A reconstruction algorithm for electrical capacitance tomography via total variation and l 0 -norm regularizations using experimental data

Transcription

1 A reconstructon alorthm for electrcal capactance tomoraphy va total varaton and l 0 -norm reularzatons usn expermental data Jaoxuan Chen 1,2, Maomao Zhan 1 and Y L 1,3 1 Graduate School at Shenzhen, Tsnhua Unversty, Shenzhen , Chna 2 Department of Automaton, Tsnhua Unversty, Bejn , Chna 3 Author to whom any correspondence should be addressed. E-mal: ly@sz.tsnhua.edu.cn Abstract Electrcal capactance tomoraphy (ECT) has been nvestated n many felds due to ts advantaes of ben non-nvasve and low cost. Sparse alorthms wth l 1 -norm reularzaton are used to reduce the smoothn effect and obtan sharp maes, such as total varaton (TV) reularzaton. Ths paper proposed for the frst tme to solve the ECT nverse problem usn an l 0 -norm reularzaton alorthm, namely the doubly extrapolated proxmal teratve hard thresholdn (DEPIHT) alorthm. The accelerated alternatn drecton method of multplers (AADMM) alorthm, based on the TV reularzaton, has been selected to acqure the frst pont for the DEPIHT alorthm. Expermental tests were carred out to valdate the feasblty of the AADMM-DEPIHT alorthm, whch s compared wth the Landweber teraton (LI) and AADMM alorthms. The results show the AADMM-DEPIHT alorthm has an mprovement on the qualty of maes and also ndcates that the DEPIHT alorthm can be a sutable canddate for ECT n post-process. Keywords: electrcal capactance tomoraphy, l 0 -norm reularzaton, total varaton

2 1. Introducton Multphase flow man occurs n a varety of ndustral processes and plants ncludn petroleum, chemcal and power ndustres. Electrcal tomoraphy (ET), such as electrcal capactance tomoraphy (ECT) and electrcal resstance tomoraphy (ERT), s consdered a hhly promsn technque. Thus, ET has wtnessed wdespread applcaton n the past [1][2][3]. Besdes ben non-radoactve and non-nvasve, ECT provdes the advantaes of ben low cost and hh process speed. Currently, ECT s a powerful process-man technque to reconstruct the permttvty dstrbutons based on the measured capactances between each par of electrodes n an ECT sensor. However, ECT has the major drawback of offern low resoluton maes due to the nherence of ll-posedness, ll-condtonn and non-lnearty. Many alorthms have been proposed to solve ECT nverse problem [4] and the most wdely-used one-step alorthm and teratve alorthm s lnear back projecton (LBP) [5] and the Landweber teraton (LI) [6], respectvely. The nverse problem n ECT s severely ll-posed, therefore the reularzaton s needed. Tkhonov reularzaton s a typcal method to solve the ECT nverse problem based on the l 2 -norm reularzaton [7]. However, ths method leads to the reconstructed maes smoothed excessvely. Recently, a sparse reconstructon wth l 1 -norm reularzaton s used to reduce the smoothn effect and to obtan sharp maes, such as total varaton (TV) reularzaton. In the past few years, the TV method for ECT man has receved consderable attenton: Soleman and Lonheart [8] explored a reularzed Gauss-Newton scheme and found that the TV reularzaton showed dstnctve advantae n obtann sharp maes; Hosan et al [9] presented dfferent alorthms to reconstruct the hh contrast objects and found that the TV method showed better results compared wth the Tkhonov reularzaton method; Ye et al [10] desned an unconventonal bass for ECT, whch s based on an extended senstvty matrx; Chen et al [11] ntroduced two numercal methods to solve the man problems n ECT based on Rudn Osher Fatem (ROF) model wth TV reularzaton. Chen et al proposed an teratve alorthm for ECT based on TV reularzaton, namely accelerated alternatn drecton method of multplers (AADMM) [11]. They concluded that the AADMM alorthm

3 could dentfy the object from ts backround effcently and make the boundary of the object clear n several cases. However, they also ponted out that some artfacts n the maes reconstructed by the AADMM could not be removed. The l 1 -norm based approaches are capable of obtann a sparse soluton by usn a soft thresholdn operator. On the other hand, these approaches yeld loss of contrast and eroded snal peaks [12]. The l 0 -norm reularzaton has ts advantaes over l 1 -norm reularzaton n many applcatons [13][14][15].However, the feasblty of l 0 -norm based approach for mprovn mae qualty has not been assessed for ECT. Althouh the AADMM alorthm could dstnush the ede of the object effectvely, the reconstructed permttvty over the reon of the object s not homoenous. The exstn post-process method to deal wth t s bnarzaton of the maes wth settn thresholds. Ths method s rouh and sometmes may make a damae to the ornal maes. In addton, the detaled values of those area sometmes cannot be attaned,.e. the thresholds cannot be determned. The man motvaton of ths paper s to mprove the qualty of the maes reconstructed by the AADMM alorthm. In ths paper, a combned alorthm for ECT va total varaton and l 0 -norm reularzatons s proposed. The alorthm conssts of two steps: the frst step s to use the AADMM alorthm to obtan the ntal soluton; the second s to use the DEPIHT alorthm to reduce the artfacts n the maes and then enhance the ntensty over the blurred area. Ths paper s oranzed as follows: n secton 2, nspred by the prevous research [11][16], a combned alorthm for ECT s ntroduced; Secton 3 descrbes the expermental setup. Results and dscusson of expermental data are provded n secton 4 to valdate the feasblty of the proposed alorthm, and secton 5 concludes the paper. 2. Prnclple of alorthm Bao et al presented an l 0 -norm based alorthm, namely extrapolated proxmal teratve hard thresholdn (EPIHT) [16]. Inspred by ths work, we propose the doubly extrapolated proxmal teratve hard thresholdn (DEPIHT) for ECT. Snce the DEPIHT for solvn l 0 -norm reularzaton problem can merely uarantee

4 local converence, the ntal pont for DEPIHT s needed. The TV reularzaton s able to an a ood shape recovery n ECT reconstructon. Thus, the AADMM alorthm s used to acqure the ntal pont for DEPIHT. The process of the AADMM-DEPIHT alorthm s shown n fure 1. TV (AADMM) EPIHT-I DEPIHT EPIHT-II Obtan ntal results Elmnate artfacts Enhance mae Fure 1. Process daram of the AADMM-DEPIHT alorthm In ECT, the mathematcal model between the capactance and permttvty dstrbutons can be represented as [17][18][19] =S (1) where λ s a normalzed capactance, S s a normalzed matrx known as the senstvty map, and s the normalzed permttvty. The eneral equaton of the AADMM alorthm can be transformed from the equaton (1) usn an optmzaton perspectve. 2 2 mn S (2)

5 where the frst term s the fdelty term wth parameter μ, the thrd term s a TV term, ε s a smoothn parameter, s a radent operator. A full descrpton of the AADMM alorthm has been publshed n [11], therefore only a bref summary of ths alorthm s ven here. The DEPIHT alorthm conssts of two steps: the frst step s the EPIHT-I alorthm, the second s the EPIHT- II alorthm. Frstly, an optmzaton case for ECT based on l 0 -norm reularzaton s ven as below, mn S r (3) where r s a non-neatve sparsty-promotn weht parameter. Then, defne two functons H() and G(), H( ) G( ) r G ( ) S (4) And the surroate functon R q (x,y) of H() s set up as q 2 Rq ( xy, ) r Gy ( ) Gy ( ), x y x y (5) where q s a non-neatve parameter. The AADMM-DEPIHT alorthm s shown n alorthm1 explctly.

6 Alorthm1. AADMM-DEPIHT AADMM step: 1. Obtan an ntal soluton by usn the AADMM alorthm, e.. tv EPIHT-I step: 2. Inputs: senstvty matrx S, capactance measurements λ, a parameter used n the extrapolaton step w, the number of teratons k max and two parameters r, q. 3. Intalze -1 = 0 = tv and k=0. 4. Whle k < k max end whle y w( ) k1 k k k1 H( y ) H( ) f k 1 y end f k1 k k1 q k1 k ar mn R (, y ) (6) k k 1 5. Let tv k max EPIHT-II step: 6. repeat the steps from 2 to 4 except for the equaton (6): ar mn R' (, y ) (7) k1 q k1 (the relatonshp between R q and R ' q wll be concerned n the follown.) 7. Output: k max.

7 The equaton (6) s ven by 1 k y G( y ) 1 2r k1 k1 q q (8) where ψ a ( ) denotes the hard thresholdn operator, whch s defned as below. () b a [ b], [ b] [ a] 0, else (9) where [ ] denotes the th component of a vector. The equaton (7) s ven by 1 k y G( y ) 1 2r k1 k1 q q (10) where υ a ( ) s analo wth the hard thresholdn operator, whch s expressed as () b [ ] f [ ] [ ] a b b a (11) In fact, the relatonshp between R q and R ' q has lttle dfference except for the meanns of the thresholdn operator. However, ths leads to a snfcantly dfferent effect on the ECT reconstructon. 3. Expermental setup Fure 2 llustrates a typcal ECT system, whch comprses manly of three subsystems: a typcal ECT sensor wth eht electrodes, a data acquston devce and a computer. In the test, the dameter of the ECT sensor was 76 mm and the anular span of each electrode was 30. The data acquston speed of the data acquston system was about 350 Hz,.e. the data acquston system can acqure about 350 sets of capactance data of

8 28 electrodes pars per second. The snal-to-nose rato (SNR) of capactance data for each of 28 electrodes pars raned from 30 db to 40 db. In order to avod the system errors and noses, the averae of thousands of frames was employed as the capactance data. The man was completed usn MATLAB R2015b on the computer wth an Intel Core GHz CPU and 4 GB of RAM. Four dstrbutons are set for the test: cross-shaped, V -shaped, two rectanular-shaped and three crcularshaped, as shown n Fure 3. Ar and dry sand were used as the low and hh permttvty materals (relatve permttvty 1 and 4 respectvely) to calbrate the system. Paper-made contaners n dfferent shapes are flled wth dry sand, whch represent for each tested dstrbuton. In addton, a mesh wth 2304 (48 48) square elements s used to enerate the senstvty matrx S wth an ar n the measurn space. Data acquston devce ECT sensor Dry sand Tested permttvty dstrbutons Fure 2. ECT system and materals used n the expermental test.

9 (a) (b) (c) (d) Fure 3. Real permttvty dstrbutons used n the expermental test : (a) cross-shaped, (b) V -shaped, (c) two rectanular-shaped and (d) three crcular-shaped Parameter selecton s essental to the qualty of reconstructed maes. Some ntal parameters of the three alorthms are stated here. The relaxaton factor and the number of teraton for LI are chosen as and 3000 respectvely. Ths can ensure the LI alorthm convere at a ood pont. The parameter selecton for the AADMM alorthm can be suested n [11] and wll not be dscussed n ths paper. In the AADMM-DEPIHT alorthm, the parameter selecton for the EPIHT-I and EPIHT-II alorthms s dfferent. In the EPIHT-I alorthm, the parameter r s set to 0.01 whle the parameter w s chosen to be In the EPIHT-II alorthm, the parameter r s set to 1 whle the parameter w s chosen to be 1. Moreover, the parameter q used n the EPIHT-I and EPIHT-II alorthms vares for the dfferent dstrbutons, whch can be rearded as a controlln parameter. 4. Results and dscusson Fure 4 provdes a 2D maes reconstructed from the expermental data. It can be found that the AADMM alorthm can dentfy the objects from the backround effcently, especally n the test 2. In the test 4, althouh the LI and AADMM alorthms can both dstnush the three crcular objects from the backround, there are some dstnct artfacts n the result of the LI alorthm. The exstn bnary process for ECT s to use a threshold method, e.. threshold operator (TO), whch can be expressed as below.

10 0 TO( thr) 1 thr thr (12) where thr denotes the value of the threshold. test True dstrbutons LI LI-TO AADMM TO AADMM- AADMM- DEPIHT (a) (b) (c) (d) (e) (f) Fure 4. 2D maes reconstructed from the expermental data : (a) the true dstrbutons, (b) the maes reconstructed from the LI alorthm, (c) the maes reconstructed from the LI alorthm wth a threshold, (d) the maes reconstructed from the AADMM alorthm, (e) the maes reconstructed from the AADMM alorthm wth a threshold and (f) the maes reconstructed from the AADMM-DEPIHT alorthm.

11 As s shown n fure 4, the cases (c) and (e) show the results of the LI and AADMM alorthms wth the operator of TO, respectvely. The parameter thr used n the TO s chosen to be 0.1 for all dstrbutons. It s worth notn that the ornal maes are normalzed between 0 and 1. Hence, the only pror knowlede s the rane of the reconstructed permttvty whle usn the threshold method. Fure 4 provdes that althouh the threshold method can make the objects n the maes more clear, ths method sometmes makes a damae to the ornal objects, shown the threshold method s rouh. Naturally, the selecton for the value of the threshold has a deep nfluence on the fnal results, whch shows ths method s unpredcted. Reardn to the results of the AADMM-DEPIHT alorthm, the maes are very vvd. To quanttatvely evaluate the fluctuaton of each man result, the standard devaton (SD) n equaton (13) s calculated. From the fure 5, t can be found that the SD of the AADMM-DEPIHT alorthm s far less than that of the LI and AADMM alorthms, ndcatn the superorty of ths alorthm. 1 M 0 1 M M 1 M 1 ( ) 0 2 (13) where σ s the standard devaton, M s the total number of pxels n an mae, s the reconstructed permttvty value and 0 s the mean value of reconstructed mae.

12 Fure 5. The standard devaton of the results reconstructed by the LI, AADMM and AADMM-DEPIHT alorthms. Fure 6 provdes a 3D maes correspondn to the fure 4. From the fure 6, t can be found that a man advantae of the DEPIHT alorthm s to enhance the maes despte the artfacts. Ths alorthm ntends to lft the non-zero permttvty to the smlar heht,.e., t turns the hlls nto pllars. Thus, t s of mportance to reduce the artfacts n the maes durn the process of the DEPIHT alorthm. The EPIHT- I alorthm s used to reduce the artfacts. It seems ths step plays a substantal role n the test 1 and 4 whle has a lttle effect n the test 2 and 3. However, compared to the AADMM alorthm, the AADMM-DEPIHT alorthm has an mprovement on the qualty of maes to some deree. Perhaps, removn the artfacts n the maes could be nvestated n the future, enabln better results n the AADMM-DEPIHT alorthm. In fact, the DEPIHT can be rearded as a post-process method for ECT. For nstance, the LI alorthm s used to acqure the frst pont and then the method becomes the LI-DEPIHT alorthm.

13 test True dstrbutons LI LI-TO AADMM TO AADMM- AADMM- DEPIHT (a) (b) (c) (d) (e) (f) Fure 6. 3D maes reconstructed from the expermental data : (a) the true dstrbutons, (b) the maes reconstructed from the LI alorthm, (c) the maes reconstructed from the LI alorthm wth a threshold, (d) the maes reconstructed from the AADMM alorthm, (e) the maes reconstructed from the AADMM alorthm wth a threshold and (f) the maes reconstructed from the AADMM-DEPIHT alorthm. Table 1: Elapsed tme (n seconds) Dstrbutons LI AADMM AADMM-DEPIHT Cross-shaped V -shaped Two rectanular-shaped Three crcular-shaped

14 Table 1 shows the elapsed tme of the three alorthms. Snce the mesh used n ths study s 2304 (48 48) square elements, the man tme of the LI and AADMM alorthms decreases a lot compared n [11]. As shown n table 1, the elapsed tme of the DEPIHT alorthm s very short, whch s benefted from the hard thresholdn operator. Ths makes the DEPIHT alorthm a sutable canddate for ECT n post-process. 5. Concluson A combned alorthm for ECT va total varaton and l 0 -norm reularzatons, namely the AADMM- DEPIHT alorthm, s presented and valdated usn expermental data. The results show the AADMM- DEPIHT alorthm does have an mprovement on the qualty of maes compared to the AADMM alorthm, e.. remove some artfacts n several cases. Furthermore, the DEPIHT alorthm, based on l 0 -norm reularzaton, has a very sutable applcaton n bnarzaton, e.. a post-process for ECT. However, the DEPIHT alorthm cannot dentfy the object from backround whle enhancn the mae. Thus, the step of reducn major artfacts s necessary. It s antcpated that l 0 -norm reularzaton methods such as the DEPIHT alorthm, can be combned wth other lobal converence alorthms, enabln better reconstructon n ECT. Acknowledements The authors would lke to thank the Natonal Natural Scence Foundaton of Chna (No ) for supportn ths work. References [1] E. A. Hosan, M. M. Zhan and M. Soleman, "A Lmted Reon Electrcal Capactance Tomoraphy for Detecton of Deposts n Ppelnes," IEEE Sensors J., vol. 15, no. 11, pp , Nov [2] C. Tan, W. Da, H. Wu and F. Don, "A Conductance Rn Coupled Cone Meter for Ol-Water Two- Phase Flow Measurement," IEEE Sensors J., vol. 14, no. 4, pp , Apr [3] S. Thele, M. J. D. Slva and U. Hampel, "Capactance Planar Array Sensor for Fast Multphase Flow Iman," IEEE Sensors J., vol. 9, no. 5, pp , May [4] W. Q. Yan and L. Pen, Imae reconstructon alorthms for electrcal capactance tomoraphy, Meas. Sc. Technol., vol. 14, no. 1, pp. R1 R13, 2003.

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