CN114494477A - Electrical imaging dynamic image reconstruction method based on prior dimension reduction Kalman filtering - Google Patents

Electrical imaging dynamic image reconstruction method based on prior dimension reduction Kalman filtering Download PDF

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CN114494477A
CN114494477A CN202210108991.XA CN202210108991A CN114494477A CN 114494477 A CN114494477 A CN 114494477A CN 202210108991 A CN202210108991 A CN 202210108991A CN 114494477 A CN114494477 A CN 114494477A
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梁光辉
杜云丹
董峰
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Tianjin University
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Abstract

The invention relates to an electrical imaging dynamic image reconstruction method based on prior dimension reduction Kalman filtering, which comprises the steps of constructing a time-varying target dynamic reconstruction model by utilizing a Kalman filtering method, and constructing a spatial projection dimension reduction matrix of a state parameter by utilizing a prior covariance matrix of the state parameter; and (3) carrying out post-processing on the Kalman filtering result by utilizing a Kalman smoothing method under an offline condition.

Description

Electrical imaging dynamic image reconstruction method based on prior dimension reduction Kalman filtering
Technical Field
The invention belongs to the technical field of electrical tomography, and relates to a dynamic visualization method for time-varying target distribution in a closed pipeline.
Background
Dynamic imaging problems are widely present in the fields of industrial production control and biomedical diagnostics, for example: reconstructing the cross section phase distribution of gas-liquid two-phase flow in the pipeline, reconstructing the temperature distribution of a combustion field, monitoring the ventilation state in the lung breathing process and the like. The high-precision and high-robustness dynamic imaging method has very important significance for ensuring production safety, improving production efficiency, improving disease diagnosis accuracy and the like. Electrical Tomography (EIT) is a non-invasive, non-radiative, low-cost technique for visualizing and detecting process parameters. The method obtains the electrical response characteristic of an object field by applying electrical excitation on the boundary of the field domain, and then inverts the distribution of electrical parameters in the field domain to realize two-dimensional/three-dimensional imaging of the measured field domain. The technology has the advantages of non-invasion, quick response, low equipment cost and the like, and is widely concerned in the industrial and biomedical fields.
In 1998, the university of Kuopio, finland, j.p. kaipio, et al, in IEEE Transactions on biological Engineering, volume 45, page 486 493, published a state evolution equation, entitled "a Kalman filter on board to track fast impedance changes in electrical impedance spectroscopy, using a Kalman filter method to dynamically estimate electrical impedance parameters, using a random walk model to construct a state evolution equation of the electrical impedance parameters, using an EIT measurement model to construct an observation equation of the electrical impedance parameters, to achieve the reconstruction of the rapid change process of the pulmonary electrical impedance, and simulation tests show that the proposed method has higher time resolution and imaging precision.
In 2002, U.Schmitt et al, university of Sael, Germany, published "effective algorithm for the regularization of dynamic Inverse schemes" I.Theorm "and" effective algorithm for the regularization of dynamic Inverse schemes: II.applications, Inverse schemes ", constructed an EIT dynamic imaging regularization framework based on the dynamic reconstruction targets on the basis of prior constraint information of the dynamic reconstruction targets in time and space, and adopted the equal interval time discretization process to perform time decoupling on the dynamic imaging targets, convert the target dynamic reconstruction problem into a discrete quasi-static reconstruction problem, and simplify the solution of the dynamic Inverse problem.
In 2004, k.y.kim et al, university of ji, korea, published on "Mathematics and Computers in Simulation" 66, page 399-408, "Dynamic inversion scheme with electronic impedance mapping", and proposed an ERT-based real-time Dynamic estimation method for electrical conductivity parameters, which uses a kinematic model to construct an evolution equation of state of electrical conductivity parameters under the condition of given prior information of object shape and position, to reduce the uncertainty of the model and improve the Dynamic estimation performance of the ERT electrical conductivity parameters. In 2006, k.y.kim et al published Dynamic electrical impedance estimation of a chemical phase using the Kalman filter at pages 81-91, volume 27 of physical measurement, and proposed a Dynamic reconstruction method of complex thoracic impedance based on linear Kalman filtering, where the EIT positive problem is solved by an analytical method based on variable separation and fourier series, the inverse problem is dynamically estimated by Kalman filtering, and online real-time estimation of complex impedance parameters is achieved by off-line pre-calculating a gain matrix.
In 2009, j.p. kaipio et al extended the estimation of flow State parameters in State estimation in process mobility-correlation of velocity field using EIT published in Inverse schemes, volume 25, constructed a concentration field and velocity field joint State evolution equation using a convection diffusion model, constructed an observation model of conductivity and concentration field using an EIT measurement model, uniformly correlated the conductivity distribution and velocity field distribution to be estimated to the concentration field distribution, and performed the synchronous estimation of conductivity distribution and velocity field using extended kalman filtering. In 2011, j.p.kaipio et al published "a reduced-order filtering application for 3D dynamic electrical conductivity morphology tomograph" in Measurement Science and Technology "volume 22, and performed a reduced-order characterization on the electrical conductivity by using a parameter dimension reduction idea, constructed an EIT electrical conductivity dynamic estimation model based on a low-order orthogonal basis characterization, and performed the estimation of the electrical conductivity parameter by using an extended kalman filter algorithm, thereby improving the speed of electrical conductivity parameter estimation.
In 2012, in Measurement & Technology volume 6, pages 63-77, by s.liu et al, north China power university, Dynamic inversion in electrical capacitance tomography using the ensemble Kalman filter, a stochastic walk model is used to construct a permittivity state evolution equation, and an ECT Measurement model is used to construct a permittivity observation equation, thereby providing an ECT permittivity Dynamic estimation method based on constraint Kalman set filtering, and improving the time and spatial resolution of ECT permittivity.
In 2017, J.Lei et al, who published in the Measurement Science and Technology volume 28, used dynamic evolution information of time-varying imaging objects and the characteristics of low-rank tensors and sparse tensors to convert imaging tasks in ECT Measurement into reconstruction problems of three-order image tensors, and proposed a new loss function considering ECT Measurement information and dynamic reconstruction object evolution information by means of a Tikhonov regularization theory and a Tensor-based multi-path data analysis method. The imaging method based on tensor can improve the imaging quality by fully utilizing the space-time correlation of the multi-linear data structure and the three-dimensional image body, has simple numerical value realization and has stronger robustness to the input data with inaccuracy.
Due to the increasing demand of practical engineering application, research on dynamic imaging algorithms is receiving more and more attention in recent years, and higher requirements are put forward on the improvement of imaging speed while high-quality reconstructed images are required to be obtained. Having a faster image reconstruction speed is not a simple task in electrical tomography measurements, and finding a fast and efficient imaging method remains an open and critical issue.
Disclosure of Invention
The invention provides a dynamic image reconstruction method for electrical tomography, which is characterized in that a time-varying target dynamic reconstruction model is constructed by using a Kalman filtering method, a spatial projection dimension reduction matrix of state parameters is constructed by using a prior covariance matrix of the state parameters, and the computational complexity of a Kalman filter is reduced. And post-processing the Kalman filtering result by using a Kalman smoothing method under an offline condition, so that the imaging quality of dynamic image reconstruction is improved. The technical scheme is as follows:
an electrical imaging dynamic image reconstruction method based on prior dimension reduction Kalman filtering is characterized in that a time-varying target dynamic reconstruction model is constructed by utilizing a Kalman filtering method, and a spatial projection dimension reduction matrix of state parameters is constructed by utilizing a prior covariance matrix of the state parameters; the Kalman smoothing method is used for post-processing the Kalman filtering result under the offline condition, and comprises the following steps:
(1) performing state space modeling on a dynamically imaged time-varying target, and performing state space modeling on the target state change process by using a pair of state evolution and state observation equations:
Xt＝Ht-1Xt-1+Wt-1
Yt＝AtXt+Et
wherein t represents the t-th excitation;
Figure BDA0003494430460000031
representing the spatial distribution of the electrical parameters in the field at the time of the tth excitation, namely state parameters;
Figure BDA0003494430460000032
representing field boundary detection data at the time of the t-th excitation; m represents the dimension of the measured data in a complete cycle excitation period; n represents an unknown electrical parameter dimension; le represents the total number of electrodes; ht-1Representing a state transition matrix, H when a random walk model is employedt-1Is an identity matrix;
Figure BDA0003494430460000033
representing the corresponding electrical tomography sensitivity matrix at the t excitation;
Figure BDA0003494430460000034
representing measurement noise; w is a group oft-1Representing state noise; if the observed noise and the state noise are zero mean Gaussian white noise, then
Figure BDA0003494430460000035
ΓE,tAnd gammaW,tCovariance representing measurement noise and state noise, respectively;
(2) projecting the state parameters onto a prior covariance matrix ΓXIn the subspace spanned by the principal eigenvectors:
Figure BDA0003494430460000041
in the formula, gtRepresenting the projection coefficients;
Figure BDA0003494430460000042
a priori estimates representing state parameters at the current time; gvRepresenting a projection matrix by means of an a priori covariance matrix ΓXThe characteristic value decomposition structure of (1); gamma-shapedXRepresents a state parameter XtThe prior covariance matrix is constructed by a Gaussian smooth prior model;
(3) state parameter X of prediction phasetThe prior mean value and the prior error covariance matrix are obtained by predicting the posterior mean value and the posterior error covariance matrix of the projection coefficient at the previous moment:
Figure BDA0003494430460000043
CX,t|t-1＝(Ht-1Gv)Cg,t-1|t-1(Ht-1Gv)T+ΓW,t-1
in the formula
Figure BDA0003494430460000044
And CX,t|t-1A prior mean and a prior error covariance matrix representing the state parameters,
Figure BDA0003494430460000045
and Cg,t-1|t-1Representing a posterior mean and a posterior error covariance matrix of the projection coefficients at the t-1 th state;
Figure BDA0003494430460000046
representing the prior mean value of the state parameter at the t-1 th state; superscript T denotes transposed symbols;
(4) in the observation update phase, the projection coefficient gtThe update is performed by the following a posteriori estimation model:
Figure BDA0003494430460000047
in the formula, pi (· |) represents conditional probability; oc represents proportional to the symbol; solving to obtain a projection coefficient gtThe posterior estimate of (a) is:
Figure BDA0003494430460000048
Figure BDA0003494430460000049
in the formula (I), the compound is shown in the specification,
Figure BDA00034944304600000410
and Cg,t|tRepresenting a posterior mean value and a posterior error covariance matrix of the projection coefficient at the current moment;
(5) the projection coefficient g in (4)tIs used for estimating and calculating the state parameter XtA posteriori estimation of (c):
Figure BDA00034944304600000411
Figure BDA00034944304600000412
in the formula (I), the compound is shown in the specification,
Figure BDA00034944304600000413
and CX,t|tRepresenting a posterior mean value and a posterior error covariance matrix of the state parameter at the current moment;
to posterior mean value
Figure BDA00034944304600000414
Displaying the visual image to obtain the image reconstruction result of the target space distribution at the current moment;
(6) performing off-line smoothing processing on the posterior estimation of the state parameters in the step (5) to obtain smooth estimation:
Figure BDA0003494430460000051
in the formula (I), the compound is shown in the specification,
Figure BDA0003494430460000052
representing a smoothed estimate of the state parameter at the previous time;
Figure BDA0003494430460000053
a smooth estimate representing the state parameter at the current time;
for smooth estimation
Figure BDA0003494430460000054
And displaying the visual image to obtain a smooth estimation result of the target space distribution at the time t-1.
Further, a state parameter X constructed by a Gaussian smooth prior modeltOf (d) a prior covariance matrix rXThe following were used:
Figure BDA0003494430460000055
in the formula (X)t)i＝(x,y)iAnd (X)t)j＝(x,y)jRespectively represent XtThe spatial coordinates of the ith and jth elements of (a); eta2Representing variance parameters for adjusting the unknown quantity XtThe variation range of (a); i | · | | represents the euclidean norm; b represents the correlation length, representing the desired size information of the imaging target; deltaijRepresents a kronecker function, 1 when i ≠ j, and 0 when i ≠ j; iota is a small normal number that guarantees the covariance matrix ΓXIt is reversible.
Drawings
FIG. 1 is a schematic diagram of data acquisition of a moving object electrical tomography system;
FIG. 2 is a dynamic simulation image reconstruction result based on the method described in the embodiment of the present invention;
FIG. 3 is a dynamic experimental model for simulating a gas-liquid two-phase medium based on the method in the embodiment of the invention;
fig. 4 shows a dynamic image reconstruction result obtained based on an experimental model of the method according to the embodiment of the present invention.
Detailed Description
The following detailed description of the implementation steps of the method according to the present invention is intended to describe the implementation steps of the present invention as an embodiment, and is not intended to be the only form of implementation of the present invention, and other embodiments capable of implementing the same structure and function are also included in the scope of the present invention.
In a specific embodiment, an electrical tomography system is adopted, and a test object is a bubble image reconstruction in a moving state in gas-liquid two-phase flow. The system data acquisition involved is shown in fig. 1, when EIT electrode excitation switches from t-1 to t-5, the target also moves from position P1Move to P5Mainly comprises the following steps:
(1) state space modeling of dynamically imaged time-varying targets
In the problem of dynamic imaging image reconstruction by using electrical tomography, a target can change along with the switching of an excitation electrode, and a state space modeling is carried out on the state change process of the current variable target by using a pair of state evolution and state observation equations:
Xt＝Ht-1Xt-1+Wt-1
Yt＝AtXt+Et
wherein t represents the t-th excitation;
Figure BDA0003494430460000061
representing the spatial distribution of the electrical parameter in the field at the time of the t-th excitation,
Figure BDA0003494430460000062
representing field boundary detection data at the time of the t-th excitation; m represents a full-cycle electrical measurement data dimension; n represents an unknown electrical parameter dimension; le represents the total number of electrodes; ht-1Representing a state transition matrix, H when a random walk model is employedt-1Is an identity matrix;
Figure BDA0003494430460000063
representing the corresponding electrical tomography sensitivity matrix at the t excitation;
Figure BDA0003494430460000064
representing measurement noise; wt-1Representing state noise; assuming that the observed noise and the state noise are zero mean Gaussian white noise, then
Figure BDA0003494430460000065
ΓE,tAnd ΓW,tRepresenting the covariance of the measurement noise and the state noise, respectively.
(2) Projecting the state parameters to a subspace for prior dimensionality reduction
Construction of a prior covariance matrix Γ of a state parameter X by a Gaussian smooth prior modelX：
Figure BDA0003494430460000066
In the formula Xi＝(x,y)iAnd Xj＝(x,y)jRespectively representing the spatial coordinates of the ith and jth elements in X; eta2A variance parameter for adjusting the variation range of the unknown quantity X; | | · | | represents the euclidean norm, b represents the correlation length, which implies the expected size information of the imaging target; deltaijRepresents a Kronecker (Kronecker Delta) function, 1 when i ═ j, 0 when i ≠ j; iota is a small normal number that guarantees the covariance matrix ΓXIt is reversible.
The state parameters are then projected onto a prior covariance matrix ΓXIn the subspace spanned by the main feature vectors, the state parameters can be parameterized and characterized by prior estimation and space projection:
Figure BDA0003494430460000067
in the formula, gtRepresenting the projection coefficients;
Figure BDA0003494430460000068
a priori estimate of a state parameter representing the current timeCounting; g denotes a projection matrix, which can be constructed by eigenvalue decomposition of a prior covariance matrix.
Eigenvalue decomposition of the prior covariance matrix with ΓX＝UBUTThe unitary matrix U is composed of the columns of the eigenvectors, the diagonal matrix B is composed of the eigenvalues, and the projection matrix is composed of the first v eigenvectors of the prior covariance matrix:
Figure BDA0003494430460000069
in the formula (I), the compound is shown in the specification,
Figure BDA0003494430460000071
(v < N) represents a projection matrix composed of eigensubspaces obtained by a prior covariance matrix eigen decomposition.
(3) Obtaining prior mean and prior error covariance matrix of state parameters after prior dimension reduction
State parameter X of state prediction stage based on prior dimensionality reduction and Kalman filtering methodtThe prior mean and the prior error covariance matrix of (a) can be obtained by predicting the posterior mean and the posterior error covariance matrix of the projection coefficient at the previous moment:
Figure BDA0003494430460000072
CX,t|t-1＝(Ht-1Gv)Cg,t-1|t-1(Ht-1Gv)T+ΓW,t-1
in the formula
Figure BDA0003494430460000073
And CX,t|t-1A prior mean and a prior error covariance matrix representing the state parameters,
Figure BDA0003494430460000074
and Cg,t-1|t-1Representing the posterior mean and the posterior of the projection coefficients at the t-1 th stateAn error covariance matrix is checked;
Figure BDA0003494430460000075
representing the prior mean value of the state parameter at the t-1 th state; the superscript T denotes the transposed symbol.
(4) A posteriori point estimates of projection coefficients calculated by a transform process
After the dimension of the parameter is reduced, the projection coefficient g can be obtained by observing the updated modeltA posteriori estimation of (c):
Figure BDA0003494430460000076
in the formula, pi (· |) represents conditional probability; and oc represents a proportional to the symbol.
Directly embedding spatial prior constraint of state parameters into a projection matrix G by using a Kalman filtering reconstruction method of prior dimension reductionvThe a posteriori point estimates of the projection coefficients can therefore be calculated directly from the above equation. Because the projection coefficient of the state parameter accords with Gaussian distribution, and the maximum posterior estimation of the Gaussian distribution model is equivalent to the posterior mean value, the posterior mean value and the posterior error covariance matrix of the projection coefficient can be directly obtained without iterative solution:
Figure BDA0003494430460000077
Figure BDA0003494430460000078
in the formula (I), the compound is shown in the specification,
Figure BDA0003494430460000079
and Cg,t|tAnd representing the posterior mean value and the posterior error covariance matrix of the projection coefficient at the current moment.
To further increase the computation speed, the method can be applied to
Figure BDA00034944304600000710
The calculation of (A) is transformed by first converting CX,t|t-1The writing is as follows:
CX,t|t-1＝(Ht-1Gv)Cg,t-1|t-1(Ht-1Gv)T+ΓW,t-1
＝(Ht-1GvKg,t-1)(Ht-1GvKg,t-1)T+ΓW,t-1
wherein the content of the first and second substances,
Figure BDA00034944304600000711
is represented by Cg,t-1|t-1The square root of (i), i.e.
Figure BDA00034944304600000712
For simplifying the presentation, writing
Figure BDA00034944304600000713
Applying Sherman-Morrison-Woodbury (SMW) matrix inverse theory
Figure BDA00034944304600000714
The calculation of (a) can be simplified as:
Figure BDA0003494430460000081
in the formula IvRepresenting a v x v identity matrix. Due to the fact that
Figure BDA0003494430460000082
Therefore, it is
Figure BDA0003494430460000083
Dimension of the middle inversion operation is reduced to v × v and ΓW,t-1Is generally a diagonal matrix, therefore
Figure BDA0003494430460000084
The calculation of (2) is simpler.
(5) Calculating the posterior mean and the posterior error covariance matrix of the state parameters
In the dimension reduction characterization model, the state parameters can be parameterized and characterized through prior estimation and space projection, and then a projection coefficient g is obtainedtAfter the posterior mean value, the full space back projection is carried out to obtain the state parameter XtA posterior mean of, and a state parameter XtThe covariance matrix of the posterior errors can also be determined from the projection coefficients gtA posteriori error covariance matrix Cg,t|tObtaining:
Figure BDA0003494430460000085
Figure BDA0003494430460000086
in the formula (I), the compound is shown in the specification,
Figure BDA0003494430460000087
and CX,t|tThe posterior means and the posterior error covariance matrix of the state parameters are represented.
To posterior mean value
Figure BDA0003494430460000088
And displaying the visual image to obtain the image reconstruction result of the target space distribution at the current moment. (6) Off-line smoothing post-processing on reconstruction result by using Kalman smoothing method
When the state parameter posterior estimation does not need to be calculated in real time, the state parameter posterior estimation is smoothly processed from the last result of the prior-check dimensionality reduction Kalman filtering estimation through a recursive estimation method, and the initial smooth estimation is defined
Figure BDA0003494430460000089
TT is the total number of states in dynamic imaging. Calculating a smooth estimate of the state parameter from back to front:
Figure BDA00034944304600000810
in the formula (I), the compound is shown in the specification,
Figure BDA00034944304600000811
is a smooth estimation of the state parameter at the previous moment, the covariance matrix of the posterior error at the previous moment of the state parameter is
Figure BDA00034944304600000812
For smooth estimation
Figure BDA00034944304600000813
And displaying the visual image to obtain a smooth estimation result of the target space distribution at the time t-1.
The implementation results are as follows: and (4) carrying out simulation and experimental test on the scheme. In the simulation test, a typical straight-line track is taken as the track of a simulated moving object, and the implementation effect is shown in fig. 2, wherein a red area represents the moving object, and a blue area represents a background medium. According to the dynamic simulation image reconstruction result, the electrical dynamic image reconstruction method based on the prior dimensionality reduction and the Kalman filtering can better reconstruct the moving target, and the electrical dynamic image reconstruction method based on the prior dimensionality reduction and the Kalman smoothing can further improve the image reconstruction quality. The target motion track in the experimental test is shown in fig. 3, and the dynamic experimental reconstruction result is shown in fig. 4. The dynamic experiment reconstruction result also verifies that the electrical dynamic image reconstruction method based on the prior dimension reduction and the Kalman filtering can better reconstruct the moving target, and the electrical dynamic image reconstruction method based on the prior dimension reduction and the Kalman smoothing can further improve the quality of image reconstruction.

Claims (2)

1. An electrical imaging dynamic image reconstruction method based on prior dimension reduction Kalman filtering is characterized in that a time-varying target dynamic reconstruction model is constructed by utilizing a Kalman filtering method, and a spatial projection dimension reduction matrix of state parameters is constructed by utilizing a prior covariance matrix of the state parameters; the Kalman smoothing method is used for post-processing the Kalman filtering result under the offline condition, and comprises the following steps:
(1) performing state space modeling on a dynamically imaged time-varying target, and performing state space modeling on the target state change process by using a pair of state evolution and state observation equations:
Xt＝Ht-1Xt-1+Wt-1
Yt＝AtXt+Et
wherein t represents the t-th excitation;
Figure FDA0003494430450000011
representing the spatial distribution of the electrical parameters in the field at the time of the tth excitation, namely state parameters;
Figure FDA0003494430450000012
representing field boundary detection data at the time of the t-th excitation; m represents the dimension of the measured data in a complete cycle excitation period; n represents an unknown electrical parameter dimension; le represents the total number of electrodes; ht-1Representing a state transition matrix, H when a random walk model is employedt-1Is an identity matrix;
Figure FDA0003494430450000013
representing the corresponding electrical tomography sensitivity matrix at the t excitation;
Figure FDA0003494430450000014
representing measurement noise; wt-1Representing state noise; if the observed noise and the state noise are zero mean Gaussian white noise, then
Figure FDA0003494430450000015
ΓE,tAnd ΓW,tCovariance representing measurement noise and state noise, respectively;
(2) projecting the state parameters onto a prior covariance matrix ΓXIn the subspace spanned by the principal eigenvectors:
Figure FDA0003494430450000016
in the formula, gtRepresenting a projection coefficient;
Figure FDA0003494430450000017
a priori estimate representing a state parameter at a current time; gvRepresenting a projection matrix by means of an a priori covariance matrix ΓXThe characteristic value decomposition structure of (1); gamma-shapedXRepresents a state parameter XtThe prior covariance matrix is constructed by a Gaussian smooth prior model;
(3) state parameter X of prediction phasetThe prior mean value and the prior error covariance matrix are obtained by predicting the posterior mean value and the posterior error covariance matrix of the projection coefficient at the previous moment:
Figure FDA0003494430450000018
CX,t|t-1＝(Ht-1Gv)Cg,t-1|t-1(Ht-1Gv)T+ΓW,t-1
in the formula
Figure FDA0003494430450000019
And CX,t|t-1A prior mean and a prior error covariance matrix representing the state parameters,
Figure FDA00034944304500000110
and Cg,t-1|t-1Representing a posterior mean and a posterior error covariance matrix of the projection coefficients at the t-1 th state;
Figure FDA00034944304500000111
representing the prior mean value of the state parameter at the t-1 th state; superscript T denotes transposed symbols;
(4) at observation update stageSegment, projection coefficient gtThe update is performed by the following a posteriori estimation model:
Figure FDA0003494430450000021
in the formula, pi (· |) represents conditional probability; oc represents proportional to the symbol; solving to obtain a projection coefficient gtIs estimated as:
Figure FDA0003494430450000022
Figure FDA0003494430450000023
in the formula (I), the compound is shown in the specification,
Figure FDA0003494430450000024
and Cg,t|tRepresenting a posterior mean value and a posterior error covariance matrix of the projection coefficient at the current moment;
(5) the projection coefficient g in (4)tIs used for estimating and calculating the state parameter XtA posteriori estimation of (c):
Figure FDA0003494430450000025
Figure FDA0003494430450000026
in the formula (I), the compound is shown in the specification,
Figure FDA0003494430450000027
and CX,t|tRepresenting a posterior mean value and a posterior error covariance matrix of the state parameter at the current moment;
to posterior mean value
Figure FDA0003494430450000028
Displaying the visual image to obtain the image reconstruction result of the target space distribution at the current moment;
(6) performing off-line smoothing processing on the posterior estimation of the state parameters in the step (5) to obtain smooth estimation:
Figure FDA0003494430450000029
in the formula (I), the compound is shown in the specification,
Figure FDA00034944304500000210
representing a smoothed estimate of the state parameter at the previous time;
Figure FDA00034944304500000211
a smooth estimate representing the state parameter at the current time;
for smooth estimation
Figure FDA00034944304500000212
And displaying the visual image to obtain a smooth estimation result of the target space distribution at the time t-1.
2. The electrical imaging dynamic image reconstruction method according to claim 1, wherein the state parameter X is constructed by a Gaussian smooth prior modeltOf the prior covariance matrix ΓXThe following:
Figure FDA00034944304500000213
in the formula (X)t)i＝(x,y)iAnd (X)t)j＝(x,y)jRespectively represent XtThe spatial coordinates of the ith and jth elements of (a); eta2Representing variance parameters for adjusting the unknown quantity XtThe variation range of (a); | | cndot | | denotes the euclidean normCounting; b represents the correlation length, representing the desired size information of the imaging target; delta. for the preparation of a coatingijRepresents a kronecker function, 1 when i ≠ j, and 0 when i ≠ j; iota is a small normal number for ensuring the covariance matrix ΓXIt is reversible.
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CN115512172A (en) * 2022-09-30 2022-12-23 赵营鸽 Uncertainty quantification method for multi-dimensional parameters in electrical impedance imaging technology
CN115512172B (en) * 2022-09-30 2023-09-15 赵营鸽 Uncertainty quantification method for multidimensional parameters in electrical impedance imaging technology

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