CN105677937A - Method for remodeling medium objects by electromagnetic inverse scattering - Google Patents

Abstract

The invention relates to a method for remodeling medium objects by electromagnetic inverse scattering. The method comprises following steps: utilizing Maxwell's equations and constitutive equations to deduce forward and reverse integral equations for electromagnetic scattering; guessing the shape of an object to be detected and discretizing integral equations with a method in order to obtain matrix equations; utilizing a Gauss-Newton minimization method to regularize ill-conditioned equations during the reverse scattering process following discretization, wherein used regularized parameters are obtained by a multiplicative regularization method; and adopting a born iterative method to carry out iteration and obtaining wave number of detected mediums at the needed precision in order to achieve the objective of remodeling mediums.The method helps to avoid the problem in the conventional method that parameters are obtained by a lot of experiment data so that solving efficiency is increased.

Description

The method of a kind of THE INVERSE ELECTROMAGNETIC SCATTERING reconstruct dielectric object

Technical field

The invention belongs to electromagnetic technology field, it relates to electromagnetic wave analysis method, especially THE INVERSE ELECTROMAGNETIC SCATTERING Theoretical Calculation, the method for the unknown dielectric object of quick, efficient reconstruction.

Background technology

Electromagnetic field or hertzian wave are prevalent in around us, and it comprises our human body itself all can interact with any object encountered, material, structure, device, system. Research and analyze the generation (radiation) of hertzian wave, propagation and interaction with object or target and have very important significance, it is all widely used in fields such as communication, radar, navigation, EMC Design, biomedical imaging, geophysical surveys. However, it is to be understood that and the complex behavior of control hertzian wave we must solve its governing equation. The behavior of hertzian wave controls by famous Maxwell equation, and it is mathematically rank partial differential equation, solves by simple method of finite difference, but there is a lot of problem. If we introduce Green's function, and by a series of mathematical derivations, Maxwell equation can be converted into integral equation form, bring great convenience to solving of problem. Solving of electromagnetism integral equation is generally realized by moments method, moments method needs represent with suitable basis funciton or unknown current density function in EXPANSION EQUATION FOR STEEL, and Rao-Wilton-Glisson (RWG) basis funciton is the basis funciton widely used the most at present.

THE INVERSE ELECTROMAGNETIC SCATTERING problem is quite important in microwave imaging and other correlation techniques. Compared to positive scattering problem, THE INVERSE ELECTROMAGNETIC SCATTERING problem has more challenge, and reason is that inverse scattering problem to be dealt with relates to nonlinear equation, and not exclusive on Xie Bengu. Usually, iteration method can be used to carry out head it off, in time domain, widely use differential equation method, such as time-domain finite difference method (FDTD), finite element method. On frequency domain, it may also be useful to integral equation method, such as Born iterative method (BIM), distortion Born iterative method (DBIM), Rytov iterative method (RIM) etc. In the middle of software design, we use Born iterative method and distortion Born iterative method.

With the use of BIM or DBIM method, integral equation, when rebuilding medium object, will use volume integral equation, because object is unknown in imaging territory, medium background is Fei Junyun.

When reconstruction of three-dimensional medium object, it relates to THE INVERSE ELECTROMAGNETIC SCATTERING integral equation (ISIE) and the positive scattering integral equation (FSIE) of electromagnetism, we need repeatedly to calculate iteration. Take SWG as the moments method of basis funciton it is the traditional method solving FSIE problem, but if target background is Fei Junyun, body form is also unknown, then solve comparatively difficulty. SWG basis funciton is defined in one in the volume elements of four sides, known material interface is needed when geometry is discrete, and two kinds of medium contact faces of different nature exist surface charge, the distribution of surface charge needs known, moments method will be caused like this to use very complicated.

Summary of the invention

It is an object of the invention to the deficiency for present method and technology, it is proposed that the method for a kind of THE INVERSE ELECTROMAGNETIC SCATTERING fast and efficiently reconstruct dielectric object.

The technical solution adopted for the present invention to solve the technical problems is as follows:

First aspect, usesMethod solves positive scattering integral equation.

Due toMethod, to split based on single tetrahedron, instead of the tetrahedron pair of SWG basis funciton, so it is easy to use. Under regularity, a micro unit integral equation can turn to:

${\∫}_{\mathrm{\Δ}V}f\left(r^{\′}\right){\mathrm{dV}}^{\′}=\underset{j=1}{\overset{P}{\Σ}}{w}_{j}f\left({r_j}^{\′}\right)$

F (r') is smooth function, P be integration point quantity, w_jUpper volume infinitesimal Δ V jth weight, regularity is degradation Gauss's regularity.Method is applied in Line Integral usually, but, if integral kernel is regular, then this method can be applied in discrete bodies integral equation. Conventional three-dimensional volume integral equation is

∫_VF (r, r') u (r') dV'=-φ (r), r ∈ V

F (r, r') is the integral kernel relevant with scalar Green's function, and u (r') is the unknown function needing to solve. φ (r) is the incident wave in scattering problem. In inventive design, by discrete for integration territory chemical conversion some tetrahedrons, using integration point as point of observation, a method is joined in application makes volume integral equation to transform following matrix equation:

$\underset{i=1}{\overset{N}{\Σ}}\underset{j=1}{\overset{P_i}{\Σ}}{w}_{ij}F(r_{mn},{r^{\′}}_{ij})u\left({r_{ij}}^{\′}\right)=-\mathrm{\φ}\left(r_{mn}\right)$

P_iIt is the quantity of the integration point of i-th element, w_ijRepresent the weight of the Integral Rule of i-th element jth canonical point. N is the total quantity of element. Detail problem describes in detail in specific implementation.

Second aspect, utilizes multiplicative regularization method, Gauss-Newton Method for minimization is optimized, the ill problem occurred after solving inverse scattering integral equation discretize. In the present invention, by amendment canonical formula. Utilizing multiplicative regularization method, cost function is optimized for

$C\left(x\right)=\frac{1}{2}\{\mathrm{\ξ}\[||\stackrel{\‾}{B_d}\·e\left(x\right)|{|}^2-{\mathrm{\σ}}^2\]+\left|\right|\stackrel{\‾}{B_m}\·(x-x_q)\left|{|}^2\right\}$

Specific implementation is as follows:

Step 1. utilize maxwell equation group and rheological equation derive just to reverse electromagnetic scattering integral equation;

Step 2. guesses the shape of testee, initially incident field is substituted into as total field inverse scattering integral equation, and by reverse scattering integral equation discretize, but due to measured scattered field data redundancy often, then there is morbid state in gained matrix equation, so utilizing Gauss-Newton Method for minimization by ill-condition equation canonical, the regularization parameter wherein used is obtained by multiplicative regularization method, thus tries to achieve the wave beam K of medium object_n;

Step 3. is by the wave number K required by step 2_nSubstitute into direct scattering integral equation, utilizeMethod carries out discretize and obtains matrix equation, and solves total field E;

Step 4. adopts Bonn iteration method (BIM) to carry out iteration, total field that total the E tried to achieve by step 3 is updated in step 2, and second time iteration, obtains wave number K_n+1, when | K_n-K_n+1| after reaching required precision, obtain the wave number of object being measured medium, thus try to achieve the relative permittivity of medium object, it is achieved reconstruct dielectric object.

In described step 1, utilize rheological equation, can band just to reverse electromagnetic scattering integral equation, equation is as follows:

To electromagnetism integral equation it is just:

$E\left(r\right)=E^{inc}\left(r\right)+{\∫}_V\stackrel{\‾}{G}(r,r^{\′};k_b)\·\mathrm{\Δ}k\left(r^{\′}\right)E\left(r^{\′}\right){\mathrm{dV}}^{\′},r\∈V$

Reverse electromagnetism integral equation is:

$\begin{array}{cc}{E}^{sca}\left(r\right)={\∫}_V\stackrel{\‾}{G}(r,r^{\′};k_b)\·\mathrm{\Δ}k\left(r^{\′}\right)E\left(r^{\′}\right){\mathrm{dV}}^{\′}& r\∈S\end{array}$

Wherein, E^scaR () is the inverse scattering electric field of the point of observation r on the S of face, E (r') is the total electric field of source point, E^incR () is the incident electric fields that projector is launched,It is dyad Green's function, Δ k (r')=k_b ²[ε_r-1], (r') k_bIt is testee background wave number, ε_r(r') it is the relative permittivity of measured target.

In upper formula, $\stackrel{\‾}{G}(r,r^{\′};k_b)=(\stackrel{\‾}{I}+\frac{\▿\▿}{{k_b}^2})g(r,r^{\′};k_b),$ Wherein $g(r,r^{\′};k_b)=e^{{\mathrm{ik}}_{b}R}/\left(4\mathrm{\π}R\right),R=|r-r^{\′}|.$

In described step 2, usually measured target is guessed for cubes, it is desired to cubes enough surrounds measured target, then adoptsIntegral equation is carried out discretize by method, and departure process is as follows:

Under regularity, a micro unit integral equation can turn to:

${\∫}_{\mathrm{\Δ}V}f\left(r^{\′}\right){\mathrm{dV}}^{\′}=\underset{j=1}{\overset{P}{\Σ}}{w}_{j}f\left({r_j}^{\′}\right)$

F (r') is smooth function, P be integration point quantity, w_jUpper volume infinitesimal Δ V jth weight, adopts Gauss's Integral Rule.

Conventional three-dimensional volume integral equation is

∫_VF (r, r') u (r') dV'=-φ (r), r ∈ V

F (r, r') is the integral kernel relevant with scalar Green's function, and u (r') is the unknown function needing to solve. φ (r) is the incident wave in scattering problem.

The present invention is when solving integral equation, and first by discrete for integration territory chemical conversion some tetrahedrons, using integration point as point of observation, application is joined point methods and volume integral equation is converted into following matrix equation:

$\underset{i=1}{\overset{N}{\Σ}}\underset{j=1}{\overset{P_i}{\Σ}}{w}_{ij}F(r_{mn},{r^{\′}}_{ij})u\left({r_{ij}}^{\′}\right)=-\mathrm{\φ}\left(r_{mn}\right)$

P_iIt is the quantity of the integration point of i-th element, w_ijRepresent the weight of the Integral Rule of i-th element jth canonical point. N is the total quantity of element. The Integral Rule that said process uses is as shown in Fig. 1 (a), (b), if integral kernel strangeness, can not directly apply, it is necessary to revise. Local correction for self-applying element isMethod is applied in the main difficult point in volume integral. The present invention proposes effective amendment scheme, the Line Integral that can be divided into a regular line integral along tetrahedron height direction and a unusual or approximate unusual tetrahedron floorage for the derivation of the volume integral equation on the tetrahedron of high strangeness core, as shown in Fig. 1 (c).

In described step 3, due to the redundancy of take off data, then can cause matrix equationUncomfortable fixed, ill-condition equation occurs. In the present invention, by amendment canonical formula. In traditional BIM or DBIM regularization scheme, it is necessary to solve x value, cost function is made to reach minimum value

$C\left(x\right)=\left|\right|\stackrel{\‾}{Z}\·x-b|{|}^2+\mathrm{\γ}||\stackrel{\‾}{W}\·x|{|}^2$

So when reaching minimum value, $\[{\stackrel{\‾}{Z}}^{*}\·\stackrel{\‾}{Z}+\mathrm{\γ}{\stackrel{\‾}{W}}^{*}\·\stackrel{\‾}{W}\]\·x={\stackrel{\‾}{Z}}^{*}\·b$

Then $x={\[{\stackrel{\‾}{Z}}^{*}\·\stackrel{\‾}{Z}+\mathrm{\γ}{\stackrel{\‾}{W}}^{*}\·\stackrel{\‾}{W}\]}^{-1}\·{\stackrel{\‾}{Z}}^{*}\·b$

In the above-described conventional approaches, the regularization scheme used strictly relies on choosing of regularization factors, and choosing of regularization factors does not have general universal method, it is necessary to a large amount of numerical experiment. In inventive design, it may also be useful to multiplicative regularization method.

Upper formula can be optimized for cost function:

$C\left(x\right)=\frac{1}{2}\{\mathrm{\ξ}\[||\stackrel{\‾}{B_d}\·e\left(x\right)|{|}^2-{\mathrm{\σ}}^2\]+\left|\right|\stackrel{\‾}{B_m}\·(x-x_q)\left|{|}^2\right\}$

Wherein, x is parameter to be asked and relative permittivity x=[ε_r1,ε_r2,…,ε_rN]^T, N is the number of discrete point in image field, and e (x) is error function, has reacted the mismatch between calculated value and observed value, and ξ is Lagrangian multiplier, and its inverse matrix is regularization parameter, and σ is the prior estimate of observed value noise,For designated model parameter x_qCovariance matrix inverse of confidence level,For estimating the inverse of the probabilistic covariance matrix of observed value. Concrete steps are shown in Fig. 3.

In described step 4, when completing steps 1,2,3, just can obtaining just to the matrix form equation with reverse electromagnetic scattering integral equation, Bonn iterative process is as follows:

Matrix equation after a incident field is discrete as the inverse scattering integral equation of total field substitution obtains testee wave number k_n;

B is by wave number k_nSubstitute into positive scattering integral equation discrete matrix equation, and then obtain total field, substitute into inverse scattering integral equation discrete matrix equation by total tried to achieve, obtain testee wave number k_n+1;

C until | k_n+1-k_n| convergence reaches required precision, otherwise returns step a.

Owing to adopting such scheme, the invention has the beneficial effects as follows:

Method is very effective for solution integral equation, and the present invention takes the lead in using this method in THE INVERSE ELECTROMAGNETIC SCATTERING problem.The SWG basis funciton used traditionally is defined in one in the volume elements of four sides, needing known material interface when geometry is discrete, and two kinds of medium contact faces of different nature exist surface charge, the distribution of surface charge needs known, moments method will be caused like this to use very complicated, andThe advantage of method, except not needing the distribution of surface charge, also comprises Quick Pretreatment, and mesh quality requires low, does not use basis funciton and trial function. During inverse scattering equation solves, owing to being ill-condition equation, it is necessary to canonical, Gauss-Newton minimumization is used to realize canonical, when choosing regularization parameter, it may also be useful to multiplicative regularization method, avoid in traditional method and must obtain parameter by a large amount of experimental data, thus improve solution efficiency.

Accompanying drawing explanation

Fig. 1 (a) is for using in the inventive methodThe Integral Rule (one point quadrature rule) of method;

Fig. 1 (b) is for using in the inventive methodThe Integral Rule (four point rules) of method;

The volume integral equation on tetrahedron that Fig. 1 (c) is high strangeness core is divided into the Line Integral schematic diagram of a regular line integral along tetrahedron height direction and a unusual or approximate unusual tetrahedron floorage.

Fig. 2 is the schematic flow sheet of the inventive method embodiment.

Fig. 3 uses multiplicative regularization method schema in the inventive method step 3.

Fig. 4 (a) is the sectional view of the x/y plane of an embodiment of the present invention medium ball. (imaging territory is chosen as 140mmx140mmx140mm cubes, medium ball radius a=40mm, relative permittivity ε_r=3.0, the center of ball and the center superposition in imaging territory).

Fig. 4 (b) is the sectional view of the xz plane of a kind of embodiment medium ball. (imaging territory is chosen as 140mmx140mmx140mm cubes, medium ball radius a=40mm, relative permittivity ε_r=3.0, the center of ball and the center superposition in imaging territory)

Fig. 5 (a) is the sectional view of the x/y plane of a kind of embodiment medium ball. (imaging territory is chosen as 140mmx140mmx140mm cubes, and medium ball is centrally located at imaging territory (-25mm, 0,0) place, medium ball radius a=40mm, relative permittivity ε_r=3.0).

Fig. 5 (b) is the sectional view of the x/y plane of a kind of embodiment medium ball. (imaging territory is chosen as 140mmx140mmx140mm cubes, and medium ball is centrally located at imaging territory (-25mm, 0,0) place, medium ball radius a=40mm, relative permittivity ε_r=3.0).

The sectional view of the x/y plane of the distance imaging center 20mm that Fig. 6 (a) is a kind of embodiment medium ball. (imaging territory is chosen as 140mmx140mmx140mm cubes, medium ball radius a=40mm, relative permittivity ε_r=3.0, the center of ball and the center superposition in imaging territory).

The sectional view of the xz plane of the distance imaging center 20mm that Fig. 6 (b) is a kind of embodiment medium ball. (imaging territory is chosen as 140mmx140mmx140mm cubes, medium ball radius a=40mm, relative permittivity ε_r=3.0, the center of ball and the center superposition in imaging territory).

Fig. 7 (a) for a kind of embodiment medium ball be positioned at x-axis to the distribution plan that changes along with x-axis of the specific inductivity of imaging territory centerline. (imaging territory is chosen as 140mmx140mmx140mm cubes, and medium ball is centrally located at imaging territory (-25mm, 0,0) place, medium ball radius a=40mm, relative permittivity ε_r=3.0. )

Fig. 7 (b) be parallel to for a kind of embodiment medium ball is positioned at x-axis to imaging territory medullary ray and distance imaging territory center 20mm, the distribution plan that the specific inductivity at distance xz plane 70mm place changes along with x-axis.(imaging territory is chosen as 140mmx140mmx140mm cubes, and medium ball is centrally located at imaging territory (-25mm, 0,0) place, medium ball radius a=40mm, relative permittivity ε_r=3.0. )

Fig. 8 (a) is the sectional view of the x/y plane of a kind of embodiment medium cubes. (imaging territory is chosen as 140mmx140mmx140mm cubes, and medium cubes is centrally located at center, imaging territory, medium cubes length of side s=40mm, relative permittivity ε_r=3.2. )

Fig. 8 (b) is the sectional view of the xz plane of a kind of embodiment medium cubes. (imaging territory is chosen as 140mmx140mmx140mm cubes, and medium cubes is centrally located at center, imaging territory, medium cubes length of side s=40mm, relative permittivity ε_r=3.2. )

Fig. 9 (a) is the sectional view of the x/y plane of a kind of embodiment two same media balls. (imaging territory is chosen as 140mmx140mmx140mm cubes, medium ball radius a=25mm, relative permittivity ε_r=2.8, two medium ball centers lay respectively at imaging territory (-25mm, 0,0) and (25mm, 0,0) place. )

Fig. 9 (b) is the sectional view of the xz plane of a kind of embodiment two same media balls. (imaging territory is chosen as 140mmx140mmx140mm cubes, medium ball radius a=25mm, relative permittivity ε_r=2.8, two medium ball centers lay respectively at imaging territory (-25mm, 0,0) and (25mm, 0,0) place. )

Embodiment

Below in conjunction with accompanying drawing illustrated embodiment, the present invention is further illustrated.

The three-dimensional medium body chosen, for receiving antenna and reflecting antenna, its ball surface measurements radius is 1.796m, and image area chooses the cubes of 140mm × 140mm × 140mm. In measuring surface, Yuan Chang positionFrom 30 ° to 150 °, step-length chooses 15 °. θ^sFrom 20 ° to 340 °, step-length is 40 °. Received bit is set toIt is 90 °, θ^sFrom 0 ° to 350 °, step-length is 10 °. Image area is divided into 42156 tetrahedron elements.

When specifically implementing, its method flow is as follows:

1. utilize maxwell equation group to derive electromagnetic field direct scattering integral equation (FSIE) and reverse electro magnetic scattering integral equation (ISIE); 2. guess the shape of object, generally can select the cubes that can as far as possible surround object under test, and make volume enough comprise testee, then utilizeMethod, by positive scattering integral equation discretize, obtains matrix equation; 3. with incident field as total field substitute into inverse scattering integral equation discrete after matrix equation obtain testee wave number k_n; 4. by wave number k_nSubstitute into positive scattering integral equation discrete matrix equation, and then obtain total field; 5 substitute into inverse scattering integral equation discrete matrix equation by total tried to achieve, and obtain testee wave number k_n+1; 6. until | k_n+1-k_n| convergence reaches required precision, returns step 4 if do not reached; If 7. required result does not restrain, then again guess that body form returns in step 1 described in step.

Utilize rheological equation, can obtain just to reverse electromagnetic scattering integral equation, equation is as follows:

To electromagnetism integral equation it is just:

$E\left(r\right)=E^{inc}\left(r\right)+{\∫}_V\stackrel{\‾}{G}(r,r^{\′};k_b)\·\mathrm{\Δ}k\left(r^{\′}\right)E\left(r^{\′}\right){\mathrm{dV}}^{\′},r\∈V$

Reverse electromagnetism integral equation is:

$\begin{array}{cc}{E}^{sca}\left(r\right)={\∫}_V\stackrel{\‾}{G}(r,r^{\′};k_b)\·\mathrm{\Δ}k\left(r^{\′}\right)E\left(r^{\′}\right){\mathrm{dV}}^{\′}& r\∈S\end{array}$

Wherein, E^scaR () is the inverse scattering electric field of the point of observation r on the S of face, E (r') is the total electric field of source point, E^incR () is the incident electric fields that projector is launched, G (r, r'; k_b) it is dyad Green's function, Δ k (r')=k_b ²[ε_r-1], (r') k_bIt is testee background wave number, ε_r(r') it is the relative permittivity of measured target.

In upper formula, $\stackrel{\‾}{G}(r,r^{\′};k_b)=(\stackrel{\‾}{I}+\frac{\▿\▿}{{k_b}^2})g(r,r^{\′};k_b),$ Wherein $g(r,r^{\′};k_b)=e^{{\mathrm{ik}}_{b}R}/\left(4\mathrm{\π}R\right),R=|r-r^{\′}|.$

Usually measured target is guessed for cubes, it is desired to cubes enough surrounds measured target, then adoptsIntegral equation is carried out discretize by method, and departure process is as follows:

Under regularity, a micro unit integral equation can turn to:

${\∫}_{\mathrm{\Δ}V}f\left(r^{\′}\right){\mathrm{dV}}^{\′}=\underset{j=1}{\overset{P}{\Σ}}{w}_{j}f\left({r_j}^{\′}\right)$

F (r') is smooth function, P be integration point quantity, w_jUpper volume infinitesimal Δ V jth weight, adopts Gauss's Integral Rule.

Conventional three-dimensional volume integral equation is

∫_VF (r, r') u (r') dV'=-φ (r), r ∈ V

F (r, r') is the integral kernel relevant with scalar Green's function, and u (r') is the unknown function needing to solve. φ (r) is the incident wave in scattering problem.

The present invention is when solving integral equation, and first by discrete for integration territory chemical conversion some tetrahedrons, using integration point as point of observation, point methods is joined in application makes volume integral equation to obey following matrix equation:

$\underset{i=1}{\overset{N}{\Σ}}\underset{j=1}{\overset{P_i}{\Σ}}{w}_{ij}F(r_{mn},{r^{\′}}_{ij})u\left({r_{ij}}^{\′}\right)=-\mathrm{\φ}\left(r_{mn}\right)$

P_iIt is the quantity of the integration point of i-th element, w_ijRepresent the weight of the Integral Rule of i-th element jth canonical point. N is the total quantity of element. The Integral Rule that said process uses is as shown in Fig. 1 (a), (b), if integral kernel strangeness, can not directly apply, it is necessary to revise. Local correction for self-applying element isMethod is applied in the main difficult point in volume integral. The present invention proposes effective amendment scheme, the Line Integral that can be divided into a regular line integral along tetrahedron height direction and a unusual or approximate unusual tetrahedron floorage for the derivation of the volume integral equation on the tetrahedron of high strangeness core, as shown in Fig. 1 (c).

Next solve inverse scattering integral equation, due to the redundancy of take off data, then can cause matrix equationUncomfortable fixed, in inventive design, it may also be useful to multiplicative regularization method.

Upper formula can be optimized for cost function:

$C\left(x\right)=\frac{1}{2}\{\mathrm{\ξ}\[||\stackrel{\‾}{B_d}\·e\left(x\right)|{|}^2-{\mathrm{\σ}}^2\]+\left|\right|\stackrel{\‾}{B_m}\·(x-x_q)\left|{|}^2\right\}$

Wherein, x is parameter to be asked and relative permittivity x=[ε_r1,ε_r2,…,ε_rN]^T, N is the number of discrete point in image field, and e (x) is error function, has reacted the mismatch between calculated value and observed value, and ξ is Lagrangian multiplier, and its inverse matrix is regularization parameter, and σ is the prior estimate of observed value noise,For designated model parameter x_qCovariance matrix inverse of confidence level,For estimating the inverse of the probabilistic covariance matrix of observed value. Concrete steps are shown in Fig. 3.

Next carrying out Bonn iteration, process is as follows:

A. with incident field as total field substitute into inverse scattering integral equation discrete after matrix equation obtain testee wave number k_n;

B. by wave number k_nSubstitute into positive scattering integral equation discrete matrix equation, and then obtain total field, substitute into inverse scattering integral equation discrete matrix equation by total tried to achieve, obtain testee wave number k_n+1;

C. until | k_n+1-k_n| convergence reaches required precision, otherwise returns steps A.

As Fig. 4 (a), (b) show medium ball radius a=40mm, relative permittivity ε_r=3.0, the center of ball and the center superposition in imaging territory. Fig. 4 (a), (b) are the sectional view of xy and xz plane respectively.

In imaging territory, the asymmetric distribution of specific inductivity is more common, moves the center of medium ball to (-25mm, 0,0), if Fig. 4 is the reconstruction figure xy of off-centered medium ball and the sectional view of xz plane.

Fig. 5 (a), (b) are xy axle and the xz shaft section view data of the medium ball reconstruction of deviation center 20mm.

Fig. 6 (a), (b) depict medium ball be positioned at x-axis to the distribution plan that changes along with x-axis of the specific inductivity of imaging territory centerline, be positioned at medium ball be parallel to x-axis to imaging territory medullary ray and distance imaging territory center 20mm, the distribution plan that the specific inductivity at distance xz plane 70mm place changes along with x-axis.

Fig. 7 (a), (b) are length of side a=80mm, ε_rThe medium cubes of=3.2, xy and the xz sectional view of the reconstruction image of cubes center and imaging territory center superposition.

Fig. 8 (a), (b) are radius a=25mm, DIELECTRIC CONSTANT ε_r=2.8, its centre of sphere relative coordinate is respectively (-25mm, 0,0), xy and the xz sectional view of the reconstruction image of two medium balls of (25mm, 0,0).

The above-mentioned description to embodiment can understand and apply the invention for ease of those skilled in the art. These embodiments obviously easily can be made various amendment by person skilled in the art, and General Principle described herein are applied in other embodiments and need not pass through creative work. Therefore, the invention is not restricted to the above embodiments, those skilled in the art, according to the announcement of the present invention, do not depart from improvement that category of the present invention makes and amendment all should within protection scope of the present invention.

Claims (6)

1. the method for a THE INVERSE ELECTROMAGNETIC SCATTERING reconstruct dielectric object, it is characterised in that: comprise the steps:

Step 1. utilize maxwell equation group and rheological equation derive just to reverse electromagnetic scattering integral equation;

Step 2. guesses the shape of testee, is utilized by integral equationMethod carries out discretize and obtains matrix equation;

In the reverse scattering equation of step 3. after discretize, utilizing Gauss-Newton Method for minimization by ill-condition equation canonical, the regularization parameter wherein used is obtained by multiplicative regularization method;

Step 4. adopts Bonn iteration method to carry out iteration, after reaching required precision, obtains the wave number of object being measured medium, thus realizes reconstruct dielectric object.

2. THE INVERSE ELECTROMAGNETIC SCATTERING according to claim 1 reconstruct dielectric object method, it is characterised in that: in step 1, utilize rheological equation, can band just to as follows with reverse electromagnetic scattering integral equation:

To electromagnetism integral equation it is just:

$E\left(r\right)=E^{inc}\left(r\right)+{\∫}_V\stackrel{\‾}{G}(r,r^{\′};k_b)\·\mathrm{\Δ}k\left(r^{\′}\right)E\left(r^{\′}\right){\mathrm{dV}}^{\′},r\∈V$

Reverse electromagnetism integral equation is:

$\begin{array}{cc}{E}^{sca}\left(r\right)={\∫}_V\stackrel{\‾}{G}(r,r^{\′};k_b)\·\mathrm{\Δ}k\left(r^{\′}\right)E\left(r^{\′}\right){\mathrm{dV}}^{\′}& r\∈S\end{array}$

Wherein, E^scaR () is the inverse scattering electric field of the point of observation r on the S of face, E (r') is the total electric field of source point, E^incR () is the incident electric fields that projector is launched,It is dyad Green's function,k_bIt is testee background wave number, ε_r(r') it is the relative permittivity of measured target;

Wherein, $\stackrel{\‾}{G}(r,r^{\′};k_b)=(\stackrel{\‾}{I}+\frac{\▿\▿}{{k_b}^2})g(r,r^{\′};k_b),$ Wherein $g(r,r^{\′};k_b)=e^{{\mathrm{ik}}_{b}R}/\left(4\mathrm{\π}R\right),$ R=| r-r'|.

3. the method for THE INVERSE ELECTROMAGNETIC SCATTERING according to claim 1 reconstruct dielectric object, it is characterized in that: in step 2,3, during owing to step 2 using against scattering integral equation, first use direct scattering integral equation discrete method by its discretize, therefore first describe in step 3 just to integral equationDiscrete integration equation method, departure process is as follows:

Under regularity, a micro unit integral equation can turn to:

${\∫}_{\mathrm{\Δ}V}f\left(r^{\′}\right){\mathrm{dV}}^{\′}=\underset{j=1}{\overset{P}{\Σ}}{w}_{j}f\left({r_j}^{\′}\right)$

Wherein, f (r') is smooth function, and P is the quantity of integration point, w_jUpper volume infinitesimal Δ V jth weight, adopts Gauss's Integral Rule;

Conventional three-dimensional volume integral equation is:

${\∫}_{V}F(r,r^{\′})u\left(r^{\′}\right){\mathrm{dV}}^{\′}=-\mathrm{\φ}\left(r\right),r\∈V$

Wherein, F (r, r') is the integral kernel relevant with scalar Green's function, and u (r') is the unknown function needing to solve; φ (r) is the incident wave in scattering problem.

4. the method for THE INVERSE ELECTROMAGNETIC SCATTERING according to claim 1 reconstruct dielectric object, it is characterised in that: by discrete for integration territory chemical conversion some tetrahedrons, using integration point as point of observation, point methods is joined in application makes volume integral equation be converted into following matrix equation:

$\underset{i=1}{\overset{N}{\Σ}}\underset{j=1}{\overset{P_i}{\Σ}}{w}_{ij}F(r_{mn},{r^{\′}}_{ij})u\left({r_{ij}}^{\′}\right)=-\mathrm{\φ}\left(r_{mn}\right)$

Wherein: P_iIt is the quantity of the integration point of i-th element, w_ijRepresent the weight of the Integral Rule of i-th element jth canonical point; N is the total quantity of element; If integral kernel strangeness, can not directly apply, it is necessary to revise; The Line Integral of a regular line integral along tetrahedron height direction and a unusual or approximate unusual tetrahedron floorage can be divided into for the derivation of the volume integral equation on the tetrahedron of high strangeness core.

5. the method for THE INVERSE ELECTROMAGNETIC SCATTERING according to claim 1 reconstruct dielectric object, it is characterised in that: in step 2, owing to using take off data redundancy, then can cause the matrix equation of direct scattering integral equation institute discretizeUncomfortable fixed, occur that ill-condition equation can not be separated;By amendment canonical formula,

Use multiplicative regularization method, it be optimized for cost function:

$C\left(x\right)=\frac{1}{2}\{\mathrm{\ξ}\[||\stackrel{\‾}{B_d}\·e\left(x\right)|{|}^2-{\mathrm{\σ}}^2\]+\left|\right|\stackrel{\‾}{B_m}\·(x-x_q)\left|{|}^2\right\}$

Wherein, x is parameter to be asked and relative permittivity x=[ε_r1,ε_r2,…,ε_rN]^T, N is the number of discrete point in image field, and e (x) is error function, has reacted the mismatch between calculated value and observed value, and ξ is Lagrangian multiplier, and its inverse matrix is regularization parameter, and σ is the prior estimate of observed value noise,For designated model parameter x_qCovariance matrix inverse of confidence level,For estimating the inverse of the probabilistic covariance matrix of observed value.

6. the method for THE INVERSE ELECTROMAGNETIC SCATTERING according to claim 1 reconstruct dielectric object, it is characterised in that: in step 4, when completing steps 1,2,3, just can obtaining just to the matrix form equation with reverse electromagnetic scattering integral equation, Bonn iterative process is as follows:

A. with incident field as total field substitute into inverse scattering integral equation discrete after matrix equation obtain testee wave number k_n;

B. by wave number k_nSubstitute into positive scattering integral equation discrete matrix equation, and then obtain total field, substitute into inverse scattering integral equation discrete matrix equation by total tried to achieve, obtain testee wave number k_n+1;

C. until | k_n+1-k_n| convergence reaches required precision, otherwise returns step a.