Embed Size (px)
Citation preview
Inverse Problems in Science and EngineeringVol. 14, No. 1, January 2006, 65–74
Fast post-processing algorithm for improving electrical
capacitance tomography image reconstruction
ALEXANDRE GREBENNIKOV*y and CARLOS GAMIOz
yBenemerita Universidad Autonoma de Puebla, Av. San Claudio y Rio verde,Ciudad Universitaria, CP 72570, Puebla, Pue, Mexico
zPrograma de Ductos, Instituto Mexicano de Petroleo Eje Central,L. Cardenas 152, CP 07730, Mexico D.F., Mexico
(Received 15 December 2003; in final form 31 October 2005)
Electrical tomography is the reconstruction of an image of the interior of a body, based onmeasurements made on its surface. Mathematically it can be described as a coefficient inverseproblem for the Laplace equation, written in the divergent form. The coefficient is a functionof the space variables and characterizes the electrical properties of the medium. One of thebest-known and most developed approaches is electrical capacitance tomography (ECT).This method has some advantages, but because of the non-linear structure of the mathematicalmodel, the results of the regularization procedures for image reconstruction are not perfect. Wepropose here a fast post-processing algorithm to improve the quality of the electricaltomography images, obtained by any method. This algorithm is based on the explicit spline-approximation method theoretically justified for restoring some classes of functions.Numerical experiments on simulated model problems confirm its good properties.
Keywords: Electrical capacitance tomography; Image reconstruction; Regularization; Splineapproximation
1. Introduction
Electrical tomography is the reconstruction of an image of the interior of a body,based on the electrical measurements made on its surface [1,2]. One of the best-known and most developed approaches is electrical capacitance tomography (ECT),which can be mathematically described as a coefficient inverse problem for theLaplace equation, written in the divergent form
@
@x"ðx, yÞu0xðx, yÞ� �
þ@
@y"ðx, yÞu0yðx, yÞ� �
¼ 0, ð1Þ
*Corresponding author. Email: [email protected]
Inverse Problems in Science and Engineering
ISSN 1741-5977 print: ISSN 1741-5985 online � 2006 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/17415970500272874
where x, y 2� is some region on a plane, u(x, y) is the electrostatic potential, thefunction "(x, y) characterizes the electrical permittivity of a medium. The goal ofECT is to reconstruct the electrical permittivity distribution of the interior ofthe object based on the knowledge of measurements of the voltages and chargeson the surface. This approach leads to a non-linear and ill-posed problem [2]. Due tothe linearization of the inverse coefficient problem it is possible to use the linearback-projection (LBP) algorithm [3–5]. To get a proper image quality it is necessaryto apply LBP iteratively or use some regularization such as the Landweber orTikhonov schemes [6,7]. These methods have some advantages, but their algorithmicand software realization for image reconstruction is rather complicated because ofthe non-linear structure of the corresponding mathematical model (1). As a result,some essential errors appear in the reconstructed image [7], and, as mentioned in [7],new fast algorithms for improving images are desirable.
Here, we consider the image obtained by any algorithm as pre-reconstructed andpropose to realize an additional fast post-processing procedure to improve the result.This scheme is a part of the ‘three steps’ concept, developed and justified for thesolution of ill-posed problems by the spline-approximation method [8], and its general-ization, the full-spline-approximation method [9,10]. This concept consists in dividingthe ill-posed problem with noisy input data into three steps: (1) pre-processingthe input data; (2) pre-reconstruction of the required functions by calculations withgiven formulas or by numerical solution of the equation describing the process; and(3) post-processing, including post-smoothing and projecting the pre-reconstructionresults on the set that characterizes the special properties of the exact solution ofthe inverse problem. In each step, different qualitative and quantitative a prioriinformation is used. This is the main difference between our approach and traditionalregularization, particularly schemes in [6], where all a priori information is used inthe context of the one basic reconstruction algorithm (i.e., it is used only in step 2).We underline that the separation of the regularization process into three steps givesus an important possibility to construct simpler algorithms. The theoretical and numer-ical justification of this concept for sufficiently general classes of ill-posed problems isgiven in [8–11]. Post-processing is an important step of the above concept because itrealizes the simplified variant of the descriptive regularization, proposed in [11], seealso [12]. Here we stress the fast realization of the post-processing, and its applicationto improving the solution of the inverse problem of capacitance tomography imaging,for simulated two-phase flow regimes. This approach and the developed algorithm arejustified with numerical experiments on simulated model problems.
2. Measurement scheme of capacitance tomography and
the form of the pre-reconstructed image
Oil wells typically produce not just oil, but a complex multi-phase mixture havingvariable amounts of oil, gas, and water. The determination of the quantity of eachcomponent actually being produced by each specific well is of the greatest importancefor the efficient exploitation of oil reservoirs. The conventional way of doing this isby separating the mixture and measuring each individual component using single-phase flow meters. However, the three-phase separators needed are excessively bulkyand expensive. Among the most promising alternative approaches currently under
66 A. Grebennikov and C. Gamio
investigation is one based on multiphase flow visualization using tomographymethods, in particular ECT. The main advantage of the tomography methods liesin their inherently non-invasive and flow-regime independent operation. ECT is anemerging technique aimed at the non-invasive internal visualization of electricallynon-conducting mixtures in industrial processes like mixing, separation, and multi-phase flow [1,2]. Only the application to flow imaging is considered here. The basicprinciple of this method is to place a sensor containing an array of between 8 and 16contiguous sensing electrodes around the pipe carrying the process fluids, at thecross-section to be investigated [13]. The pipe wall should be electrically non-conducting in the zone of the electrodes, which are typically 10 cm long (see figure 1).The sensor has an outer cylindrical metallic screen covering the whole assembly, whichis always kept at an electric potential of zero volts. The sensing electrodes are connectedto an apparatus that allows all the mutual capacitances between the different electrodepairs to be measured, and from this set of measurements the electrical permittivitydistribution inside the sensor is obtained using a suitable inversion algorithm. Thepermittivity distribution reflects the spatial arrangement of the phases in the flow.Image reconstruction can thus be regarded as an inverse permittivity problem.
For two-phase flows like gas–oil, the permittivity distribution directly determinesthe distribution of each phase, whereas for three-phase flows like gas–oil–water, thedistribution of an additional parameter (i.e., conductivity, etc.) must be obtainedfirst through a different tomography modality in order to resolve each phasedistribution. This article however, deals only with the two-phase problem, and thethree-phase problem is intended to be addressed in a future report.
The use of the cylindrical guards at the ends of the sensing electrodes (and theassumption that the phase distribution changes slowly in the axial direction) allowsthe sensor to be represented by a two-dimensional (2D) model [3]. Assuming too thatthe flow changes negligibly during the time required for one set of measurements,and that the frequency of the excitation voltage is so small that the correspondingwavelength is much larger than the sensor dimensions, a static model can be considered.
The corresponding ECT inverse problem consists in reconstructing the permittivitydistribution based on the measured mutual capacitances. This is an ill-posed problemthat has numerical instability. The traditional formulation leads to a non-linearproblem and the Tikhonov regularization scheme is used to solve it. One of thealgorithmic and program realizations of such an approach is presented in the systemfor the EIDORS project [14] that uses the MATLAB software package. An essentialelement of this approach is the use of the finite element method for the solution of the
Figure 1. View of a typical capacitance tomography sensor.
Fast post-processing algorithm 67
forward problem and an iterative Tikhonov-regularized scheme for solving the inverseproblem. As a rule, there are some essential errors in the reconstructed images, andtheir improvement is possible by post-processing if the values of the permittivity "1and "2 of the mixture components are known. Another important aspect in theEIDORS system is the use of Delaunay triangulation of the considered domain �.Using this type of spatial discretization, the structure of the input and output datais not appropriate for fast post-processing algorithms. In the considered problem wehave a simple domain �, a circle. To construct the fast post-processing algorithms it ispossible to simplify the data structure by some re-calculations, which are presented next.
3. Post-processing algorithm
We use the simulated output (image) data from the EIDORS system as the noisydata on a triangular mesh, at points pi¼ (xi, yi), i¼ 1, . . . ,N, in the considereddomain �. That is, the function of two variables "(x, y) is defined as the approximatevalues "i¼ "( pi)þ �i, which consist of the exact values of function " at the points pi,with errors �i, i¼ 1, . . . ,N, such that we know the estimation ��maxifj�ijg.
The first part of the proposed algorithm consists in recalculating the originalfunction "(x, y) obtained on the stage of the solution of the inverse problem, on aregular grid {xi, yi} in �. For this purpose, we use the irrational interpolation IR(")of the function "(x, y):
IRðx, y, "Þ ¼
PNi¼1 riðx, yÞ"ðpiÞPN
i¼1 riðx, yÞ, ð2Þ
riðx, yÞ ¼ ðx� xiÞ2 þ ðy� yiÞ2� ��2
, i ¼ 1, . . . ,N: ð3Þ
The function IR(", x, y) interpolates the values "( pi) at the points pi. It is exact onconstant functions and, hence, has approximating properties. We shall supposefurther that the set of the points pi is dense in �, d¼max1�i�N minj 6¼i | pi� pj |� �for known error estimation �, and a condition holds: d! 0, N!1. In this caseif k f kC [�]�M¼ const, then k IR(", x, y) kC [�]�M, which guarantees that a uniformerror � in the input data will not increase with irrational approximation.
The second part of the algorithm is recursive smoothing by explicit formulasfor two-dimensional splines constructed on the regular uniform grid {xi, yi}on �¼ [�1, 1]� [�1, 1], xi¼�1þ h(i� 1), i¼�2, . . . , nþ 2; yj¼�1þ h( j� 1),j¼�2, . . . , nþ 2; h¼ 2/(n� 1). Let si,3(u) be a local basic cubic spline, constructedon the units ui�2, . . . , uiþ2; i¼ 0, . . . , nuþ 1; where u is x or y. We use theformulas of the recursive smoothing spline-method [9] for the case of functions oftwo variables:
Skðx, y, "Þ ¼Xni¼1
Xnj¼1
Sk�1ðxi, yi, "ÞsiðxÞsjðyÞ, ð4Þ
k ¼ 1, 2, . . . ,K; ð5Þ
S0ðxi, yj, "Þ ¼ IRðxi, yj, "Þ: ð6Þ
68 A. Grebennikov and C. Gamio
The number of smoothes K is the regularization parameter, which can be chosenhere in accordance with the residual principle using the discrete root-mean-squareestimation � of the errors, i.e., K is chosen as the maximum among all k for whichthe following inequality is fulfilled:
Xni¼1
Xnj¼1
Skðxi, yj, "Þ � IRðxi, yj, "Þ�� ��2 � c�2n2, ð7Þ
where c¼ const>1. Theoretical and numerical justification of the regularizationproperties of this type of smoothing is presented in [9,11].
The third part of the algorithm is the projection of the smoothed data to the set�"" ¼ f"1, "2g of the known values. We underline that the presented algorithmsare based on explicit approximation formulas (2)–(6) that do not require solving anyequations. It makes this algorithm fast in its numerical realization.
Let us present the results of the numerical experiments. The domain � is the unit circle.The triangular Delaunay mesh with N¼ 144 is constructed by the adaptmesh programof the MATLAB PDE toolbox [15]. Exact simulated data are chosen as the piecewiseconstant distribution of the basic permittivity value "1¼ 1 in the main part of theunit circle, and "2¼ 3 as the inhomogeneity. Graphs (a) in figures 2–4 demonstratethese distributions of " and the triangular initial mesh. Approximated (noisy) values"i at the points of the mesh are obtained by disturbing the exact values with casualrandom errors, with error estimation �¼ 0.5. These values simulate the output datathat can be obtained by any method of reconstruction of the desired permittivity,particularly by the method realized in the EIT2D program suite of the EIDORS project[14]. Graphics (b) in figures 2–4 present the recalculation of these noisy data ontothe regular mesh {xi, yi} with n¼ 21, 51, 101. Graphs (c) in all these figures explainthe result of post-processing by the MATLAB filtering program medfilt2. Graphs(d) present post-processing by our proposed algorithm. Comparing graphs (c) and(d) in figures 2–4 we see the advantages of the algorithm developed.
In figure 5 we have an example of the EIT2D application: the upper graph showsthe distribution of the exact values ", whereas the lower graph shows " recuperatedby EIT2D algorithms. To compare the results of EIT2D reconstruction with thepost-processed ones using the proposed algorithm, we chose a finer mesh withn¼ 31, c¼ 1.2, and calculated for the considered example the estimation �¼ 0.54using images of the exact and reconstructed distribution of 1/" .
In figure 6 we present: (a) 1/" recuperated by the EIT2D algorithms, (b) recuperationby formula (2) and smoothing by (4)–(7) without projection on �"", (c) projection onlyto the set �"" without smoothing, and finally (d) 1/" recuperated by the developedalgorithm (number of smoothing K ¼ 5).
4. Image pre-reconstruction and data post-processing in ECT experiments
In figures 7–9 we present photographs of capacitance measurement experimentsrealized at the Instituto Mexicano del Petroleo (IMP) and the results afterpost-processing the LBP-reconstructed images using the proposed algorithm.
Fast post-processing algorithm 69
1
1.5
2
2.5
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1(a)
1
1.5
2
2.5
3
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1(b)
0.4
0.6
0.8
1
1.2
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1(c)
1
1.5
2
2.5
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1(d)
Figure 2. Post-processing the model tomography image: n ¼ 21, "2 ¼ 3.
1
1.5
2
2.5
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1(a)
1
1.5
2
2.5
3
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1(b)
1
1.5
2
2.5
3
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1(c)
1
1.5
2
2.5
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1(d)
Figure 3. Post-processing the model tomography image: n ¼ 51, "2 ¼ 3.
70 A. Grebennikov and C. Gamio
1
1.5
2
2.5
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1(a)
1
1.5
2
2.5
3
1 0.5 0 0.5 1−1
−0.5
0
0.5
1(b)
1
1.5
2
2.5
3
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1(c)
1
1.5
2
2.5
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1(d)
Figure 4. Post-processing the model tomography image: n ¼ 101, "2 ¼ 3.
Figure 5. Reconstruction of the permittivity distribution by the EIT2D program.
Fast post-processing algorithm 71
Figure 7. The experiment.
Figure 6. Post-processing EIT2D image: n ¼ 31, "2 ¼ 3.
72 A. Grebennikov and C. Gamio
Using an actual capacitance tomography system, various objects were placed insidethe sensor and real capacitance measurements were taken. From those capacitancemeasurements, ‘pre-reconstructed’ images were generated using the LBP algorithm.The proposed post-processing methods were then applied on the images in order toimprove them.
Figure 8. Pre-reconstructed by LBP image of the experiment.
Figure 9. Post-processed image of the experiment.
Fast post-processing algorithm 73
5. Conclusions
A fast algorithm for improving electrical capacitance tomography images is proposedfor the case of two-phase flow regimes with known values of the permittivity of themixture components. Its regularization properties are demonstrated in numericalexperiments for simulated model examples.
Acknowledgements
The authors wish to thank the Instituto Mexicano del Petroleo for the support receivedfor this research.
References
[1] Beck, M.S. and Brown, B.H., 1996, Process tomography: a European innovation and its application.Measurement Science and Technology, 7, 215–224.
[2] Williams, R.A. and Beck, M.S., 1995, Process Tomography: Principles, Techniques and Applications(Oxford: Butterworth-Heinemann).
[3] Xie, C.G., Plaskovski, A. and Beck, M.S., 1989, 8-electrode capacitance system for two-component flowidentification. Part 1: Tomographic flow imaging. IEE Proceedings A, 136(4), 173–183.
[4] Xie, C.G., Huang, S.M., Holey, B.S., Thorn, R., Lenn, C., Snowden, D. and Beck, M.S., 1992, Electricalcapacitance tomography for flow imaging: system model for development of image reconstructionalgorithms and design of primary sensors. IEE Proceedings G, 139(1), 89–98.
[5] Yang, W.Q., Beck, M.S. and Byars, M., 1995, Electrical capacitance tomography: from design toapplications. Measurement and Control, 28, 261.
[6] Byars, M., 2001, Developments in electrical capacitance tomography. In: Proceedings of the 2ndWorld Congress on Industrial Process Tomography, Hannover, Germany, 29–31 August, pp. 542–549.
[7] Yang, W.Q. and Peng, L., 2003, Image reconstruction algorithms for electrical capacitance tomography.Measurement Science and Technology, 14, R1–R13.
[8] Grebennikov, A.I., 1988, Spline approximation method for solving some incorrectly posed problems.Doclady Akademiya Nauk SSSR, 298(3), 533–537.
[9] Grebennikov, A.I., 2002, Regularization of applied inverse problems by full spline-approximationmethod. WSEAS Transaction on Systems, 1(2), 124–129.
[10] Grebennikov, A.I., 2003, Regularization algorithms for electric tomography images reconstruction.WSEAS Transactions on Systems, 2(2), 487–493.
[11] Morozov, V.A. and Grebennikov, A.I., 1992, Methods of Solving Ill-posed Problems: Algorithmic Aspect(Moscow: Moscow State Univ. Pub. House).
[12] Gilyazov, S.F. and Goldman, N.L., 2000, Regularization of Ill-posed Problems by Iteration Methods(Netherlands: Kluwer Academic Publishers).
[13] Fraguela, A., Gamio, C. and Hinestroza, D., 2002, The inverse problem of electrical capacitancetomography and its application to gas-oil 2-phase flow imaging. WSEAS Transactions on Systems,1(2), 130–137.
[14] Vauhkonen, M., Lionhert, W.R., Heikkienen, L.M., Vauhkonen, P.J. and Kaipio, J.P., 2001,A MATLAB package for the EIDORS project to reconstruct two-dimensional EIT images.Physiological Measurement, 22, 107–111.
[15] [www.mathworks.com]
74 A. Grebennikov and C. Gamio
