Application of Dynamic Time Warping on Kalman Filtering Framework for Abnormal ECG Filtering

Transcription

1 Applcaton of Dynamc Tme Warpng on Kalman Flterng Framework for Abnormal ECG Flterng Abstract. Mohammad Nknazar, Bertrand Rvet, and Chrstan Jutten GIPSA-lab (UMR CNRS 5216) - Unversty of Grenoble Grenoble, France Exstng nonlnear Bayesan flterng frameworks serve as an effectve tool for the model-based flterng of nosy ECG recordngs. However, snce these methods are based on lnear phase assumpton, for some heart defects where abnormal waves only appear n certan cycles of the ECG, they are unable to smultaneously flter the normal and abnormal ECG segments. In ths paper, a new method based on Dynamc Tme Warpng (DTW), whch benefts nformaton of all channels for nonlnear phase state calculaton s presented. Results on real and synthetc data show that the new method can be successfully appled for flterng normal and abnormal ECG segments smultaneously. Keywords: ECG denosng, Kalman flterng, nonlnear Bayesan flterng, lnear phase, dynamc tme warpng. 1 Introducton The extracton of hgh-resoluton pathologcal cardac sgnals from a multchannel nosy electrocardogram (ECG) remans an mportant problem for the bomedcal engneerng communty. Despte of the rch lterature n the feld of ECG processng, there are stll many clncal applcatons that lack relable sgnal processng tools to extract pathologcal ECG beats contamnated wth background nose. In [1], Bayesan flters such as the Extended Kalman Flter (EKF) and Extended Kalman Smoother (EKS) have been proposed for ECG denosng. The state-space model used for these flters s nspred from [2], whch suggests the use of Gaussan mxtures to model realstc synthetc ECGs. The basc dea s to approxmate the PQRST waves by the sum of 5 weghted Gaussan shape functons. In [1], the synthetc ECG generator proposed n [2], transferred nto polar coordnates from Cartesan coordnates. Ths modfcaton and some other modfcatons make t smpler and more straghtforward n nterpretaton [1]. The modfed model n ts dscrete form, wth the assumpton of a small samplng perod of δ s: θ k+1 = (θ k + ωδ)mod(2π) z k+1 = δ α ω {P,Q,R,S,T } b 2 θ,k exp( θ2,k (1) 2b 2 ) + z k + η 139

2 where θ and z are the state varables n polar coordnates and k denotes the dscrete tme ndex. The α and b correspond to the peak ampltude n mllvolts and center parameters of the Gaussan functons used for modelng each of the ECG components. We defne θ,k = (θ k ψ )mod(2π), n whch, ψ corresponds to the center of the th Gaussan functon. ω s the angular velocty of the trajectory as t moves around the lmt cycle n the x y plane [1], and η z s a random addtve nose. The system state and process vectors are defned as: { x k = [θ k, z k ] T w k = [α P,..., α T, b P,..., b T, ψ P,..., ψ T, ω, η z ] T (2) wth Q k = E { } w k wk T as process nose covarance matrx. The nosy ECG s assumed as observaton of the Kalman flter. In addton, by detectng the R- peaks of ECG sgnal, an addtonal observaton s acheved. In ths model t s assumed that the phase values are strctly lnear between 0 and 2π n the ntermedate samples of two R-peaks. The addtonal phase observaton φ and the nosy ECG measurements, s, are related to the state vector as follows: [ ] [ ] [ ] [ ] φk 1 0 θk uk =. + (3) s k 0 1 z k v k where u k and v k are the correspondng observaton noses, and the observaton nose covarance matrx s gven as R k = E { [u k, v k ] T [u k, v k ] }. In ths model a strctly lnear phase has been assumed n the state equaton of the model. However, t s not always a vald assumpton. Applcaton of ths model to many of the common ECG abnormaltes s rather straghtforward, snce the model parameters may be smply recalculated and used n the flter model. However, for some heart defects such as the Premature Ventrcular Contracton (PVC), where the abnormal wave only appears n certan cycles of the ECG, some modfcatons n the state equatons are necessary to smultaneously flter the normal and abnormal segments. In order to do so, n ths paper we ntend to modfy the phase equaton usng nformaton of dfferent channels. The rest of the paper s organzed as follows: In secton 2 equatons and theory supportng our proposed method are descrbed. In secton 3 results of the proposed method appled on dfferent data and dscusson about the results are presented. Fnally, our concluson s stated n secton 4. 2 Method The frst modfcaton of the phase state can be addng a random addtve nose η θ to the phase state equaton (n ths paper we refer to t as flexble lnear). Therefore, the phase model would no longer be strctly lnear and slght fluctuatons around lnear phase are allowed. Hence, state equatons are: θ k+1 = (θ k + ωδ)mod(2π) + η θ z k+1 = δ α ω {P,Q,R,S,T } b 2 θ,k exp( θ2,k (4) 2b 2 ) + z k + η z 140

3 Although ths modfcaton may mprove performance of the flter, t stll assumes that all beats are almost smlar and no beat dffers much from the others. Moreover, t only uses nformaton of the current channel to make ECG phase. The second modfcaton, whch benefts nformaton of all channels s usng Dynamcal Tme Warpng (DTW) [3] for phase state calculaton. DTW s a method for measurng smlarty between two sequences or matrces, whch may vary n tme or speed. Ths method s wdely used n speech recognton to recognze a unque word when t s pronounced fast or slowly. In ths method an optmal match between two gven sequences or matrces wth certan restrctons s found [3]. For our problem of nterest, a multchannel ECG beat reference E(l) R M s frstly selected and a lnear phase s assgned to t, then current multchannel ECG beat s(k) R M and the reference ECG beat are nonlnearly warped to optmze ther smlarty of ther nonlnear varatons. Fnally, as t s llustrated n Fgure 1, the phase observaton of the current ECG beat s acheved by algnng lnear phase of the reference ECG beat, accordng to optmal match of the reference and current ECG beats. Computatonal cost of the method s low and DTW algorthm can be mplemented easly. Ths model of phase state can also be further modfed by addng a random addtve nose to make t more flexble (n ths paper we refer to t as flexble DTW). Estmaton of phase state based on DTW methods s especally valuable when n some beats one or more ECG waves (P, Q, R, S and T) appear sooner or later than normal ones. In those cases, snce DTW methods search for optmal match between reference and current beats, premature or delayed occurrence of the ECG waves are compensated n the phase state. Therefore, the EKF flter can better follow premature or delayed ECG waves. Another parameter that may also affect flterng performance s expanson or contracton of each ECG wave n some dssmlar beats. Here agan, t s possble to compensate the devaton from lnear phase usng DTW methods. 3 Results Fgure 2 shows results of proposed methods on a part of the record 116 of the MIT-BIH Arrhythma Database [4], [5]. Ths database conssts of two-channel ambulatory ECG recordngs, n whch some beats sgnfcantly dffer from other beats. Mean ECG has been adopted as reference ECG beat of DTW methods. Mean ECG and other parameters of the method have been calculated accordng to [1]. As t s seen n Fgure 2, the best result s provded by flexble DTW. Although flexble-lnear phase provdes better results n comparson to strctlnear phase, t s stll unable to follow a beat, whch s dssmlar to other beats. In order to have better comparson, resdual results whch are subtracton of the orgnal sgnal from fltered sgnals are plotted on the rght column of Fgure 2. As t s seen, some ECG parts are deterorated by strct and flexble-lnear phases, whle, DTW methods are able to follow the sgnal n these scenaros. In order to study the performance of the methods n dfferent stuatons, 141

4 Fgure 1: A typcal example of DTW method for fndng optmal match between reference ECG beat and current ECG beat. synthetc ECG data have been generated to model these dssmlartes. In (1) ψ denotes locaton of Gaussan functons, so premature and delayed occurrence of the ECG waves can be modeled wth varyng ψ around ther values. Expanson and contracton of ECG waves can also be modeled wth varyng b. Fgure 3 shows results of dfferent methods for dfferent range of ψ varatons, where 100% corresponds to 2π. The synthetc data consst of eght channels and nput sgnal to nose rato (SNR) s equal to 15 db. For each value of ψ varatons, ffty trals have been carred out to have statstcally relable results. As t s seen, when ψ varatons are very low and all beats are very smlar, lnear methods provde better results, snce nose cannot affect them. However, DTW methods are affected by the nose, nevertheless, they dd not deterorate nput sgnals, because ther output SNRs are stll more than 15 db. As ψ varatons become larger, the dfference between performance of lnear and DTW methods become lower. For varaton equal to 0.5%, same performance s acheved and from ths pont, DTW methods dramatcally outperform lnear ones. Smlar trals have been carred out for varatons of b, wdth of Gaussan functons. As t can be seen n Fgure 4, here gan, for low values of b varatons, lnear methods outperform DTW methods. However, as b varatons become larger, DTW methods sgnfcantly outperform lnear methods. Fgures 3 and 4 show that addng nose to phase state equaton can lead to mprove results of DTW methods for large sgnal dstortons. Practcally, for very slght varatons of ψ or b, flexble lnear method provdes the best results, whle, for larger values of ψ or b varatons, flexble DTW method outperforms other methods. 142

5 Fgure 2: Illustraton of proposed method on real data. Left, Up to Down: Orgnal record 116 of the MIT-BIH Arrhythma Database, strct lnear, flexble lnear, DTW, flexble DTW outputs. Rght, Up to Down: Subtracton of the orgnal ECG from strct lnear, flexble lnear, DTW, flexble DTW outputs. 25 Output SNR (db) Lnear 10 Flexble Lnear DTW Flexble DTW Varaton of Center of Each ECG Wave (ψ) [%] Fgure 3: Mean value of EKF output SNR for dfferent range of ψ varatons 143

6 24 Output SNR (db) Lnear Flexble Lnear DTW Flexble DTW Varaton of Wdth of Each ECG Wave (b) [%] Fgure 4: Mean value of EKF output SNR for dfferent range of b varatons 4 Concluson Dfferent versons of nonlnear Bayesan flterng frameworks have been presented to flter nosy ECG recordngs. However, due to the lnear phase assumpton, they have not been able to flter normal and abnormal ECG segments smultaneously. In ths work a new method based on DTW for phase state calculaton has been presented. Results on real and synthetc data show that DTW methods provde more relable phase state when dssmlarty between current beat and other beats s large, because ths dssmlarty s compensated n phase state. Ths method may therefore serve as an effectve tool for smultaneously flterng normal and abnormal ECG segments. Moreover, optmal match between reference and current beats, provded by DTW method, may be used n future works as a feature to classfy normal and abnormal beats. References [1] R. Samen, M. B. Shamsollah, C. Jutten, and G. D. Clfford, A Nonlnear Bayesan Flterng Framework for ECG Denosng, IEEE Trans. Bomed. Eng., vol. 54, no. 12, pp , December [2] P. E. McSharry, G. D. Clfford, L. Tarassenko, and L. A. Smth, A Dynamc Model for Generatng Synthetc Electrocardogram Sgnals, 50, pp , mar [3] C. S. Myers, A comparatve study of several dynamc tme warpng algorthms for speech recognton, Ph.D. thess, Massachusetts Insttute of Technology, [4] The MIT-BIH Arrhythma Database, Avalable Onlne: physobank/database/mtdb/. [5] A. L. Goldberger, L. A. N. Amaral, L. Glass, J. M. Hausdorff, P. Ch. Ivanov, R. G. Mark, J. E. Metus, G. B. Moody, C-K. Peng, and H. E. Stanley, Physobank, physotoolkt, and physonet : Components of a new research resource for complex physologc sgnals, Crculaton, vol. 101, no. 23, pp. e215 e220,