ECG COMPRESSION VIA MATRIX COMPLETION Luisa F. Polania ⋆ , Rafael E. Carrillo ⋆ , Manuel Blanco-Velasco † and Kenneth E. Barner ⋆ ⋆ Electrical and Computer Engineering Dept., University of Delaware, Newark, DE, 19716, USA {polania, carrillo, barner}@ee.udel.edu †Dept. Teor´ıa de la Se ˜ nal y Comunicaciones, Universidad de Alcal´ a, Madrid, Spain manuel.blanco@uah.es ABSTRACT Matrix completion is a new paradigm in signal process- ing that seeks to recover a low-rank matrix based on small number of observations. We propose an ECG compression scheme that organizes the ECG data in a low-rank matrix via a period normalization stage. By using matrix completion, we can recover the ECG data matrix from a few number of entries, and therefore achieve high compression ratios com- parable to the ones obtained by existing compression tech- niques. The proposed scheme offers a low-complexity en- coder and good tolerance to quantization noise, which allows a significant reduction in the number of bits per sample and, yet achieve good quality in the reconstruction. 1. INTRODUCTION An ECG is an important physiological signal for cardiac dis- ease diagnostics. With the increasing use of modern electro- cardiogram monitoring devices that generate vast amounts of data and require huge storage capacity, ECG compression becomes mandatory to efficiently store and retrieve this data from medical database. ECG compression techniques are classified into three categories [1]: Direct methods, trans- form methods and other compression methods. In the first category, the ECG samples are processed directly paying at- tention to the redundancy among them. Several schemes such as the AZTEC (Amplitude Zone Epoch Coding), FAN, TP (Tunning Point) and CORTES (Coordinate Reduction Time Encoding System) [2] have been specifically devel- oped for ECG data compression. In the second category, the wavelet transform-based methods play an interesting role due to their easy implementation and efficiency. The lat- ter works in this area are characterized by hierarchical tree structures, such as embedded zero-tree wavelet (EZW) [3] and set partitioning in hierarchical tree (SPIHT) [4] proto- cols, which make use of the self-similarity of the wavelet transform across scales within a hierarchically decomposed wavelet tree. Compressive sensing (CS) is a new approach for the ac- quisition and recovery of sparse signals that enables sam- pling rates significantly below the classical Nyquist rate [5]. CS can be applied in distributed scenarios [6, 7], where the objective is to independently compress several signals that are characterized by presenting a sparse correlation and then, jointly decompress them exploiting such signal correlations. Based on these ideas, we proposed in [8] a CS-based ECG compression framework that utilizes distributed compressed sensing to exploit the inter-beat correlation structure. An interesting line of research focuses on transform- ing the original one-dimensional ECG waveforms into two- dimensional information, followed by a processing stage us- ing image processing tools. The idea of these methods is to exploit both intra-beat and inter-beat correlations. For ex- ample, Lee and Buckley [9] applied DCT transform to an ECG data matrix composed of regular heartbeats, and Bil- gin [10] applied JPEG2000 to the similar constructed matrix. Following this line of thought, we propose a bidimensional ECG compression scheme based on matrix completion, that exploits both intra and inter-beat correlations. Matrix completion refers to the problem of reconstruct- ing a low-rank matrix from a small set of observed entries possibly corrupted by noise. Cand` es and Recht in [11] ex- tend the CS ideas (recovery of sparse signals) to the comple- tion of low-rank matrices, in which the ℓ 1 norm is replaced with the nuclear norm of a matrix. They introduce a convex relaxation to the NP-hard problem of finding the matrix with minimum rank that matches the observed entries. They also introduce the concept of incoherence property and prove that solving the convex relaxation problem for a low-rank matrix that satisfies the incoherence property, reaches exact recov- erability with high probability provided that the number of observed entries is large enough. Directly solving the rank minimization problem requires solving a semidefinite program whose complexity is cubic in the matrix size. Recently, many authors have proposed ef- ficient algorithms for solving the low-rank matrix comple- tion problem, such as Singular Value Threshholding [12], OptSpace [13], and Fixed Point Continuation algorithms (FPC) [14]. For our method, we decide to use FPC due to its robustness, efficiency and great recoverability. The main assumption of the matrix completion prob- lem is that the matrix to be recovered has to be low-rank. To meet this requirement, our method starts by forming a low-rank matrix where each column corresponds to an ECG cycle whose length has been normalized. The subsequent step refers to the sampling process where the observed set is drawn uniformly at random from the formed matrix. The decoding process is performed using matrix completion fol- lowed by period de-normalization. The proposed scheme of- fers a low-complexity encoder and good tolerance to quanti- zation noise. The performance of the proposed algorithm, in terms of reconstructed signal quality and compression ratio, is evaluated using the MIT-BIH Arrhythmia Database. 2. REVIEW OF MATRIX COMPLETION Matrix completion refers to the problem of reconstructing a low-rank matrix M from a small set of observed entries possibly corrupted by noise. A full rank matrix of dimensions (n × n) has n 2 degrees of freedom and therefore, it is not possible to estimate all its values from a small subset of all the entries in the matrix.