesian Focusing (IBF) [26]. These methods differ from each other in the hypotheses and in other algorithmic aspects. The method described in this paper shares the theoretical bases with the aforementioned ones and is adapted to volumetric mapping. The optimization problem of Equation (10) has no general analytic solution but can be solved iteratively using the following consideration:  
\[\|\mathbf{q}\|_{p}^{p}=\sum_{n=1}^{N}|q_{n}|^{p}=\sum_{n=1}^{N}w_{sp,n}^{2}|q_{n }|^{2}=\|\mathbf{W}_{sp}\mathbf{q}\|_{2}^{2}\;. \tag{11}\]  
Each iteration is a regularized weighted least-square problem in the general Tikhonov formulation or the equivalent Bayesian one, thus leading to the IRLS approach, as the \(L_{p}\) optimization is achieved making use of weighted least squares and a diagonal weighting matrix \(\mathbf{W}_{sp}\) to converge to a sparse solution. Weights depend on the result of the previous iteration according to the following expression:  
\[w_{sp,n}^{(it)}=\left|\hat{q}_{n}^{(it-1)}\right|^{\frac{p-2}{2}} \tag{12}\]  
where the superscript \((it)\) is the current iteration and \(w_{sp,n}\) is the \(n\)-\(th\) generic diagonal element.