阅读说明：本技术 基于布尔运算的电阻抗成像方法 (Electrical impedance imaging method based on Boolean operation ) 是由 刘�东 顾丹萍 杜江峰 于 2020-11-11 设计创作，主要内容包括：一种基于布尔运算的电阻抗成像方法,包括以下步骤：建立完备电极模型,通过有限元方法求解得到观测模型；建立基于B样条曲线形式的基本单元形状的矩阵表示形式；对B样条曲线进行布尔运算；基于B样条曲线与布尔运算相结合的形状重构算法；执行重构算法,进行图像重构,以实现基于布尔运算的电阻抗成像。本发明所提出的方法可以重建得到多相、内嵌等复杂形状,得到的重建电阻抗图像成像精度、空间分辨率高,并且对电阻抗成像中普遍存在的模型误差、测量噪音等具有很好的鲁棒性。(An electrical impedance imaging method based on Boolean operation comprises the following steps: establishing a complete electrode model, and solving by a finite element method to obtain an observation model; establishing a matrix representation form of a basic unit shape based on a B spline curve form; performing Boolean operation on the B-spline curve; a shape reconstruction algorithm based on the combination of a B spline curve and Boolean operation; and executing a reconstruction algorithm to reconstruct the image so as to realize the electrical impedance imaging based on Boolean operation. The method provided by the invention can reconstruct and obtain complex shapes such as multiphase shapes, embedded shapes and the like, and the obtained reconstructed electrical impedance image has high imaging precision and spatial resolution and has good robustness to model errors, measurement noises and the like commonly existing in electrical impedance imaging.)

1. An electrical impedance imaging method based on Boolean operation is characterized by comprising the following steps:

establishing a complete electrode model, and solving by a finite element method to obtain an observation model;

establishing a matrix representation form of a basic unit shape based on a B spline curve form;

performing Boolean operation on the B-spline curve;

a shape reconstruction algorithm based on the combination of a B spline curve and Boolean operation;

and executing a reconstruction algorithm to reconstruct the image so as to realize the electrical impedance imaging based on Boolean operation.

2. The electrical impedance imaging method of claim 1, wherein the method is adapted for absolute imaging, multi-phase imaging and differential imaging.

3. Electrical impedance imaging method according to claim 1, characterized in that the complete electrode model is as follows:

wherein, sigma (x) is the conductivity distribution, x belongs to omega and is a space coordinate, and z_lAs contact resistance, U_lAnd I_lAre respectively an electrode e_lThe voltage and current on, n represents the outer unit normal.

4. Electrical impedance imaging method according to claim 1, characterized in that the general expression of the observation model is:

V＝U(σ)+e；

where V is the measured voltage, U (σ) is the positive problem solution solved using the finite element method, i.e., the calculated voltage, and e is the additive gaussian noise.

5. The electrical impedance imaging method of claim 1, wherein the establishing a matrix representation of the basic cell shape based on a B-spline curve form comprises:

representing each basic unit boundary as a B spline curve form;

carrying out recursive processing on the B spline curve form;

a matrix representation in the form of a B-spline curve is obtained.

6. An electrical impedance imaging method according to claim 1, wherein the operation of the basic cells by the boolean operation comprises combining, intersecting and subtracting.

7. The electrical impedance imaging method of claim 1, wherein the shape reconstruction algorithm based on a B-spline curve combined with boolean operations comprises:

each basic unit is made into a new medium boundary shape through Boolean operation, then the corresponding conductivity is respectively defined for the medium, the conductivity distribution is related to the control point of the B-spline curve and can be expressed as

8. Electrical impedance imaging method according to claim 2, characterized in that, in case the electrical impedance imaging method is applied to absolute imaging, the observation model is specifically represented as:

wherein, V is the measured voltage, U is the voltage obtained by solving the positive problem, e is the additive noise, delta is the discrete degree parameter of the finite element grid, and omega is the solving domain.

9. An electrical impedance imaging method according to claim 2, wherein in the case that the electrical impedance imaging method is applied to absolute imaging, based on a least square method and a regularization technique, the shape reconstruction of the medium boundary is achieved by solving a minimization problem represented by the following expression using a gauss-newton method:

in the formula, L_eFor observing the covariance matrix of the noiseThe Cholesky factor of (1), satisfyReg (·) is a regularization term.

10. An electrical impedance imaging method according to claim 9, wherein the gauss-newton method is used to solve the minimization problem by iteration, which continuously modifies the conductivity values and the control point parameters of the elementary cells; and/or

In the iterative process, the Jacobian matrix needs to be solved

Wherein the content of the first and second substances,can be solved by a standard method to obtain,can pass throughCalculating by a dynamic method; and/or

And the iteration process adopts a linear search method, and the iteration is terminated until the iteration step length is less than a positive value or equal to zero, so that the final reconstructed image is obtained.

Technical Field

The invention relates to the technical field of electrical impedance tomography, in particular to a method for reconstructing an impedance distribution image in an object body by combining Boolean operation and electrical impedance imaging technology.

Background

Electrical Impedance Tomography (EIT) is a novel non-invasive, non-radiative, low-cost, and continuously-monitored functional imaging method, and the medical imaging technology applies a relatively small safe excitation current (voltage) to the outside of a human body based on the characteristic that each tissue of the human body has different Electrical impedance values, and measures the voltage (current) through electrodes placed on the surface of the body, thereby reconstructing a two-dimensional or three-dimensional Electrical impedance distribution image in the human body. As the electrical conductivity of each organ and tissue of the human body is different, the electrical conductivity image obtained by the EIT technology not only contains rich anatomical information, but also can reflect the physiological, pathological state and functional information corresponding to the electrical conductivity of the organ and the tissue. The EIT technology has attractive application prospect in the directions of lung ventilation detection, early detection and diagnosis of various diseases such as tumors, epilepsy and the like.

Since 12 months in 2019, people in the world face huge tests on novel coronary pneumonia, and the method can find suspected patients in early stage and effectively treat and monitor confirmed patients and has important significance for controlling development of epidemic situations. In the new diagnostic protocol for coronavirus pneumonia (seventh edition of trial), it is mentioned that severely infected patients can rapidly progress to acute respiratory distress syndrome, EIT can monitor infected patients, and EIT technology can be used for retentivity evaluation before patients are treated with pulmonary retentions. Meanwhile, the development of the pneumonia infected person can be monitored through EIT imaging, the treatment effect is judged, and the possibility of cross infection between an isolation room and a radiology department during examination is reduced to the maximum extent. EIT imaging technology plays an important role in monitoring, diagnosing and the like of new coronary pneumonia.

The electrical impedance technology has the advantages of no wound, no radiation, rich functional information and the like, but the electrical impedance technology is essentially the inverse problem of pathological nonlinearity and unsuitability, the current imaging cannot reach the imaging precision of the technologies such as X-ray Computed Tomography (CT), Magnetic Resonance Imaging (MRI) and the like, the reconstruction capability of the EIT technology on complex media such as multiple phases, embedded media and the like is limited due to the unsuitability nature of the EIT technology, and the reconstruction condition of a large number of complex media exists in medical and industrial applications. Therefore, research and development of high-precision stable-performance EIT systems and algorithms can improve reconstruction capability of the EIT systems and algorithms for complex media such as multi-phase media and embedded media, and application of the EIT systems and algorithms in clinical medicine and industrial fields is explored, and the EIT systems and algorithms are the current hot and difficult problems.

In carrying out the present invention, the applicant has discovered several prior art techniques and their existing disadvantages.

1. The prior art proposes applying B-spline curves to EIT. The method converts the EIT image reconstruction problem into the shape reconstruction problem, approaches the target shape by adjusting the control points by using the B-spline curve, improves the retention capacity of the reconstruction algorithm on sharp features, but the algorithm does not have the topological evolution capacity, and needs to obtain the prior information of the number of objects in the region before reconstruction. For the condition that the number of objects in the region cannot be known, the algorithm cannot be reconstructed or cannot be reconstructed to obtain a reliable reconstruction result, which greatly limits the popularization and application of the algorithm.

2. In the prior art, a plurality of MMCs are given for each target object as initial iteration values, the shape of the iteration process is changed by optimizing geometric characteristic parameters of components and changing connectivity or distribution among the components, and the conductivity distribution and a voltage value obtained by calculation of a positive problem model are changed by iteration through a Gauss-Newton method, so that a target functional is minimized through an inverse problem, and shape reconstruction is finally realized. However, the present invention is only applicable to reconstruction in the case of two phases, and cannot be applied to reconstruction in complex shapes such as multi-phase and embedded shapes.

3. In the prior art, a bioelectrical impedance imaging method based on a particle swarm and a regularization Gaussian-Newton iterative algorithm exists. According to the method, non-uniform subdivision is adopted in the positive problem calculation so as to improve the imaging precision. And in the solving of the bioelectrical impedance imaging problem, a particle swarm algorithm is adopted to generate an initial value close to a true value to serve as an initial value of the regularized Gauss-Newton iterative algorithm, and then the regularized Gauss-Newton iterative algorithm is utilized to solve the inverse problem. The method adopts a non-uniform subdivision and a standard particle swarm method to generate an initial value for a regularized Gauss-Newton iterative algorithm, can improve the imaging accuracy and overcome the problem that the Newton algorithm is sensitive to the initial value.

Disclosure of Invention

In view of the above, the present invention provides a method of electrical impedance imaging based on boolean operations, in order to solve at least one of the above technical problems.

To achieve the above object, as an aspect of the present invention, there is provided a boolean operation-based electrical impedance imaging method including the steps of:

establishing a complete electrode model, and solving by a finite element method to obtain an observation model;

establishing a matrix representation form of a basic unit shape based on a B spline curve form;

performing Boolean operation on the B-spline curve;

a shape reconstruction algorithm based on the combination of a B spline curve and Boolean operation;

and executing a reconstruction algorithm to reconstruct the image so as to realize the electrical impedance imaging based on Boolean operation.

Wherein the method is suitable for absolute imaging, multi-phase imaging and differential imaging.

Wherein the complete electrode model is as follows:

wherein, sigma (x) is the conductivity distribution, x belongs to omega and is a space coordinate, and z_lAs contact resistance, U_lAnd I_lAre respectively an electrode e_lThe voltage and current on, n represents the outer unit normal.

Wherein, the general expression of the observation model is as follows:

V＝U(σ)+e；

where V is the measured voltage, U (σ) is the positive problem solution solved using the finite element method, i.e., the calculated voltage, and e is the additive gaussian noise.

Wherein the establishing a matrix representation of the basic cell shape based on the B-spline curve form comprises:

representing each basic unit boundary as a B spline curve form;

carrying out recursive processing on the B spline curve form;

a matrix representation in the form of a B-spline curve is obtained.

Wherein the operation performed on the basic unit by the Boolean operation comprises union, intersection and subtraction.

The shape reconstruction algorithm based on the combination of the B spline curve and the Boolean operation comprises the following steps:

each basic unit is made into a new medium boundary shape through Boolean operation, then the corresponding conductivity is respectively defined for the medium, the conductivity distribution is related to the control point of the B-spline curve and can be expressed as

Wherein, in the case that the electrical impedance imaging method is applied to absolute imaging, the observation model is specifically represented as:

wherein, V is the measured voltage, U is the voltage obtained by solving the positive problem, e is the additive noise, delta is the discrete degree parameter of the finite element grid, and omega is the solving domain.

Under the condition that the electrical impedance imaging method is suitable for absolute imaging, based on a least square method and a regularization technology, a Gaussian Newton method is used for solving a minimization problem shown by the following expression to realize the shape reconstruction of the medium boundary:

in the formula, L_eFor observing the covariance matrix of the noiseThe Cholesky factor of (1), satisfyReg (·) is a regularization term.

Solving the minimization problem through iteration, wherein the conductivity value and the control point parameters of the basic unit are continuously corrected in the iteration process; and/or

In the iterative process, the Jacobian matrix needs to be solved

Wherein the content of the first and second substances,can be solved by a standard method to obtain,can be calculated by a perturbation method; and/or

And the iteration process adopts a linear search method, and the iteration is terminated until the iteration step length is less than a positive value or equal to zero, so that the final reconstructed image is obtained.

Based on the technical scheme, compared with the prior art, the electrical impedance imaging method based on Boolean operation has at least one or part of the following beneficial effects:

(1) b-spline curves are used for representing basic units, and the shapes of the basic units are locally and flexibly adjusted by adjusting the positions of control points in the iteration process, so that the algorithm has the capability of reconstructing detailed characteristics of objects;

(2) the control points of the curve are used as design variables, so that the image reconstruction problem is converted into the shape reconstruction problem, and the number of unknowns, the calculation time and the cost are effectively reduced;

(3) the algorithm has the topological evolution capability by combining Boolean operation, so that the number of objects contained in a region does not need to be known before reconstruction, and the algorithm can be suitable for solving more practical problems;

(4) the basic units are combined, intersected, subtracted and the like by utilizing Boolean operation to construct complex shapes such as multiple phases, embedded and the like;

(5) the algorithm provided by the invention can reconstruct complex shapes such as multiple phases, embedded shapes and the like, and the obtained reconstructed electrical impedance image has high imaging precision and spatial resolution and has good robustness to model errors, measurement noises and the like commonly existing in electrical impedance imaging.

Drawings

FIG. 1 is a schematic diagram of an operation process combining union and subtraction operations according to an embodiment of the present invention;

FIG. 2 is a flow chart of a method provided by an embodiment of the present invention;

FIG. 3 is a schematic diagram of a 3-degree B-spline closed curve provided by an embodiment of the present invention;

FIG. 4 is a schematic diagram of the conductivity distribution of each region after Boolean operation provided by the embodiment of the present invention;

fig. 5 is a flowchart of an algorithm provided by an embodiment of the present invention.

Detailed Description

With the development of science and technology, the living standard of people is continuously improved, people begin to pay more attention to physical health, and more people hope to obtain better medical services. Medical imaging technology is one of the most important ways in various medical examinations and medical diagnoses. The currently common imaging modes mainly include MRI, CT, ultrasound and the like, and doctors can clearly see the imaging of internal organs and tissues of human bodies through the technology, so that accurate diagnosis can be made. However, CT, ultrasound, MRI, etc. are known to have many drawbacks, which are expensive, and radiation, which causes significant damage to the human body after long-term use.

The electrical impedance imaging method has the advantages of no wound, no radiation, convenient use, relatively low equipment price and the like, becomes a hot point of research at home and abroad in recent years, has great development potential in the field of medical imaging, and simultaneously plays an important role in monitoring, diagnosing and the like of new coronary pneumonia by the EIT imaging technology. The traditional EIT algorithm has been developed for more than 30 years, but the imaging precision, real-time performance, stability, complex shape reconstruction and other aspects of the traditional EIT algorithm still have many limitations. In medicine and industry, the situations that the interior of a reconstruction region is complex and the number of objects contained in the reconstruction region is not clear exist in large quantity, so that the electrical impedance imaging algorithm which has topological evolution capability, can reconstruct complex shapes such as multiple phases and embedded shapes and has high resolution and robustness has great practical requirements.

The invention discloses an electrical impedance imaging technology based on Boolean operation. The method fully utilizes the advantages of local modifiability and flexible control characteristics of the B-spline curve, topological evolution of Boolean operation, complex shape construction capability and the like, further improves imaging quality on the basis of the original imaging algorithm, has reconstruction capability of complex shapes such as multiple phases and embedding, and has good robustness and stability on model errors, measurement noises and the like which are ubiquitous in electrical impedance imaging.

Specifically, the invention discloses an electrical impedance imaging method based on Boolean operation, which comprises the following steps:

establishing a complete electrode model, and solving by a finite element method to obtain an observation model;

establishing a matrix representation form of a basic unit shape based on a B spline curve form;

performing Boolean operation on the B-spline curve;

a shape reconstruction algorithm based on the combination of a B spline curve and Boolean operation;

and executing a reconstruction algorithm to reconstruct the image so as to realize the electrical impedance imaging based on Boolean operation.

Wherein the method is suitable for absolute imaging, multi-phase imaging and differential imaging.

Wherein the complete electrode model is as follows:

wherein, sigma (x) is the conductivity distribution, x belongs to omega and is a space coordinate, and z_lAs contact resistance, U_lAnd I_lAre respectively an electrode e_lVoltage and current on, n denotesThe unit is normal.

Wherein, the general expression of the observation model is as follows:

V＝U(σ)+e；

where V is the measured voltage, U (σ) is the positive problem solution solved using the finite element method, i.e., the calculated voltage, and e is the additive gaussian noise.

Wherein the establishing a matrix representation of the basic cell shape based on the B-spline curve form comprises:

representing each basic unit boundary as a B spline curve form;

carrying out recursive processing on the B spline curve form;

a matrix representation in the form of a B-spline curve is obtained.

Wherein the operation performed on the basic unit by the Boolean operation comprises union, intersection and subtraction.

The shape reconstruction algorithm based on the combination of the B spline curve and the Boolean operation comprises the following steps:

each basic unit is made into a new medium boundary shape through Boolean operation, then the corresponding conductivity is respectively defined for the medium, the conductivity distribution is related to the control point of the B-spline curve and can be expressed as

Wherein, in the case that the electrical impedance imaging method is applied to absolute imaging, the observation model is specifically represented as:

wherein, V is the measured voltage, U is the voltage obtained by solving the positive problem, e is the additive noise, delta is the discrete degree parameter of the finite element grid, and omega is the solving domain.

Under the condition that the electrical impedance imaging method is suitable for absolute imaging, based on a least square method and a regularization technology, a Gaussian Newton method is used for solving a minimization problem shown by the following expression to realize the shape reconstruction of the medium boundary:

in the formula, L_eFor observing the covariance matrix of the noiseThe Cholesky factor of (1), satisfyReg (·) is a regularization term.

Solving the minimization problem through iteration, wherein the conductivity value and the control point parameters of the basic unit are continuously corrected in the iteration process; and/or

In the iterative process, the Jacobian matrix needs to be solved

Wherein the content of the first and second substances,can be solved by a standard method to obtain,can be calculated by a perturbation method; and/or

And the iteration process adopts a linear search method, and the iteration is terminated until the iteration step length is less than a positive value or equal to zero, so that the final reconstructed image is obtained.

In order that the objects, technical solutions and advantages of the present invention will become more apparent, the present invention will be further described in detail with reference to the accompanying drawings in conjunction with the following specific embodiments.

The invention aims to provide an electrical impedance imaging method based on the combination of a B-spline curve and Boolean operation, which is used for reconstructing multi-phase, embedded and other complex media in medicine and industry and solves the problems of low electrical impedance reconstruction image spatial resolution and low imaging precision caused by noise, error and the like to a certain extent.

According to the method, a plurality of B-spline curves are given as initial iteration curves, the shapes of basic units represented by the B-spline curves are adjusted by changing the positions of corresponding control points, the basic units are operated by single or multiple Boolean operations in a combined form, the change of the graph shapes in the iteration process is realized, the conductivity distribution and the voltage value obtained by calculation of a positive problem model are changed by iteration through a Gauss-Newton method, so that a minimized target functional is solved through an inverse problem, and the image reconstruction is finally realized.

And constructing to obtain a complex medium shape by performing Boolean operation such as combination, intersection, subtraction and the like on the basic units. Taking the lung shape with embedded substances as an example, the lung shape can be formed by combining a plurality of shapes, and then the lung shape is subtracted to obtain an embedded object, wherein a schematic diagram of a relevant operation process is shown in fig. 1.

The invention provides an electrical impedance imaging technology based on Boolean operation, which not only has reconstruction capability of complex shape, but also has good robustness to model errors, measurement noises and the like commonly existing in electrical impedance imaging.

The invention adopts the B-spline curve form to represent the basic unit, and combines with Boolean operation to reconstruct the EIT shape, and fully utilizes the local adjustability of the B-spline curve and the topological evolution of the Boolean operation. The method has the advantages that: the method comprises the following steps of firstly, representing a basic unit by using a B-spline curve, and carrying out local flexible control on the basic unit by adjusting the position of a control point, so that the method has the capability of reconstructing detailed characteristics of an object; secondly, control points of the B spline curve are used as design variables, so that the dimension of the unknown number, the calculation cost and the calculation time are reduced to a great extent; combining Boolean operation to make the algorithm have topology evolution capability, and the number of objects in the region does not need to be known during reconstruction; and fourthly, combining, intersecting, subtracting and other operations are carried out on the basic units by utilizing Boolean operation to construct complex shapes such as multiphase shapes, embedded shapes and the like. Based on the above advantages, an idea has been created to combine it with electrical impedance imaging techniques.

The invention is mainly innovated aiming at the inverse problem of EIT technology, a plurality of B-spline curves (basic units) are initially given and used as the initial values of a Gauss-Newton iteration method, and Boolean operation (combination, intersection, subtraction and the like) is carried out on the basic units by solving a target functional to obtain the iteration shape, thereby changing the conductivity distribution and calculating the voltage value, and finally realizing the shape reconstruction.

The flow diagram of the present invention is shown in fig. 2.

The specific technical scheme of the invention is as follows:

1. the positive problem is that: and (4) carrying out finite element subdivision on the area to be solved, and establishing a (FEM) finite element model.

The method comprises the steps of uniformly and equidistantly placing 16 electrodes on the boundary of an interested area, exciting two electrodes in the interested area by taking turns at constant current to cause voltage inside the interested area, solving a positive problem by establishing a Complete Electrode Model (CEM), and measuring corresponding voltage values on the other electrodes.

The CEM model is as follows:

wherein, sigma (x) is the conductivity distribution, x belongs to omega and is a space coordinate, and z_lAs contact resistance, U_lAnd I_lAre respectively an electrode e_lThe voltage and current on, n represents the outer unit normal.

The finite element method approximation can be used to obtain an observation model of the positive problem:

V＝U(σ)+e

wherein V is the measured voltage, U (sigma) is the positive problem solution obtained by using the finite element method to solve, namely the calculated voltage, e is additive Gaussian noise, and the mean value is e^*Covariance of Γ_e。

2. Implicit expression of basic unit shape based on B-spline curve form:

each elementary cell boundary is represented in the form of a B-spline curve. With n +1 vertices on a 2-dimensional plane as control vertices of the B-spline curve, the corresponding uniform 3-degree B-spline curve C can be represented as a linear representation of the B-spline basis function

Wherein N is_i，k(t) denotes the ith k (k is usually set to 3) B-spline basis function, defined in recursive form as follows:

wherein i is 0, 1, …, n, andrepresents a uniform node vector, defined as

The B-spline basis function and the B-spline curve control vertex are defined independently of each other, and therefore, the B-spline curve can be expressed in the form of matrix multiplication

C＝NP

Here, N is a matrix composed of B-spline basis functions, and P is a matrix composed of the horizontal and vertical coordinates of the control points of the B-spline curve.

As shown in fig. 3, the diagram is a schematic diagram of a 3-degree B-spline closed curve, in which black points are control points of the B-spline curve, a black solid line is the B-spline curve, and points encircled by circles indicate repetitive control points.

3. Performing Boolean operation on the B-spline curve:

expressing the basic cell boundaries by a B-spline curve, defining a level set function f (x), wherein f (x) is more than 0, f (x) is 0 on the B-spline curve, and f (x) is less than 0 outside the B-spline curve. Thus, for N_cB-spline curveCan obtain N_cCorresponding level set function { f_j，j＝1，2，…，N_c}. It is also an advantage of the algorithm proposed by the present invention that initially given B-spline curves may be overlapped two by two or not overlapped with the rest of B-spline curves, and during the optimization process, two originally overlapped B-spline curves may not be overlapped any more, and two originally non-overlapped B-spline curves may become overlapped with each other: topology evolution capability. Performing Boolean operations on pairwise overlapping B-spline curves is equivalent to performing a level set functionAnd calculating by taking max or min. The method can adopt single Boolean operation for a plurality of B-spline curves, and can also adopt a plurality of Boolean operation combination forms to carry out max and min taking combination operation on the corresponding level set function.

B-spline curve representation based shape reconstruction algorithm combined with Boolean operation:

the basic units form a new medium boundary shape through Boolean operation, then corresponding conductivities are respectively defined for the medium, and the following figures show the conductivity distribution schematic diagrams of a plurality of simple shapes. It should be noted that the schematic diagram shows the case of a single B-spline curve, the case of a single-form boolean operation performed on two B-spline curves, and the case of a multiple-form boolean operation performed on a plurality of B-spline curves. In the actual algorithm design, various shapes and Boolean algorithm combinations also exist, which also embodies the flexibility of the electrical impedance reconstruction algorithm proposed by the present invention. As shown in fig. 4, the conductivity distribution of each region after boolean operation is schematically shown.

As can be seen from the above, the conductivity distribution is related to the control points of the B-spline curve and can be expressed asFrom the mathematical theory, the solution of the electrical impedance imaging technology belongs to the solution of an elliptic partial differential equation, and an observation model of the electrical impedance imaging technology under an absolute imaging frame can be expressed as follows:

wherein, V is the measured voltage, U is the voltage obtained by solving the positive problem, e is the additive noise, delta is the discrete degree parameter of the finite element grid, and omega is the solving domain.

The observation model under the differential imaging framework can be expressed as:

ΔV≈JΔσ+Δe

wherein, DeltaV is the difference value of the front and the back groups of measurement voltages,to assume the jacobian matrix of voltage versus conductivity at the initial background conductivity value,for the two sets of conductivity differences before and after the solution, Δ e is the difference between the two sets of additive noise.

For the sake of simple representation, absolute imaging is taken as an example here, and based on the least square method and the regularization technique, the following minimization problem is solved by using the gauss-newton method to realize the shape reconstruction of the medium boundary:

in the formula, L_eFor observing the covariance matrix of the noiseThe Cholesky factor of (1), satisfyReg (·) is a regularization term.

And iteratively solving the minimization problem by a Gauss-Newton method, and continuously correcting the conductivity value and the control point parameters of the basic unit in the iterative process.

In the iterative process, the Jacobian matrix needs to be solved

Wherein the content of the first and second substances,can be obtained by solving the problem by a standard method,the calculation can be performed by a perturbing method.

And the iteration process adopts a linear search method, and the iteration is terminated until the iteration step length is less than a small positive value or equal to zero, so that the final reconstructed image is obtained. Fig. 5 shows a specific flowchart of the algorithm.

The above-mentioned embodiments are intended to illustrate the objects, technical solutions and advantages of the present invention in further detail, and it should be understood that the above-mentioned embodiments are only exemplary embodiments of the present invention and are not intended to limit the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the protection scope of the present invention.