ECG signal reconstruction from undersampled measurement using a trained overcomplete dictionary

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Introduction
Electrocardiogram (ECG) is one of the most significant types of signals for cardiac analyses and diagnostics. Proper analyses of ECG signals requires sizable data which demand more storage and bandwidth to transmit for wearable and mobile types of health monitoring devices, The amount of signal data has seen explosive growth while pressures on energy efficiency increase.
Traditional compression is performed after the whole data is obtained. This offline approach can be efficient in analyzing the data to reduce redundancies in the signal, but storing data can incur high power. Compressive Sensing (CS) [1]- [4] shows the underlying signal can be reconstructed precisely from fewer samples than Shannon's so long as the signal is sparse or compressible in a certain domain, though the reconstructed signal will contain small distortions. In this paper, we propose a new compressive sensing and reconstruction technique for ECG signals by using a trained overcomplete dictionary to be constructed by K-SVD [5]. A comparison is presented with conventional compressive sensing signal reconstruction method using MIT-BIH arrhythmia database [6].

Compressive Sensing Framework
Compressive sensing (CS) uses a linear transform to compress a signal and reconstructs it by exploiting its sparsity, which means that most of the components are zero or almost zero to represent the signal. Mathematically speaking, it assumes an N-dimensional signal can be represented as where ∈ ℝ , and ∈ ℝ × . By sensing system, the observation can be represented as where ∈ ℝ , and ∈ ℝ × , which is called the sensing matrix. If ≪ , the signal sensed by the system is undersampled. That means Eq.(2) is underdetermined, where is the observed signal from the sensing system and is the original signal to be reconstructed. There is no unique solution due to the ill-posed condition. However, if it is assumed that the original vector is a sparse or compressible signal, which has a few coefficients in the transformed domain by a basis matrix , we can find sparse approximation by adding sparsity constraint to the ill-posed problem, which is represented as Algorithms for finding sparse approximation include orthogonal matching pursuit (OMP) [7], and least absolute shrinkage and selection operator (LASSO) [8].

Dictionary Learning
The approximation performance to find sparse approximation in terms of the approximation quality and the sparsity of the sparse vector depend on the dictionary as well as the signal itself. If the dictionary contains well-estimated atoms or basis vectors, then the sparse vector will have a small number of non-zero elements. Universally used dictionaries, such as those for discrete cosine transform, Fourier transform, and wavelet transform, are orthogonal dictionaries. They have mathematical simplicity and few redundancies to represent dictionaries themselves, but they are not suitable to represent signals with few redundancies. Due to the limitation of the orthogonal dictionaries, researchers proposed overcomplete dictionaries by data-driven learning. K-SVD is one of the popular algorithms for constructing overcomplete dictionaries by learning. Given a set of -dimensional signals = [y ! y ! … y ! ] ∈ ℝ !×! , the goal of K-SVD is to find the optimal dictionary ∈ ℝ !×! , where is an overcomplete dictionary (K > ).
! is sparsity, which means the maximum number of non-zero coefficients to represent given signal.

Proposed Algorithm
Our proposed algorithm is a new approach to reconstructing a signal from compressive sensing system. Our proposed algorithm has a preprocessing stage for constructing a dictionary, which can make the signal sparse or compressible in the transformed domain. Fig.1 shows the full procedure of the proposed algorithm.  Fig.1 shows the flow of our dictionary construction procedure for the overcomplete dictionary from the given dataset. For constructing a set of training data, our algorithm performs normalization. To remove the effects of the bias term, we also make the training data zero-mean. After normalization and removal of the bias term, a set of normalized signal vectors can be presented to Y = [y ! y ! … y ! ] ∈ ℝ !×! as a data set for training. Using the training dataset Y, the optimized dictionary to represent the data under the sparsity constraint is obtained by K-SVD dictionary learning. The trained dictionary performs well when representing signals with a few coefficients, but not signals with generality due to reflecting the characteristics of the signal used at the training stage. To compensate for this shortcoming, our proposed method uses the combined dictionary, which is made up of general basis matrices, such as DCT or wavelet basis matrix, as atoms of the dictionary.

Reconstruction Strategy
Our proposed algorithm assumes that the signal would be obtained by random sampling, which means that the signal is selectively digitized in a time window. The sensing mechanism can be represented as a sensing matrix, Φ ∈ ℝ !×! (m < ), which can select random samples. The observation can be presented using the sensing matrix Φ. The key point of our proposed algorithm is the signal model. We assume that the original signal can be presented as The signal is composed of two parts: the signal shape and the bias term, which defines vector c. Our target ECG signal has a changeable baseline, so it is needs to separate the bias term to estimate the shape part more accurately. We can take the sensing matrix and our proposed signal model to derive the equation of observation y, thereby obtaining the following equation: The observation y also has two parts: the shape (i.e., the zero-mean signal part) and the bias term. Since the sensing matrix is a simply random selection matrix, we can easily estimate the bias term by taking the average of the observation y for the parameter estimation step in Fig.1. The bias term, called vector c, is where 1 means a vector whose elements are all 1. After estimating the parameter and subtracting it from observation, the simply zero-mean signal part remains.
The zero-mean observation part can be a simple linear equation with ill-posed conditions. In this paper, LASSO (Least Absolute Shrinkage and Selection Operator) is used as a solver for the convex problem. In this paper, we use MATLAB toolbox provided by sparselab [10] as a solver.

Experimental Results
For comparing the proposed algorithm with a conventional CS-based signal reconstruction algorithm, we have chosen the MIT-BIH arrhythmia database as the test signals. It provides 48 sets of 2-lead ECG signals at 360 11-bit samples/sec. Our metrics for evaluating signal reconstruction are percentage root-mean-squared distortion (PRD) and compression ratio (CR). PRD [9] shows the reconstruction error as a percentage and is defined as where is the number of samples, and x ! and x ! are respectively the original data and the reconstructed data from the proposed algorithm. CR is defined as where b !"#$ and b !"#$ represent the number of bits required for the original and compressed signals, respectively.

Dictionary Construction
We have chosen 24 out of 48 signals from the MIT-BIH arrhythmia database for training our algorithm and constructing the dictionary. Another 24 signals except for the signals used in training are utilized to evaluate the performance of our proposed algorithm. Table 1 depicts the signals included in training dataset.
User-specific parameters on the procedures of dictionary learning include sparsity, the size of dictionary, and the size of minimum signal block. The proposed algorithm uses = as sparsity constraint for dictionary construction. It also uses 64 samples as a minimum signal block, and 5012 atoms as a dictionary size, which means that the trained dictionary is a 64×5012 matrix. After obtaining the trained dictionary, we combine it with 64×64 DCT basis and wavelet basis, and the final size of the combined dictionary is 64×5140.

Comparison of the reconstruction results
Figs. 2(a) and 2(b) show the results using conventional DCT and wavelet bases (daubechies4) with half of the randomly selected samples from original signal, i.e., CR=50%. Our method using the combined dictionary achieves PRD=1.34% when CR=50%. Fig. 3 depicts the performance over the MIT-BIH arrhythmia database. It shows high CR leads to high distortion. According to the dictionary or basis used into the reconstruction method, the performance is shown differently. Our proposed method shows less distortion than the other conventional methods do.

Conclusion
We propose reconstructing ECG signals from undersampled data by compressive sensing. The signal reconstruction from undersampled data is  performed by finding a sparse solution using a given dictionary, and the overall performance depends on the dictionary utilized in the approximation process. Dictionaries constructed by our method achieve good approximation performance by combining the trained dictionary by K-SVD with universal types of dictionaries such as DCT or wavelet transform. Our algorithm shows good performance with low distortion in terms of the CR and PRD over MIT-BIH arrhythmia database. Since the algorithm is learning-based method, the performance can be improved if the training dataset is organized well.