⇦Chinese Physics Letters, 2016, Vol. 33, No. 1, Article code 018402⇨ Efficient Solution to Electromagnetic Scattering Problems of Bodies of Revolution by Compressive Sensing * Meng Kong(孔勐)1,2, Ming-Sheng Chen(陈明生)2**, Liang Zhang(张量)2, Xin-Yuan Cao(曹欣远)2, Xian-Liang Wu(吴先良)1,2 Affiliations 1School of Electronic and Information Engineering, Anhui University, Hefei 230601 2School of Electronic and Information Engineering, Hefei Normal University, Hefei 230601 Received 12 August 2015 ^*Supported by the National Natural Science Foundation of China under Grant Nos 51477039 and 51207041, the Program of Hefei Normal University under Grant Nos 2014136KJA04 and 2015TD01, and the Key Project of Provincial Natural Science Research of University of Anhui Province of China under Grant No KJ2015A174.
^**Corresponding author. Email: chenms@ustc.edu.cn Citation Text: Kong M, Chen M S, Zhang L, Cao X Y and Wu X L 2016 Chin. Phys. Lett. 33 018402 Abstract Under the theory structure of compressive sensing (CS), an underdetermined equation is deduced for describing the discrete solution of the electromagnetic integral equation of body of revolution (BOR), which will result in a small-scale impedance matrix. In the new linear equation system, the small-scale impedance matrix can be regarded as the measurement matrix in CS, while the excited vector is the measurement of unknown currents. Instead of solving dense full rank matrix equations by the iterative method, with suitable sparse representation, for unknown currents on the surface of BOR, the entire current can be accurately obtained by reconstructed algorithms in CS for small-scale undetermined equations. Numerical results show that the proposed method can greatly improve the computational efficiency and can decrease memory consumed. DOI:10.1088/0256-307X/33/1/018402 PACS:84.40.-x © 2016 Chinese Physics Society Article Text Fast and accurate analysis of electromagnetic scattering problems by the method of moment (MOM) is an interesting topic. Based on MOM, many fast algorithms have been used in scattering analysis, such as fast multipole method (FMM) and conjugate gradient fast Fourier transform (CG-FFT) and adaptive integral method (AIM). Recently, a new signal processing technique called compressive sensing (CS) is introduced to solving scattering problems combined with MOM, in which a new incident source that includes different incident angle information is designed for effectively analyzing monostatic scattering characteristics over a wide incident angle.[1,2] For scattering problems of the body of revolution (BOR), according to the axial symmetry of the structure, the vector surface integral equation is transformed into two scalar–linear integral equations, which can be solved by MOM for the so-called 2$\frac{1}{2}$-dimensional objects.[3-8] To improve the efficiency of method of moment for BOR (BOR-MOM), a sparse impedance matrix was obtained by fast wavelet transform.[9,10] However, the full-scale dense impedance matrix must be generated before sparse transform. In this Letter, we present a new scheme to obtain an underdetermined equation for BOR-MOM based on the CS theory, in which only several rows of the impedance matrix need to be computed to serve as the measurement matrix in CS. Considering the vector surface integral equation in BOR is transformed into two scalar–linear integral equations, it will be easier to find a suitable sparse transform for scalar currents, we choose the second kind of Chebyshev polynomials and db4 wavelet transform to construct a sparse representation. With the orthogonal matching pursuit (OMP) technique, the real current value can be accurately reconstructed from the underdetermined equation. Differently shaped objects are presented and discussed to demonstrate the effectiveness of the proposed method. The BOR-MOM construct a specific vector method to reduce the number of unknowns dramatically.[7] On the BOR surface, the current density ${\boldsymbol J}_{\rm s}$ is induced by the incident magnetic field intensity ${\boldsymbol H}^{\rm i}$, and the magnetic field integral equation (MFIE) can be expressed as $$\begin{align} {\boldsymbol J}_{\rm s} -{\boldsymbol n}\times {\boldsymbol K}({\boldsymbol J}_{\rm s} )={\boldsymbol n}\times {\boldsymbol H}^{\rm inc},~~ \tag {1} \end{align} $$ where ${\boldsymbol n}$ denotes the outward unit vector normal to the BOR surface. The integro-differential operator $K$ is given as $$\begin{align} {\boldsymbol K}(X)=-\int {{\boldsymbol X}\times \nabla {\boldsymbol G}d{S}'},~~ \tag {2} \end{align} $$ where ${\boldsymbol G}$ is the Green function in free space. According to axial symmetry of the BOR structure, $J_{\rm s}$ can be expanded as $$\begin{align} {\boldsymbol J}_{\rm s}={\boldsymbol J}_{\rm s}^t +{\boldsymbol J}_{\rm s}^\phi.~~ \tag {3} \end{align} $$ Through the Fourier series, Eq. (3) can be expressed as $$\begin{align} {\boldsymbol J}_{\rm s}={\boldsymbol J}_{\rm s}^{t}+{\boldsymbol J}_{\rm s} ^\phi=\sum\limits_{n=-\infty}^\infty {\{{\boldsymbol J}_{sn}^t +{\boldsymbol J}_{sn}^\phi \}} e^{-jn\phi}.~~ \tag {4} \end{align} $$ Rewrite it as $$\begin{alignat}{1} {\boldsymbol J}_{\rm s}=\sum\limits_{n=-\infty}^\infty \Big\{\sum\limits_{i=1}^N I_{ni}^t {\boldsymbol f}_{ni}^t +\sum\limits_{i=1}^N I_{ni}^\phi {\boldsymbol f}_{ni}^\phi \Big\} e^{-jn\phi},~~ \tag {5} \end{alignat} $$ where $N$ is the number of discrete segments of the generatrix on BOR, $I_{ni}^t$ and $I_{ni}^\phi$ are the two components of the unknown current in $t$ and ${\rm \phi}$ directions, respectively; ${\boldsymbol f}_{ni}^t$ and ${\boldsymbol f}_{ni}^\phi$ are the basis functions used to expand $t$ and $\phi$ components, and it can be formulated as $$\begin{alignat}{1} \!\!\!\!\!\!{\boldsymbol f}_{ni}^t={\boldsymbol t}\frac{T_i}{d}e^{jn\phi}, ~~{\boldsymbol f}_{ni}^\phi={\boldsymbol \phi}\frac{T_i}{d}e^{jn\phi},~~i=1,2,\ldots,N,~~ \tag {6} \end{alignat} $$ where $T_i$ is the triangular functions, and $d$ is the distance between any two segment points. Choosing the test functions as $$\begin{alignat}{1} \!\!\!\!{\boldsymbol W}_{nq}^p={\boldsymbol p}\frac{T_q}{d}e^{-jn\phi}~~~~~({\boldsymbol p}={\boldsymbol t},{\boldsymbol \phi};~q=1,2,\ldots,M).~~ \tag {7} \end{alignat} $$ Substituting Eq. (5) into Eq. (1), and weighted by test functions Eq. (7), simultaneously, a linear algebraic equation ${\boldsymbol ZI}={\boldsymbol V}$ is yielded as $$\begin{align} &\left[\begin{matrix}{[Z^{tt}]_{M\times N}}& {[Z^{t\phi}]_{M\times N}}\\ {[Z^{\phi t}]_{M\times N}}& {[Z^{\phi \phi}]_{M\times N}}\end{matrix}\right]_{2M\times 2N} \left[\begin{matrix} {[I^t]_{N\times 1}}\\ {[I^\phi ]_{N\times 1}}\end{matrix}\right]_{2N\times 1}\\ =\,&\left[\begin{matrix} {[V^t]_{M\times 1}}\\ {[V^\phi ]_{M\times 1}} \end{matrix}\right]_{2M\times 1}.~~ \tag {8} \end{align} $$ In the traditional BOR-MOM, $M$ is usually equal to $N$, and the right-hand incident vector can be described by $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!\!\![V_n^{\rm p}]_q=\langle {\boldsymbol W}_{nq}^{\rm p},{\boldsymbol H}^{\rm inc}\rangle,~~({\boldsymbol p}={\boldsymbol t},{\boldsymbol \phi},~q=1,2,\ldots,M),~~ \tag {9} \end{alignat} $$ and the impedance matrix can be computed by $$\begin{align} Z_{iq}^{tt}=\,&\frac{1}{2}\int_{ti} {\int_0^{2\pi} {T(t)}} \frac{T(t')}{\rho '}e^{jn(\phi '-\phi )}d\phi dt\\ &+\frac{1}{4\pi}\int_{ti} {\int_{tq} {\int_0^{2\pi} {\int_0^{2\pi} {T(t)}}}} T(t')e^{jn(\phi '-\phi)}\\ &\cdot\Big\{[(\rho '-\rho )\cos \gamma '-(z'-z)\sin \gamma ']\cos (\phi '-\phi )\\ &-2\rho \cos \gamma '\sin ^2\Big(\frac{\phi '-\phi}{2}\Big)\Big\}\\ &\cdot(1+jkr)\frac{e^{-jkr}}{r^3}d\phi d\phi 'dtdt',\\ Z_{iq}^{t\phi}=\,&\frac{1}{4\pi}\int_{ti} {\int_{tq} {\int_0^{2\pi} {\int_0^{2\pi} {T(t)}}}} T(t')e^{jn(\phi '-\phi )}(z'\\ &-z)\sin (\phi '-\phi )(1+jkr)\frac{e^{-jkr}}{r^3}d\phi d\phi 'dtdt',\\ Z_{iq}^{\phi t}=\,&\frac{1}{4\pi}\int_{ti} {\int_{tq} {\int_0^{2\pi} {\int_0^{2\pi} {T(t)}}}} T(t')e^{jn(\phi '-\phi )}\\ &\cdot[\rho '\cos \gamma '\sin \gamma -\rho \cos \gamma \sin \gamma '\\ &-(z'-z)\sin \gamma \sin \gamma ']\sin (\phi '-\phi )\\ &\cdot(1+jkr)\frac{e^{-jkr}}{r^3}d\phi d\phi 'dtdt', \\ Z_{iq}^{\phi \phi}=\,&\frac{1}{2}\int_{ti} {\int_0^{2\pi} {T(t)}} \frac{T(t')}{\rho '}e^{jn(\phi '-\phi )}d\varphi dt\\ &+\frac{1}{4\pi}\int_{ti} {\int_{tq} {\int_0^{2\pi} {\int_0^{2\pi} {T(t)}}}} T(t')e^{jn(\phi '-\phi )}\\ &\cdot\Big\{[(\rho '-\rho )\cos \gamma-(z'-z)\sin \gamma ]\cos (\phi '-\phi )\\ &+2\rho '\cos \gamma '\sin ^2\Big(\frac{\phi '-\phi}{2}\Big)\Big\}\\ &\cdot(1+jkr)\frac{e^{-jkr}}{r^3}d\phi d\phi 'dtdt',~~ \tag {10} \end{align} $$ where $\gamma$ is the angle between $t$ and $z$. Differently from the traditional BOR-MOM, according to the CS technique, only several random rows of the impedance matrix in Eq. (8) are extracted from the dense full rank matrix to form a measurement matrix ${\boldsymbol Z}_{2M\times 2N}(M\ll N)$, and the underdetermined equation can be expressed as $$\begin{alignat}{1} {\boldsymbol Z}_{2M\times 2N} {\boldsymbol I}_{2N\times 1}={\boldsymbol V}_{2M\times 1},~~(M\ll N),~~ \tag {11} \end{alignat} $$ where the right-hand vector ${\boldsymbol V}_{2M\times 1}$ is just the known measurements of unknown currents ${\boldsymbol I}_{2N\times 1}$. Hence, in the CS theory,[11] the unknown currents can be reconstructed from ${\boldsymbol V}_{2M\times 1}$ and ${\boldsymbol Z}_{2M\times 2N}$. If ${\boldsymbol I}_{2N\times 1}$ in Eq. (11) is sparse, the CS technique described above can be directly applied. Otherwise, we must seek a suitable sparse transform matrix ${\it {\boldsymbol \Psi}}_{2N\times 2N}$ to compress the unknown response ${\boldsymbol I}_{ 2N\times 1}$ as $$\begin{align} &{\boldsymbol \psi}_{2N\times 2N} \times {\boldsymbol I}_{2N\times 1}={\boldsymbol A}_{2N\times 1},~~ \tag {12} \end{align} $$ $$\begin{align} &{\boldsymbol I}_{2N\times 1}={\boldsymbol \psi}^T_{2N\times 2N} \times {\boldsymbol A}_{2N\times 1},~~ \tag {13} \end{align} $$ where ${\boldsymbol A}_{2N\times 1}=(\alpha _{1}, \alpha _{2}, \alpha _{3},{\ldots}, \alpha _{2N})^{\rm T}$ is a $K$-sparse vector. Substituting Eq. (13) into Eq. (11), $$\begin{align} {\boldsymbol V}_{2M\times 1}={\boldsymbol Z}_{2M\times 2N}^{\rm CS} \times {\boldsymbol A}_{2N\times 1},~~ \tag {14} \end{align} $$ where ${\boldsymbol Z}_{2M\times 2N}^{\rm CS}={\boldsymbol Z}_{2M\times 2N} \times {\it {\boldsymbol \Psi}}_{2N\times 2N}^{\rm T}$. Rewriting Eq. (14) as a matrix form, one will obtain $$\begin{align} &\left(\begin{matrix} V_1 \\ V_2 \\ \vdots \\ V_{2M} \\ \end{matrix}\right)_{2M\times 1}={\boldsymbol Z}^{\rm CS}_{2M\times 2N} {\boldsymbol A}_{2N\times 1}\\ =\,&\left(\begin{matrix} {Z^{\rm CS}_{1,1}}& {Z^{\rm CS}_{1,2}}& \ldots& {Z^{\rm CS}_{1,2N}}\\ {Z^{\rm CS}_{2,1}}& {Z^{\rm CS}_{2,2}}& \ldots& {Z^{\rm CS}_{2,2N}}\\ && \vdots&\\ {Z^{\rm CS}_{2M,1}}& {Z^{\rm CS}_{2M,2}}& \ldots& {Z^{\rm CS}_{2M,2N}}\\ \end{matrix}\right)_{2M\times 2N} \\ &\cdot \left(\begin{matrix} {\alpha _1}\\ {\alpha _2}\\ \vdots\\ {\alpha _{2N}}\\ \end{matrix}\right)_{2N\times 1},~~ \tag {15} \end{align} $$ where ${\boldsymbol Z}_{2M\times 2N}$ can be constructed by the interaction between the source of random $M$ field points and the source of all the $N$ points in both $t$ and ${\rm \phi}$ directions, and ${\boldsymbol V}_{i}$ ($i=1,2,3,{\ldots},2M$) represent the corresponding excitations on field points. In this work, OMP is used to reconstruct the sparse ${\boldsymbol A}_{2N\times 1}$, thus the real induced current ${\boldsymbol I}_{2N\times 1}$ can be obtained by Eq. (13). To obtain an accurate reconstructed induced current with high probability, the new method must be restricted by the following conditions. First, the selection of the measurement matrix must satisfy the restricted isometry property (RIP).[12] Generally, Green's function plays the most important role in computing impedance matrix elements, and the Toeplitz property is a characteristic of Green's function.[13,14] Thus, the impedance matrix ${\boldsymbol Z}_{2M\times 2N}$ in the proposed method has the Toeplitz property. According to Refs. [15–17], the Toeplitz matrix or the quasi-Toeplitz matrix have been proved to meet RIP. Secondly, the reconstruction accuracy is improved following with the increase of the sparsity ($M/N$),[18] and the suitable sparse transform is a key issue to ensure the computational efficiency and accuracy.

Fig. 1. The sphere illuminated by the plane wave.

Fig. 2. Relationship between $M/N$ and the reconstructed error in different discrete steps.

The total computational complexity of the proposed method can be divided into two parts: one is to generate the measurement matrix, and the other is to reconstruct the original induced current. As we know, the complexity of solving a matrix equation in the traditional MOM is $O(PN^{2})$, where $P$ is the number of iteration steps, and the complexity of impedance matrix filling is $O(N^{2})$ in the traditional MOM for BOR. In the proposed method, the complexity of OMP is $O(KMN)$,[19] while the computation complexity of generating measurement matrix is $O(MN)$. Meanwhile, based on the CS theory, $K\ll M\ll N$. Therefore, the complexity of the new method will be far less than the traditional BOR-MOM. Scattering analysis of differently shaped BOR objects is presented to test the effectiveness of the proposed method. To compare the error between real currents and the reconstructed currents, we define the relative error as $$\begin{align} {\it \Delta}=\frac{||\tilde {\boldsymbol I}-{\boldsymbol I}||_2}{||{\boldsymbol I}||_2}\times 100\%,~~ \tag {16} \end{align} $$ where ${\tilde{\boldsymbol I}}$ is reconstruction currents by OMP, and ${\boldsymbol I}$ is original currents by the traditional BOR-MOM. Numerical experiments were performed on an Intel core i7 personal computer.

Fig. 3. Current distribution of the PEC sphere at different segments ($M/N=0.3$, $N=150$, and $\phi=0^{\circ}$).

Fig. 4. Current distribution of the PEC sphere at different segments ($M/N=0.3$, $N=150$, and $\phi=90^{\circ}$).

Fig. 5. A cone-sphere object illuminated by the plane wave.

A perfect electrical conducting (PEC) sphere with its radius of 1 m, which was illuminated by an axially incident plane wave of 300 MHz, is considered as the first example shown in Fig. 1. The 2nd kind of Chebyshev polynomial was selected to construct sparse transform of the proposed method.[20] To further discuss characteristics of the proposed method, the reconstructed error with different discrete steps is described in Fig. 2. Therefore, we select $M/N=0.3$ and discrete step=($\pi\cdot\lambda$)/150 ($N=150$) to balance the reconstructed error and the computational complexity. Due to the universality of the proposed method, the result of Fig. 2 has a certain reference value for other BOR objects. Figures 3 and 4 show the magnitudes of computed currents at $\phi=0^{\circ}$ and $\phi=90^{\circ}$ obtained by CS in comparison with that by the traditional BOR-MOM. As listed in Table 1, the total computing time of CS is about one third of BOR-MOM following the matrix size decreasing to 30% of their original scale, and the accuracy of the proposed method meets the requirement. As shown in Fig. 5, a PEC cone-sphere object with its radius of 0.2 m and the cone angle $\alpha=20^{\circ}$ is considered, and the object is illuminated by a plane wave of 300 MHz. As shown in Figs. 6 and 7, when we set $M/N=0.3$ and $N=128$ and use the db4 wavelet as sparse transform, the reconstructed currents by CS agree well with those of BOR-MOM for $\phi=0^{\circ}$ and $\phi=90^{\circ}$.

**Table 1.** Comparison of the calculation data between BOR-MOM and CS for the PEC sphere.
Example	Algorithm	Matrix size	Total computing time (s)	Reconstructed error (%)	Reduction in time (%)
Fig. 3	BOR-MOM	300$\times$300	156.03
Fig. 3	CS	90$\times$300	54.26	1.17	34.78
Fig. 4	BOR-MOM	300$\times$300	159.87
Fig. 4	CS	90$\times$300	55.56	0.94	34.75

**Table 2.** Comparison of the calculation data between BOR-MOM and CS for the PEC cone-sphere object.
Example	Algorithm	Matrix size	Total computing time (s)	Reconstructed error (%)	Reduction in time (%)
Fig. 6	BOR-MOM	256$\times$256	53.09
Fig. 6	CS	85$\times$256	18.34	2.44	34.54
Fig. 7	BOR-MOM	256$\times$256	53.22
Fig. 7	CS	85$\times$256	18.60	2.21	34.95

Fig. 6. Current distribution of the PEC cone-sphere object at different segments ($M/N=0.3$, $N=128$, and $\phi=0^{\circ}$).

Fig. 7. Current distribution of the PEC cone-sphere object at different segments ($M/N=0.3$, $N=128$, and $\phi=90^{\circ}$).

As listed in Table 2, the matrix size in CS is 30% of the original size as well as example 1, and the proposed method has about 35% reduction in computing time compared with BOR-MOM. In summary, an efficient method based on the CS technique has been proposed to analyze the electromagnetic scattering characteristic of BOR objects, in which random rows of the impedance matrix are computed to form an underdetermined equation. Instead of the traditional iteration method, the underdetermined equation can be quickly solved by a reconstruction algorithm such as OMP with the help of suitable sparse transform. Relationship between the discrete step and reconstructed error has been discussed to obtain a balance. Finally, the high efficiency and accuracy of the proposed method has been validated by scattering analysis of differently shaped BOR objects. References Method of Moments Based on Prior Knowledge for Solving Wide Angle EM Scattering ProblemsAccurate PMCHWT Solutions for Scattering from Arbitrarily-Shaped Penetrable Bodies of Revolution [Open Problems in CEM]Radiation and scattering from bodies of revolutionScattering from arbitrarily-shaped lossy dielectric bodies of revolutionUse of wavelets for an efficient solution of electromagnetic scattering by conducting bodies of revolutionA fast-wavelet solution of electromagnetic scattering by dielectric bodies of revolutionSparse Transform Matrices and Their Application in the Calculation of Electromagnetic Scattering Problems

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