 |
International Journal of Bioelectromagnetism Vol. 5, No. 1, pp. 236-237, 2003. |
 |
 |
www.ijbem.org |
|
ECG Averaging Based on Hausdorff Metric Leonid S. Fainzilberg International Research and Training Center of Information Technologies and Systems, Kiev, Ukraine Correspondence: L.S.Fainzilberg, IRTC ITS,
Prospect Academica Glushkova 40, Kiev-680, 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
2. Basic Results Let's assume that observed ECG signal Let
We suppose also that any
where In this case, the nonlinear stochastic model to simulate real ECG signal may be obtained:
where
and 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 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
where
and The average trajectory may be easy estimated by points placed near
corresponding point of  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 ECG-images. In proceedins of the 6th European Congress on Intelligent Techniques and Soft Computing (EUFIT ’98, Germany), 1998, v. 3, 1878-1882.
© International Society for Bioelectromagnetism
|
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. 
be a result
of distortions of some periodic process 
by random perturbation
, where 
- unknown function.
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 
-th fragment (
) on the 
-th ECG cycle is a
result of operator transformation to corresponding fragments of
: 
are random parameters
of perturbation (by amplitude and time) and 
is the parameter of
time shift.
,
, 
are sequences of the
limited on a level random variables with zero average (see Fig.
1).
etalons 
which generate
-th ECG cycle according
to probabilities 
.
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:
- Euclid distance.
. Its projection gives
good estimate of etalon cycle 
in the time domain
(see Fig. 2). 