CN113052927B - Imaging detection method for improving spatial resolution - Google Patents
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Abstract
The invention discloses an imaging detection method for improving spatial resolution, and provides a novel processing function for processing conductivity distribution coefficients. And then reconstructing the optimal conductivity distribution coefficient obtained by the processing by using an operator for reconstructing the conductivity distribution, obtaining the optimal conductivity distribution and imaging. The method effectively solves the problems that sharp edges in the reconstructed image of the Tikhonov regularization method are too smooth and redundant artifacts exist in the background, improves the spatial resolution of the reconstructed image, has better anti-noise performance, and provides a new method for the field of tumor detection.
Description
Technical Field
The invention belongs to the technical field of electrical tomography, and particularly relates to an imaging detection method for improving spatial resolution, which is used for realizing the reconstruction of a conductivity distribution image of tumors.
Background
The electrical tomography is a technique for realizing the reconstruction of the distribution of parameters inside a region based on the measurement of electrical parameters (conductivity/permeability/permittivity) of the region to be measured, and includes resistance tomography (Electrical Resistance Tomography, ERT), electrical impedance tomography (Electrical impedance tomography, EIT), electromagnetic tomography (Electromagnetic Tomography, EMT) and capacitance tomography (Eletrical Capacitance Tomography, ECT). Electrical tomography has received increasing attention for its advantages of non-invasiveness, non-radiation, low cost, real-time, etc., wherein EIT has been widely used in the fields of industrial process detection, geophysical exploration, biomedical imaging, etc.
The EIT image reconstruction process is a pathological inverse problem, which causes a problem of low spatial resolution of the reconstructed image, and hinders development and application of EIT. Regularization is an effective method to solve the inverse problem pathology, with the total variation regularization method based on the L1 norm (Total variation regularization method, TV) and the Tikhonov regularization method based on the L2 norm (Tikhonov Regularization method, TR) being the two most commonly used regularization methods. Studies have shown that TV can preserve boundary discontinuities, reconstruct sharp edges of images, but at the same time can also create serious artifacts in the background of the reconstructed images. In contrast, TR has better performance for a measured object with continuously distributed conductivities, and artifacts in the background of the reconstructed image are reduced, but when the conductivities of the boundaries of the measured object change sharply, the result of reconstructing the image is not accurate enough, resulting in a reduced spatial resolution of the reconstructed image. Therefore, the invention provides an imaging detection method for improving the spatial resolution, which is used for improving the problems that sharp edges in a TR reconstructed image are too smooth and redundant artifacts exist in a background, so that the spatial resolution of the reconstructed image is improved.
Disclosure of Invention
The invention aims at solving the problem of EIT image reconstruction of TR, and provides an imaging detection method for improving spatial resolution, which is used for optimizing the conductivity distribution of EIT reconstructed tumors. In order to increase the spatial resolution of TR, a processing function is proposed to process the conductivity distribution coefficients. In addition, in order to promote sparsity of the processing functions, a non-convex function is proposed as a non-convex regularization term of the processing functions; meanwhile, in order to ensure that the processing function has a unique optimal solution, the range of limiting parameters in the non-convex function is constrained so as to ensure the convexity of the processing function; furthermore, in order to solve the optimal solution of the processing function, converting the unconstrained optimization problem into a constraint problem, and decomposing the constraint problem into sub-problems according to the augmented Lagrangian function for iterative solution; and finally, carrying out inverse transformation on the optimal conductivity distribution coefficient obtained by solving to obtain optimal conductivity distribution. Compared with the TR method, the imaging detection method for improving the spatial resolution effectively solves the problems that sharp edges in the reconstructed image are too smooth and redundant artifacts exist in the background, and obviously improves the spatial resolution of the reconstructed image.
The invention adopts the following technical scheme for realizing the purposes: an imaging detection method for improving spatial resolution is characterized by comprising the following specific steps: taking the image reconstruction process of tumor conductivity distribution as a linear discomfort inverse problem Sg=b, wherein S is a sensitivity matrix, g is conductivity distribution, b is a boundary relative voltage measurement value, and setting the optimal conductivity distribution asThe proposed optimal conductivity distribution +.>The optimization model of (2) is:
in the method, in the process of the invention,is the initial conductivity distribution, J (g) is the objective function, S * Is a normalized sensitivity matrix obtained by normalizing the sensitivity matrix S, and lambda is a regularization parameter used for balancing the fidelity term +.>And regularization termRepresenting the L2 norm. By solving the minimum value of J (g), the initial conductivity distribution is obtained>
The sensitivity matrix S can be obtained according to a sensitivity theory, elements in the sensitivity matrix are called as sensitivity coefficients, and a calculation formula of the sensitivity coefficients is as follows:
wherein s is ij Is the sensitivity coefficient of the jth electrode pair to the ith electrode pair, phi i 、φ j Respectively the ith electrode pair and the jth electrode pair, when the excitation current is I i 、I j The field potential distribution at the time, the distribution form of the sensitivity matrix is as follows:
the sensitivity matrix S is normalized by adopting the following mode:
the normalized sensitivity coefficient and the sensitivity matrix are respectively s * ij And S is * A representation;
a processing function M (a) is proposed, the specific expression of which is as follows:
wherein a is a conductivity distribution coefficient; w is an operator of the decomposition conductivity distribution; w (W) T Is an operator to reconstruct the conductivity distribution;non-convex regular term lambda e Non-convex regular term parameters，/>Is a non-convex function, a e,w Is the conductivity distribution coefficient, c, at scale e and time scale w e Is a limiting parameter; lambda (lambda) o ||DW T a|| 1 Is a convex regular term, where λ o Is a convex regular term parameter that is used to determine, D is a first order differential matrix of which, I 1 Represents an L1 norm;
in the processing function M (a), a non-convex functionThe specific forms proposed are as follows:
when limiting parameter c e The value range of (C) is 0 to or less than c e <1/λ e At this time, the non-convex functionThe sparsity of the non-convex regularization term can be maintained, and the processing function M (a) can be a strict convex function. When the processing function M (a) is a convex function, it can be ensured that it has a unique optimal solution.
To solve for the optimal conductivity distribution coefficientProblem of need to be unconstrained->Conversion to the corresponding constraint problem:
s.t.r＝a
in U 1 (a) And U 2 (r) are respectively:
U 2 (r)＝λ o ||DW T r|| 1
the extended Lagrangian function of the constraint problem is expressed as:
where μ is the Lagrangian penalty parameter.
According to the augmented lagrangian function, the constraint problem is decomposed into the following sub-problems:
where k is the number of iterations, the initial value of k is set to 0, and the maximum value of k is set to 30; r is (r) k Representing the value of the kth iteration of r, the initial value of rd k The value representing the kth iteration of d, the initial value d of d 0 ＝0。
The overall steps of the reconstruction method are as follows:
[1] object field information is acquired.
In the circular measuring field without object, the adjacent mode is adopted to conduct current excitation and voltage measurement to the electrode pairs uniformly distributed on the boundary of the measuring field so as to obtain the blank field boundary voltage measurement value b 1 . Then, the object is placed inside the measuring field area to obtain a full-field boundary voltage measurement value b 2 . Known b 1 And b 2 In the case of (b) can be according to b 2 -b 1 The boundary relative voltage measurement b required for calculation is obtained.
The sensitivity matrix S is obtained according to a sensitivity theory, elements in the sensitivity matrix are called as sensitivity coefficients, and the calculation formula of the sensitivity coefficients is as follows:
wherein s is ij Is the sensitivity coefficient of the jth electrode pair to the ith electrode pair, phi i 、φ j Respectively the ith electrode pair and the jth electrode pair, when the excitation current is I i 、I j Field potential distribution at that time. The sensitivity matrix is distributed as follows:
[2] and (5) normalizing the sensitivity matrix.
The sensitivity matrix is normalized in the following manner:
the normalized sensitivity coefficient and the sensitivity matrix are respectively s * ij And S is * And (3) representing.
[3] And (5) setting parameters.
Regularization parameter λ=0.55, non-convex regularization term parameter λ e =0.95, convex regularization term parameter λ o =0.16, limiting parameter c e ＝0.99/λ e 。
[4]Solving for the initial conductivity distribution
[5]Distributing the initial conductivitySolving the optimal conductivity distribution coefficient by introducing the proposed processing function M (a)>
[6]Operator W for reconstructing conductivity distribution T For a pair ofThe reconstruction is carried out and the image is reconstructed,obtaining optimal conductivity distribution->
[7]With optimum conductivity distributionImaging is performed.
The beneficial effects of the invention are as follows: an imaging detection method for improving the spatial resolution by optimizing the initial conductivity distribution based on the traditional TRTo increase the spatial resolution of the reconstructed image. Compared with TV and TR, the imaging detection method for improving the spatial resolution has good effects in the aspects of reconstructing sharp edges of images, reducing background redundancy artifacts and the like.
Drawings
Fig. 1 is a block flow diagram of an imaging detection method for improving spatial resolution according to the present invention.
Fig. 2 and 3 show examples of the present invention, 2 typical models (model a and model B) were selected to simulate tumors, using three methods: TV, TR and the imaging detection method for improving the spatial resolution provided by the invention are used for setting different noise environments (a noise-free environment and an environment with 5% noise level), reconstructing an image, and the result of reconstructing the image is shown in the figure.
Fig. 4 and 5 are quantitative estimates of relative reconstruction model spatial resolution in a noise-free environment and an environment with a noise level of 5%, respectively—values of relative error and correlation coefficient.
Detailed Description
An imaging detection method for improving spatial resolution of the present invention is described with reference to the accompanying drawings and examples.
The imaging detection method for improving the spatial resolution is used for optimizing the conductivity distribution of EIT reconstructed tumor images. To improve the problem of too smooth sharp edges in the TR reconstructed image, the TR method is chosen to solve for the initial conductivity distribution.
Fig. 1 is a flowchart of an imaging detection method for improving spatial resolution according to the present invention.
The embodiment comprises the following specific steps: the method takes the image reconstruction process of the tumor conductivity distribution as a linear discomfort inverse problem sg=b. Where S is the sensitivity matrix, g is the conductivity distribution, and b is the difference between the boundary voltage measurements.
The inverse problem can be represented by an objective function in the form of a least squares optimization:where f (g) is an objective function. Regularization is an effective method to solve the inverse problem discomfort. The general form of the regularization method can be expressed as: />Where λ is a regularization parameter used to balance the least squares termAnd regularization term R (g).
The standard TR method may replace the regularization term R (g) with a regularization termCan be described as:where E is the identity matrix. However, when TR is discontinuously distributed on the measured medium, an excessively smooth phenomenon occurs on the boundary, resulting in inaccurate reconstructed images and reduced spatial resolution; redundancy artifacts in the reconstructed image background are yet another major factor affecting spatial resolution.
In order to improve the spatial resolution of the TR reconstructed image, the present invention solves for the initial conductivity distributionOn the basis of (a) further optimizing the initial conductivity distribution +.>An imaging detection method for improving spatial resolution is provided.
An imaging detection method for improving the spatial resolution takes an image reconstruction process of tumor conductivity distribution as a linear discomfort inverse problem sg=b. Where S is the sensitivity matrix, b is the boundary relative voltage measurement, and g is the conductivity distribution. The proposed optimal conductivity distributionThe optimization model of (2) is:
in the method, in the process of the invention,is the initial conductivity distribution; j (g) is an objective function; s is S * The sensitivity matrix is normalized, and is obtained by normalizing the sensitivity matrix S; lambda is a regularization parameter used to balance the fidelity term +.>And regularization termRepresenting the L2 norm. />Is the optimal conductivity distribution coefficient, and can be obtained by solving the minimum value of the processing function M (a).
The invention proposes a processing function M (a), the specific expression of which is as follows:
wherein a is a conductivity distribution coefficient; w is an operator of the decomposition conductivity distribution; w (W) T Is an operator to reconstruct the conductivity distribution;non-convex regular term lambda e Non-convex regular term parameters,/>Is a non-convex function, a e,w Is the conductivity distribution coefficient, c, at scale e and time scale w e Is a limiting parameter; lambda (lambda) o ||DW T a|| 1 Is a convex regular term, where λ o Is a convex regular term parameter that is used to determine, D is a first order differential matrix of which, I 1 Representing the L1 norm.
In the processing function M (a), the non-convex function of the inventionThe specific forms proposed are as follows:
when limiting parameter c e The value range of (C) is 0 to or less than c e <1/λ e At this time, the non-convex functionThe sparsity of the non-convex regularization term can be maintained, and the processing function M (a) can be a strict convex function. When the processing function M (a) is a convex function, it can be ensured that it has a unique optimal solution.
The specific implementation steps are as follows:
the first step: object field information is acquired.
In the circular measuring field without object, the adjacent mode is adopted to conduct current excitation and voltage measurement to the electrode pairs uniformly distributed on the boundary of the measuring field so as to obtain the blank field boundary voltage measurement value b 1 . Then, the object is placed inside the measuring field area to obtain a full-field boundary voltage measurement value b 2 . Known b 1 And b 2 In the case of (b) can be according to b 2 -b 1 The boundary relative voltage measurement b required for calculation is obtained.
The sensitivity matrix S is obtained according to a sensitivity theory, elements in the sensitivity matrix are called as sensitivity coefficients, and the calculation formula of the sensitivity coefficients is as follows:
wherein s is ij Is the sensitivity coefficient of the jth electrode pair to the ith electrode pair, phi i 、φ j Respectively the ith electrode pair and the jth electrode pair, when the excitation current is I i 、I j Field potential distribution at that time. The sensitivity matrix is distributed as follows:
and a second step of: and (5) normalizing the sensitivity matrix.
The sensitivity matrix is normalized in the following manner:
the normalized sensitivity coefficient and the sensitivity matrix are respectively s * ij And S is * And (3) representing.
And a third step of: and (5) setting parameters.
Regularization parameter λ=0.55, non-convex regularization term parameter λ e =0.95, convex regularization term parameter λ o =0.16, limiting parameter c e ＝0.99/λ e 。
Fourth step: solving for the initial conductivity distribution
Let the derivative of J (g) be equal to zero, then the minimum value of J (g) can be expressed as:
(S T S+λE)g＝S T b
where E is the identity matrix. Because of S T The inverse matrix of s+λe exists, then the solution of the initial conductivity distribution g is:
fifth step: distributing the initial conductivitySolving the optimal conductivity distribution coefficient by introducing the proposed processing function M (a)
To solve for the optimal conductivity distribution coefficientUnconstrained problem->Can be translated into corresponding constraint problems:
s.t.r＝a
where r is a constraint variable. U (U) 1 (a) And U 2 (r) are respectively:
U 2 (r)＝λ o ||DW T r|| 1
the augmented Lagrangian function of the constraint problem can be expressed as:
where μ is a lagrangian parameter, μ=0.7; d is an iteration variable.
According to the augmented lagrangian function, the constraint problem is decomposed into the following sub-problems:
where k is the number of iterations, the initial value of k is set to 0, and the maximum value of k is set to 30; r is (r) k Representing the value of the kth iteration of r, the initial value of rd k The value representing the kth iteration of d, the initial value d of d 0 ＝0。
By an augmented Lagrangian function, combined with the definition of the L2 norm, a k+1 The solution of (2) can be written in the form:
where p is a first auxiliary variable,
to solve for r k+1 Defining a second auxiliary variable v by using the proximity operator and the semi-orthogonal linear transformation thereof, and using v k Representing v value at kth iteration of algorithm and setting v k ＝a k+1 -d k . According to the sub-problem r k+1 ，v k The adjacency operator of (2) may be defined as:
in the method, in the process of the invention,z is a third auxiliary variable. Using proximity operators and their semi-orthogonal linear transforms, W T v k Is defined as:
r is then k+1 The solution of (2) is finally expressed as:
where TVD represents the TV denoising method, which can be calculated by the chordwise algorithm.
When the iteration number k satisfies k=30, the iteration process is terminated, and the optimal conductivity distribution coefficient is obtainedIs a solution to (a).
Sixth step: obtaining optimal conductivity distribution
According to the obtained optimal conductivity distribution coefficientOperator W for reconstructing conductivity distribution T For->Reconstruction is performed, i.e.)>Thereby obtaining an optimal conductivity distribution->
Seventh step: with optimum conductivity distributionImaging is performed.
Fig. 2 and 3 show examples of the present invention, 2 typical model simulated tumors were selected, and images were reconstructed using TV, TR and the present method, respectively. From the results, it can be seen that the sharp edges of the TR reconstructed image are too smooth and exhibit a pattern shape and size that is not sufficiently accurate. In contrast, the reconstructed image of the TV keeps sharp edges, the shape and the size of the reconstruction are more accurate, and the reconstruction effect is better. However, serious artifacts appear in the background of the reconstructed image. Compared with TV and TR, the method can effectively improve the spatial resolution of the reconstructed image in the reconstruction process. The reconstructed image with clear background and more accurate shape and size can more truly describe the real situation of the tumor.
In the research of the medical imaging field, reconstructing tumor images by using an EIT technology has very important significance. Clear background, accurate size and shape, help doctor to judge the development condition of patient's illness well. As can be seen from the reconstructed image at 5% noise level shown in fig. 3, the proposed method is more noise resistant than TV and TR, and still can present a reconstructed image of relatively high quality in a noisy environment. This also shows that the feasibility and practicality of the method in real life are better.
To better evaluate the performance effect of the method, the quality of the reconstructed image is quantitatively evaluated by adopting Relative Error (RE) and correlation coefficient (Correlation Coefficient, CC):
the smaller the relative error of the image and the larger the correlation coefficient, the higher the spatial resolution of the reconstructed image and the better the quality. Where g' is the calculated conductivity distribution of the reconstruction region, g * Is the actual conductivity distribution, g' q Andthe calculated conductivity distribution and the actual conductivity distribution of the q-th grid, respectively,/o>And->The average calculated conductivity distribution and the average actual conductivity distribution of the q-th grid, respectively. FIGS. 4 and 5 show the TV, TR and the present inventionThree different proposed methods reconstruct the relative error and correlation coefficients of the image in a noise-free environment and in a 5% noise level environment for 2 typical models. The data result shows that compared with TV and TR, the imaging detection method for improving the spatial resolution provided by the invention has the lowest relative error and the highest correlation coefficient, and can accurately reconstruct a real image under the condition of a noise-free environment and a noise level of 5%, thereby effectively improving the spatial resolution of the TR reconstructed image.
The foregoing description of the preferred embodiments of the invention is not intended to be limiting, but rather to enable any modification, equivalent replacement, improvement or the like to be made within the spirit and principles of the invention.
Claims (1)
1. An imaging detection method for improving spatial resolution is characterized in that: taking the image reconstruction process of tumor conductivity distribution as a linear discomfort inverse problem Sg=b, wherein S is a sensitivity matrix, b is a boundary relative voltage measurement value, g is conductivity distribution, and the optimal conductivity distribution is proposedThe optimization model of (2) is:
in the method, in the process of the invention,is the initial conductivity distribution, J (g) is the objective function, S * The sensitivity matrix is normalized, and is obtained by normalizing the sensitivity matrix S; lambda is a regularization parameter used to balance the fidelity term +.>And regularization term-> Represents L2 norm->Is the optimal conductivity distribution coefficient, and is obtained by solving the minimum value of the processing function M (a);
a processing function M (a) is proposed, the specific expression of which is as follows:
wherein a is a conductivity distribution coefficient; w is an operator of the decomposition conductivity distribution; w (W) T Is an operator to reconstruct the conductivity distribution;non-convex regular term lambda e Non-convex regular term parameters,/>Is a non-convex function, a e,w Is the conductivity distribution coefficient, c, at scale e and time scale w e Is a limiting parameter; lambda (lambda) o ||DW T a|| 1 Is a convex regular term, where λ o Is a convex regular term parameter that is used to determine, D is a first order differential matrix of which, I 1 Represents an L1 norm;
in the processing function M (a), a proposed non-convex functionThe specific form of (2) is as follows:
when limiting parameter c e The value range of (C) is 0 to or less than c e <1/λ e At this time, the non-convex functionThe sparsity of the non-convex regular term is reserved, the processing function M (a) is made to be a strict convex function, and when the processing function M (a) is a convex function, the unique optimal solution is ensured;
the overall process of the imaging method is as follows:
the first step: acquiring object field information
In the circular measuring field without object, the adjacent mode is adopted to conduct current excitation and voltage measurement to the electrode pairs uniformly distributed on the boundary of the measuring field so as to obtain the blank field boundary voltage measurement value b 1 Then, the object is placed inside the measurement field to obtain a full-field boundary voltage measurement value b 2 B is known to be 1 And b 2 In the case of (b) 2 -b 1 Obtaining a boundary relative voltage measured value b required by calculation;
the sensitivity matrix S is obtained according to a sensitivity theory, elements in the sensitivity matrix are called as sensitivity coefficients, and a calculation formula of the sensitivity coefficients is as follows:
wherein s is ij Is the sensitivity coefficient of the jth electrode pair to the ith electrode pair, phi i 、φ j The excitation current of the ith electrode pair and the jth electrode pair is I i 、I j The field potential distribution at the time, the distribution form of the sensitivity matrix is as follows:
and a second step of: sensitivity matrix normalization
The sensitivity matrix is normalized in the following manner:
the normalized sensitivity coefficient and the sensitivity matrix are respectively s * ij And S is * A representation;
and a third step of: parameter setting
Regularization parameter λ=0.55, non-convex regularization term parameter λ e =0.95, convex regularization term parameter λ o =0.16, limiting parameter c e ＝0.99/λ e ；
Fourth step: solving for the initial conductivity distribution
Let the derivative of J (g) be equal to zero, then the minimum value of J (g) is expressed as:
(S T S+λE)g＝S T b
where E is an identity matrix, because S T The inverse matrix of S+λE exists, the initial conductivity distributionThe solution of (2) is:
fifth step: distributing the initial conductivityCarrying out a processing function M (a), and solving the optimal conductivity distribution coefficient +.>
To solve for the optimal conductivity distribution coefficientUnconstrained problem->Conversion to the corresponding constraint questionThe following problems:
s.t.r＝a
wherein r is a constraint variable, U 1 (a) And U 2 (r) are respectively:
U 2 (r)＝λ o ||DW T r|| 1
the extended Lagrangian function of the constraint problem is expressed as:
where μ is a lagrangian parameter, μ=0.7; d is an iteration variable;
according to the augmented lagrangian function, the constraint problem is decomposed into the following sub-problems:
where k is the number of iterations, the initial value of k is set to 0, and the maximum value of k is set to 30; r is (r) k Representing the value of the kth iteration of r, the initial value of rd k The value representing the kth iteration of d, the initial value d of d 0 ＝0；
By an augmented Lagrangian function, combined with the definition of the L2 norm, a k+1 The solution of (a) is as follows:
where p is a first auxiliary variable,
to solve for r k+1 Defining a second auxiliary variable v by using the proximity operator and the semi-orthogonal linear transformation thereof, and using v k Representing the value of v at the kth iteration, and let v be k ＝a k+1 -d k According to the sub-problem r k+1 ，v k Is defined as:
where h (r) =φ (W T r)，z is a third auxiliary variable, W is a semi-orthogonal linear transformation using a neighborhood operator T v k Is defined as:
r is then k+1 The solution of (2) is finally expressed as:
in the formula, TVD represents a TV denoising method, and is calculated through a string stretching algorithm, when the iteration number k meets k=30, the iteration process is terminated, and the optimal conductivity distribution coefficient is obtainedSolution of (2);
sixth step: obtaining optimal conductivity distribution
According to the obtainedOptimal conductivity distribution coefficientOperator W for reconstructing conductivity distribution T For->Performing reconstruction, i.e.Obtaining an optimal conductivity distribution->
Seventh step: with optimum conductivity distributionImaging is performed.
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|---|
| Reduction of Staircase Effect With Total Generalized Variation Regularization for Electrical Impedance Tomography;Yanyan Shi等;IEEE SENSORS JOURNAL;第19卷(第21期);全文 * |
| Sparse image reconstruction of intracerebral hemorrhage with electrical impedance tomography;Yanyan Shi等;Journal of medical imaging;第8卷;全文 * |
| 两相流测量中环形电导传感器特性研究;施艳艳等;中国电机工程学报;第30卷(第17期);全文 * |
| 动态电阻抗成像时空相关性重建方法研究;王琦等;电子测量与仪器学报;第32卷(第2期);全文 * |
| 脑阻抗断层成像的图像重构算法;帅万钧等;国外医学生物医学工程分册;第27卷(第6期);全文 * |
| 高分子基质作用下多孔碳酸钙的仿生合成;佘岚;杨秀清;马晓明;杨林;;化学研究与应用(10);全文 * |
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