Application of Gradient Projection for Sparse Reconstruction to Compressed Sensing for Image Reconstruction of Electrical Capacitance Tomography

Transcription

1 Journal of Electrical and Electronic Engineering 08; 6(): doi: 0.648/j.jeee ISS: (Print); ISS: (Online) Application of Gradient Projection for Spare Recontruction to Compreed Sening for Image Recontruction of Electrical Capacitance omography Lifeng Zhang *, Yajie Song Department of Automation, orth China Electric Power Univerity, Baoding, China addre: * Correponding author o cite thi article: Lifeng Zhang, Yajie Song. Application of Gradient Projection for Spare Recontruction to Compreed Sening for Image Recontruction of Electrical Capacitance omography. Journal of Electrical and Electronic Engineering. Vol. 6, o., 08, pp doi: 0.648/j.jeee Received: April, 08; Accepted: June, 08; Publihed: June 0, 08 Abtract: he compreed ening algorithm baed on gradient projection for pare recontruction (CS-GPSR) i applied to electrical capacitance tomography (EC) image recontruction in thi paper. Firt, uing the orthogonal bai of FF tranformation, the grey ignal of original image can be pare. Secondly, the obervation matrix of EC ytem wa deigned by rearranging the exciting-meauring order, and the capacitance meaurement and correponding enitivity matrix can be obtained. Finally, the recontructed image can be obtained uing CS-GPSR algorithm. Simulation experiment were carried out and the reult howed that the recontructed image with higher quality can be obtained uing the preented CS-GPSR algorithm, compared with conventional linear back projection (LBP) and Landweber iterative algorithm. Keyword: Electrical Capacitance omography, Image Recontruction, Compreed Sening, Gradient Projection for Spare Recontruction. Introduction Electrical capacitance tomography (EC) technique wa developed in 980, which i widely ued for on-line monitoring multi-phae flow in indutrial pipeline. EC technique baed on capacitance enitivity mechanim ha many advantage, uch a non-radiation, non-intruion, imple tructure, low cot, eay intallation and quick repone. At preent, it ha been widely applied in the field of modern indutrial production, uch a chemical, petroleum and electric power [-5]. he chematic diagram of EC ytem can be een in Figure. he proce of image recontruction of the EC ytem i a follow: the data acquiition ytem generate a inuoidal voltage ignal of a certain frequency and tranmit it to the exciting electrode of the enor array. And then, the capacitance value between different electrode pair are meaured. Finally, the cro-ectional permittivity ditribution image of media in the pipe can be obtained uing ome kind of image recontruction algorithm. Figure. Recontructed image and binary image of EC. EC image recontruction i a typical ill-poed problem and it olution i untable. A a reult, the accuracy of image recontruction algorithm will limit the application of EC in indutrial field. Image algorithm of EC can be divided into non-iterative and iterative algorithm. on-iterative algorithm i imple and fat, uch a linear back projection (LBP) algorithm, but the quality of the recontructed image i low, which cannot meet the requirement of indutrial

2 47 Lifeng Zhang and Yajie Song: Application of Gradient Projection for Spare Recontruction to Compreed Sening for Image Recontruction of Electrical Capacitance omography production. Iterative algorithm, uch a Landweber iterative algorithm, ha higher preciion for image recontruction. However, it convergence peed i low and it ha the emi-convergence characteritic. Recently, a variety of algorithm have been tudied to improve the viualization quality, which may be ummarized a follow: () Reearch on the recontruction model of EC. he recontruction model of EC i naturally nonlinear. Some reearcher would like to find ome kind of nonlinear recontruction model or modify the enitivity matrix to obtain better recontructed image. () Making full ue of priori information. EC image recontruction problem i often converted to an imization problem, in which the priori information can be integrated into the regularization term, and the quality of recontructed image can be greatly improved. However, the definition of lo function i crucial for thi kind of algorithm, which will directly determine the recontruction reult. (3) Spare repreentation baed algorithm. In recent year, parity regularization ha been widely applied to image proceing and image recontruction a the parity i one of ditinct characteritic of natural image. In 009, Wright et al. repreented the tet ample baed on the overcomplete dictionary with the training ample a the bae element [6]. Ye et al. preented a bai baed on which the permittivity ditribution to be recontructed for EC i naturally pare in 05 [7]. With recent development of compreed ening (CS), everal image recontruction algorithm for EC baed on CS were propoed, in which the parity proce i indipenable and the meaurement matrix needed to be deigned to meet the condition of CS [8-0]. hee will be complicated and time-conuming, and the recontructed image will be better when the real ditribution tend to be parer. In 004, Donoho and Candè propoed compreed ening (CS) theory []. It break through the limitation of the traditional ampling theorem on the ignal bandwidth and make the original ignal can be preciely recontructed with a mall amount of ampling data, which bring a revolutionary breakthrough for the ignal acquiition technology. he theory tate that if the ignal i pare or compreible in a certain tranform domain, the random obervation matrix can be ued to obtain the low dimenional meaurement ignal. By olving an imization problem, the original ignal can be accurately recontructed from the low dimenional meaurement ignal. At preent, the compreed ening theory ha been widely ued in radar, communication, compreed imaging, medical imaging and other field. In thi paper, the compreed ening algorithm baed on gradient projection for pare recontruction (CS-GPSR) wa applied to EC image recontruction. Firtly, the grey ignal of original image were tranformed into pare ignal by uing the Fat Fourier ranformation bai. Secondly, the obervation matrix wa deigned by rearranging the row of the enitivity matrix of EC in a random order and the meaurement projection vector wa deigned by rearranging the row of the capacitance value vector in the ame order. hirdly, the mathematical model of EC ytem wa etablihed. Finally, recontructed image were carried out uing LBP, Landweber iterative algorithm and CS-GPSR algorithm, repectively. And alo the image binarization of recontruction image were done baed on the imal threhold method. Experimental reult how that the recontructed image with higher accuracy can be obtained uing CS-GPSR algorithm.. Principle of Compreed Sening he premie condition of the compreed ening theory i that the ignal mut be pare or compreible. However, the ignal in reality are often not pare. o olve thi problem, an orthogonal tranformation i uually applied to parely repreent the ignal and make it atify the requirement of CS theory. For a dicrete ignal x of length, it can be linearly expreed by a et of dimenional bai vector x = = i ψ i i= Where Ψ i an orthogonal matrix of, Ψ = [ ψ ψ ψ ]. i the coefficient vector of. If only ha K ( K << ) non-zero coefficient, x i pare under the bai Ψ and it pare degree i K. Several orthogonal bae can be ued to parely repreent the ignal x, uch a Fat Fourier tranform bai (FF), dicrete coine tranform bai (DC), dicrete ine tranform bai (DS) and dicrete wavelet tranform bai (DW). he core of the compreed ening theory i linear meaurement. hrough deigning appropriate linear obervation ytem, M ( M << ) meaurement value of the original ignal are obtained, from which the original ignal x of can be recontructed. Ψ c () y = Φx = ΦΨ = Α () Where y i the meaurement value vector of M. Φ i the obervation matrix of M. CS A ( A CS = ΦΨ ) i the compreed ening matrix of M. Recontructing x from the given y according to equation () i a linear programming problem. Since the number of equation i le than the number of unknown ( M << ), equation () i ill-conditioned and uually ha no definite olution. However, uing the prior condition of K pare, equation () ha a chance to be olved. According to compreed ening theory, when the compreed ening matrix iometry property (RIP) criterion, K nonzero coefficient of can be accurately recontructed from M obervation value by uing a variety of CS recontruction algorithm, while CS A atifie the retricted

3 Journal of Electrical and Electronic Engineering 08; 6(): enuring the convergence of the recontruction proce. he equivalent condition of RIP i that the obervation matrix Φ i uncorrelated to pare matrix Ψ. It wa already proved that when Gauian random matrix i ued a the obervation matrix and arbitrary orthogonal bai i ued a the pare matrix, the compreed ening matrix CS A atifie the RIP condition under great probability, which mean that the ill-conditioned equation () can be olved. he mot direct approach to make ue of the pare or compreible prior condition i uing the minimum L0 norm model. x = arg min = Ψ 0.t. y = ΦΨ Becaue the L0 norm i non-convex, the numerical computation problem of equation (3) i P-hard and very untable. Up to now, variou alternative model and correponding CS recontruction algorithm have been propoed, in which the bai puruit algorithm, the matching puruit algorithm, the iterative threhold algorithm and the minimum total variation algorithm are commonly ued algorithm. 3. CS-GPSR Algorithm for EC he Fat Fourier tranform (FF) ha a better effect on the binary image ignal, which can be ued a a pare bai for EC ytem to tranform the image gray ignal into the pare ignal. (3) g = Ψ (4) FF where g i the normalized gray vector of. ΨFF i FF bai of. i the pare coefficient vector of. he compreed ening theory require that the obervation proce mut be linear. he linear model of EC ytem i a follow: λ = Sg (5) Subtituting (4) into (5), the mathematical model of EC ytem baed on compreed ening theory can be obtained a expreed in (6) EC λ = Sg = SΨ A (6) FF = where λ i the meaurement projection vector of M. S of M i the enitivity matrix of EC ytem and obervation matrix of CS. g i grey vector of. EC A i the compreed ening matrix of EC ytem. he number of the independent capacitance meaurement value which are acquired by the 6-electrode EC ytem i M = 6 5/ = 0. In order to enure the imaging accuracy of the EC ytem, the image area for the recontructed image i divided into 3 3 quare grid and the number of pixel in the imaging area of pipe i 8. According to compreed ening theory, under the condition that when the compreed ening matrix EC A atifie the RIP, the element value of the image grey vector g can be recontructed accurately from the M independent capacitance meaurement of the projection vector λ, that i to ay, the obervation matrix mut be Gauian random matrix. However, due to the limitation of the EC ytem hardware and inherent ampling method, it i very difficult to realize random obervation and acquire the correponding capacitance meaurement data. In order to improve the randomne of the obervation matrix, in thi paper, the random obervation matrix wa deigned by rearranging the row of the enitivity matrix S in a random order and the correponding random meaurement projection vector wa obtained by rearranging the row of the capacitance value vector λ in the ame order. By doing that, the random obervation matrix i uncorrelated to pare matrix, which atifie the RIP under a certain probability. In reference [], it wa pointed out that when the obervation matrix and pare matrix are not related, the minimum L norm problem and the minimum L0 norm problem have the ame olution. he minimum L norm problem i a convex imization problem, which can be eaily tranformed into a linear programming problem. g = arg min = Ψ FF.t. λ = S In order to olve equation (7), the contrained convex imization problem i tranformed into an uncontrained convex imization problem. Meanwhile, the L norm regularization model can be ued to overcome the ill-condition of the EC ytem. g = arg min S = Ψ FF Ψ FF λ Ψ FF + τ In thi paper, the gradient projection for pare recontruction algorithm baed on compreed ening theory (CS-GPSR) i ued to olve the model expreed in equation (8) and to realize the image recontruction of EC ytem. he GPSR algorithm i a claical algorithm for the ignal recontruction of CS [3]. By plitting the variable into it poitive and negative part (i.e. = u v, where u = () + = max(0,) 0, v = ( ) + 0 ), the equation (9) of the L norm regularization model can be tranformed into the quadratic programming problem which i expreed in equation (0). min (7) (8) λ A + τ (9)

4 49 Lifeng Zhang and Yajie Song: Application of Gradient Projection for Spare Recontruction to Compreed Sening for Image Recontruction of Electrical Capacitance omography where min u,v [,,,] λ A(u v) + τ u + τ v (0) = i the column vector coniting of one. After deduction, problem (0) can be rewritten a more tandard quadratic programming form a howed in equation (). where minc z + z.t. z 0 Bz F(z) () z u =, λ b v b = A, c = τ + and b A A A A B =. A A A A he GPSR algorithm bae on the teepet decent direction g and ue Armijo rule performing a backtracking line earch to update the iteration tep α 0. ( F(z )) k, z > 0 or ( F(z )) k < 0 Where gk =. 0, otherwie Along the negative gradient direction g, the teepet decent tep α 0 for each update i atified, which can be computed explicitly a (g ) g α 0 = () (g ) Bg he iterative procedure of GPSR algorithm i a follow: (0) Initialization: elect initial value of z, and chooe parameter β (0,) and µ (0,/ ), and et i = 0. Step : compute α 0 according the equation (). Step : elect α from equence α 0, βα 0, β 0,... in turn, until the following condition i met ( i+ ) (i+ ) F(z ) F(z ) µ F(z ) (z z ), where z (i+ ) = (z α F(z )) + Step 3: Check whether the termination condition i atified. If the termination condition expreed in equation (3) i atified, then tep. (i ) z + F(z α will be output. Otherwie, return back to (i+ ) F(z ) F(z ) ) tolp (3) he grey value of the recontructed image are not binary. In order to compare the quality of recontructed image obtained by LBP, Landweber and CS-GPSR algorithm, the imal threhold method wa ued to binarize the initial recontructed image. he imal threhold i determined by conidering all poible threhold and chooing the threhold which ha the minimum error from the meaured value, which can be expreed a follow: th min Sg λ (4) th where g i the gray value vector of the image obtained by the imal threhold method. 4. Simulation Experiment In thi paper, four kind of typical flow pattern were et to imulate the oil-ga two-phae flow. he relative permittivity of oil and ga were et to be 3 and, repectively. COMSOL 3.5a (finite element analyi oftware) wa ued to create EC enor imulation model and calculate the imulated capacitance meaurement. MALAB oftware wa ued to develop the EC image recontruction algorithm and image proceing. he imaging area of pipe i divided into 8 pixel by quare grid. In the imulation proce, the initial image were recontructed by LBP algorithm, Landweber iterative algorithm and the CS-GPSR algorithm, which can be een in Figure (a). After that, the imal threhold method wa ued to obtain binary image, which can be hown in Figure (b). From Figure (a), it can be een that the quality of the recontructed image by the CS-GPSR algorithm i obviouly improved compared with that of LBP algorithm and Landweber iterative algorithm. he CS-GPSR algorithm can clearly recontruct the location information and edge information of the object, that i to ay, the object recontructed by the CS-GPSR algorithm ha an accurate ditribution, a regular edge and a high hape fidelity. On the contrary, the object recontructed by LBP algorithm and Landweber iterative algorithm ha many obviou artifact, which make the edge of object inditinguihable. From the recontructed image of the flow pattern 3 and 4, it can be een that for relatively complex flow pattern, the image recontructed by LBP algorithm i completely ditorted and the image accuracy recontructed by Landweber iterative algorithm i alo low, however, the recontructed image with good accuracy and hape fidelity can alo be obtained uing CS-GPSR algorithm. It can alo be een from Figure (b), after imal threhold proceing, the recontructed image by CS-GPSR algorithm are more cloe to the original ditribution and have no artifact. he hape fidelity of CS-GPSR i bet for the tudied three algorithm.

5 Journal of Electrical and Electronic Engineering 08; 6(): (a) Recontructed image (b) Binary image Figure. Recontructed image and binary image of EC. In order to quantitatively evaluate the performance of thi algorithm, the relative error (RE) and the correlation coefficient (CC) were ued a the evaluating index for the recontruction algorithm, which are defined a follow [4]:

6 5 Lifeng Zhang and Yajie Song: Application of Gradient Projection for Spare Recontruction to Compreed Sening for Image Recontruction of Electrical Capacitance omography CC = RE g g = (5) i= g ( g g)( g g) i M ( g g) ( g g) i= i i= i i (6) where g and g are the true permittivity ditribution of the tet object and the recontructed permittivity ditribution, repectively. g and g are the mean value of g and g, repectively. i the number of the pixel in the imaging area and equal 8 in our imulation. he relative error (RE) and the correlation coefficient (CC) for the recontructed image in Figure (a) and (b) were calculated and were lited in able to 4. able. Calculated RE of recontructed image in Figure (a). Flow pattern LBP Landweber CS-GPSR Pattern Pattern Pattern Pattern able. Calculated RE of recontructed image in Figure (b). Flow pattern Pot-proceing Pot-proceing Pot-proceing LBP Landweber CS-GPSR Pattern Pattern Pattern Pattern able 3. Calculated CC of recontructed image in Figure (a). Flow pattern LBP Landweber CS-GPSR Pattern Pattern Pattern Pattern able 4. Calculated CC of recontructed image in Figure (b). Flow pattern Pot-proceing Pot-proceing Pot-proceing LBP Landweber CS-GPSR Pattern Pattern Pattern Pattern For the evaluating index RE and CC, it can be een from their definition in equation (5) and (6) that maller RE and larger CC imply the recontructed image with higher quality for EC. hrough analyzing the evaluating index RE and CC in able to 4, it can be een that the relative error (RE) of the recontructed image uing CS-GPSR algorithm i the mallet compared that of LBP and Landweber iterative algorithm. Meanwhile, the correlation coefficient (CC) of the recontructed image uing GPSR algorithm i the larget compared with that of LBP and Landweber iterative algorithm, which how that the recontructed image baed on CS-GPSR are more cloe to the real ditribution. After the imal threhold method proceing, the accuracy of the three algorithm are improved obviouly, but the recontruction effect of the CS-GPSR algorithm i better than the other two algorithm. herefore, the algorithm dicued in thi paper can be applied to the EC image recontruction, by uing which the recontructed image with higher accuracy can be obtained. 5. Concluion In thi paper, the compreed ening algorithm baed on gradient projection for pare recontruction (CS-GPSR) wa applied to EC image recontruction. Becaue the CS theory can accurately recontruct the original ignal with le obervation data, it i effective to olve the ill-conditioned problem of EC image recontruction. Meanwhile, the CS-GPSR algorithm can effectively recontruct the location information and edge information of the object in the field. Simulation reult how that the propoed algorithm can recontruct a high preciion image with le obervation data, which provide a way and method for EC image recontruction. Acknowledgement he author thank the ational atural Science Foundation of China (o ) and the Fundamental Reearch Fund for the Central Univeritie (o. 07MS3) for upporting thi reearch. Dr. Lifeng Zhang would alo like to thank the China Scholarhip Council for upporting hi viit to the Univerity of Mancheter. Reference [] H. S. app, A. J. Peyton, E. K. Kemley and R. H. Wilon, Chemical engineering application of electrical proce tomography, Senor Actuat. B-Chem., 003, vol. 9 pp [] I. Imail, J. C. Gamio, S. F. A. Bukhari, and W. Q. Yang, omography for multi-phae flow meaurement in the oil indutry, Flow Mea. Intrum., 005, vol. 6 pp [3] H. X. Wang and L. F. Zhang, Identification of two-phae flow regime baed on upport vector machine and electrical capacitance tomography, Mea. Sci. echnol., 009, vol. 0 pp [4] G. McKenzie and P. Record, Prognotic monitoring of aircraft wiring uing electrical capacitive tomography, Rev. Sci. Intrum., 0, vol. 8 pp [5] W. Q. Yang and S. Liu, Role of tomography in ga/olid flow meaurement, Flow Mea. Intrum., 000, vol. pp [6] J. Wright, A. Y. Yang, A. Ganeh, S. S. Satry, and Y. Ma, Robut face recognition via pare repreentation, IEEE ran. Pattern Anal. Mach. Intell., 009, vol. 3 pp. 0 7.

7 Journal of Electrical and Electronic Engineering 08; 6(): [7] J. M. Ye, H. G. Wang, and W. Q. Yang a, Image recontruction for electrical capacitance tomography baed on pare repreentation, IEEE. Intrum. Mea., 05, vol. 64 pp [8] H. C. Wang, I. Fedchenia, S. L. Shihkin, A. Finn, L. L. Smith, and M. Colket, Sparity-inpired image recontruction for electrical capacitance tomography, Flow Mea. Intrum., 05, vol. 43 pp [9] L. F. Zhang, Z. L. Liu, and P. ian, Image recontruction algorithm for electrical capacitance tomography baed on compreed ening, Acta Electronica Sinica, 07, vol. 45 pp for electrical capacitance tomography, Mea. Sci. echnol., 999, vol. 0 pp. L37 L39. [] D. L. Donoho, Compreed ening, IEEE. Inform. heory, 006, vol. 5 pp [] S. S. Chen, D. L. Donoho, M. A. Saunder, Atomic decompoition by bai puruit, SIAM J. Sci. Comput., 00, vol. 43 pp [3] M. A. Figueiredo, R. D owak, S. J. Wright, Gradient Projection for Spare Recontruction : Application to Compreed Sening and Other Invere Problem, IEEE J-SSP, 007, vol. pp [0] W. Q. Yang, D. M. Spink,. A. York, and H. McCann, Optimization of an iterative image recontruction algorithm