Nonlinear State-space Projections Research Articles

Clustering based multiple state–space projections

The method of Nonlinear State–Space Projections (NSSP) is very effective in suppression of white Gaussian noise; however, in colored noise environment its operation is not satisfactory. In this paper, we propose an idea of multiple weighted state–space projections aimed to increase projective filtering performance in this demanding environment.We introduce a new method of state–space points clustering which combines a linguistically defined initialization of prototypes with possibilistic approach to memberships calculation and prototypes refinement. For each cluster created, a signal subspace is formed, and each state–space point is projected on the assumed number of the selected signal subspaces. Weighting the corrections resulting from such projections, according to the distances between the projected point and the clusters prototypes, allows to limit errors caused by wrong construction of the subspaces involved. Automatic calculation of the subspaces dimensions allows for further improvement of the colored noise suppression.The proposed methods are applied to electrocardiograms (ECG) processing and compared to a few previously developed modifications of nonlinear projective filtering. Tested in the colored electromyographic (EMG) noise environment, they significantly outperform the reference methods. Their possible applications are discussed and an exemplary one, concerning separation of the maternal and the fetal ECG, is illustrated.

On clustering based nonlinear projective filtering of biomedical signals

Abstract We propose to modify the method of nonlinear state-space projections (NSSP) by application of the technique of k-means clustering. NSSP performs reconstruction of the state-space representation of the processed signals using the Taken's method of delays. Then it projects each state-space point on the appropriately constructed signal subspace and recovers the one-dimensional signal by averaging the results of all projections. The k-means clustering is applied to form so-called neighborhoods on the basis of which the signal subspaces are created. Within these neighborhoods, local density around each state-space point is estimated, to make construction of the signal subspaces more immune to high energy electromyographic noise. The developed method is applied to process different types of ECG signals. For reference, the original NSSP method and its previously developed modifications are used. In different types of noise environment, the proposed method appears more effective than the original one and in most cases than the other reference methods. Moreover, visual results of fetal phonocardiogram and electronystagmogram processing show the wide range of its possible applications.

Fetal heart beat detection by Hilbert transform and non‐linear state‐space projections

This study describes an integrated approach for fetal heart beat detection from abdominal ECG signals. The signal is first processed by the first-order difference operation, and then, the envelope of the differentiated signal is given by the Hilbert transform, which indicates both the maternal and fetal heart beats. After that, the non-linear state-space projections is applied to the envelope signal to remove the maternal heart beats, obtaining the fetal heart beats envelope. The validity and performance of the described method are confirmed through experiments on real ECG data. A comparison with the well-known independent component analysis method has also been presented.

Single-lead fetal electrocardiogram estimation by means of combining R-peak detection, resampling and comb filter

Minimal detecting electrodes are preferred to miniaturise fetal electrocardiogram (fECG) monitoring devices for application in non-clinical environments. In this paper, a new method to estimate the fECG using a single-lead abdominal signal is introduced. In this method, for a preprocessed abdominal ECG recording, we follow a multi-step procedure to estimate the fECG signal. First, the locations of the maternal R-peaks are detected. Each R–R interval in the abdominal signal is resampled to have the same number of samples by changing its corresponding sampling frequency. A comb filter, which has teeth that coincide with the harmonics of the maternal electrocardiogram (mECG), is applied to the resampled signal. Each R–R interval in the filtered signal is resampled again to recover its original sampling frequency, and the mECG signal is obtained. This mECG signal is subtracted from the abdominal signal, and the residual signal is considered to be a primary estimate of the fECG signal. The same procedure can be applied to the residual signal to enhance the fECG signal. Compared to two other single-lead-based methods, singular value decomposition and nonlinear state-space projection, the proposed method has shown improved robustness and fidelity in restoration of the fECG during testing with synthetic ECG signals and a real fetal ECG database from MIT-BIH PhysioBank.

Projective filtering of time warped ECG beats

The paper proposes a modification of the nonlinear state-space projections (NSSP) method. Our approach, when applied to ECG signal processing, considerably improves the method's performance. One of the crucial operations in NSSP is the search for neighborhoods of the state-space trajectory points. The modification proposed is based on imposing a few restrictions on the time location of the neighborhood points. Dynamic time warping, a technique which allows for nonlinear alignment of time series or sequences of vectors, is applied as a straightforward solution to the task of neighborhoods determination with the restrictions imposed. The influence of nonlinear alignment on the distributions of the determined neighborhoods is presented, and the resulting method of ECG enhancement is investigated.

Measurement method for the fetal electrocardiogram

A fetal electrocardiogram (ECG) provides clinically significant information concerning the electrophysiological states of a fetus. Since the first attempt by Cremer in 1906, for about 100 years nobody has been able to non‐invasively extract the fetal ECG popularly used in clinical situations. The low amplitude of the fetal signals and the high background noise and the non‐linearity of the ECG signals mainly have been interfering with the noninvasive extraction of the fetal ECG. The aim of this paper is to introduce a new nonlinear filtering method to extract the fetal electrocardiogram detected from the mother's abdominal wall. This method consists of two steps: The rough extraction step performed by blind source separation (BSS) with references and the precise extraction step with fast non‐linear state space projection (FNSSP) based on a nonlinear state space projection (NSSP), which needs several complicated conditions or has limitations. So, it takes a lot of time to calculate. Our new method does not require such conditions, and it has about a thirty‐fold speedup when compared with the previous method. Our new method has been successfully applied to on‐line recordings with fetal components and will be useful for monitoring the fetal electrocardiogram in clinical applications.

Fetal ECG extraction with nonlinear state-space projections.

We describe a method to suppress maternal and noise contaminations in single-lead fetal ECG recordings. A nonlinear state space projection technique originally developed for noise reduction in deterministically chaotic signals is used. The method is successfully applied to recordings with fetal components and noise of comparable amplitude.