## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Idea and Processing Scheme

#### 2.2. The ECG Trace Conspicuity as an Estimate of Local Information Density

#### 2.3. ECG Waves Delineation

^{®}by Aspel S.A., Zabierzow, Poland) software that meets the performance requirements of international IEC standard [26] and on the other hand allows for access to interpretation of metadata. The wave delimitation procedure embedded in Ascard 6 software uses the first and second derivatives of various versions of filtered signal in order to determine the point where the wave energy emerges from the background noise [31]. The algorithm also uses wave-specific features individual for each wave’s onset and endpoint.

#### 2.4. Piecewise Adaptation of the Local Relevance Function

_{m}, f

_{s}) accordingly to the linear relationship:

_{s}= 500 Hz and the minimum usable value of local sampling frequency tested in two experiments was set to f

_{m}

_{1}= 100 Hz (Figure 4) and f

_{m}

_{2}= 50 Hz respectively.

#### 2.5. ECG Signal Resampling

_{j}): j ∈ J.}, where J. is a countable indexing set, and f satisfies some a priori constraints [33]. Extension of the classical Shannon theory to the non-uniform sampling of bandlimited functions specifies that for the exact and stable reconstruction of such function f from its samples {f(x

_{j}): x

_{j}∈ X}, it is sufficient that the Beurling density,

- Given a generator φ, conditions on X have to be defined, usually in the form of a density, such that the norm equivalence (6) holds.$${c}_{p}{\Vert f\Vert}_{{L}_{v}^{p}}\le {\left({\displaystyle \sum _{{x}_{j}\in X}{\left|f({x}_{j})\right|}^{p}{\left|v({x}_{j})\right|}^{p}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}\le {C}_{p}{\Vert f\Vert}_{{L}_{v}^{p}}$$Then, at least in principle, $f\in {V}_{v}^{p}(\varphi )$ is uniquely and stably determined by ${f|}_{X}$.
- Reconstruction procedures useful and efficient in practical applications have to be designed as fast numerical algorithms which recover f from its samples ${f|}_{X}$, when (6) is satisfied.

^{2}(φ), alternative reconstruction procedures based on Neumann series have been designed [34]. In [33] Aldroubi presented the example iterative algorithm with the proof of the convergence of results. The reconstruction of the uniform biosignal from an incomplete time series was also developed by Candes et al. [36] and Needell and Tropp [37].

_{j}({n, v(n)}) to the continuous space with the use of 3rd order polynomial function,

_{n}, t

_{n}

_{+1}], n ∈ {0, 1, … N − 1} is best fitted to the time series N

_{j}. Next, the output signal representation is obtained by sampling the S

_{n}(t) at desired time points m:

#### 2.6. Implementation Details

## 3. Evaluation of the Method

- Selecting a set of test signals complying with international standards (the CSE Database);
- Selecting tools (the ECG interpretive software) and error measures (PRD, local PRD and WDD);
- Selecting the range of method parameters (gMRF, sMRF, sampling interval);
- Comparing differences between original and processed records with error measures (PRD compares discrete values of samples, whereas the WDD compares values of diagnostic results);
- Statistical processing of error values estimated for each file with each combination of parameters.

#### 3.1. The Test Signal Set

#### 3.2. The Error Metrics

_{int.}, QRS

_{dur.}, QT

_{int.}, QTp

_{int.}, P

_{dur.}, PR

_{int.}, QRS

_{peaks_no.}, Q

_{wave_exsist.}, Δ

_{wave_exsist.}, T

_{shape}, P

_{shape}, ST

_{shape}, QRS(+)

_{amp.}, QRS(−)

_{amp.}, P

_{amp.}, T

_{amp.}, ST

_{elevation}, ST

_{slope}) of compared beats and Λ is a diagonal matrix of weights heuristically set to [42],

#### 3.3. Performance Assessment

## 4. Discussion

_{s}–f

_{m}) available for adaptive sampling.

- On a PC platform i7 3770 (
^{®}Intel, Santa Clara, CA, USA), 3400 MHz, 8 GB RAM—0.943% (i.e., 106 times faster than the ECG acquisition); - on a mobile platform PXA-270 (
^{®}Toradex AG., Horw, Switzerland), 624 MHz, 64 MB RAM—7.518% (i.e., 13.3 times faster than the ECG acquisition).

_{s}between the adjacent heart beats. Otherwise, the CR will be lower than estimated by ΣMRF in the case of fast rhythms because main shortenings in the ECG pattern take place out of the waves (i.e., the accelerating heart first reduces its inactivity periods). A separate approach should be studied in cases of abnormal ECG when wave borders cannot be determined.

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**(

**a**) The example of expert’s eyeglobe trajectory over a 12-lead ECG plot (CSE-Mo-001); (

**b**) Corresponding bar graph of the attention density (each bar collects gaze time in a 32 ms interval in ECG record)—the background for gMRF (modified from [20]).

**Figure 3.**Projection of generalized medical relevance function (gMRF) to local positions of the heartbeat sections: (

**a**) gMRF (with the bar graph it stems from), (

**b**) aMRF (adapted medical relevance function) calculated for CSE-Mo001 record, (

**c**) aMRF calculated for CSE-Mo003 record.

**Figure 4.**Interval of consecutive samples in a non-uniform representation (f

_{m}

_{1}= 100 Hz) calculated for the reference beat from file CSE001.

**Figure 5.**Regular and irregular representations of the same heartbeat (

**a**) global scale, (

**b**) local scale (terminal section of QRS). Vertical axes represent the ECG voltage, approximately 2.44 μV per unit.

**Figure 6.**Comparison of bar graphs for scanpath-determined gMRF (blue) and example infarct or a conduction defect-oriented sMRF (red). Wave borders are: P-onset: 106, P-end: 196, QRS-onset 246, QRS-end: 338 and T-end 672.

Parameter | gMRF (Perceptual) | sMRF (with ROI) | |||
---|---|---|---|---|---|

f_{s} Range (Hz) | 100 … 500 | 50 … 500 | 100 … 500 | 50 … 500 | |

Compression ratio | 3.01 | 3.61 | 3.83 | 4.72 | |

PRD [%] ([μV] *) | global | 3.11 (46.6) | 3.73 (55.9) | 3.96 (59.3) | 4.88 (73.1) |

within P-wave | 0.16 (2.4) | 0.18 (2.7) | 0.35 (5.3) | 0.41 (6.2) | |

within QRS complex | 0.22 (3.3) | 0.22 (3.3) | 0.22 (3.3) | 0.24 (3.6) | |

within T-wave | 0.37 (5.6) | 0.41 (6.2) | 0.44 (6.7) | 0.47 (7.1) | |

out of waves | 1.11 (16.6) | 1.77 (26.5) | 2.70 (40.4) | 3.93 (58.8) | |

WDD [%] | 0.21 | 0.23 | 0.37 | 0.41 | |

RR interval std [ms] | 1.5 | 1.5 | 1.5 | 1.5 | |

P-wave duration std [ms] (15) ** | 10.3 | 10.7 | 12.4 | 14.1 | |

PQ interval length std [ms] (10) | 7.1 | 7.2 | 8.6 | 9.7 | |

QRS duration std [ms] (10) | 7.6 | 7.6 | 7.6 | 7.6 | |

QT interval length std [ms] (30) | 14.7 | 16.5 | 16,1 | 18.2 | |

P axis std [deg] | 7.5 | 8.8 | 7.8 | 9.7 | |

QRS axis std [deg] | 2.1 | 2.7 | 2.4 | 3.0 | |

T axis std [deg] | 3.1 | 3.5 | 3.3 | 3.5 |

**Table 2.**Performance of the proposed adaptive sampling method compared to recent landmark systems for ECG compressed sensing; NB authors use different data sets for testing.

Work (Test Set) | Method | Scenario 1 | Scenario 2 | ||
---|---|---|---|---|---|

CR | PRD [%] | CR | PRD [%] | ||

Mamaghanian [14] (MIT-BIH) | wavelet db10 | 3.70 | 2.00 | 10.00 | 9.00 |

CS | 2.04 | 2.00 | 3.45 | 9.00 | |

Mishra [53] (10 ECGs custom set) | db | 2.00 | 1.31 | 6.00 | 17.37 |

rbio3.9 | 2.00 | 0.32 | 6.00 | 10.84 | |

Craven [13] (MIT-BIH) | SPIHT | 6.10 | 1.95 | 12.00 | 4.00 |

AD-Q6 | 6.75 | 3.20 | 11.10 | 4.50 | |

Polania [15] (subset of MIT–BIH) | MMB–IHT | 6.40 | 3.76 | ||

MMB–CoSaMP | 6.40 | 3.96 | |||

Polania [51] (European QT DB) | RBM-OMP-like | 2.00 | 1.20 | 5.00 | 6.30 |

BPON | 2.00 | 1.30 | 5.00 | 8.90 | |

Chen [54] (MIT-BIH) | rbio5.5 | 2.00 | 10.03 | 5.00 | 46.63 |

rbio5.5-JBHI | 2.00 | 3.85 | 5.00 | 9.10 | |

this work (CSE) | perceptual | 3.01 | 3.11 | 4.72 | 4.88 |

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