CN106296766B - A kind of image reconstructing method of the capacitance chromatography imaging based on ROF model - Google Patents
A kind of image reconstructing method of the capacitance chromatography imaging based on ROF model Download PDFInfo
- Publication number
- CN106296766B CN106296766B CN201610630259.3A CN201610630259A CN106296766B CN 106296766 B CN106296766 B CN 106296766B CN 201610630259 A CN201610630259 A CN 201610630259A CN 106296766 B CN106296766 B CN 106296766B
- Authority
- CN
- China
- Prior art keywords
- formula
- matrix
- image
- algorithm
- norm
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T11/00—2D [Two Dimensional] image generation
- G06T11/003—Reconstruction from projections, e.g. tomography
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T2211/00—Image generation
- G06T2211/40—Computed tomography
- G06T2211/424—Iterative
Landscapes
- Physics & Mathematics (AREA)
- General Physics & Mathematics (AREA)
- Engineering & Computer Science (AREA)
- Theoretical Computer Science (AREA)
- Image Processing (AREA)
Abstract
The invention discloses a kind of image reconstructing methods of capacitance chromatography imaging based on ROF model, by ROF model use in the image reconstruction of the capacitance chromatography imaging, using single step or the restructing algorithm of iteration, in terms of overcoming the shortcomings of that existing algorithm keeps with anti-noise at flow pattern edge, improve the quality of reconstructed image.
Description
Technical field
The present invention relates to electrical capacitance tomography field, more particularly to a kind of capacitance chromatographic based on ROF model at
The image reconstructing method of picture.
Background technique
In multiphase flow detection field, capacitance chromatography imaging (Electrical Capacitance Tomography,
It ECT is) that a kind of can carry out visual novel monitoring technology to dielectric distribution in pipeline space.The basic principle of ECT is in work
Industry pipeline disposes capacitance sensor array, to measure capacitance of the capacitance sensor two-by-two between electrode.According to capacitor
Value and specific algorithm, the dielectric distribution in the restructural monitoring of the sensor out region of ECT system, so that it is determined that the flow pattern inside pipeline
Feature.ECT technology has many advantages, such as that radiationless, non-intruding, at low cost and speed are fast, therefore is widely used in petroleum, chemical industry, electricity
The industrial circles such as power and metallurgy.
The superiority and inferiority of ECT restructing algorithm to reconstruction result it is accurate whether play a crucial role.Due to ECT image weight
Structure has " soft field " effect, so that dielectric distribution and capacitance are a kind of nonlinear relationships.And ECT image reconstruction is " disease
State ", it is meant that it is very sensitive to measurement error and noise.Existing ECT restructing algorithm be mostly based on dielectric distribution and
Capacitance is realized in the hypothesis of certain linear relationship.Therefore than sharp (such as rectangle) and there are the feelings of noise jamming at flow pattern edge
Under condition, the result reconstructed using these algorithms is not satisfactory.Therefore, a kind of holding flow pattern edge more information is found
It is particularly important with the ECT restructing algorithm for having stronger noise immunity.
Rudin-Osher-Fatemi (ROF) model is employed for image denoising field for the first time, and the model can be preferably
Keep the minutias such as the Edge texture of image.If therefore can by ROF model use in the image reconstruction of ECT, must can greatly
The quality of ground improvement reconstructed image.However ECT image reconstruction be it is ill posed, that is, the dimension of the capacitance data measured be much smaller than to
The dimension of reconstructed image results in the multi-solution of image reconstruction result.Recently as the proposition of " compressed sensing " concept, it is
The research of ECT image reconstruction algorithm is filled with new vitality.Briefly, " compressed sensing " utilizes less data sample weight
Build out original signal.Therefore based on " compressed sensing " concept and ROF model ECT image reconstructing method will be one have choose
The problem of war property.In view of the complexity for producing flow pattern in pipeline in practice is different, also for reduce as far as possible ECT imaging when
Between, propose that a kind of single step and iteration ECT restructing algorithm seems very necessary respectively.
Summary of the invention
The object of the invention is exactly to solve image quality issues in the image reconstruction of ECT.
Technical problem of the invention is resolved by technical solution below:
A kind of image reconstructing method of the capacitance chromatography imaging based on ROF model, by Rudin-Osher-Fatemi model
Apply in the image reconstruction of the capacitance chromatography imaging, using single step or the ECT restructing algorithm of iteration, improves reconstructed image
Quality, the image reconstruction of the capacitance chromatography imaging are to be obtained under the premise of known sensitivity field matrix S by measurement capacitance λ
Dielectric constant g distributed image；The following steps are included:
S1, sensitive field matrix and capacitance are inputted；
S2, it is calculated using single step algorithm or iterative algorithm；
S3, output medium distributed image.
Another specific aspect according to the present invention, it is described to input sensitive field matrix and capacitance is establish ECT linear
Change model:
λ=Sg (1)
In formula: λ is the normalization capacitor vector of m × 1, and S is the normalization sensitivity field matrix of m × n, and g is the normalizing of n × 1
Change dielectric constant vector；
Another specific aspect according to the present invention, single step algorithm described in step S2, comprising the following steps:
S21, parameter initialization；
S22, dielectric constant is directly calculated by single step formula.
Another specific aspect according to the present invention, parameter initialization described in step S21 is:
Based on ROF modular concept, 2- norm and 1- norm are introduced respectively in formula (1), then dielectric constant is solved in formula (1)
The problem of being distributed g is converted into following convex optimization problem:
In formula: μ is fidelity term parameter,For gradient operator.
Defconstant；Without loss of generality, it is assumed that be the matrix of a N × N to reconstructed image, be denoted as f (x, y)；f(x,y)
N × 1 a vector g (i), n=N × N can be converted into；Based on forward-difference method, it is utilized respectively companion matrix GxAnd GyCalculate g's
Horizontal and vertical gradient, wherein Gx,Gy∈Rn×nIt is calculated by formula (3)；For 1≤i, j≤N,
Laplacian Matrix Δ is determined by formula (4)；
G (i) is the vector representation of dielectric constant g.
Another specific aspect according to the present invention, single step formula is in step S22
In formula:WithThe respectively Fourier transformation of g and Fourier inversion, STFor the transposition of sensitive field matrix
Matrix, r are a normal number.
Another specific aspect according to the present invention, iterative algorithm described in step S2, comprising the following steps:
S31, parameter initialization；
S32, it solves to obtain dielectric constant using conjugate gradient method；
S33, correlation formula is updated；
S34, judgement | | gk-gk-1||2Whether less than zero, if so, carrying out S35；If it is not, then carrying out S32；
S35, output dielectric constant.
Another specific aspect according to the present invention, step S31 parameter initialization the following steps are included:
S311, it is based on ROF modular concept, introduces 2- norm and 1- norm respectively in formula (1), then solved and be situated between in formula (1)
Electric constant is distributed the problem of g, is converted into following convex optimization problem:
In formula: μ is fidelity term parameter, and ε is smooth item parameter,For gradient operator；
S312, defconstant and new space；Without loss of generality, it is assumed that it is the matrix of a N × N to reconstructed image, note
For f (x, y)；F (x, y) can be converted into n × 1 a vector g (i), n=N × N；Theorem in Euclid space RN’×N’It is defined as SPACE V, then g
∈V；Based on forward-difference method, it is utilized respectively companion matrix GxAnd GyThe horizontal and vertical gradient of g is calculated, wherein Gx,Gy∈Rn×n
It is calculated by formula (3)；
For 1≤i, j≤N,
Laplacian Matrix Δ is determined by formula (4)；
If the inner product and 2- norm of SPACE V are respectively defined as ()VWith | | | |V；A new space W is now defined,
Its inner product and 2- norm is expressed as ()wWith | | | |w；It is all the matrix of n × 2 that T, P, which might as well be taken, and T, P
∈ W, then W inner product is expressed as formula:
(T, P)W=(T1, P1)V+(T2, P2)V (6)
In formula: T1And T2It is the first row of T and the vector that secondary series represents, P respectively1And P2It is same；The 2- norm of W
It is provided by formula (7)；
Another specific aspect according to the present invention, step S32 solve to obtain dielectric constant using conjugate gradient method, specifically
The following steps are included:
S321, using alternating direction Multiplier Algorithm, propose the alternating direction Multiplier Algorithm accelerated；Formula (5) are converted first
For constrained convex optimization problem；
In formula: | | p | |1WFor the 1- norm of p under the space W,WhereinUtilize alternating side
To Multiplier Algorithm, formula (8) is decomposed into 2 subproblems；
In formula: d and s is auxiliary variable, and τ is a normal number；
It can be seen that formula (9) right side of the equal sign is a quadratic programming problem, it is easy to get its optimal solution are as follows:
In formula:For divergence operator；
Formula (10) can regard a typical linear equation (such as Ax=b form) as；
S322, it waits for that the dimension of reconstructed image is higher due to ECT, conjugate gradient method is selected to solve gkIt can accelerate to solve speed
Degree.
Another specific aspect according to the present invention updates correlation formula in step S33, specifically includes the following steps:
Formula (8) another subproblem is for solving pk；
Formula (11), which can use with typical contraction operator, to be solved；
In formula: i-th of element of contraction operator " shrink " is provided by formula (13)；
Auxiliary variable d and s is determined by following four formula；
Another specific aspect according to the present invention, step S34 are that iterative algorithm is in an iterative process in judgment step S33
Whether the difference of the dielectric constant of connected 2 generations output is less than setting value tol under 2- norm measure；Step is returned to if being less than
S32；If more than the end value for then exporting dielectric constant.
The beneficial effect of the present invention compared with the prior art is:
A kind of image reconstructing method of capacitance chromatography imaging based on ROF model of the invention, can be preferably using ROF model
Ground keeps the minutias such as Edge texture, the denoising of image, by ROF model use in the image reconstruction of ECT, and is based on this side
Method proposes a kind of single step and iteration ECT restructing algorithm respectively, overcomes " soft field " effect of ECT image reconstruction, by flow pattern from
Distinguish in background, improves the quality of reconstructed image.
Detailed description of the invention
Fig. 1 is flow chart of the invention；
Fig. 2 is to tie up ECT model schematic for the 2 of emulation；
Fig. 3 is the flow pattern schematic diagram for emulation；
Fig. 4 is experimental system block diagram；
Fig. 5 a is the practical flow pattern of image reconstruction of emulation；
Fig. 5 b is the image reconstruction LI algorithm reconstruction result figure of emulation；
Fig. 5 c is the image reconstruction SAL algorithm reconstruction result figure of emulation；
Fig. 5 d is the image reconstruction AADMM algorithm 1st generation reconstruction result figure of emulation；
Fig. 5 e is image reconstruction AADMM algorithm the 3rd generation reconstruction result figure of emulation；
Fig. 5 f is image reconstruction AADMM algorithm the 6th generation reconstruction result figure of emulation；
Fig. 6 a is the practical flow pattern of image reconstruction for emulating capacitance data and being added after Gaussian noise；
Fig. 6 b is the image reconstruction LI algorithm reconstruction result figure for emulating capacitance data and being added after Gaussian noise；
Fig. 6 c is the image reconstruction SAL algorithm reconstruction result figure for emulating capacitance data and being added after Gaussian noise；
Fig. 6 d is the image reconstruction AADMM algorithm reconstruction result figure for emulating capacitance data and being added after Gaussian noise；
Fig. 7 a is the practical flow pattern of image reconstruction of experiment；
Fig. 7 b is the image reconstruction LI algorithm reconstruction result figure of experiment；
Fig. 7 c is the image reconstruction SAL algorithm reconstruction result figure of experiment；
Fig. 7 d is the image reconstruction AADMM algorithm reconstruction result figure of experiment.
Specific embodiment
Flow chart of the invention as shown in Figure 1, the present invention the following steps are included:
S1, sensitive field matrix and capacitance are inputted；
S2, it is calculated using single step algorithm or iterative algorithm；
S3, output medium distributed image.
The sensitive field matrix of the input and capacitance are to establish the inearized model of ECT:
λ=Sg (1)
In formula: λ is the normalization capacitor vector of m × 1, and S is the normalization sensitivity field matrix of m × n, and g is the normalizing of n × 1
Change dielectric constant vector；
Single step algorithm described in step S2, comprising the following steps:
S21, parameter initialization；
S22, dielectric constant is directly calculated by single step formula.
Iterative algorithm described in step S2, comprising the following steps:
S31, parameter initialization；
S32, it solves to obtain dielectric constant using conjugate gradient method；
S33, correlation formula is updated；
S34, judgement | | gk-gk-1||2Whether less than zero, if so, carrying out S35；If it is not, then carrying out S32；
S35, output dielectric constant.
It is emulated as shown in Fig. 2, establishing 2 dimension ECT model shown in Fig. 2 inside COMSOL software, 2 dimensions of emulation
ECT model is made of 101 copper electrodes, 102PVC pipeline, 103 air, 104 insulation screens and 105 blending agents.Model uses
8 traditional electrode circular sensors, flow pattern use double square flow pattern, as shown in Figure 3.Emulating the height medium used is empty respectively
Gas and oil, relative dielectric constant are 1 and 2.3 respectively.
The sensor of emulation is 8 electrode sensors, therefore can get a measurement capacitance in 28 (7 × 8/2).By reconstruct image
As subdivision is 4096 (64 × 64) a units.Sensitivity field square when the sensitive field matrix of emulation is barnyard (medium is air entirely)
Battle array.
Analog simulation is carried out below.
Step 1: since ECT image reconstruction has " soft field " effect, so that dielectric distribution and capacitance are a kind of non-linear
Relationship.Therefore in order to simplify reconstruction model, it is necessary first to establish the inearized model of ECT:
λ=Sg (1)
In formula: λ be 28 × 1 normalization capacitor vector, S be 28 × 4096 normalization sensitivity field matrix, g be 4096 ×
1 normalization dielectric constant vector；
The target of ECT image reconstruction is exactly to obtain dielectric by measurement capacitance λ under the premise of known sensitivity field matrix S
Constant is distributed g；
Method one handles the simulation result of step 1 with single step algorithm, comprising the following steps:
Step 2: being based on ROF modular concept, introduce 2- norm and 1- norm respectively in formula (1), then solves in formula (1)
The problem of dielectric constant distribution g, can be converted into following convex optimization problem:
In formula: μ is fidelity term parameter, is taken as 1,For gradient operator.
Step 3: for the needs of subsequent algorithm statement, here defconstant and new space；It is one to reconstructed image
64 × 64 matrix is denoted as f (x, y)；F (x, y) can be converted into 4096 × 1 vector g (i)；Theorem in Euclid space RN’×N’It is defined as
SPACE V, then g ∈ V；Based on forward-difference method, it is utilized respectively companion matrix GxAnd GyThe horizontal and vertical gradient of g is calculated, wherein
Gx,Gy∈R4096×4096It is calculated by formula (3)；
For 1≤i, j≤64,
Laplacian Matrix Δ is determined by formula (4)；
If the inner product and 2- norm of SPACE V are respectively defined as ()VWith | | | |V；A new space W is now defined,
Its inner product and 2- norm is expressed as ()wWith | | | |w；It is all one 4096 × 2 matrix that T, P, which might as well be taken, and
T, P ∈ W, then W inner product is expressed as formula:
(T, P)W=(T1, P1)V+(T2, P2)V (5)
In formula: T1And T2It is the first row of T and the vector that secondary series represents, P respectively1And P2It is same；The 2- norm of W
It is provided by formula (7)；
Step 4: utilizing and simplify the Lagrangian method (augmented Lagrangian, AL) of augmentation, proposes simplification
Augmentation Lagrangian method (simplified augmented Lagrangian, SAL) algorithm；First convert formula (2) to about
The convex optimization problem of beam；
In formula: μ is taken as 1, | | q | |1WFor the 1- norm of q under the space W,Wherein
Corresponding Augmented Lagrangian Functions can be exported by formula (7):
In formula: L ∈ W, r are a normal numbers, are taken as 1；
The problem of q given for one, solution g, is equivalent to convex optimization problem shown in solution formula (9)；
DefinitionFor the Fourier transformation of g；According to Fourier transformation relevant knowledge, can obtain:
In formula:WithThe respectively Fourier transformation of g and Fourier inversion, STFor the transposition of sensitive field matrix
Matrix, r are a normal number, L=(L1,L2), q=(q1,q2)；It is one multiple that formula (10), which solves obtained dielectric constant distribution g,
Number, and the value of imaginary part is much smaller than the value of real part, therefore can export the real part of g as dielectric constant；
The influence of L and q to formula (10) is not considered, and formula (10) is reduced to following formula:
Method two handles the emulation of step 1 with iterative algorithm, comprising the following steps:
Step 2: being based on ROF modular concept, introduce 2- norm and 1- norm respectively in formula (1), then solves in formula (1)
The problem of dielectric constant distribution g, can be converted into following convex optimization problem:
In formula: μ is fidelity term parameter, and being taken as 0.5, ε is smooth item parameter, is taken as 0.01,For gradient operator；
Step 3: defconstant and new space；Without loss of generality, it is assumed that be one 64 × 64 square to reconstructed image
Battle array, is denoted as f (x, y)；F (x, y) can be converted into the vector g (i) of n × 1, n=64 × 64；Theorem in Euclid space RN’×N’It is defined as sky
Between V, then g ∈ V；Based on forward-difference method, it is utilized respectively companion matrix GxAnd GyThe horizontal and vertical gradient of g is calculated, wherein Gx,
Gy∈Rn×nIt is calculated by formula (13)；
For 1≤i, j≤64,
Laplacian Matrix Δ is determined by formula (4)；
Wherein, Gx, Gy are the matrixes of a n × n；Rn×nIndicate the two-dimensional matrix of a n × n.
If the inner product and 2- norm of SPACE V are respectively defined as ()VWith | | | |V；A new space W is now defined,
Its inner product and 2- norm is expressed as ()wWith | | | |w；It is all one 4096 × 2 matrix that T, P, which might as well be taken, and
T, P ∈ W, then W inner product is expressed as formula:
(T, P)W=(T1, P1)V+(T2, P2)V (5)
In formula: T1And T2It is the first row of T and the vector that secondary series represents, P respectively1And P2It is same；The 2- norm of W
It is provided by formula (7)；
Step 4: alternating direction Multiplier Algorithm (alternating direction method of is utilized
Multipliers, ADMM), propose alternating direction Multiplier Algorithm (the Accelerated alternating accelerated
Direction method of multipliers, AADMM) algorithm；Constrained convex optimization is converted by formula (12) first to ask
Topic；
In formula: | | p | |1WFor the 1- norm of p under the space W,WhereinUtilize the side ADMM
Formula (13) is decomposed into 2 subproblems by method；
In formula: d and s is auxiliary variable, and τ is a normal number, is taken as 5；
It can be seen that formula (14) right side of the equal sign is a quadratic programming problem, it is easy to get its optimal solution are as follows:
In formula:For divergence operator；
Formula (15) can regard a typical linear equation (such as Ax=b form) as；Since ECT waits for the dimension of reconstructed image
It is higher, select conjugate gradient method (conjugate gradient, CG) to solve gkIt can accelerate solving speed；
Formula (13) another subproblem is for solving pk；
Formula (16), which can use with typical contraction operator, to be solved；
In formula: i-th of element of contraction operator " shrink " is provided by formula (18)；
Auxiliary variable d and s is determined by following four formula；
Wherein, z indicates that a matrix, v indicate a scalar；Zi indicates the value of the i-th row in z matrix；C and d is one
Companion matrix；α is an auxiliary scalar；K indicates the number of iterations.
Step 5: the difference of the AADMM algorithm dielectric constant that connected 2 generations export in an iterative process exists in judgment step four
Whether it is less than setting value tol under 2- norm measure；If more than the end value of dielectric constant is then exported, if being less than, S32 is carried out.
The flow chart of AADMM algorithm is provided by table 1；
The flow table of table 1AADMM algorithm
Fig. 5 a, Fig. 5 b, Fig. 5 c, Fig. 5 d, Fig. 5 e and Fig. 5 f are the image reconstruction result figures of emulation.By Fig. 5 b, Fig. 5 c and figure
5f is it is found that two kinds of algorithms of SAL and AADMM can preferably maintain the edge feature of rectangle, and Landweber iteration
(LI) two rectangles of algorithm have been interconnected on together.It can be seen that part occurs in the reconstruction result of SAL algorithm by Fig. 5 c
Distortion, causes the lower edges of flow pattern and background to be blended in together.And Fig. 5 d and Fig. 5 f embody AADMM algorithm can be preferably
Flow pattern is distinguished in background.
In order to further verify the noise immunity of three kinds of algorithms, Gaussian noise is added in the capacitance data of emulation, makes noise
Than reaching 17dB, result is as shown in Fig. 6 a, Fig. 6 b, Fig. 6 c, Fig. 6 d.As seen from the figure, result and reality that LI algorithm reconstructs
Flow pattern differs greatly.According to the reconstruction result of SAL and AADMM algorithm, it still can substantially be inferred to the Partial Feature of practical flow pattern,
For example practical flow pattern should be 2 symmetrical figures.It is therefore seen that SAL and AADMM algorithm has stronger noise immunity.Comparison diagram
The phenomenon that 6c, Fig. 6 d, flow pattern preferably can be distinguished in background, be distorted image not by discovery AADMM algorithm.
Actual experiment is carried out below, it is identical when the parameter of each algorithm is kept with analog simulation in experimentation, such as Fig. 4 institute
Show, the sensor diameter of experiment is 76mm, there are 8 electrodes, and the width of each electrode is 30 °.Flow pattern equally uses double square
Flow pattern, specific size is the width of rectangle and height is 10mm and 30mm respectively.Two rectangular symmetricals are placed in center sensor, they
The distance between be 15mm.Testing the height medium used is air and dry sand respectively, and relative dielectric constant is 1 and 4 respectively.
Rectangular vessel is made using cardboard, dry sand is tested with carrying.
Likewise, the sensor of experiment is also 8 electrode sensors, therefore it can get a measurement capacitance in 28 (7 × 8/2).
It is 4096 (64 × 64) a units by reconstructed image subdivision.Experiment is when also with sensitive field matrix, to be barnyard i.e. medium be air entirely
Sensitive field matrix.
Fig. 7 a, Fig. 7 b, Fig. 7 c, Fig. 7 d are the image reconstruction result figures of experiment.Comparison diagram 5a, Fig. 5 b, Fig. 5 c, Fig. 5 f and figure
7a, Fig. 7 b, Fig. 7 c, Fig. 7 d, it is available with essentially identical conclusion when emulation, it was demonstrated that 2 kinds of algorithms proposed by the present invention
Superiority.
The above content is combine it is specific/further detailed description of the invention for preferred embodiment, cannot
Assert that specific implementation of the invention is only limited to these instructions.General technical staff of the technical field of the invention is come
It says, without departing from the inventive concept of the premise, some replacements or modifications can also be made to the embodiment that these have been described,
And these substitutions or variant all shall be regarded as belonging to protection scope of the present invention.
Claims (10)
1. a kind of image reconstructing method of the capacitance chromatography imaging based on ROF model, which is characterized in that by ROF model use in
In the image reconstruction of the capacitance chromatography imaging, using single step or the capacitance chromatography imaging ECT restructing algorithm of iteration, improve reconstruct
The quality of image, the image reconstruction of the ECT are to be situated between under the premise of known sensitivity field matrix S by measurement capacitance λ
The distributed image of electric constant g；The following steps are included:
S1, sensitive field matrix and capacitance are inputted；
S2, it is based on ROF modular concept, is calculated using single step algorithm or iterative algorithm；
S3, output medium distributed image.
2. image reconstructing method according to claim 1, which is characterized in that input sensitive field matrix and capacitance in step S1
It is:
Establish the inearized model of ECT:
λ=Sg (1)
In formula: λ is the normalization capacitor vector of m × 1, and S is the normalization sensitivity field matrix of m × n, and g is that the normalization of n × 1 is situated between
Electric constant vector.
3. image reconstructing method according to claim 1, which is characterized in that single step algorithm described in step S2, including it is following
Step:
S21, parameter initialization；
S22, dielectric constant is calculated by single step formula.
4. image reconstructing method according to claim 3, which is characterized in that parameter initialization described in step S21 include with
Lower step:
S211, it is based on ROF modular concept, introduces 2- norm and 1- norm respectively in formula (1), then it is normal to solve dielectric in formula (1)
The problem of number distribution g, is converted into following convex optimization problem:
In formula: μ is fidelity term parameter,For gradient operator；
S212, defconstant；If being the matrix of a N × N to reconstructed image, it is denoted as f (x, y)；A n is converted by f (x, y)
× 1 vector g (i), n=N × N；Based on forward-difference method, it is utilized respectively companion matrix GxAnd GyCalculate the horizontal and vertical ladder of g
Degree:
For 1≤i, j≤N,
Wherein, Gx, Gy are the matrixes of a n × n；Rn×nIndicate the two-dimensional matrix of a n × n；G (i) is dielectric constant g
Vector representation；
Wherein Gx,Gy∈Rn×nIt is calculated by formula (3)；Laplacian Matrix Δ is determined by formula (4).
5. image reconstructing method according to claim 4, which is characterized in that single step formula is by the formula in step S22
(4) result obtained substitutes into following single step formula, obtains dielectric constant g:
In formula:WithThe respectively Fourier transformation of g and Fourier inversion, STFor the transposition square of sensitive field matrix
Battle array, r are a normal number.
6. image reconstructing method according to claim 1, which is characterized in that iterative algorithm described in step S2, including it is following
Step:
S31, parameter initialization；
S32, it solves to obtain dielectric constant using conjugate gradient method；
S33, correlation formula is updated；
S34, judgement | | gk-gk-1||2Whether less than zero, if so, carrying out S35；If it is not, then carrying out S32；
S35, output dielectric constant.
7. image reconstructing method according to claim 6, which is characterized in that step S31 parameter initialization the following steps are included:
S311, it is based on ROF modular concept, introduces 2- norm and 1- norm respectively in formula (1), then it is normal to solve dielectric in formula (1)
The problem of number distribution g, it is converted into following convex optimization problem:
In formula: μ is fidelity term parameter, and ε is smooth item parameter,For gradient operator；
S312, defconstant and new space；If being the matrix of a N × N to reconstructed image, it is denoted as f (x, y)；
N × 1 a vector g (i), n=N × N are converted by f (x, y)；Theorem in Euclid space RN’×N’It is defined as SPACE V, then g ∈ V；Base
In forward-difference method, it is utilized respectively companion matrix GxAnd GyThe horizontal and vertical gradient of g is calculated, wherein Gx,Gy∈Rn×nBy formula (3)
It is calculated；
For 1≤i, j≤N,
Laplacian Matrix Δ is determined by formula (4)；
Wherein, Gx, Gy are the matrixes of a n × n；Rn×nIndicate the two-dimensional matrix of a n × n；
If the inner product and 2- norm of SPACE V are respectively defined as ()VWith | | | |V；A new space W is now defined, it
Inner product and 2- norm are expressed as ()wWith | | | |w；It is all the matrix of n × 2 that T, P, which might as well be taken, and T, P ∈ W,
Then W inner product is expressed as formula:
(T, P)W=(T1, P1)V+(T2, P2)V (6)
In formula: T1And T2It is the first row of T and the vector that secondary series represents, P respectively1And P2It is same；
Wherein: the 2- norm of W is provided by formula (7).
8. image reconstructing method according to claim 6, which is characterized in that step S32 solves to obtain using conjugate gradient method
Dielectric constant, specifically includes the following steps:
S321, using alternating direction Multiplier Algorithm, propose the alternating direction Multiplier Algorithm accelerated；First convert formula (5) to
The convex optimization problem of constraint；
In formula: | | p | |1WFor the 1- norm of p under the space W,WhereinUtilize alternating direction multiplier
Formula (8) is decomposed into 2 subproblems by algorithm；
In formula: d and s is auxiliary variable, and τ is a normal number；
Formula (9) right side of the equal sign is a quadratic programming problem, optimal solution are as follows:
In formula:For divergence operator；
Formula (10) can regard a typical linear equation as；
S322, it waits for that the dimension of reconstructed image is higher due to ECT, conjugate gradient method is selected to solve gkIt can accelerate solving speed.
9. image reconstructing method according to claim 6, which is characterized in that update correlation formula in step S33, specifically include
Following steps:
Formula (8) another subproblem is for solving pk；
Formula (11) can use typical contraction operator to solve；
In formula: i-th of element of contraction operator " shrink " is provided by formula (13)；
By following four formula:
Determine auxiliary variable d and s；
Wherein, z indicates that a matrix, v indicate a scalar；Zi indicates the value of the i-th row in z matrix；C and d is an auxiliary
Matrix；α is an auxiliary scalar；K indicates the number of iterations.
10. image reconstructing method according to claim 6, which is characterized in that step S34 is that iteration is calculated in judgment step S33
Whether the difference of the method dielectric constant that connected 2 generations export in an iterative process is less than setting value tol under 2- norm measure；If small
In then returning to step S32；If more than the end value for then exporting dielectric constant.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201610630259.3A CN106296766B (en) | 2016-08-03 | 2016-08-03 | A kind of image reconstructing method of the capacitance chromatography imaging based on ROF model |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201610630259.3A CN106296766B (en) | 2016-08-03 | 2016-08-03 | A kind of image reconstructing method of the capacitance chromatography imaging based on ROF model |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN106296766A CN106296766A (en) | 2017-01-04 |
| CN106296766B true CN106296766B (en) | 2019-01-25 |
Family
ID=57664742
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201610630259.3A Active CN106296766B (en) | 2016-08-03 | 2016-08-03 | A kind of image reconstructing method of the capacitance chromatography imaging based on ROF model |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN106296766B (en) |
Families Citing this family (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN107424158B (en) * | 2017-08-14 | 2019-11-26 | 清华大学深圳研究生院 | A kind of image binaryzation method and equipment of capacitance chromatography imaging |
| CN108120698B (en) * | 2017-11-22 | 2020-05-19 | 南京航空航天大学 | Optical fiber tomography method for monitoring the load distribution of flexible sheet structures |
| CN109215093B (en) * | 2018-07-27 | 2022-12-23 | 深圳先进技术研究院 | Low-dose PET image reconstruction method, device, equipment and storage medium |
| CN111337547B (en) * | 2020-03-10 | 2021-06-08 | 深圳市联恒星科技有限公司 | Complex multi-frequency real-time capacitance tomography method based on multiple measurement vectors |
| CN111413376B (en) * | 2020-04-10 | 2021-06-22 | 燕山大学 | Imaging Method of Coplanar Array Capacitive Sensor |
| CN112614204A (en) * | 2020-12-29 | 2021-04-06 | 哈尔滨理工大学 | Capacitance tomography image reconstruction method based on improved least square |
Citations (9)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| EP0294089A2 (en) * | 1987-05-30 | 1988-12-07 | Shimadzu Corporation | Data acquisition circuit |
| US6577700B1 (en) * | 2001-06-22 | 2003-06-10 | Liang-Shih Fan | Neural network based multi-criteria optimization image reconstruction technique for imaging two- and three-phase flow systems using electrical capacitance tomography |
| CN1538168A (en) * | 2003-10-21 | 2004-10-20 | 浙江大学 | Method and device for measuring oil-gas two-phase flow based on electrical capacitance tomography system |
| CN2695964Y (en) * | 2003-10-21 | 2005-04-27 | 浙江大学 | Oil-gas two-phase flow investigating device based-on capacitive chromatographic imaging system |
| US7187794B2 (en) * | 2001-10-18 | 2007-03-06 | Research Foundation Of State University Of New York | Noise treatment of low-dose computed tomography projections and images |
| CN101241094A (en) * | 2008-03-12 | 2008-08-13 | 天津大学 | Non-contact electrical impedance sensor and image reconstruction method based on the sensor |
| CN101520478A (en) * | 2009-03-13 | 2009-09-02 | 北京航空航天大学 | Direct image reconstruction method based on capacitance tomography of round sensor |
| CN102620855A (en) * | 2012-03-30 | 2012-08-01 | 华南理工大学 | Method and system for measuring polymer melt temperature field based on capacitance tomography |
| CN105374016A (en) * | 2015-10-30 | 2016-03-02 | 沈阳工业大学 | Three-dimensional electrical capacitance tomography imaging image reconstruction method |
-
2016
- 2016-08-03 CN CN201610630259.3A patent/CN106296766B/en active Active
Patent Citations (9)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| EP0294089A2 (en) * | 1987-05-30 | 1988-12-07 | Shimadzu Corporation | Data acquisition circuit |
| US6577700B1 (en) * | 2001-06-22 | 2003-06-10 | Liang-Shih Fan | Neural network based multi-criteria optimization image reconstruction technique for imaging two- and three-phase flow systems using electrical capacitance tomography |
| US7187794B2 (en) * | 2001-10-18 | 2007-03-06 | Research Foundation Of State University Of New York | Noise treatment of low-dose computed tomography projections and images |
| CN1538168A (en) * | 2003-10-21 | 2004-10-20 | 浙江大学 | Method and device for measuring oil-gas two-phase flow based on electrical capacitance tomography system |
| CN2695964Y (en) * | 2003-10-21 | 2005-04-27 | 浙江大学 | Oil-gas two-phase flow investigating device based-on capacitive chromatographic imaging system |
| CN101241094A (en) * | 2008-03-12 | 2008-08-13 | 天津大学 | Non-contact electrical impedance sensor and image reconstruction method based on the sensor |
| CN101520478A (en) * | 2009-03-13 | 2009-09-02 | 北京航空航天大学 | Direct image reconstruction method based on capacitance tomography of round sensor |
| CN102620855A (en) * | 2012-03-30 | 2012-08-01 | 华南理工大学 | Method and system for measuring polymer melt temperature field based on capacitance tomography |
| CN105374016A (en) * | 2015-10-30 | 2016-03-02 | 沈阳工业大学 | Three-dimensional electrical capacitance tomography imaging image reconstruction method |
Non-Patent Citations (4)
| Title |
|---|
| Image reconstruction algorithms for electrical capacitance tomography;W Q Yang等;《Measurement Science and Technology》;20021211;第14卷(第1期);第1-13页 |
| Optimizing the Geometry of Three-Dimensional Electrical Capacitance Tomography Sensors;Yi Li等;《IEEE SENSORS JOURNAL》;20150331;第15卷(第3期);第1567-1574页 |
| 基于本原对偶算法的图像去噪问题;吴君;《中国优秀硕士学位论文全文数据库 信息科技辑》;20160515(第5期);第5-6页 |
| 用于两相流测量的ECT图像重构技术研究;胡红利等;《工业仪表与自动化装置》;20100405(第2期);第100-103页 |
Also Published As
| Publication number | Publication date |
|---|---|
| CN106296766A (en) | 2017-01-04 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN106296766B (en) | A kind of image reconstructing method of the capacitance chromatography imaging based on ROF model | |
| Burkhart et al. | Properties of interstellar turbulence from gradients of linear polarization maps | |
| Dines et al. | Analysis of electrical conductivity imaging | |
| CN108490496B (en) | Gravitational field inversion of Density method based on pseudo-radial basis function neural network | |
| Achitouv et al. | Improving reconstruction of the baryon acoustic peak: The effect of local environment | |
| CN107424158B (en) | A kind of image binaryzation method and equipment of capacitance chromatography imaging | |
| Zhang et al. | Image reconstruction of electrical capacitance tomography based on an efficient sparse Bayesian learning algorithm | |
| Tessore et al. | GLASS: Generator for Large Scale Structure | |
| Zdun et al. | Fast MPI reconstruction with non-smooth priors by stochastic optimization and data-driven splitting | |
| Chen et al. | Image reconstruction algorithms for electrical capacitance tomography based on ROF model using new numerical techniques | |
| Bao et al. | Integral equation fast solver with truncated and degenerated kernel for computing flaw signals in eddy current non-destructive testing | |
| Tong et al. | Regularization iteration imaging algorithm for electrical capacitance tomography | |
| Koch et al. | Designing quantum many-body matter with conditional generative adversarial networks | |
| Cao et al. | Iterative reconstruction algorithm for electrical capacitance tomography based on Calderon’s method | |
| CN118091774A (en) | A hybrid regularization-based inversion method for airborne electromagnetic data | |
| Ntampaka et al. | The importance of being interpretable: Toward an understandable machine learning encoder for galaxy cluster cosmology | |
| Zhao et al. | A novel computational imaging algorithm based on split Bregman iterative for electrical capacitance tomography | |
| Hamilton et al. | Fast absolute 3D CGO-based electrical impedance tomography on experimental tank data | |
| CN113267287B (en) | Three-dimensional space-time field reconstruction method for shock wave overpressure | |
| Heinrich et al. | CMB delensing with neural network based lensing reconstruction in the presence of primordial tensor perturbations | |
| Alves et al. | Extra polarization states of cosmological gravitational waves in alternative theories of gravity | |
| Chen | Subspace-based optimization method for inverse scattering problems with an inhomogeneous background medium | |
| Wiaux et al. | Compressed sensing reconstruction of a string signal from interferometric observations of the cosmic microwave background | |
| Adamek et al. | Ray tracing the integrated Sachs-Wolfe effect through the light cones of the dark energy universe simulation-full universe runs | |
| CN110687535B (en) | Rapid microwave imaging method |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| C06 | Publication | ||
| PB01 | Publication | ||
| C10 | Entry into substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |