## 3 The Proposed S-LCMV Cost Function
Meanwhile, as \(A_{p}\), \(D_{s}\), and \(B_{q}\), are all normal constants in \(g_{s,ap}(t,d_{m})\), \(g_{s,ap}(t,d_{m})\) will follow \(C_{1}\) to become infinitesimally small when \(p\) becomes large. Then, the sum of a finite number of infinitesimal quantities, namely, \(g_{p}(t,d_{m})=\sum_{s=1}^{S}\sum_{q=1}^{Q}g_{s,ap}(t,d_{m})\), also becomes an infinitesimal quantity. The above analysis shows that the value of \(g_{s,ap}(t,d_{m})\) is dominated by only a small subset of \(g_{s,ap}(t,d_{m})\), and the term \(\mathbf{g}(t,d_{m})\), containing the wall thickness information, possesses an obvious sparse property. The sparse structure of \(\mathbf{g}(t,d_{m})\) is presented in the experimental section for ease of understanding  
Based on the above discussion, it can be concluded that for large \(p\), \(g_{p}(t,d_{m})\) is an infinitesimal quantity that minimally contributes to the received signal \(U_{m}(t,z_{m},\,d_{m})\). Consequently, sparse constraints corresponding to the dominant \(g_{p}(t,d_{m})\) are imposed and the S-LCMV cost function is formulated as  
\[\left\{ \begin{array}{l}\min\,\mathbf{w}_{s}^{T}\mathbf{R}_{us}\mathbf{w}_{s} \\ s.t.\,\,\mathbf{w}_{s}^{T}\mathbf{V}_{K}=\mathbf{f}_{K}\end{array} \right. \tag{21}\]  
where \(\mathbf{V}_{K}=[\mathbf{v}^{\prime}(z_{1}),\,\mathbf{v}^{\prime}(z_{2}),\, \ldots,\,\mathbf{v}^{\prime}(z_{M})]^{T},\,\mathbf{v}^{\prime}(z_{m})=[v_{1}(z_{ m}),\,\ldots,\,v_{k}(z_{m}),\,\ldots,\,v_{k}(z_{m})]\), \(\mathbf{f}_{K}=[1,1,\,\ldots,\,1]\in\mathbb{Z}^{1\times K}\) with \(K<P\), and \(\mathbf{R}_{us}=E[\widetilde{\mathbf{u}}\widetilde{\mathbf{u}}^{\top}]\) denotes the auto-correlation matrix of \(\widetilde{\mathbf{u}}\). Each element in \(\widetilde{\mathbf{u}}\) is expressed as  
\[\widetilde{U}_{m}(t,z_{m},d_{0})=-\frac{\xi\chi}{t_{\text{of}}}\underset{k=1}{ \overset{K}{\sum}}g_{k}(t,d_{m})v_{k}(z_{m}) \tag{22}\]  
Equation (21) is then solved as  
\[\mathbf{w}_{s}=\mathbf{R}_{us}^{-1}\mathbf{V}_{K}(\mathbf{V}_{K}^{T}\mathbf{R }_{us}^{-1}\mathbf{V}_{K})^{-1}\mathbf{f}_{K}{}^{T} \tag{23}\]  
Due to the sparsity of \(\mathbf{g}(t,d_{m})\), \(K\) is much smaller than \(P\). In particular, numerical simulations confirmed that \(K<M<P\), meaning that the number of constraints was fewer than the DoFs in the adaptive weight function. In this case, \(\mathbf{V}_{K}{}^{T}\mathbf{R}_{us}{}^{-1}\mathbf{V}_{K}\) will become well-conditioned and invertible. Accordingly, accurate weight coefficients can be calculated and the inspection effect can be improved effectively. In addition, solving Equation (23) incurs a much lower computational cost than solving Equation (5) because it involves a small-scale matrix multiplication. To evaluate the computational cost in quantity, we adopted the multiplication and division number (MDN) [31]. Note that the MDN of an \(M\)\(\times\)\(M\) matrix inverse is approximately \(2M^{3}/3\), and that of multiplying an \(M\)\(\times\)\(N\) matrix by an \(N\)\(\times\)\(H\) matrix is _MNH_. Therefore, we have  
\[MDN_{S-LCMV}=\frac{2}{3}(M^{3}+K^{3})+KM(K+M)+K^{2}+MK+M^{2} \tag{24}\]  
\[MDN_{LCMV}=\frac{2}{3}(M^{3}+P^{3})+PM(P+M)+P^{2}+MP+M^{2} \tag{25}\]  
where _MDN\({}_{S-LCMV}\)_ and _MDN\({}_{LCMV}\)_ represent the MDNs of solving Equations (23) and (5), respectively. Evidently, _MDN\({}_{S-LCMV}\)_ is considerably smaller than _MDN\({}_{LCMV}\)_ when \(K\) << \(P\). The above study shows that the PEC signal exhibits sparsity depending on the Gauss-Legendre stage \(p\), which is employed to construct an S-LCMV method that can accurately calculate the weighting coefficients and thus effectively eliminate the interference of the different TRDs on the detection effect. Moreover, the proposed method can also greatly reduce the computational cost. The practical effectiveness of the proposed approach will be verified by simulation experiments in the next section.