CN112802147A - High-precision image reconstruction method for improving background definition - Google Patents

High-precision image reconstruction method for improving background definition Download PDF

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CN112802147A
CN112802147A CN202110187974.5A CN202110187974A CN112802147A CN 112802147 A CN112802147 A CN 112802147A CN 202110187974 A CN202110187974 A CN 202110187974A CN 112802147 A CN112802147 A CN 112802147A
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value
objective function
image reconstruction
imaging
measurement voltage
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施成成
张波
魏国栋
王蕾
亓震
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Abstract

The invention discloses a high-precision image reconstruction method for improving background definition, which comprises the following steps: (1) and acquiring boundary measurement voltage U and a sensitivity matrix A required by reconstruction according to the field to be measured. (2) And setting initialization parameters. (3) Set the circulation condition as
Figure DDA0002943910210000011
Or the number of iterations reaches 200. (4) Solution model to begin calculating objective function
Figure DDA0002943910210000012
(5) Judging whether the iteration termination condition is met or not, if so, terminating the iteration, and carrying out the next operation; if not, setting k to be k +1, jumping back to the step (3), and continuing to iteratively solve. (6) And imaging according to the finally solved gray value. The high-precision image reconstruction method for improving the background definition has better reconstructed image quality in image reconstruction, and has the advantage of removing step artifacts compared with a total variation algorithmThe method has the advantages of being more advantageous, and enabling the parameters to be selected with obvious objectivity and simplicity through the selection of the adaptive parameters.

Description

High-precision image reconstruction method for improving background definition
Technical Field
The invention relates to a resistance tomography image reconstruction technology, in particular to a high-precision image reconstruction method for improving background definition, and belongs to the technical field of resistance tomography.
Background
Electrical Resistance Tomography (ERT) is a novel measurement technology developed in recent years, and has a wide application prospect in the fields of medical clinical monitoring, industrial measurement and the like because the technology has the advantages of no radiation, non-invasion, quick response, simple structure, low cost and the like. In the last two decades, the ERT technology has been developed rapidly, but the quality of the reconstructed image is not ideal because the solution of the ERT inverse problem has the difficulties of nonlinearity, undercharacterization, ill-posed property and the like.
The image reconstruction of the electrical resistance tomography is a nonlinear ill-defined inverse problem, and the nonlinear problem can be converted into a linear problem to be solved through linearization processing. Because the number of the acquired boundary voltages of the measured field is far less than the pixel value of the solving field, the inverse problem is solved without qualification, and for the inverse problem solving without qualification, a regularization method is usually adopted to find a solution to approximate a real solution. Typical algorithms widely used at present include a non-iterative Linear Back Projection (LBP) algorithm, an iterative Tikhonov algorithm and a Total Variation (TV) regularization algorithm in order to improve the accuracy of the reconstructed image. However, the linear back projection algorithm belongs to one-step imaging, and a large amount of artifacts are generated in the process of reconstructing an image, so that the image definition is low; the Tikhonov algorithm has improved imaging quality, but the Tikhonov regularization method adopts L2The norm is a regular term, so that the measured medium has good performance when being continuously distributed, and excessive smoothness is applied to the boundary when the measured medium is not continuously distributed, so that the resolution of a reconstructed image is reduced; to further improve the image quality problem, full-variational algorithms were subsequently employed to better improve image quality, which preserved the discontinuity of the boundary while allowing sharp edges to be reconstructed to produce sharper images, but which nevertheless fully-variational algorithmsThe method has certain defects, and generates a step effect phenomenon while preserving edges, so that the quality of reconstructed images has new problems.
Aiming at the problem of step artifacts generated in the image reconstruction process by the algorithm, the invention provides a high-precision image reconstruction method for improving the background definition, which can effectively keep sharp edges, well inhibit the step artifacts in the image reconstruction process and has good robustness in the image reconstruction process.
Disclosure of Invention
The invention aims to provide a high-precision image reconstruction method for improving background definition, which can effectively reduce step artifacts, improve background definition and improve anti-noise performance. Compared with an LBP algorithm, a Tikhonov algorithm and a total variation algorithm, the method provided by the invention has an obvious effect on the aspect of improving the imaging quality of the resistance tomography reconstructed image.
The invention adopts the following technical scheme for realizing the purpose: a high-precision image reconstruction method for improving background definition considers resistance tomography as a linear ill-posed problem Ag-U. Wherein A is a sensitivity matrix, U is a relative boundary measurement voltage value, and g is an imaging gray value. The minimum objective function I can be established by the sensitivity matrix A and the relative boundary measurement voltage U as
Figure BDA0002943910190000021
Then, the minimization of the objective function I can obtain a corresponding solution model as
Figure BDA0002943910190000022
On the basis of the solving result of the objective function I each time, the following optimization process is further carried out: first, an objective function II is defined as
Figure BDA0002943910190000023
Figure BDA0002943910190000024
For the value of the imaging gray-scale to be solved,α is an optimization parameter I; its minimization objective function is in the form of
Figure BDA0002943910190000025
To solve the minimization problem in the objective function II, the following threshold function is introduced to perform a defined solution thereto:
Figure BDA0002943910190000026
u is the relative boundary measurement voltage value,
Figure BDA0002943910190000027
represents the minimum value within a threshold range, wherein
Figure BDA0002943910190000028
Updating variables according to each iteration of the objective function I
Figure BDA0002943910190000029
The iterative form of the solution for objective function II can be listed as:
Figure BDA00029439101900000210
then judging whether the iteration meets the iteration termination condition
Figure BDA00029439101900000211
Or whether the maximum iteration number is 150 times is reached, so that the optimal solution, namely the optimal imaging gray value for imaging can be obtained.
The invention has the beneficial effects that: compared with a Tikhonov algorithm and a total variation algorithm, the high-precision image reconstruction method for improving the background definition has better reconstructed image quality in image reconstruction, has greater advantages in removing step artifacts compared with the total variation algorithm, and has obvious objectivity and simplicity in parameter selection through self-adaptive parameter selection.
Drawings
Fig. 1 is a flow chart of a high-precision image reconstruction method for improving background definition according to the present invention.
FIG. 2 shows the circular single-section field to be measured, the modes of exciting current and measuring voltage and the electrode distribution of the electrical resistance tomography system of the present invention.
Fig. 3 is an image reconstruction diagram of a high-precision image reconstruction method for improving background definition by using a Tikhonov algorithm, a total variation algorithm and one of the two true model distributions according to an embodiment of the present invention.
Fig. 4 shows the relative image error and correlation coefficient of two real models reconstructed by the three methods under the same conditions.
In the figure: 1-electrode, 2-field to be measured, 3-measurement voltage, 4-excitation current.
Detailed Description
The high-precision image reconstruction method for improving the background definition of the invention is described with reference to the accompanying drawings and embodiments.
The invention provides a high-precision image reconstruction method for improving background definition, which aims at the problems of step artifacts and unclear background generated by the traditional regularization algorithm, takes the solution result of an objective function I as the basis, combines the provided image reconstruction optimization method, and adopts L2Norm as a fidelity term, L1The norm is used as a punishment item, an optimal value is selected by self-adaptively selecting a regularization parameter and a weight factor, and the objective function provided by the invention is solved by a limiting function with an accurate threshold range, so that the optimal imaging gray value solving is completed.
Fig. 1 is a flowchart of a high-precision image reconstruction method for improving background sharpness according to the present invention. The optimal imaging gray value can be solved according to the flow chart, and the specific implementation steps are as follows:
the method comprises the following steps: firstly, solving a relative boundary measurement voltage value and a sensitivity matrix, adopting an adjacent excitation mode, injecting excitation current into an electrode to obtain a null field voltage U of boundary measurement1When a target object exists in the field, the boundary voltage obtained by measurement is the full-field voltage U2Full field voltage U of the object2And a null field voltage U not containing the target1Making a difference to obtain a relative boundary measureMagnitude voltage value U, namely: u is equal to U2-U1Then, combining with the sensitivity theory, obtaining a sensitivity matrix through calculation, wherein the calculation formula is as follows:
Figure BDA0002943910190000031
in the formula, AijIs the sensitivity coefficient, phi, of the jth electrode pair to the ith electrode pairi，φjThe ith electrode pair and the jth electrode pair respectively have excitation current of Ii，IjThe field potential distribution of time, all the sensitivity coefficients A obtained by calculationijJointly forming a sensitivity matrix A;
step two: the relationship between the relative boundary measurement voltage value and the conductivity distribution is nonlinear, and can be expressed as f (σ) ═ U, where σ denotes conductivity, and the nonlinear form of the relative boundary measurement voltage value and the conductivity distribution is converted into a linear form Ag Δ σ ═ Δ U, where Δ σ denotes a disturbance value of the conductivity, Δ U denotes a change in voltage difference value caused by a change in the conductivity, and can be expressed as an imaging gray scale value, i.e., the linear form Ag Δ σ ═ Δ U is further expressed as Ag ═ U, and g denotes an imaging gray scale value;
step three: according to the linear form of the relative boundary measurement voltage value and the conductivity distribution obtained in the step two, an objective function I of the electrical resistance tomography can be established as follows:
Figure BDA0002943910190000032
by minimizing the objective function I, a solution model thereof can be obtained as
Figure BDA0002943910190000033
Then, an objective function II is established on the basis of the objective function I:
Figure BDA0002943910190000034
Figure BDA0002943910190000035
for the imaging gray value to be solved, the solution model can be obtained by minimizing the objective function II
Figure BDA0002943910190000036
Figure BDA0002943910190000037
The optimal imaging gray value finally used for imaging, p represents an exponential parameter, and alpha is an optimization parameter I, so that the result of the global optimal solution can be corrected. According to the obtained solving model, and the relative boundary measurement voltage value U and the sensitivity matrix A are combined, the solving result of the objective function I can be obtained
Figure BDA0002943910190000038
The solution form of the objective function II can be expressed as:
Figure BDA0002943910190000041
Figure BDA0002943910190000042
is a defined function and can be expressed as
Figure BDA0002943910190000043
U is the relative boundary measurement voltage value,
Figure BDA0002943910190000044
represents the minimum value within a threshold range, wherein
Figure BDA0002943910190000045
Step four: according to the solving form of the objective function I and the objective function II in the third step, the algorithm process of the optimal imaging gray value is as follows: (1) initialization: an initial value g is given0，α＝0.2，
Figure BDA0002943910190000046
(2) Updating preliminary variables
Figure BDA0002943910190000047
Figure BDA0002943910190000048
(3) Furthermore, the utility modelNewly optimized imaging gray scale value
Figure BDA0002943910190000049
Figure BDA00029439101900000410
(4) Judging whether the iteration meets the iteration termination condition
Figure BDA00029439101900000411
Or whether the maximum iteration times is reached, if so, the iteration is terminated, and the obtained result is obtained
Figure BDA00029439101900000412
As the optimum imaging gray value
Figure BDA00029439101900000413
If not, setting k to be k +1, jumping back to the step (2) of the step four, and continuing to iteratively solve.
Step five: obtaining the optimal imaging gray value according to the step four
Figure BDA00029439101900000414
And (6) imaging.
Fig. 2 shows a schematic diagram of a sensor array in electrical resistance tomography, which includes basic excitation current 4 and measurement voltage 3 portions and sixteen electrode 1 distribution forms.
Two medium models with different distributions are selected as an embodiment, the real distribution of the target objects in the field is shown in one column on the left side of the figure 3, and the other three columns are respectively represented as a Tikhonov algorithm, a total variation algorithm and a regularization algorithm provided by the invention from left to right. In order to better embody the algorithm of the present invention differently from the other two algorithms, the imaging results of the three reconstruction algorithms are shown in fig. 3, respectively. It can be seen that, in the two typical models, when the Tikhonov algorithm is adopted, the obvious phenomenon that the edge is too smooth exists in the background and the artifact phenomenon exists at the same time, so that the quality of image reconstruction is seriously influenced; compared with the Tikhonov algorithm, the image reconstruction quality of the total variation algorithm is improved, but the phenomenon of the step effect still exists, the background of the algorithm provided by the text is clearer in the imaging effect, the boundary of the target object is more complete, the step effect and the artifact are removed, and the reconstruction result is far better than the reconstruction results of the Tikhonov algorithm and the total variation algorithm.
In electrical resistance tomography, an image Relative Error (RE) and Correlation Coefficient (CC) evaluation algorithm are generally adopted to quantify the image reconstruction quality, and an expression is shown in (i) and (ii), wherein the smaller the image Relative Error is, the larger the Correlation Coefficient is, and the better the image reconstruction quality is.
Figure BDA00029439101900000415
Figure BDA0002943910190000051
Where σ is the calculated conductivity of the reconstructed region, σ*Is the actual conductivity, t represents the number of pixels,
Figure BDA0002943910190000052
and
Figure BDA0002943910190000053
represents sigma and sigma*Average value of (a) ("sigmaiAnd σi *Expressed are σ and σ*The ith triangle cell of (1).
Fig. 4 shows relative errors and correlation coefficients of the three methods for reconstructing images of two models, and it can be seen that the high-precision image reconstruction method for improving background definition provided by the invention has the lowest relative error and the highest correlation coefficient, compared with the Tikhonov algorithm and the total variation algorithm, can accurately reconstruct the distribution in the measured field 2, and obviously improves the solving precision of the resistance tomography inverse problem.
The above description is only exemplary of the present invention and should not be taken as limiting the invention, as any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (1)

1. A high-precision image reconstruction method for improving background definition is characterized by comprising the following specific steps:
the method comprises the following steps: firstly, solving a relative boundary measurement voltage value and a sensitivity matrix, adopting an adjacent excitation mode, injecting excitation current into an electrode to obtain a null field voltage U of boundary measurement1When a target object exists in the field, the boundary voltage obtained by measurement is the full-field voltage U2Full field voltage U of the object2And a null field voltage U not containing the target1And (3) obtaining a relative boundary measurement voltage value U by difference, namely: u is equal to U2-U1Then, combining with the sensitivity theory, obtaining a sensitivity matrix through calculation, wherein the calculation formula is as follows:
Figure FDA0002943910180000011
in the formula, AijIs the sensitivity coefficient, phi, of the jth electrode pair to the ith electrode pairi，φjThe ith electrode pair and the jth electrode pair respectively have excitation current of Ii，IjField potential distribution of time, all sensitivity coefficients A obtained by the above calculationijJointly forming a sensitivity matrix A;
step two: the relationship between the relative boundary measurement voltage value and the conductivity distribution is nonlinear, and can be expressed as f (σ) ═ U, where σ denotes conductivity, and the nonlinear form of the relative boundary measurement voltage value and the conductivity distribution is converted into a linear form Ag Δ σ ═ Δ U, where Δ σ denotes a disturbance value of the conductivity, Δ U denotes a change in voltage difference value caused by a change in the conductivity, and can be expressed as an imaging gray scale value, i.e., the linear form Ag Δ σ ═ Δ U is further expressed as Ag ═ U, and g denotes an imaging gray scale value;
step three: according to the linear form of the relative boundary measurement voltage value and the conductivity distribution obtained in the step two, an objective function I of the electrical resistance tomography can be established as follows:
Figure FDA0002943910180000012
by minimizing the objective function I, a solution model thereof can be obtained as
Figure FDA0002943910180000013
Then, an objective function II is established on the basis of the objective function I:
Figure FDA0002943910180000014
Figure FDA0002943910180000015
for the imaging gray value to be solved, the solution model can be obtained by minimizing the objective function II
Figure FDA0002943910180000016
Figure FDA0002943910180000017
The method is characterized in that the optimal imaging gray value finally used for imaging is obtained, p represents an exponential parameter, alpha is an optimization parameter I, the result of the global optimal solution can be corrected, and the solution result of the objective function I is obtained according to the obtained solution model and by combining a relative boundary measurement voltage value U and a sensitivity matrix A
Figure FDA0002943910180000018
The solution form of the objective function II is expressed as:
Figure FDA0002943910180000019
Figure FDA00029439101800000110
is a limiting function, expressed as
Figure FDA00029439101800000111
U is the relative boundary measurement voltage value,
Figure FDA00029439101800000112
represents the minimum value within a threshold range, wherein
Figure FDA00029439101800000113
Step four: according to the solving form of the objective function I and the objective function II in the third step, the algorithm process of the optimal imaging gray value is as follows: (1) initialization: an initial value g is given0，α＝0.2，
Figure FDA00029439101800000114
(2) Updating preliminary variables
Figure FDA00029439101800000115
Figure FDA00029439101800000116
(3) Updating optimized imaging gray values
Figure FDA00029439101800000117
Figure FDA0002943910180000021
(4) Judging whether the iteration meets the iteration termination condition
Figure FDA0002943910180000022
Or whether the maximum iteration times is reached, if so, the iteration is terminated, and the obtained result is obtained
Figure FDA0002943910180000023
As the optimum imaging gray value
Figure FDA0002943910180000024
If not, setting k to be k +1, jumping back to the step (2) of the step four, and continuing to iteratively solve;
step five: obtaining the optimal imaging gray value according to the step four
Figure FDA0002943910180000025
And (6) imaging.
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