CN112401865A - Electrical impedance imaging method based on super-shape - Google Patents
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Abstract
An electrical impedance imaging method based on a super-shape comprises the following steps: establishing a complete electrode model, and solving by a finite element method to obtain an observation model; expressing the basic unit by using a super shape; expressing a topological description function by using a super shape; performing Boolean operation on the super-shape; a shape reconstruction algorithm based on a combination of hyper-shape representation and Boolean operations; and executing a reconstruction algorithm to perform image reconstruction so as to realize the electrical impedance imaging based on the super-shape. The algorithm provided by the invention can represent objects in different shapes through the same expression, can reconstruct complex shapes such as multiple phases and embedded shapes, and has good robustness on model errors, measurement noises and the like commonly existing in electrical impedance imaging.
Description
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 a super shape (super shape) and an electrical impedance imaging technology.
Background
Electrical Impedance Tomography (EIT), which has been started in the 80 th 20 th century, is a new direction of medical imaging technology, and based on the characteristic that various tissues of a human body have different Electrical impedance values, a relatively small safe excitation current (voltage) is applied to the outside of the human body, and the voltage (current) is measured by electrodes placed on the surface of the body, so that a two-dimensional or three-dimensional Electrical impedance distribution image in the human body is reconstructed. After the technology appears, the technology has received wide attention in the fields of industry, medicine, geological exploration and the like due to the advantages of non-invasive type, no radiation, high imaging speed, simple equipment, low price, visualization and the like, and has attractive application prospects in the directions of imaging, complex process monitoring and the like.
The EIT image reconstruction problem is highly non-linear due to the "soft-field" nature of EIT techniques. At present, the imaging precision of technologies such as x-ray Computed Tomography (CT), Magnetic Resonance Imaging (MRI) and the like cannot be achieved. In recent decades, EIT imaging techniques have been studied and a number of imaging methods have been proposed by many researchers. Aiming at the problem that the EIT technology is not suitable, the EIT imaging method generally uses prior information as regularization constraint of a solution, so that the reconstruction process is stabilized, and the resolution of a reconstruction result is improved. Such methods are referred to as shape-based reconstruction methods, including fourier methods, level set methods, B-spline-based methods, decomposition methods, monotonicity-based methods, deep learning-based methods, and the like. The traditional Fourier method can accurately reconstruct smooth boundary objects such as circles, ellipses and the like, but can not accurately reconstruct sharp features of the objects, and has a certain smoothing effect on reconstruction of the sharp features. In view of the practical problem, the B-spline method is applied to EIT, the capability of the imaging method for maintaining sharp features of an object is successfully improved, but the B-spline curve cannot accurately express a circular shape, so that the popularization and the application of the B-spline-based electrical impedance imaging method are limited to a certain extent. Therefore, it is a current hotspot and difficult problem to provide an EIT system and an algorithm with high precision and stable performance, which can simultaneously reconstruct a smooth boundary and a sharp feature of an object, improve the reconstruction capability of the EIT system on complex shapes, and explore the application of the EIT system in clinical medicine and industrial fields.
In carrying out the present invention, applicants have discovered that the following prior art exists:
1. the prior art proposes to apply a combination of B-spline curves and joint operations among boolean operations to EIT. The algorithm represents a basic unit by using a B spline curve form, a reconstructed object is constructed through the combined operation of Boolean operation, and the shape of a target is approximated by adjusting the position of a control point of the spline curve. The algorithm keeps the reconstruction capability of the sharp features of the B spline method, realizes topology evolution through joint operation, and finally can realize the reconstruction of complex shapes. The algorithm combines the Boolean operation combined operation with the B-spline curve method, and has the topological evolution capability on the basis of the original B-spline method. However, the algorithm mainly aims at the condition of two phases in EIT, and only adopts the combined operation in Boolean operation, so that reliable results cannot be obtained for multi-phase and other complex shapes through reconstruction; in addition, in the iterative solution process by using the Gauss-Newton method, the Jacobian matrix of the voltage relative to the control point of the B-spline curve is solved by a disturbance method, so that the Jacobian matrix obtained by the solution has certain errors, and certain influence is generated on the result.
2. The prior art is based on an electrical impedance imaging method of a movable deformable component (MMC). According to the method, a plurality of MMCs are given as initial iteration values for a target object, the shape change in the iteration process is realized by optimizing the geometric characteristic parameters of the components and changing the connectivity and distribution among the components, the conductivity distribution is iteratively updated by a Gauss-Newton method, the voltage value obtained by calculation of a positive problem model is used, the target functional is minimized by an inverse problem, and finally the shape reconstruction is realized.
The MMC-based electrical impedance imaging technology not only has high resolution image reconstruction capability, but also has good robustness to model errors, measurement noises and the like which commonly exist in electrical impedance imaging. However, the present invention is only applicable to reconstruction in the case of two phases, but is not applicable to reconstruction in the case of complicated shapes such as multi-phase shapes and embedded shapes.
3. The prior art has a bioelectrical impedance imaging method based on a particle swarm and a regularization Gaussian-Newton iterative algorithm. The invention adopts non-uniform subdivision in the positive problem calculation 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, it is a primary object of the present invention to provide a method of electrical impedance imaging based on a super-shape, which is intended to partially 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 method of electrical impedance imaging based on a super shape, comprising the steps of:
establishing a complete electrode model, and solving by a finite element method to obtain an observation model;
expressing the basic unit by using a super shape;
expressing a topological description function by using a super shape;
performing Boolean operation on the super-shape;
a shape reconstruction algorithm based on a combination of hyper-shape representation and Boolean operations;
and executing a reconstruction algorithm to perform image reconstruction so as to realize the electrical impedance imaging based on the super-shape.
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 zlAs contact resistance, UlAnd IlAre respectively an electrode elThe 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 super-shape describes the shape and size of the basic unit by using geometric parameters, and is expressed as follows:
where q represents a point on the boundary of the super-shape, q0Representing the coordinates of the centre point of the super-shape, theta ∈ [ -pi, pi]，η＝(q0，s，m，a，b，n1，n2，n3Phi) represents a vector composed of the geometric parameters of the elementary cells, s is the scaling factor, m controls the rotational symmetry, n1，n2，n3For controlling the curvature of the boundary, a, b control the size of the super-shape, phi denotes the rotation angle of the super-shape.
Wherein, in the step of expressing the topology description function by using the super-shape, the topology description function is defined as follows:
f(x，η)＝μ(x)·|d(x，η)|＝μ(x)·||x-q(θ*，η)||；
where μ (x) is a sign function, and when a point within a region is located inside a super-shape, μ (x) is 1; when a point within a region is located outside the super-shape, μ (x) ═ 1; when a point of the region is located on the boundary of the super-shape, μ (x) is 0; | d (x, η) | represents the shortest distance from a point x within the region to a boundary point of the super-shape expressed by the geometric parameter η, q (θ)*Eta) represents the corresponding boundary point, where theta*The angle of the parameter corresponding to the nearest boundary point can be represented by
And (4) calculating.
Wherein the operation performed on the basic unit by the Boolean operation comprises union, intersection and subtraction.
Wherein, in the case that the electrical impedance imaging method is applied to absolute imaging, the observation model is specifically represented as:
V＝Uδ，Ω(σ(x，η))+e；
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, LeFor observing the covariance matrix of the noise
The Cholesky factor of (1), satisfy
Reg (·) is a regularization term.
Solving the minimization problem through iteration, wherein the conductivity value and the geometric parameter value 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,
the calculation can be performed by numerical derivation, as follows:
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 has at least one or part of the following beneficial effects:
the invention provides an electrical impedance imaging technology based on a super-shape. The super-shape has wide application in the fields of ecology, geometric modeling, topological optimization and the like. The invention combines the Boolean operation with the medical imaging field and has the following advantages:
(1) the super-shape can represent objects with different shapes, such as shapes with smooth boundaries and sharp features, by adopting different parameter values through a simple expression;
(2) by means of super-shape expression, the geometric parameters are used as design variables, and the number of unknowns, calculation time and cost are effectively reduced;
(3) the super-shape represents the reconstruction shape through an explicit form, so that more geometric information can be applied to the shape reconstruction process;
(4) the algorithm has the topological evolution capability by combining with Boolean operation, so that the number of objects contained in a region does not need to be known before the shape reconstruction is carried out, and the algorithm can be suitable for solving more practical problems;
(5) the basic units are subjected to Boolean operation operations such as combination, intersection, subtraction and the like to construct complex shapes such as multiple phases and embedded shapes;
(6) the algorithm provided by the invention can represent objects in different shapes through the same expression, can reconstruct complex shapes such as multiple phases and embedded shapes, and has good robustness on model errors, measurement noises and the like commonly existing in electrical impedance imaging.
Drawings
FIG. 1 is a schematic flow chart of a method provided by an embodiment of the present invention;
FIG. 2 is a schematic diagram of a super shape provided by an embodiment of the present invention;
FIG. 3 is an example flow chart provided by an embodiment of the present invention.
Detailed Description
The super-shape, also called the Gielis curve, was first proposed by Gielis in 2003, and was originally used to describe the shape of plants and other organisms, and then was applied in the fields of ecology, geometric modeling, topological optimization, etc. A super-shape may represent a variety of shapes by a simple expression, including asymmetric shapes, shapes with smooth boundaries (e.g., circles, ellipses, etc.), shapes with sharp corner features (e.g., triangles, rectangles, etc.), and the like. In view of the flexible expression of shapes by the super-shape method and the topological evolution capability of Boolean operation, we consider developing an electrical impedance shape reconstruction algorithm combining super-shape representation and Boolean operation. The shapes of the elementary cells are described using a hyper-shape, the flexibility of which is exploited to allow the elementary cells to be transformed into various shapes in an iterative process, and the object shapes are reconstructed by boolean operations (including intersection, subtraction, union, etc.).
The invention combines the super-shape method and Boolean operation to be applied to EIT, utilizes the flexible expression capability of the super-shape to various shapes and the topological evolution capability of the Boolean operation, combines the prior information, realizes the accurate reconstruction of complex shapes, and improves the accuracy, stability and robustness of the electrical impedance shape reconstruction algorithm.
Specifically, the invention discloses an electrical impedance imaging method based on a super-shape, which comprises the following steps:
establishing a complete electrode model, and solving by a finite element method to obtain an observation model;
expressing the basic unit by using a super shape;
expressing a topological description function by using a super shape;
performing Boolean operation on the super-shape;
a shape reconstruction algorithm based on a combination of hyper-shape representation and Boolean operations;
and executing a reconstruction algorithm to perform image reconstruction so as to realize the electrical impedance imaging based on the super-shape.
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 zlAs contact resistance, UlAnd IlAre respectively an electrode elThe 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 super-shape describes the shape and size of the basic unit by using geometric parameters, and is expressed as follows:
where q represents a point on the boundary of the super-shape, q0Representing the coordinates of the centre point of the super-shape, theta ∈ [ -pi, pi]，η＝(q0，s，m，a，b，n1，n2，n3Phi) represents a vector composed of the geometric parameters of the elementary cells, s is the scaling factor, m controls the rotational symmetry, n1，n2，n3For controlling the curvature of the boundary, a, b control the size of the super-shape, phi denotes the rotation angle of the super-shape.
Wherein, in the step of expressing the topology description function by using the super-shape, the topology description function is defined as follows:
f(x，η)＝μ(x)·|d(x，η)|＝μ(x)·||x-q(θ*，η)||；
where μ (x) is a sign function, and when a point within a region is located inside a super-shape, μ (x) is 1; when a point within a region is located outside the super-shape, μ (x) ═ 1; when a point of the region is located on the boundary of the super-shape, μ (x) is 0; | d (x, η) | represents the shortest distance from a point x within the region to a boundary point of the super-shape expressed by the geometric parameter η, q (θ)*Eta) represents the corresponding boundary point, where theta*The angle of the parameter corresponding to the nearest boundary point can be represented by
And (4) calculating.
Wherein the operation performed on the basic unit by the Boolean operation comprises union, intersection and subtraction.
Wherein, in the case that the electrical impedance imaging method is applied to absolute imaging, the observation model is specifically represented as:
V＝Uδ，Ω(σ(x，η))+e；
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, LeFor observing the covariance matrix of the noise
The Cholesky factor of (1), satisfy
Reg (·) is a regularization term.
Solving the minimization problem through iteration, wherein the conductivity value and the geometric parameter value 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,
the calculation can be performed by numerical derivation, as follows:
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 the super-shape and Boolean operation, which can accurately reconstruct various shapes of different types in a region, keep the smoothness and sharp characteristics of a reconstructed object and have good robustness on model errors, measurement noises and the like commonly existing in electrical impedance imaging.
According to the invention, a plurality of super-shapes (basic units) are given as initial iteration curves, and the characteristic that the super-shapes can represent objects with various shapes (including smooth boundaries or objects with sharp angles and symmetrical or asymmetrical objects) through a simple expression is utilized, so that the problem that the smooth and objects with sharp features cannot be simultaneously and accurately represented by the existing EIT imaging method is solved; and the basic unit is operated by single or multiple Boolean operations in a combined form, so that the change of the graph shape in the iterative process is realized, the condition that the number of objects in the region is unknown in practical application is solved, and the reconstruction of a complex shape is realized.
The invention adopts the super-shape form to represent the basic unit, and combines with Boolean operation to reconstruct the EIT shape, and fully utilizes the capability of the super-shape to represent various shapes through a single expression and the capability of topological evolution and the like of the Boolean operation. The method has the advantages that: expressing a basic unit by using a super-shape, and simultaneously reconstructing by using the same expression to obtain a shape with a smooth boundary and a sharp feature; secondly, the basic unit is expressed by the super-shape through a display method, so that more geometric information can be directly applied to shape reconstruction; the shapes of the super shapes can express symmetrical and asymmetrical shapes, so that the shapes expressed by the method are more various; 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; constructing and obtaining complex shapes such as multiphase and embedded shapes by performing Boolean operation operations such as combination, intersection, subtraction and the like on the basic units.
The invention is mainly innovated aiming at the inverse problem in the EIT technology, a plurality of super-shapes (basic units) are initially given and used as the initial value of a Gauss-Newton iteration method, single or a plurality of Boolean operation operations (combination, intersection, subtraction and the like) in a combination form are carried out on the basic units to obtain the iteration shape, and the conductivity distribution, the geometric parameter values of the super-shapes and the calculation voltage value are changed by solving a target functional, so that the shape reconstruction is realized.
The flow diagram of the present invention is shown in FIG. 1.
The specific technical scheme of the invention is as follows:
1. the positive problem is that: and carrying out finite element subdivision on the area to be solved, and establishing an FEM 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 zlAs contact resistance, UlAnd IlAre respectively an electrode elThe 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 Γe。
2. The basic unit is expressed by a super shape:
the basic cell boundaries are represented by expressions of a super-shape. The super-shape describes the shape, size and other information of the basic unit by using some geometrical parameters, and is expressed as follows:
where q represents a point on the boundary of the super-shape, q0Representing the coordinates of the centre point of the super-shape, theta ∈ [ -pi, pi]，η＝(q0，s，m，a，b，n1，n2，n3Phi) represents a vector composed of the geometric parameters of the elementary cells, s is the scaling factor, m controls the rotational symmetry, n1，n2，n3For controlling boundary curvesThe ratio, a, b, controls the size of the super-shape, phi denotes the angle of rotation of the super-shape.
Exponential parameter n, considering that parameters s, a, b are all used to control the size of the super-shape1，n2，n3Both for controlling the curvature of the boundary, so we fix the parameter s to 3, n1＝n2＝n3And (n), fixing or unifying the parameters of the same function into the same value, thereby reducing the parameters and reducing the dimensionality of the unknown number. Thus, the parameter η (q) required for the super-shape to express one basic unit is equal to0M, a, b, n, phi) is reduced, thereby alleviating the ill-posed nature of the inverse problem.
A schematic diagram of a super-shape based on the above representation is shown in fig. 2.
3. Expressing the topology description function using a hyper-shape:
the present invention adopts signed distance Function to describe the Topological Description Function (TDF) of the super-shape, which is defined as follows
f(x，η)＝μ(x)·|d(x，η)|＝μ(x)·||x-q(θ*，η)||
Where μ (x) is a sign function, and when a point within a region is located inside a super-shape, μ (x) is 1; when a point within a region is located outside the super-shape, μ (x) ═ 1; when a point of the region is located on the boundary of the super-shape, μ (x) is 0. In addition, | d (x, η) | represents the shortest distance from a point x within the region to the boundary point of the super-shape expressed by the geometric parameter η, q (θ)*Eta) represents the corresponding boundary point, where theta*The angle of the parameter corresponding to the nearest boundary point can be represented by
And (4) calculating.
4. Performing Boolean operation on the super-shape:
the basic cell boundaries are expressed by a hyper-shape, defining the corresponding topology description function f (x, η). Thus, for NcBar super shape
Can obtain the productNcCorresponding topology description functions, i.e. level set functions { f }j，j＝1，2，...，Nc}. It is also an advantage of the algorithm proposed by the present invention that the initial given hyper-shape may be overlapped two by two or not overlapped with the rest of the hyper-shapes, and during the iterative process, the positions of the hyper-shapes may be changed, so that the two originally overlapped hyper-shapes may not be overlapped any more, and the two originally non-overlapped hyper-shapes may become overlapped with each other: topology evolution capability. Performing boolean operations on two overlapping hyper-shapes is equivalent to performing max or min calculations on the level set function. The method can adopt single Boolean operation for a plurality of super shapes, and also can adopt a Boolean operation combination form, namely, composite operation of max and min is carried out on the corresponding level set function.
5. Shape reconstruction algorithm based on the combination of hyper-shape representation and Boolean operation:
as can be seen from the above, the conductivity distribution is related to the geometric parameters of the super-shape, and can therefore be expressed as σ ═ σ (η),
from 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:
V＝Uδ，Ω(σ(x，η))+e
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 be the Jacobian matrix of voltage versus conductivity under the assumption of an initial background conductivity value, Δ σ ═ Δ σ (η),
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, LeFor observing the covariance matrix of the noise
The Cholesky factor of (1), satisfy
Reg (·) is a regularization term.
And (4) iteratively solving the minimization problem by a Gauss-Newton method, and continuously correcting the conductivity value and the geometric parameter value 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 numerical derivation, as follows:
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.
The overall example process of the present invention is shown in fig. 3.
The traditional EIT algorithm has been developed for more than 30 years, but still has many limitations in various aspects such as imaging precision, real-time performance, stability and complex shape reconstruction. A number of existing EIT imaging methods are well able to reconstruct shapes with smooth boundaries, while relatively smooth reconstruction results are obtained for sharp features of the object. Meanwhile, in medicine and industry, the number of objects contained in a reconstruction region is not clear, so that the electrical impedance imaging algorithm which has topological evolution capability, can reconstruct complex shapes such as multiple phases and embedding and has urgent practical requirements on different shapes with smooth boundaries and sharp features and high resolution and robustness is provided.
The invention provides an electrical impedance imaging technology based on a super-shape. The method fully utilizes the advantages of the flexible expression capability of the super-shape to different shapes, the topological evolution of Boolean operation, the complex shape construction capability and the like, further improves the imaging quality on the basis of the original imaging algorithm, has the reconstruction capability of complex shapes such as multiple phases and embedding, and has good robustness and stability to model errors, measurement noises and the like commonly existing in electrical impedance imaging.
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.
Claims (10)
1. An electrical impedance imaging method based on a super-shape is characterized by comprising the following steps:
establishing a complete electrode model, and solving by a finite element method to obtain an observation model;
expressing the basic unit by using a super shape;
expressing a topological description function by using a super shape;
performing Boolean operation on the super-shape;
a shape reconstruction algorithm based on a combination of hyper-shape representation and Boolean operations;
and executing a reconstruction algorithm to perform image reconstruction so as to realize the electrical impedance imaging based on the super-shape.
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 zlAs contact resistance, UlAnd IlAre respectively an electrode elThe 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. An electrical impedance imaging method according to claim 1, wherein the super-shape describes the shape and size of the elementary cells by using geometrical parameters, expressed as follows:
where q represents a point on the boundary of the super-shape, q0Representing the coordinates of the centre point of the super-shape, theta ∈ [ -pi, pi]，η＝(q0，s，m，a，b，n1，n2，n3Phi) represents a vector composed of the geometric parameters of the elementary cells, s is the scaling factor, m controls the rotational symmetry, n1，n2，n3For controlling the curvature of the boundary, a, b control the size of the super-shape, phi denotes the rotation angle of the super-shape.
6. Electrical impedance imaging method according to claim 1, wherein in the step of expressing a topological description function using a hyper-shape, the topological description function is defined as follows:
f(x，η)＝μ(x)·|d(x，η)|＝μ(x)·||x-q(θ*，η)||；
where μ (x) is a sign function, and when a point within a region is located inside a super-shape, μ (x) is 1; when a point within a region is located outside the super-shape, μ (x) ═ 1; when a point of the region is located on the boundary of the super-shape, μ (x) is 0; | d (x, η) | represents the shortest distance from a point x within the region to a boundary point of the super-shape expressed by the geometric parameter η, q (θ)*Eta) represents the corresponding boundary point, where theta*The angle of the parameter corresponding to the nearest boundary point can be represented by
And (4) calculating.
7. 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.
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:
V＝Uδ，Ω(σ(x，η))+e；
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, LeFor observing the covariance matrix of the noise
The Cholesky factor of (1), satisfy
Reg (·) 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 geometric parameter values 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,
the calculation can be performed by numerical derivation, as follows:
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.
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