CN104614411B - The electricity tomography Lp regularization reconstruction method being declined based on p vector equal difference - Google Patents

Abstract

The present invention relates to a kind of electricity tomography Lp regularization reconstruction method being declined based on p vector equal difference, it is applied to bubble flow tomography, carry out updating according to gained solution the p vector being made up of the p value on each pixel in image in the often step iteration of Lp regularization inverse problem solving using Gauss Newton iteration, obtain the p distribution with field domain object space distribution character, it is finally completed calculating and obtains reconstruction image, step is as follows：Obtain and rebuild required retive boundary measured value vector b and sensitivity matrix A；Set up the object function of Lp regularization；Calculate equal difference descending factors；Solved using Gauss Newton iterative formula；In each iteration, vectorial using being solved renewal p；Imaging.The present invention is conducive to the accurate solution of electricity tomography inverse problem, improves image reconstruction quality.

Description

The electricity tomography Lp regularization reconstruction method being declined based on p vector equal difference

Technical field

The invention belongs to electricity chromatography technical field of imaging, relate to the use of Lp regularization method realize image reconstruction and Gauss-Newton alternative manner.

Background technology

Multiphase flow refers to comprise substantially interfacial fluid system, such as the liquid (gas) of bubbles (drop), immiscible Liquid, containing the gas of solid particle or liquid etc., they frequently appear in power, chemical industry, oil, nuclear energy, metallurgical engineering waited Cheng Zhong, has highly important effect to commercial production and scientific research.The flow pattern of multiphase flow refers to present in its pipeline The geometry nowed forming different with dynamic characteristic, it can be common in two phase flow by the form of component or phase come qualitative description Flow pattern include bubble flow, slug flow, annular flow etc..

Electricity chromatography imaging technique (Electrical Tomography, ET) is that occur from the later stage eighties in last century A kind of new process tomographic imaging technology based on electrical characteristics sensitive mechanism, the medium that its physical basis are different has difference Electrical characteristics (electrical conductivity/dielectric coefficient/complex admittance/pcrmeability), by judge object in sensitivity field electrical characteristics be distributed just may be used Deduce the distribution situation of this middle medium.Electricity chromatography imaging technique mainly includes Electrical Resistance Tomography (Electrical Resistance Tomography, ERT), capacitance chromatography imaging (Electrical Capacitance Tomography, ECT), electrical impedance tomography (Electrical Impedance Tomography, EIT) and electromagnetic chromatographic (Electrical Magnetic Tomography,EMT).Electricity chromatography is imaged on multiphase flow and biomedical sector has extensively Application prospect, it is possible to achieve long-term, continue to monitor.

Electricity tomography inverse problem (i.e. image reconstruction problem) solution has non-linear.By linearization process, permissible Problem is converted into linear reverse temperature intensity.For the ill-posedness of reverse temperature intensity, generally choose regularization method process inverse Problem.The thought of regularization method is to find one to carry out approaching to reality solution by the stable disaggregation that prior information constrains.Prior information Different and Regularization function form differences of choosing make regularization method have different application forms, such as with the 2 of solution Norm realizes the L2 regularization method of the stable solution of inverse problem for Regularization function：Vauhkonen M et al. sent out in 1998 Table in《IEEE medical imaging》(Medical Imaging, IEEE Transactions) volume 17, the 285-293 page, entitled 《Tikhonov regularization based on electrical impedance tomography and prior information select》(Tikhonov regularization And prior information in electrical impedance tomography) article；With 1 norm solving it is Regularization function realizes the L1 regularization method that inverse problem stably solves：Jin, Bangti et al. were published in 2012《Engineering In numerical computations》(International Journal For Numerical Methods In Engineering) the 89th Volume, the 337-353 page, entitled《Electrical impedance tomography algorithm for reconstructing based on sparse regularization》(A reconstruction algorithm for electrical impedance tomography based on sparsity Regularization article).

But inverse problem gained solution is solved using L2 regularization smooth phenomenon occurred, become image has larger tail Shadow；And L1 regularization is solved to the field domain with smooth object distribution and sparse problem was occurred it is impossible to fully demonstrate field The actual size of domain object, therefore introduces the Lp regularization method with p norm as Regularization function and (typically chromatographs in electricity P ∈ [1,2] is taken) in picture.Daubechies I et al. was published in 2004《Mathematics and Applied Mathematics》(Communications On Pure and Applied Mathematics) volume 57, the 1413-1457 page, entitled《Linear inverse problem is sparse The iteration threshold algorithm of constraint》(An iterative thresholding algorithm for linear inverse Problems with a sparsity constraint) article provide solve Lp regularization iterative algorithm.Zhang Lingling Et al. be published in 2013《Multiphase flow detection and instrument and meter》(Flow Measurement and Instrumentation) volume 33, the 244-250 page, entitled《Electrical Resistance Tomography inverse problem iteration threshold algorithm》(An iterative thresholding algorithm for the inverse problem of electrical Resistance tomography) article iteration threshold algorithm is applied in Electrical Resistance Tomography, and to during p=1.5 Imaging results are discussed, and demonstrate the effectiveness of method.

But existing research in, using Lp regularization carry out electricity tomography reverse temperature intensity generally to whole field domain select Select the p value of a fixation, and the field domain of different objects distribution needs given different p value, to obtain more preferable stable solution.This Method have ignored the spatial character of the field domain of different objects distribution it is impossible to project the feature of field domain itself, and the regulation of p value is asked The solution of topic brings extra work amount, is unfavorable for the popularization of method.

Content of the invention

It is an object of the invention to overcoming the above-mentioned deficiency of prior art, a kind of electricity tomography Lp regularization weight is proposed Construction method, the present invention, based on Lp regularization, in conjunction with Gauss-Newton iteration, solves that L2 Regularization Solution is excessively smooth and L1 The excessively sparse problem of Regularization Solution, improves solving precision and the image reconstruction quality of electricity tomography inverse problem.The present invention's Technical scheme is as follows：

A kind of electricity tomography Lp regularization reconstruction method being declined based on p vector equal difference is it is adaptable to bubble flow chromatographs Imaging, the method regards electricity chromatography imaging problem as linear ill-posed problem Ax=b, and wherein A is sensitivity matrix, b For retive boundary measured value vector, x be and field domain material electrical properties be distributed corresponding imaging gray value vectors, be called solution to Amount, the method using Lp regularization inverse problem solving carries out image reconstruction.It is characterized in that,

Carry out using Gauss-Newton iteration being updated according to gained solution in the often step iteration of Lp regularization inverse problem solving The p vector being made up of the p value on each pixel in image, obtains the p distribution with field domain object space distribution character, finally Complete to calculate and obtain reconstruction image, step is as follows：

(1) according to the measurement to tested field domain, obtain and rebuild required retive boundary measured value vector b and sensitivity matrix A；

(2) set up the object function of Lp regularization；

(3) initiation parameter is set：The initial value x of solution vector x₀=0, p vector initial value p₀=2, p vector stop value p_stop=1；Set iterationses N；

(4) calculate equal difference descending factors α=(p₀-p_stop)/N；

(5) solved using Gauss-Newton iterative formula；

(6) vectorial using being solved renewal p：Judge in solution vector, whether each element is zero, if then corresponding pixel points P value keep back p value constant；If it is not, then the p value of corresponding pixel points is updated to back p value and equal difference descending factors Difference；

(7) judge whether iteration completes, if then iteration ends, carry out next step operation, if it is not, then rebound (5th) step Continue to solve；

(8) solve gained gray value according to final, be imaged.

Preferably, the object function of described Lp regularization is：Wherein λ ＞ 0 is regularization coefficient, | | | | for Euclid norm, p vector meets either element and belongs to [1,2]；During in view of p=1 Object function non-differentiability, utilizesApproximately above-mentioned object function, wherein n be solution to The dimension of amount x, j is the counting integer from 1 to n, x_jFor j-th element in solution vector x, β is small adjustable parameter, meets β ＞ 0.

Using Gauss-Newton iterative formula it is：

Wherein k is current iterationses, meets 1≤k≤N；x_kIt is the solution that kth time iteration obtains, x_k-1It is (k-1) The solution that secondary iteration obtains；It is to work as x=x_k-1When object function first differential,For x=x_k-1When object function Second-order differential, obtained by following two formula respectively：

Wherein p_k-1It is the p vector that back is (k-1) secondary iteration updates acquisition；Diag () is to be constructed by vector The function of diagonal matrix, each element of vector constitutes the diagonal element of diagonal matrix, and the off-diagonal element of diagonal matrix is zero.

The invention has the beneficial effects as follows based on the process using Gauss-Newton iterative Lp regularization, proposing a kind of Declined using the equal difference of p in iterative process and realize Lp adaptive electricity tomographic image method for reconstructing, be reconstruction image Each pixel offer of smooth domain is worth for 2 or the p value that is close to 2；Each pixel of object area is had to provide for reconstruction image It is worth for 1 or the p value that is close to 1, obtain the p distribution with field domain object space distribution character, complete finally to solve.The present invention Effectively overcome L2 regularization and the shortcoming of L1 regularization, improve reverse temperature intensity precision and image reconstruction quality, have Higher robustness and the bigger suitability.

Brief description

Fig. 1 is a kind of flow chart element of electricity tomography Lp self adaptation method for reconstructing being declined based on p equal difference of the present invention Figure；

Fig. 2 is the Electrical Resistance Tomography system circular list tested field domain in section and the distribution of electrodes of the present invention；

Fig. 3 is the true distribution of three models of example selection of the present invention：A () is that two roundlets model (b) are three Circle model (c) is four roundlet models

Fig. 4 be the present invention example in three models imaging results schematic diagram under L2 regularization solution：Wherein (a-c) Correspond to the model (a-c) in Fig. 3 respectively；

Fig. 5 be the present invention example in three models imaging results schematic diagram under L1 regularization solution：Wherein (a-c) Correspond to the model (a-c) in Fig. 3 respectively；

Fig. 6 be the present invention example in the imaging under the self adaptation Lp regularization that this method proposes solves of three models tie Fruit schematic diagram：Wherein (a-c) corresponds to the model (a-c) in Fig. 3 respectively；

In figure：

1st, tested field domain 2, electrode

Specific embodiment

In conjunction with the accompanying drawings and embodiments a kind of of the present invention is rebuild based on the electricity tomography Lp self adaptation that p equal difference declines Method is illustrated.

A kind of electricity tomography Lp self adaptation method for reconstructing being declined based on p equal difference of the present invention, is turned to Lp canonical Basis, the solution tried to achieve for L2 regularization is excessively smooth and problem that solution that L1 regularization is tried to achieve is excessively sparse, in conjunction with Gauss- Newton iterative formula, proposes to utilize back result of calculation to update by the p on each pixel in image in an iterative process The p vector that value is constituted, and complete currently to calculate using the p vector after updating, until the method for reconstructing of iteration ends.

As shown in figure 1, a kind of electricity tomography Lp self adaptation method for reconstructing stream being declined based on p equal difference for the present invention Cheng Tu.It is illustrated in figure 2 the Electrical Resistance Tomography system circular list tested field domain in section of one of electricity tomography and electrode divides Cloth, is evenly distributed on field domain outer wall using 16 electrodes.Choosing three typical blister flow models is embodiment, and in field domain, object is true Real distribution is as shown in Fig. 3 (a-c).In order to preferably embody the regularization of self adaptation Lp and L2 regularization and L1 regularization in the present invention Difference, provide solving result under these three regularization methods for three models respectively.Embodiment comprises the following specific steps that：

A kind of electricity tomography Lp self adaptation method for reconstructing being declined based on p equal difference it is adaptable to bubble flow tomography, The method regards electricity chromatography imaging problem as linear ill-posed problem Ax=b, and wherein A is sensitivity matrix, and b is relatively Boundary survey value vector, x is to be distributed corresponding imaging gray value vectors with field domain material electrical properties.

The object function of Lp regularization is：

Wherein λ ＞ 0 is regularization coefficient, | | | | for Euclid norm, p is by each pixel of reconstruction image The constant vector that p value is constituted.Consider object function non-differentiability during p=1, using new object function：

Approximately former object function.Method for reconstructing includes following steps：

(1) it is directed to three typical blister flow models, obtain respectively and each rebuild required boundary survey value vector sum spirit Sensitive matrix：

Boundary survey value vector is that measurand is placed in electricity chromatography imaging measurement system, uniformly divides outside tested field domain 16 electrodes (as shown in Figure 2) of cloth, using current excitation voltage measurement and the pattern that do not measure of exciting electrode, gather cycle motivation Boundary voltage under circulation measurement, obtains the vector that 208 measured values are constituted altogether；Inverse problem right-hand vector b is without inclusions Difference (the i.e. right-hand vector relative edge of the barnyard boundary voltage vector b1 and boundary survey voltage vector b2 having thing field containing inclusions Boundary measured value vector b=b1-b2)；

Sensitivity matrix is the boundary survey voltage according to the barnyard without inclusions, in conjunction with sensitivity theory, calculates spirit Sensitive matrix, computing formula is：

Wherein A_ijIt is j-th electrode pair sensitivity coefficient to i-th electrode pair, φ_i,φ_jIt is respectively i-th electrode pair And j-th electrode pair is I in exciting current_i,I_jWhen field domain Potential Distributing, x, y are distributed as the transverse and longitudinal coordinate of field domain；

(2) initiation parameter is set：

The initiation parameter of setting includes：Preset parameter, solution vector initial value x₀=0, p vector initial value p₀=2, p vector Stop value p_stop=1；The iterationses N=5 rule of thumb choosing, regularization coefficient λ=1 × 10^-4, fine setting parameter beta=1 × 10^-16；

(3) calculate equal difference descending factors α, computing formula is：

α=(p₀-p_stop)/N；

(4) solved using Gauss-Newton iterative formula, its Iteration is：

Wherein k is the iterationses of current step, meets 1≤k≤N；x_kIt is the solution that kth time iteration obtains, x_k-1It is (k- 1) solution that secondary iteration obtains；It is to work as x=x_k-1When object function first differential,For x=x_k-1When target letter The second-order differential of number, is obtained by following two formula respectively：

Wherein p_k-1It is the p vector that back is (k-1) secondary iteration updates acquisition；Diag () is to be constructed by vector The function of diagonal matrix, each element of vector constitutes the diagonal element of diagonal matrix, and the off-diagonal element of diagonal matrix is zero.And count Calculate second-order differentialWhen eliminate higher order term；

(5) determine the p vector in iterative process：

In order to preferably compare the different of this method and L2 regularization and L1 regularization, three models in embodiment are divided Do not provide the p vector needed for L2 regularization, L1 regularization and the regularization of self adaptation Lp：

To L2 regularization, in iterative process, p vector keeps all elements to meet p=2 constant；

To L1 regularization, in iterative process, p vector keeps all elements to meet p=1 constant；

To the regularization of self adaptation Lp, using being solved renewal p vector in iterative process, more New Policy is：

Wherein p_kIt is the p vector that kth time iteration updates；L is the position of each pixel of field domain；

(6) judge whether iteration completes, if then iteration ends, carry out next step operation, if it is not, then rebound step (4) Continue to solve；

(7) solve gained gray value according to final, be imaged：

Gained Regularization Solution is corresponded on the pixel of Electrical Resistance Tomography, carries out Grey imaging.Fig. 4 is just showing L2 Then change result of calculation image, Fig. 5 show L1 regularization result of calculation image, Fig. 6 is that the regularization of self adaptation Lp calculates knot Fruit image, in figure (a-c) is corresponding with model (a-c) respectively.

As can be seen that under identical parameter setting, the solution of L2 regularization is excessively smooth, become image tail shadow is big；And L1 is just Then changed sparse it is impossible to describe the size of object well；The result of self adaptation Lp regularization is between L2 regularization and L1 canonical Change in the middle of acquired results, improve reverse temperature intensity precision and image reconstruction quality, more accurately describe object in field domain Distribution.

Embodiment described above is the several preferable model of the present invention, and the present invention is not limited to this embodiment and accompanying drawing institute is public The content opened.Every without departing from complete equivalent or modification under spirit disclosed in this invention, all in the scope of protection of the invention.

Claims (3)

1. a kind of electricity tomography Lp regularization reconstruction method being declined based on p vector equal difference is it is adaptable to bubble flow chromatographs into Picture, the method regards electricity chromatography imaging problem as linear ill-posed problem Ax=b, and wherein A is sensitivity matrix, and b is Retive boundary measured value vector, x is to be distributed corresponding imaging gray value vectors with field domain material electrical properties, is called solution vector, Image reconstruction is carried out using the method for Lp regularization inverse problem solving it is characterised in that

Carry out using Gauss-Newton iteration being updated by scheming according to gained solution in the often step iteration of Lp regularization inverse problem solving The p vector that p value on each pixel in picture is constituted, obtains the p distribution with field domain object space distribution character, is finally completed Calculate and obtain reconstruction image, step is as follows：

(1) according to the measurement to tested field domain, obtain and rebuild required retive boundary measured value vector b and sensitivity matrix A；

(2) set up the object function of Lp regularization；

(3) initiation parameter is set：The initial value x of solution vector x₀=0, p vector initial value p₀=2, p vector stop value p_stop= 1；Set iterationses N, determine regularization coefficient λ；

(4) calculate equal difference descending factors α=(p₀-p_stop)/N；

(5) solved using Gauss-Newton iterative formula；

(6) vectorial using being solved renewal p：Judge in solution vector, whether each element is zero, if the p value of then corresponding pixel points Keep back p value constant；If it is not, then the p value of corresponding pixel points is updated to the difference of back p value and equal difference descending factors；

(7) judge whether iteration completes, if then iteration ends, carry out next step operation, if it is not, then rebound (5th) step continues Solve；

(8) solve gained gray value according to final, be imaged.

2. the electricity tomography Lp regularization reconstruction method being declined based on p vector equal difference according to claim 1, it is special Levy and be, the object function of described Lp regularization is：Wherein λ ＞ 0 is regularization coefficient, | | | | for Euclid norm, p vector meets either element and belongs to [1,2]；In view of object function non-differentiability during p=1, profit WithApproximately above-mentioned object function, wherein n be solution vector x dimension, j be from 1 to The counting integer of n, x_jFor j-th element in solution vector x, β is small adjustable parameter, meets β ＞ 0.

3. the electricity tomography Lp regularization reconstruction method being declined based on p vector equal difference according to claim 2, it is special Levy and be：Using Gauss-Newton iterative formula it is：

$x_k=x_{k-1}-{\[{\▿}^{2}F\left(x_{k-1}\right)\]}^{-1}\▿F\left(x_{k-1}\right)$

Wherein k is current iterationses, meets 1≤k≤N；x_kIt is the solution that kth time iteration obtains, x_k-1It is (k-1) secondary iteration The solution obtaining；It is to work as x=x_k-1When object function first differential,For x=x_k-1When object function second order Differential, is obtained by following two formula respectively：

$\▿F\left(x_{k-1}\right)=A^T({\mathrm{Ax}}_{k-1}-b)+\mathrm{\λ}diag\left(p_{k-1}{\left(\sqrt{x_{k-1}^2+\mathrm{\β}}\right)}^{p_{k-1}-2}{x}_{k-1}\right)$

${\▿}^{2}F\left(x_{k-1}\right)=A^{T}A+\mathrm{\λ}diag\left(p_{k-1}{\left(\sqrt{x_{k-1}^2+\mathrm{\β}}\right)}^{p_{k-1}-2}\right)$

Wherein p_k-1It is the p vector that back is (k-1) secondary iteration updates acquisition；Diag () is diagonal by vector construction The function of battle array, each element of vector constitutes the diagonal element of diagonal matrix, and the off-diagonal element of diagonal matrix is zero.