International Journal of Bioelectromagnetism Vol. 5, No. 1, pp. 236237, 2003. 
www.ijbem.org 
ECG Averaging Based on Hausdorff Metric Leonid S. Fainzilberg Correspondence: L.S.Fainzilberg, IRTC ITS,
Prospect Academica Glushkova 40, Kiev680, Ukraine, 03680. Abstract. A new method
for estimation of ECG's averaged cycle is proposed. The method consist
of following steps: transformation from time domain signal to the
phase space, estimation of reference trajectory in the phase space
using Hausdorff metric, estimation of average trajectory and its
inverse transformation to the time domain. Proposed method is more
suitable for processing signal with nonlinear perturbation like
ECG in comparison with traditional methods.
Keywords: ECG Stochastic Model; Phase Space 1. Introduction When the problems of computer processing and analysis of ECG are solving the traditional representation of ECG in the time domain _{ } leads to some errors. The situation is caused by nonlinear distortions of _{ } wave, _{ } complex and _{ } segment from one cycle to other. Moreover, it is known that boundaries of fragments of real ECG usually are fuzzy. Hence, alternative approaches to the problem have to be studied. One of them based on transformation of time domain signal to a specific image in the phase space was considered in [Fainzilberg, 1998]. Now we present further study results of this method to a problem of ECG averaging. 2. Basic Results Let's assume that observed ECG signal be a result of distortions of some periodic process by random perturbation , where  unknown function. Let _{ } is a part of unobserved function _{ } on one period _{ } and _{ } have to be estimated by ECG processing. We assume that _{ } is the function consisting of _{ } fragments
We suppose also that any _{ } th fragment (_{ } ) on the _{ } th ECG cycle is a result of operator transformation to corresponding fragments of _{ } :
where _{ } are random parameters of perturbation (by amplitude and time) and _{ } is the parameter of time shift. In this case, the nonlinear stochastic model to simulate real ECG signal may be obtained:
where
and _{ } , _{ } are sequences of the limited on a level random variables with zero average (see Fig. 1).
Figure 1. Result of ECG simulation according to stochastic model (3). The nonlinear stochastic model (3) may be easy generalized to simulate ECG signal with broken morphology of beats (for example, extra systoles) by using _{ } etalons _{ } which generate _{ } th ECG cycle according to probabilities _{ } . Despite of nonlinear distortions of etalons it may be show that diagnostic features of distorted etalons have close phase coordinates. This gives following method for estimation of ECGs averaged cycle. Let we have set _{ } of vectors _{ } corresponding to _{ } cycles of observed ECG in normalized phase space. Then we may define reference cycle _{ } as trajectories having minimum sum of Hausdorff distances to other trajectories:
where
and _{ }  Euclid distance. The average trajectory may be easy estimated by points placed near corresponding point of _{ } . Its projection gives good estimate of etalon cycle _{ } in the time domain (see Fig. 2). Figure 2. ECG in the phase space (left), its fragment (middle) and averaged cycle in the time domain (right). 3. Discussion and Conclusion We use Hausdorff metric to construct the average trajectory of observed ECG in the phase space. In comparison with traditional this method is more suitable for processing signal with nonlinear perturbation like real ECG. The projection of constructed average trajectory gives good presentation of ECG average cycle in the time domain and may be used for patients’ diagnoses. References Fainzilberg L. Heart functional state diagnostic using pattern recognition of phase space ECGimages. In proceedins of the 6^{th} European Congress on Intelligent Techniques and Soft Computing (EUFIT ’98, Germany), 1998, v. 3, 18781882.
