Improved Landweber Algorithm Based on Correlation

Transcription

1 Journal of Modern Physcs, 07, 8, ISSN Onlne: 53-0X ISSN Prnt: Improved Landweber Algorthm Based on Correlaton Yqun Kang *, Sh Lu School of Control and Computer Engneerng, North Chna Electrc Power Unversty, Beng, Chna How to cte ths paper: Kang, Y.Q. and Lu, S. (07) Improved Landweber Algorthm Based on Correlaton. Journal of Modern Physcs, 8, Receved: July 0, 07 Accepted: August, 07 Publshed: August 5, 07 Copyrght 07 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY 4.0). Open Access Abstract In order to obtan good effect of mage reconstructon of ECT, ths paper proposes an ECT mage magng method of mprove Landweber algorthm based on data correlaton analyss. Usng Maxwell smulaton software, the method frst determnes the dvson of the mcro element, then calculates the capactance of each element as a hgh delectrc constant, and makes a correlaton analyss between each capactance values and the measured capactance values, a correlaton coeffcent can be obtaned. We can combne the correlaton coeffcent wth the senstve feld, and obtan a new senstve feld by usng the soft feld characterstcs of the senstve feld. Combne the new senstve feld wth the Landweber algorthm to mproved mage, we can get better magng of poston mage. Keywords Electrcal Capactance Tomography, Improved Landweber, Correlaton Analyss, Reconstructon Senstve Feld. Introducton At present, there are a large number of two-phase flow systems n many felds, such as electrcty, petroleum, chemcal ndustry, metallurgy and other natonal economc producton. In these systems, due to the complex flow mechansm of two-phase flow, randomly large, the use of tradtonal detecton technology s dffcult to be used for observaton or measurement methods. And accordng to flow pattern change crteron or flow chart to determne the flow mode wth the flow parameters s also dffcult []. So electrcal capactance tomography (ECT) developed gradually snce the 980s. ECT s an magng technque, whch can measure delectrc constant changes and materal dstrbuton n ppes or tubes []. It uses dfferental medum has DOI: 0.436/mp Aug. 5, Journal of Modern Physcs

2 Y. Q. Kang, S. Lu dfferent delectrc constant, and by measurng the capactance values between the electrodes placed on the surface of the tube and the spatal dstrbuton of the delectrc constant n the tube s calculated, combned wth the senstve feld, the dstrbuton of the workng medum n the ppelne s further deduced. Because ECT can change the poston and number of sensor, produce dfferent magng results, and t does not affect the moton of the flud, fast, cheap, so t s consdered to be the future development prospects of magng technology [3] [4]. In the era of bg data, bg data can be summarzed as four characterstcs, data volume, velocty, varety, veracty. Data mnng s a technque that looks for regulartes and rules from a large number of data by analyzng each data, there are three man steps data preparaton, regular search and regular representaton. Data preparaton s to select the requred data from the relevant data source and ntegrate t nto a data set for data mnng. The regular search s to fnd out the rules contaned n the data set n some way. The regular representaton ndcates that the rules are expressed as much as possble n a user-understandable manner (such as vsualzaton). Capactance data mnng process s: frst of all data acquston and storage, through the smulaton software to get the capactance value of the flter nto a data set when the mcro element at hgh delectrc constant; and then the data set for processng and analyss, such as the use of lnear and non-lnear statstcal analyss of data and accordng to certan rules of the data set classfcaton, and analyzng the relatonshps between data and data categores; then the classfcaton of data after data mnng, explore and fnd nternal connecton between data [5] [6]; fnally, the results wll be magng, and easy to analyze and use. Statstcs s a comprehensve dscplne, whch by searchng, collatng, analyzng data and other means to acheve the nature of the measured obect, and even predct the future of the obect [7]. The regresson equaton s a mathematcal expresson that reflects the regresson relatonshp of a varable (dependent varable) to another or a set of varables (ndependent varable) from the sample data by regresson analyss. Regresson lnear equatons are used more often, we can use the least squares method to fnd the regresson lne equaton to get the regresson lnear equaton [8]. The statstcal method used n ths study s correlaton analyss. Correlaton analyss has been wdely used n many felds because of ts ablty to quckly and effcently dscover the nterrelatonshps between varous parameters. Correlaton analyss refers to the analyss of two or more varables that are correlated to measure the degree of correlaton between the two varables [9]. The scope and areas covered by the correcton cover almost every aspect of the world we see, and there are great dfferences n the defnton of correlaton n dfferent dscplnes [0]. Conventonal ECT method s bult on the electromagnetcal theory, whch encounters dffcultes when the problem becomes nonlnear. It s therefore worthwhle to nvestgate ths problem from another angle by explorng other possble relatonshps between the mages and the measured capactance DOI: 0.436/mp Journal of Modern Physcs

3 Y. Q. Kang, S. Lu data. In ths study, t s our ntenson to nvestgate the correlaton of the permttvty dstrbutons wth capactance measurements.. The Theoretcal Bass.. Prncples of Electrcal Capactance Tomography In ths paper, a square ECT sensor wth 8 electrodes s used []. The grd s dvded nto 3 3, as shown n Fgure. Usually, a crcle ECT sensor s popular, as a result of ths paper uses dvded nto 3 3, and the edge of the crcle ECT sensor can not be dvded nto the same small square, t s not easy to fnd the correlaton. The delectrc constant of the workng medum s ε, and the rest s ε. As shown n Fgure (b), the blue mcro element and the workng medum do not concde, we can know that the correlaton s small, the red mcro element and the workng medum concde, so the correlaton s large. By the defnton of the senstve feld: l C, ( e) C, = δ ( C, C, )( ε ε) l C, ( e) C, C( e) = S e e, h l where S, electrode and the electrode; C C C h l,, e s the senstvty of the e-th test mcro element between the, e s the capactance values when the delectrc constant of the e-th element s ε and the and the delectrc constant n the h l other mcro element s ε, C, and C, respectvely are the capactance values when the ppelne flled wth delectrc constant s ε and ε ; δ ( e) s a correcton factor related to the area of the e-th unt n the ppelne. It can be seen from the Equaton () that there s a certan lnear relatonshp C e. between the S ( e ) and the normalzed capactance value, () () Fgure. Square ECT sensor wth 8 electrodes. DOI: 0.436/mp Journal of Modern Physcs

4 Y. Q. Kang, S. Lu Conventonally, the smulaton method s used to obtan the senstve feld. We usually obtan senstve feld by auto Maxwell software. By calculatng the nternal potental dstrbuton of the ppelne under each exctaton voltage, drectly usng the pont multplcaton can generally the senstve feld. The specfc formula s: where E(, ) S E ( xy, ) E( xy, ) (, ) = dd x y xy (3) pxy (, ) V V xy s the electrc feld dstrbuton nsde of the ppelne when the electrode s excted (where the voltage appled to the electrode s constant V ), pxy (, ) s the area of the pxel n (x, y). After obtanng the senstve feld by smulaton method, by formula () we can obtan the capactance value when the delectrc constant s ε n the e-th element and the delectrc constant n the other mcro element are ε [] [3]... Prncples of Correlaton Coeffcent There are two varables n a lnear regresson, n whch x s an observable and controllable common varable, whch s often called an ndependent varable or controllable varable, and y s a random varable, whch s often called a dependent varable or a response varable. There s a sgnfcant lnear correlaton between x and y by scatterng or calculatng the correlaton coeffcent, that s, the relatonshp between x and y s as follows [4] [5]: It s usually consdered that ~ N ( 0, ) y = a + bx + η (4) η σ and suppose σ has no relatonshp wth x. Substtutng the observed data ( x, y)( =,, n) nto Equaton (4), and note that the sample s a smple random sample, we obtan: (,, ) N y = a + bx + η = n η,, ηn η ~ 0, σ Because G= CS w (G s the pxel gray value), and the Equaton () can be wrtten as Equaton (6). we can smplfy Equaton (6) as T C k = ε ε C C C G δ + C (6) h l l, w k h l where A ( ) ( C C ) (5) T l C, k = AG Cw + C + η, (7) η,, ηn η ~ N ( 0, σ) = ε ε δk. The values of a and b (that s C and AG) are requred to be mnmzed by: l n ( T ) w (8) Q n = C C = C C + AG C = = The method of mnmzng the sum of squares of devatons s called the least squares method. The sum of squares of the devatons s transformed nto: DOI: 0.436/mp Journal of Modern Physcs

5 Y. Q. Kang, S. Lu Let n = T T T ( ) ( w w) ( w ) ( C ) T T T C AG Cw Cw AG C C C Cw T n T T w w w Q = C C AG C C + C C AGC n n n = + = = = + n C a bc C C C C = T ( w, ) AG = R C C = T T ( Cw Cw)( C C) T T ( Cw Cw) ( C C) (9) (0) whch reflects the degree of correlaton between the two varables. AG s n the range of to. Snce A s constant, R and G are dentcally dstrbuted. C, e C,8 e R η C, e C,8 e R η = C ( w) C8 ( w) + C04, ( e) C04,8 ( e) R04 η 04 The new senstve feld matrx constructed by R as weghts s: P e R S, e P e R S, e = P04 ( e) R04 S, ( e) P e P e G = C ( w) C8 ( w) P04 ( e).3. Prncples of Improve Landweber Algorthm The teratve process of the Landweber teraton [6]: ( k+ ) ( k) T ( k) α W () () (3) G = G + S C SG (4) The teratve process of the Landweber teraton based on the correlaton matrx generated n ths study s: ( k+ ) ( k) T ( k) α W G = G + P C PG (5) where α s teraton step, α = λmax n smulaton, P s the correlaton matrx, T and λ s the maxmum egenvalue of P P. max 3. Smulatons 3.. Image Reconstructon by Smulaton As shown n Fgure, from left to rght (that s from (a) to (c)) for three knds of work condtons, from top to bottom (that s from () to (4)) are the orgnal pc- DOI: 0.436/mp Journal of Modern Physcs

6 Y. Q. Kang, S. Lu ture, the mage map of Landweber algorthm, the mage map of correlaton algorthm, the mage map of mproved Landweber algorthm based on correlaton, the number of teraton s 50. Varous mage maps are obtaned can be seen from the Fgure. In Fgure (a) Landweber algorthm can separate two square workng medum, although the correlaton algorthm can not completely separate two square workng medum, t can clearly fgure out the shape of square. In Fgure (b) two algorthms can not be maged correctly, but can only make approxmate postons. In Fgure (c) the senstve poston of Landweber s not obvous, and the workng medum s not maged at all. In the correlaton algorthm, the ntensty of the two adacent workng medum s enhanced. The mproved Landweber algorthm based on correlaton has better magng effect. In Fgure (4)-(a), the sze of the workng medum and the poston magng results are better. In Fgure (4)-(b) the locaton and number of workng medums can be clearly represented. Fgure. The mage of workng medum. DOI: 0.436/mp Journal of Modern Physcs

7 Y. Q. Kang, S. Lu 3.. The Senstvty Analyss by Smulaton We know that due to the soft feld characterstcs of the senstve feld, the senstve feld wll be affected when the workng medum s added nto the ppelne, whch wll lead to unsatsfactory magng results. The results show that the method used n ths paper s usng the correlaton between capactance and senstve feld, make full use of the soft feld characterstc, we can reconstruct the senstve feld matrx, and the reconstructed senstve feld can be regarded as the senstve feld after puttng nto the workng medum. It can be seen from Fgure 3, from top to bottom (that s from () to (4), of these, () s the orgnal senstve feld, () to (4) correspond to the case n Fgure from (a) to (b)) are the senstve feld between electrodes, from left to rght (that s from (a) to (b)) for the senstve feld between the electrodes and, electrodes and 6, electrodes and 6. The senstve feld of the workng medum Fgure 3. The mage of the senstve feld between electrodes. DOI: 0.436/mp Journal of Modern Physcs

8 Y. Q. Kang, S. Lu part s strengthened, so that the magng effect s better Image Error and Correlaton Coeffcent Accordng to the two standards of the mage error and mage correlaton coeffcent, a further comparson s made [7]: Correlaton coeffcent = ˆ ε ε Image error = (3) ε n n ( ε ˆ ε = ) n ( ε ˆ ) ( ˆ ε ε ε) = = (4) where ε s the real permttvty dstrbuton, ˆε s the reconstructed permttvty dstrbuton, ε and ˆε are the mean values of ε and ˆε. Fgure 4 s the mage error, and Fgure 5 s the correlaton coeffcent. In the fgure, represents Landweber algorthm, represents correlaton algorthm, and 3 to 7 represent mproved Landweber algorthm based on correlaton, the number of teraton s 0, 30, 50, 80, 00. As can be seen from the fgure, the mage error of the correlaton algorthm and the mproved Landweber sgnfcantly Fgure 4. Image error. Fgure 5. Correlaton coeffcent. DOI: 0.436/mp Journal of Modern Physcs

9 Y. Q. Kang, S. Lu decreased, the correlaton coeffcent sgnfcantly ncreased. Wth the ncrease of the number of teratons, the value of mage correlaton coeffcent tends to be gradual. 4. Concluson Ths paper presents an algorthm based on data analyss and correlaton coeffcent. The algorthm frstly uses the nverse method to obtan the capactance value when a certan element s a hgh delectrc constant workng medum, whch s smpler and more effectve than usng the cyclc acquston method. Comparng wth the orgnal senstve feld, t can be found that the reconstructed senstve feld s strengthened n the presence of the workng medum, so the magng effect becomes better. By comparng the mage wth the Landweber teratve algorthm, t can be seen that the correlaton coeffcent method has the advantage of reducng the sze of the mage shape. Improved Landweber algorthm based on correlaton can mprove mage qualty better, the accuracy of the method of ths study s hgh and the error s small. It s more accurate to determne the locaton of multple workng fluds. It can be seen from the mage that the study n the paper s prone to deformaton n the dagonal drecton, and the future research work should focus on how to modfy the mage at the dagonal. And the correlaton coeffcent method can mage the workng medum wth hgh delectrc constant, how to make the delectrc constant of the hgher workng medum magng better s a key research drecton. References [] Yang, W.Q. and Peng, L.H. (003) Measurement Scence and Technology, 4, [] L, Y. and Holland, D.J. (03) IEEE Sensors Journal, 3, [3] Ba, B.F., Guo, L.J. and Chen, X.J. (00) Proceedngs of the CSEE,, [4] Le, J., Lu, S., L, Z.H. and Sun, M. (007) Proceedngs of the CSEE, 7, [5] Han, J., Kamber, M. and Pe, J. (006) Data Mnng: Concepts and Technques. Morgan Kaufmann Publshers, San Francsco. [6] L, D.R., Wang, S.L. and L, D.Y. (03) Spatal Data Mnng Theores and Applcatons. Scence Press, Beng. [7] Vapnk, V.N. (995) The Nature of Statstcal Learnng Theory. Sprnger-Verlag, New York. [8] You, J.H., Xu, Q.F. and Zhou, B. (008) Chnese Annals of Mathematcs, 9, [9] L, G.J. and Cheng, X.Q. (0) Bulletn of Chnese Academy of Scences, 7, [0] Lang, J.L., Feng, C.J. and Song, P. (06) Chnese Journal of Computers, 39, -7. [] Wang, H.X., Tang, L. and Yan, Y. (007) Chnese Journal of Scentfc Instrument, 8, [] Guo, H.X., Yu, S.S. and Zhou, J.L. (000) Journal of System Smulaton,, DOI: 0.436/mp Journal of Modern Physcs

10 Y. Q. Kang, S. Lu [3] Yang, W.Q., Spnk, D.M. and York, T.A. (999) Measurement Scence and Technology, 0, [4] Guyon, I. and Elsseeff, A. (003) Journal of Machne Learnng Research, 3, [5] Zhang, F., Xe, Z.H. and Cheng, J.T. (05) Fre Control & Command Control, 40, [6] Landweber, L. (95) Amercan Journal of Mathematcs, 73, [7] Tenou, S. and Merbout, M. (0) Measurement, 45, Submt or recommend next manuscrpt to SCIRP and we wll provde best servce for you: Acceptng pre-submsson nqures through Emal, Facebook, LnkedIn, Twtter, etc. A wde selecton of ournals (nclusve of 9 subects, more than 00 ournals) Provdng 4-hour hgh-qualty servce User-frendly onlne submsson system Far and swft peer-revew system Effcent typesettng and proofreadng procedure Dsplay of the result of downloads and vsts, as well as the number of cted artcles Maxmum dssemnaton of your research work Submt your manuscrpt at: Or contact mp@scrp.org DOI: 0.436/mp Journal of Modern Physcs