Generalized Rational Variable Projection With Application in ECG Compression

Abstract

In this paper we develop an adaptive transformdomain technique based on rational function systems. It is of general importance in several areas of signal theory, including filter design, transfer function approximation, system identification, control theory etc. The construction of the proposed method is discussed in the framework of a general mathematical model called variable projection. First we generalize this method by adding dimension type free parameters. Then we deal with the optimization problem of the free parameters. To this order, based on the well-known particle swarm optimization (PSO) algorithm, we develop the multi-dimensional hyperbolic PSO algorithm. It is designed especially for the rational transforms in question. As a result, the system along with its dimension is dynamically optimized during the process. The main motivation was to increase the adaptivity while keeping the computational complexity manageable. We note that the proposed method is of general nature. As a case study the problem of electrocardiogram (ECG) signal compression is discussed. By means of comparison tests performed on the PhysioNet MIT-BIH Arrhythmia database we demonstrate that our method outperforms other transformation techniques.

ECG Beat Representation and Delineation by Means of Variable Projection

“…Due to their simplicity and low computational complexity, linear models are frequently used to model ECG signals [12], [18], [19]. One of the main challenges in this context is to find a proper set of atoms (i.e., a dictionary) that matches the problem area.…”

“…Nonlinear parametric modeling of ECG waveforms dates back to the 1980s, when Sörnmo et al [23] introduced the dilated Hermite functions for approximating the shape of the QRS complex. Their work inspired many others: Laguna et al [10] and Kovács et al [12] compressed ECG signals; Lagerholm et al [13] utilized this approach for heartbeat clustering purposes; Haraldsson et al [14] extracted features by means of Hermite expansion to detect myocardial infarction, while Sandryhaila et al [11] improved reconstruction accuracy by using the discrete analogue of the dilated Hermite function system. These approaches utilize a single nonlinear parameter that is the dilation of atoms.…”

“…Due to Eqs. (11) and (12), these intervals are also distinct, which implies that the dot products of the QRS, T and P components are very small, that is, they are pairwise almost orthogonal (see, e.g., Fig. 3(d)).…”

“…We have recently successfully used adaptive Hermite functions for ECG wave segmentation, providing additionally a low-dimensional wave shape representation which potentially holds further important diagnostic information for the applications mentioned above [9]. In general, Hermite functions have shown to be very well suited to ECG signal processing, ECG data compression [10]- [12], QRS complex clustering [13], and detection of myocardial infarction [14]. Despite the usefulness of these functions some limitations remain, especially in relation to (baseline) noise and pathological ST segment elevations/depressions, which reveals the need to extend the basis function dictionary to achieve a correct beat delineation and representation.…”

Design and Optimization of ECG Modeling for Generating Different Cardiac Dysrhythmias

“…Another pole-zero model utilized the poles and zeros to form clusters, and the clusters can be related to the constituent waves of the ECG [ 3 ]. Another study optimized the rational functions and their poles for ECG signal using particle swarm optimization [ 4 ]. The pole-zero system’s problem is that it cannot find a satisfactory solution because it involves finding an optimum point on a (potentially) multimodal error surface [ 5 ].…”