A reconstruction algorithm for electrical capacitance tomography via total variation and l0-norm regularizations using experimental data

A r econstruction algorithm for electrical capacitance tomography via total variation and l 0 -norm regularizations using experimental data Jiaoxuan Chen 1,2 , Maomao Zhang 1 and Y i Li 1,3 1 Graduate School at Shenzhen, T singhua University , Shenzhen 5180 55, C hina 2 Department of Automation, T singhua Un iversity , Beijing 100084, China 3 Author to whom any correspondenc e should be address ed. E-mail: liyi@ sz.tsinghua.edu.c n Abstract Electrical capacitance tomography ( EC T ) h a s b e e n in v e s ti g a t ed i n m any fields due to its advanta ges of being non-invasive and low cost. Sparse algorithms with l 1 -norm regularization are used to reduce the smoothing effect and obtain sharp images, such as total variation (TV) regularization. T his paper proposed for the first time to solve the ECT inverse problem using an l 0 -norm regularization algorithm, namely the doubly extrapolated proximal iterative hard thres holding (DEPIHT) algo rithm. The accelerated al ternating direction method of multipliers (AADMM) algorithm , based on the TV regula rization, has been selected to acquire the first point for the DEPIHT algorithm. Experimental tests were c arried out to validate the feasibility of the AADMM-DEPIHT algorithm, which is com pared with the Landweber it eration (LI) and AADMM algorithms. The results show the AADMM-DEPIHT algorithm has an improvem ent on the qualit y of images and also indicates tha t the DEPIHT algorithm can be a suitable candidate for ECT in post-process. Keywor ds: electrical capacitanc e tomography , l 0 -norm r egular ization, total variatio n 1. Introduction Multiphase flow imaging occurs in a variety of industrial proce sses and plants including petroleum , chemical and power industries. Elect rical tomography (ET), such as elect rical capac itance tomography (ECT) and electrical r esistance tom ography (ER T), is considered a high ly prom ising tec hnique. Thus, ET has witness ed widespread a pplication in the past [1][2 ][3] . Bes ides b eing n on-radioactive and non-invasive, ECT provides the advantages of bein g low cost and h igh process speed. Curre ntly , ECT is a powerful process-imaging technique to reconstruct the permittivity di stributions based o n the m easured c apacitances between each pair of electrodes in an ECT sensor . However , ECT has the major draw back of offering low resolution images due to the in herence of ill-p osedness, ill-conditio ning and non -l inearity . Many algorithms have been proposed to solve ECT inverse problem [4] and the most widely-used one-step algorithm and iterative algorithm is linear back projection (LB P) [5] and the Landweber iteration (LI) [6] , respectively . The inverse problem in ECT is severely ill-posed, therefore the regularization is needed. T i k h o n o v r e g u l a r i z a t i o n i s a t y p i c a l m e t h o d t o s o l v e t h e E C T i n verse problem based on the l 2 -norm regularization [7] . However , this m ethod leads to the rec onstructed im ages smooth ed excessi vely . Recentl y , a sparse reconst ruction with l 1 -norm regularization is used to reduce the smoothi ng effect and to obtain sharp images, such as total variation (TV) regularization. In the pas t few years, the TV method for ECT imaging has received considerabl e attention: Soleim ani and Lionheart [8] expl ored a regul arized Gauss-Newton scheme and found that the TV regularization showed distinctive advantage in obtaining sharp images; Hosani et al [9] presented dif ferent algorithm s to reconstruct th e high contras t obj ects and found that the TV method showed better results compared w ith the Tikhonov regularization method; Y e et al [1 0] d e s i g n e d a n unconvention al basis for ECT , which is based on an extended sen sitivity matrix; Chen et al [11] i ntroduced two numerical methods to solve the imaging problem s in ECT base d on Rudin–Osher–Fatem i (ROF) model with TV regul arization. Chen et al propose d an iterati ve algorithm for ECT based o n TV regulariza t ion, namely accelerated alternating direction method o f multiplier s (AADMM) [1 1] . They concluded that the AADMM algorithm could identify the object from its background ef ficiently and m ake the boundary of the object clear in several cases. However , they also pointed o ut that some artifacts in th e images reconstruc ted by the AADMM could not be removed. The l 1 -norm based approaches are capable of obtaining a sparse soluti on by using a soft thresholding operator . On the other ha nd, these approaches yield loss of contrast and eroded signal p eaks [12] . The l 0 -norm regularization h as its advantages over l 1 -norm regularization in many applications [13][14][15] .However , the feasibility of l 0 -norm based approach for im proving image quality has not been assessed for ECT . Although the AADMM algorith m could distingu ish t he edg e of the object effectively , the reconstructed permittivity over the region of the object is not homogeno us. T he existing post-pr ocess method to deal with it is binarization of the i mages with setting thresholds. This method is rough and sometim es may make a damage to the original images. In addition, the detailed values of those area sometimes cannot be attained, i.e. the thresholds cannot be determined. The main motivation of this paper is to improve the quality of the images reconstructed by the AADMM algorithm . In this paper , a c ombined algorithm for ECT via total variation a nd l 0 -norm regula rizations is pr oposed. The algorithm consists of tw o steps: the first step is to use the AADMM algo rithm to obtain the in itial so lution; the second is to use the DEPIHT algorithm to reduce the artifacts in the im ages and then enhance the i ntensity over the blurr ed area. This paper is or ganized as follows: in section 2, inspired by t he previous research [1 1][16] , a combined algorithm for ECT is introduced; Section 3 describes the experimental set up. Resul ts and discussi on o f experimental data are provided in section 4 to validate the feasibility o f t he proposed algorithm, and section 5 concludes the paper . 2. Princilple of algorithm Bao et al presented an l 0 -norm based algorithm, namely extrapolated proximal iterative h ard thresholding (EPIHT) [16] . Inspired by this work, w e propose t he doubly extrapolated pro xim al iterative hard th resholding (DEPIHT) for ECT . Since the DEPIHT for solving l 0 -norm regularization problem can merely guarantee local conver gence, the initial point for DEPIHT is nee ded. The TV regularization is a ble to gain a good shape recovery in ECT reconstruction. Thus, the AADMM algorithm is us ed to acquire the initial point for DEPIHT . The process of the AADMM-DEPIHT algorithm is shown in figure 1 . TV (A A D MM) O b tain i n iti al re sults El i min a te ar ti fac ts En han c e im age EP IHT - I EP IH T - II DE PI H T Figure 1. Process diagram of the AADMM-DEPIHT algorithm In ECT , the mathematical model between the capacitanc e and perm ittivity distributions can be represented as [17][18] [19] = Sg  (1) where λ is a normalized capacitance, S is a normalized matrix known as the sensitivity map, and g i s t h e normalized pe rmittivity . The general equatio n of the AADMM algorithm can be transformed from the equation (1) u sing an optimization perspective. 22 22 1 min 22 g Sg g g       (2) where the first term is the fidelity term with parameter μ , the third term is a TV t erm , ε is a smoothi ng parameter ,  is a gradient operator . A full description of the AADMM algori thm has been publis hed in [11] , therefore only a brief sum mary o f this algorithm is given here. The DEPIHT algorithm consists of two steps: the first step is t he EPIHT -I algorithm, the second is the EPIHT - II algorithm. Firstly , an optimizatio n case for ECT based on l 0 -norm regulariza tion is given as below , 22 22 0 11 min 22 g Sg g r g    (3) where r is a non-nega tive sparsity-prom oting weight parameter . Then, define two func tions H ( g ) and G ( g ), 0 22 22 () () 11 () 22 Hg G g rg Gg S g g         (4) And the surrogate fu nction R q ( x , y ) of H ( g ) is set up as 2 02 (, ) ( ) ( ) , 2 q q R xy r g Gy Gy x y x y         (5) where q is a non-nega tive param eter . The AADMM-DEPIHT algorithm is sho wn in alg orithm1 explicitly . Algorithm1. AADMM-DEPIHT AADMM step: 1. Obtain an i nitial solution by using the AADMM algorithm, e.g. g tv EPIHT -I step: 2. Inp uts: sensitivity matri x S , c apacitance m easurements λ , a parameter used in th e extrapolati on step w , the number of it erations k max and two parameters r , q . 3. Initia lize g -1 = g 0 = g tv and k =0 . 4 . W h i l e k < k max 11 () kk k k yg w g g     if 1 () ( ) kk Hy H g   1 kk yg   end if 11 arg min ( , ) kq k g g Rg y   ( 6 ) 1 kk   e n d w h i l e 5 . L e t max tv k gg  EPIHT -II step: 6 . r epeat the steps fr om 2 to 4 except for the equat ion (6): 11 arg min ' ( , ) kq k g g Rg y   ( 7 ) (the relationship between q R a n d ' q R will be concerne d in the following.) 7. Output: max k g . The equation (6 ) is given by 11 2 1 1 () kk r q k yG y q g          (8) where ψ a (∙) denotes the hard thresholding operator , which is defined as below .  [ ] , [ ] [ ] 0 , el se () i a ii i bb a b        (9) where [∙] i denote s the i th component of a vector . T he equation (7) is given by 11 2 1 1 () kk r q k yG y q g          (10) where υ a (∙) is analo g with the hard t hresholding operat or , which is exp ressed as   [ ] if [ ] [ ] () ai i i i bb a b   ( 1 1 ) In fact, the relationship be tween q R and ' q R has little dif ference except for the meanings of t he thresholding o perator . However , this leads to a significa ntly differe nt ef fect on the ECT reconstructi on. 3. Experimental setup Figure 2 illustrates a t ypical EC T system, which com prises m ain ly of three subsyste ms: a ty pical ECT senso r with eight electrodes, a data acquisition device and a computer . I n t h e te s t , t h e d i am e t er o f t h e EC T s e n s o r was 76 mm and the angular span o f each electrode was 30°. The d ata acquisition speed of the dat a acquisiti on system was about 350 Hz, i.e. th e d a ta a c q u is it io n s y st e m c a n a c q u ir e a b ou t 3 50 se t s of c a p a citance data of 28 electrodes pairs per seco nd. T he signal-to-no ise ratio (SN R) of capacitance data for each of 28 electrod es pairs ranged from 30 dB to 40 dB. In order to avoid the system errors and noises, the average of thousands of frames was employed as the capacitance data. The imaging was completed using MA TLAB R2015b o n the computer with an Intel Core i 5-6400 2.7 GHz CPU and 4 GB of RAM. Four distributions are set for t he test: cross-shaped, ‘V’-shap ed, two rectangular- shaped and three circular- shaped, as shown in Figure 3. Air and dry sand were used as the low an d high permittivity m aterials (relati ve permittivity 1 and 4 respectively ) to calibrate the system . Pap er -made containers in different shapes are filled w i t h d r y s a n d , w h i c h r e p r e s e n t f o r e a c h t e s t e d d i s t r i b u t i o n . I n addition, a mesh with 2304 (48×48) square elements is used t o generate the sensitivity m atrix S with an air i n the measuring s pace. D at a acqui s i ti on de vice EC T s e ns or D ry s and Test ed permit t i vi t y di st r ibutions Figure 2. ECT system and materials used i n the experiment al tes t. (a) (b) (c) (d) Figure 3. Real perm ittivity distrib utions used in the experimen tal test : (a) cross-shap ed, ( b) ‘V’-shaped, (c) two rectangular -shaped and ( d) three circ ular -shaped Parameter selection is essential to the quality o f reconstructe d images. Some initial parameters of the three algorithms are sta ted h ere. The relaxati on factor a nd t he nu mbe r of iteration for LI are chosen as 0.008 and 3000 respectiv ely . This can ensure the LI algorithm conver ge a t a good point. The parameter selection for the AADMM algorithm can be suggested in [11] and will not be discussed in this paper . In the AADMM-DEPIHT algorithm, the parameter selectio n for the EPIHT -I an d EPIHT -II algorithms is different. In the EPIHT -I algorithm, the parameter r is set to 0 .01 while the parameter w i s chosen to be 1000. I n the EPIHT -II algorithm, the parameter r i s s e t t o 1 w h i l e t h e p a r a m e t e r w is chosen to be 1. Moreover , the parameter q u s e d i n t h e EPIHT -I and EPIHT -II algorithms varies for the dif ferent distri butions, which can be regarded as a controlli ng parameter . 4. Results and discussion Figure 4 provides a 2D images reconstructed from the experiment al data. It can be found that the AADMM algorithm can identify the objects from the background efficien tly , esp ecially in the te st 2. In the t est 4, although the LI and AADMM algorithms can both distinguish the t hree circular ob jects fro m the backgrou nd, there are some distin ct artifacts in the resul t of the LI algor ithm. The exis ting binary pr ocess for ECT is to use a thresh old m etho d, e.g. threshold operator (TO) , which can be expressed as below . 0 () 1 i i gt h r TO thr gt h r       (12) where thr denotes the value of the threshold. test Tr u e distributions LI LI-TO AADMM AADMM- TO AADMM- DEPIHT 1 2 3 4 ( a ) ( b ) ( c ) ( d ) ( e ) ( f ) Figure 4. 2D images reconstructed from the experimental data : (a) the true distributions, (b) the images reconstructed fro m the LI algorithm, (c) the im ages reconstructed from the LI algorithm with a threshold, (d) the images reconstructed from the AADMM algorithm, (e) the imag es reconstructed from the AADMM algorithm with a threshold and (f) the images rec onstructed from the AADMM-DEPIHT algorithm. A s i s s h o w n i n f i g u r e 4 , t h e c a s e s ( c ) a n d ( e ) s h o w t h e r e s u l t s o f t h e L I a n d A A D M M a l g o r i t h m s w i t h t h e operator of T O, respectively . The parameter thr used in the TO is chosen to b e 0.1 for all distributions . It i s worth noting that the origin al images are normalized between 0 and 1. Hence, the onl y prior knowledge is the ra nge of the recons tructed permittivity while usin g the thr esh old m ethod. Figure 4 provides that althou gh the threshold method can make the object s in the images more cl ear , this m ethod sometim es makes a damage to the original objects, showing the threshold method is rough. Naturally , the selection for the value of the threshold has a de ep influence on the final results, whi ch show s this method is unpredicted. Regarding to t he results o f the AADMM-DEPIHT a lgorithm , the images are very vivid. T o quantitativel y evaluate the fluctuat ion of each imaging result, the standard d eviation (SD) in equation ( 13) is calculated. From the figure 5, it can be found that the SD of the AADMM-DEP IHT algorithm is far less than that of the LI and AADMM algorithm s, indicating the superiority of this alg orithm. 2 0 1 0 1 1 () 1 M i i M i i gg M gg M        (13) where σ is the standard deviation, M is the total number of pixels in an image, g i is the reconstructed permittivity value and g 0 is the mean value of rec onstructed image. Figure 5. The standard de viation of the res ults reconstructed b y the LI, AADMM and AADMM-DEPIHT algorithms. Figure 6 provides a 3D images corresponding to the figure 4. Fr om the figure 6, it can be found that a main advantage of the DEPIHT algorithm is to enhance the images despite the artifacts. This algorithm intends to l ift the non-zero perm ittivity to the similar hei ght , i.e ., it turns the “hills” into “pil lars”. Thus, it is of im portance to reduc e the artifacts in the i mages during the process of the DEPIHT algorithm. The EPIHT - I algorithm is used to reduce the artifacts. It seems this step p lay s a su bs tan ti al rol e i n t he t es t 1 a nd 4 w hil e has a little effect in the test 2 and 3. However , compared to t he AADMM algorithm, the AADMM-DEPIHT algorithm has an improvement on the quality of images to some d egree. Perhaps, removing the artifacts in the images could be investigate d i n th e fut ure, e nabling better results in the AADMM-DEPIHT algorithm . In fact, the DEPIHT can be regarded as a post-process method for E CT . For instance, the L I algorithm is used to acquire the first point and then the method becomes the LI-D EPIHT algorithm. test Tr u e distributions LI LI-TO AADMM AADMM- TO AADMM- DEPIHT 1 2 3 4 ( a ) ( b ) ( c ) ( d ) ( e ) ( f ) Figure 6. 3D images reco nstructed from the experimental data : (a) the true distributions, (b) the im ages reconstructed from the LI algorit hm, (c) the im ages reconstruct ed from the LI algorithm with a threshold, (d) the images reconstructe d fro m the AADMM algorithm, (e) the images reconstructed from the AADMM algorithm with a threshold and (f) the images rec onstructed from the AADMM-DEPIHT algorithm. T able 1: Elapsed t ime (in seconds) Distributions LI AADMM AADMM-DEPIHT Cross-shaped 0.47 1.22 1.37 ‘V’-shaped 0.45 2.41 2.53 T wo rectangular -shaped 0.46 2.51 2.63 Three circular -shaped 0.47 0.71 0.85 T able 1 shows the elapsed time of the three algorithms. Since t he mesh used in this study is 2304 (48×48) square elements, the imaging time of the LI and AADMM algorithm s decreases a lot compared in [11]. As shown in table 1, the elapsed time of the DEPIHT algorithm is v ery short, which is benefited from the hard thresholding o perator . This m akes the DEPIHT algorithm a suitab le candidate for ECT in post-process. 5. Conclusion A combined algorithm for ECT via total variation and l 0 -norm regularizations, namely the AADMM- DEPIHT algorithm, is presented and validated using experimental d a t a . T h e r e s u l t s s h o w t h e A A D M M - DEPIHT algorithm does have an improvem ent on the quality of ima ges compared to the AADMM algorithm, e.g. remove some artifacts in several cases. Furthermore, the DEPIH T algorithm, based on l 0 -norm regularization, has a very suitable application in binarization , e.g. a post-process for ECT . 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