Compressive sensing exploiting wavelet-domain

Compressive sensing exploiting wavelet-domain dependencies for ECG compression Luisa F. Polaniaa , Rafael E. Carrillob , Manuel Blanco-Velascoc and Kenneth E. Barnera a Dept. of Electrical and Computer Engineering, University of Delaware, Newark, DE, 19716, USA; b Dept. of Electrical Engineering, Ecole ´ Polytechnique F´ed´erale de Lausanne, Laussane, Switzerland; c Dept. Teor´ıa de la Se˜ nal y Comunicaciones, Universidad de Alcal´a, Madrid, Spain.

ABSTRACT Compressive sensing (CS) is an emerging signal processing paradigm that enables sub-Nyquist sampling of sparse signals. Extensive previous work has exploited the sparse representation of ECG signals in compression applications. In this paper, we propose the use of wavelet domain dependencies to further reduce the number of samples in compressive sensing-based ECG compression while decreasing the computational complexity. R wave events manifest themselves as chains of large coefficients propagating across scales to form a connected subtree of the wavelet coefficient tree. We show that the incorporation of this connectedness as additional prior information into a modified version of the CoSaMP algorithm can significantly reduce the required number of samples to achieve good quality in the reconstruction. This approach also allows more control over the ECG signal reconstruction, in particular, the QRS complex, which is typically distorted when prior information is not included in the recovery. The compression algorithm was tested upon records selected from the MIT-BIH arrhythmia database. Simulation results show that the proposed algorithm leads to high compression ratios associated with low distortion levels relative to state-of-the-art compression algorithms. Keywords: Compressed sensing, ECG compression, wavelet transform, CoSaMP.

1. INTRODUCTION The electrocardiogram (ECG) is an essential physiological signal for cardiac diagnosis. The online storage and transmission of digital ECG signals are useful in many applications, including Holter recording and telemedicine. However, the amount of ECG data grows with the increase of sampling rate, sample resolution, recording time, and the number of channels and gradually becomes a problem in applications when storage space and bandwidth are limited. Therefore, efficient ECG data compression methods, capable of reducing data redundancy and preserving the necessary diagnosis information, are required. Most ECG compression algorithms belong to either of the following categories1: direct data compression methods, which detect redundancies by direct analysis of actual signals; or transform methods, which first transform the signal to some time-frequency representation better suited for removing redundancies. The latter works in this area are characterized by hierarchical tree structures, such as embedded zero-tree wavelet (EZW)2 and set partitioning in hierarchical tree (SPIHT)3 protocols, which make use of the self-similarity of the wavelet transform across scales within a hierarchically decomposed wavelet tree. Compressive sensing (CS)4 is an new framework for the acquisition of sparse signals that enables signal reconstruction from significantly fewer data samples than suggested by conventional sampling theory. Additionally, CS promises to advance the sensor technology by offering potential applications in low-power and low-complexity sensors. Motivated by this revelation, CS has been successfully applied to ECG signals5 , reducing the required number of samples to achieve an acceptable reconstruction for medical diagnosis. Furthermore, it has shown good results as an alternative for ECG compression in the context of wireless body sensor networks6. Previous works in the area also exploit the dependencies between values and locations of the signal wavelet coefficients7 , however, ignoring the high correlation between adjacent heartbeats. Compared to conventional ECG compression algorithms, CS has some important advantages: 1) It transfers the computational burden from the encoder to the decoder, and thus offers simpler and lower-power hardware implementations for the encoder; 2) the location of the largest coefficients in the wavelet domain does not need to be encoded.

Compressive Sensing, edited by Fauzia Ahmad, Proc. of SPIE Vol. 8365, 83650E © 2012 SPIE CCC code: 0277-786X/12/$18 · doi: 10.1117/12.919478 Proc. of SPIE Vol. 8365 83650E-1 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/26/2015 Terms of Use: http://spiedl.org/terms

CS algorithms have been extended to the joint recovery of multiple sparse or compressible signals. For instance, simultaneous orthogonal matching pursuit (SOMP)8 is an extension of OMP to recover a collection of sparse signals sharing the support of the nonzero coefficients. Duarte et al. proposed the CoSOMP algorithm, a modified version of CoSaMP tailored to the common sparse supports model, to further improve the reconstruction performance of multiple signals in terms of number of measurements and quality of the recovered signals9 . Regarding ECG compression, Polania et al. used the SOMP algorithm to exploit the correlation between adjacent heartbeats5 . In this paper, we propose to exploit both wavelet domain dependencies across scales and common support between adjacent heartbeats to significantly reduce the required number of samples for accurate cardiac diagnosis. Our model for the wavelet representation of consecutive heartbeats consists of a collection of components sharing the same support and an innovation part corresponding to the varying support. The proposed reconstruction algorithm starts with CoSOMP to detect the common support. Next, this information passes as input to a modified version of the CoSaMP with partially known support (CoSaMP-PKS) algorithm10 to recover each heartbeat separately. The performance of the proposed method in terms of reconstructed signal quality is evaluated using the MIT-BIH Arrhythmia Database. Of special interest is the higher quality in the reconstruction of the the QRS complex compared to other algorithms for the recovery of multiple signals that do not exploit structure sparsity models.

2. BACKGROUND 2.1 Compressed Sensing Review Let x ∈ RN be a signal that is either K-sparse or compressible in some orthogonal PK basis Ψ, then x can be well approximated by a linear combination of a small set of vectors from Ψ, i.e. x ≈ i=1 si ψi , where K ≪ N . Let Φ be an m×N sensing matrix, m < N . Compressive sensing4 deals with the recovery of x from undersampled linear measurements of the form y = Φx = ΦΨs. Compressed sensing states that when the columns of the sparsity basis Ψ cannot sparsely represent the rows of the measurement matrix Φ and the number of measurements m is greater than O(Klog(N/K)), then it is possible to recover the original signal. In practical scenarios with noise, the signal s can be recovered by solving the following convex optimization min ksk1 s. t. ky − ΦΨsk2 ≤ ǫ,

(1)

with ǫ a bound on the measurement noise. A family of iterative greedy algorithms8, 11 are shown to enjoy a similar approximate reconstruction, generally with less computational complexity. Matching pursuit, OrthogonalMatching Pursuit (OMP)8 and CoSaMP11 being examples. However, these algorithms require more measurements for exact reconstruction than the l1 minimization approach.

2.2 Wavelet tree structure Consider a signal x of length N = 2L , given a bandpass wavelet function ψ(t) and a lowpass scaling function φ(t), the wavelet representation of x can be expressed in terms of shifted versions of φ(t) and shifted and dilated versions of ψ(t) NX L NX L −1 L −1 X x= aL,i φL,i + dj,i ψj,i , (2) i=0

j=1 i=0

where j denotes the scale of analysis and L indicates the coarsest scale. Nj = N/2j corresponds to the number of coefficients at scale j ∈ {1, . . . , L} and i represents the position, 0 ≤ i ≤ 2j − 1. The wavelet transform consists of the the scaling coefficients aL,i and wavelet coefficients dj,i . The multiscale nesting structure of the wavelet atoms induces a binary tree structure on the wavelet coefficients. We say that dj,i is the parent of its two children dj+1,2i and dj+1,2i+1 . Due to this nesting property, edges and other singularities manifest themselves in the wavelet domain as chains of large wavelet coefficients along the branches of the tree. This gives rise to the definition of a connected subtree, which is a connected set of nodes Ω satisfying the condition that whenever a coefficient dj,i ∈ Ω, then its parent is in the set Ω as well.

Proc. of SPIE Vol. 8365 83650E-2 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/26/2015 Terms of Use: http://spiedl.org/terms

2.3 CoSOMP CoSOMP9 is a reconstruction algorithm for signal ensembles under a common sparse support model. It is based on model-based compressive sensing12 , a new framework for CS that captures the underlying structure of the support of the largest coefficients of the signal using a union-of-subspaces model. Let x|Ω represent the entries of x corresponding to the set of indices Ω ⊆ {1, . . . , N }, and let ΩC denotes the complement of the set Ω. A signal model MK is then defined as the union of mK canonical K-dimensional subspaces m K [ MK = Xm , Xm := {x : x|Ωm ∈ RK , x|ΩCm = 0}; (3) m=1

each subspace Xm contains all signals x with supp(x) ∈ Ωm . Signals from Mk are called K-model sparse. At each iteration, Model-based CoSaMP replaces the best K-term approximation with a best K-term model sparse approximation. A particular case of a signal model is a connected tree model for the wavelet coefficients. Now, consider an ensemble of length-N signals {x1 , . . . , xJ } and define the set of K-sparse signals with common support as SK = {X = [xT1 . . . xTN ]T ∈ RJN s.t. xj (n) = 0 for n 6∈ Ω, Ω ⊆ {1, . . . , N }, |Ω| = K}

(4)

CoSOMP extends the ideas of model-based recovery to signal ensembles by finding the best approximation of the signal X under the model SK . Interestingly, the number of CS measurements to recover each xj (∀j) decreases as the size of the ensemble, J, increases.

2.4 Iterative algorithms with partially known support Using partially known support in the reconstruction of signals can significantly reduce the number samples because the problem is recast as finding the unknown support, which corresponds to a sparser signal than the original. In this respect, greedy algorithms are ideal to incorporate partially known support since they are based on computing the support of the sparse signal x iteratively. Once the support of the signal is computed correctly, the pseudo-inverse of the measurement matrix restricted to the corresponding columns can be used to reconstruct the actual signal x. The iterative algorithms with partially known support start with removing the contribution of the components corresponding to the known support by forming the updated residual vector r = y − ΦT0 (Φ†T0 y), where ΦT0 is the matrix formed by selecting the columns of Φ in T0 . At each iteration, when the algorithm updates the support of the signal, the known support is always included in the update. These ideas can be easily incorporated into OMP and CoSaMP, giving rise to their partially known support versions OMP-PKS and CoSaMP-PKS,10 respectively.

3. ECG SIGNALS IN THE WAVELET DOMAIN AND JOINT SPARSITY MODEL In this section, we study the wavelet representation of ECG signals to motivate the application of compressive sensing in ECG compression. Three important properties are the basis of the suitability of ECG signals for the proposed CS reconstruction algorithm: heartbeats are highly sparse in the wavelet domain, wavelet coefficients of ECG signals have a connected tree structure and the wavelet representation of a set of consecutive heartbeats share a large fraction of common support.

3.1 Wavelet tree structure of ECG signals The sparsity and structure of the wavelet coefficients of ECG signals have offered many advantages in the compression of ECG signals based on transform coding2, 3 . We also want to exploit the tree structure of the wavelet coefficients but in a compressive sensing framework. Let us consider a 10 sec signal taken from the record 117 of the MIT-BIH Arrhythmia Database shown in the upper panel of Fig. 1. The lower panel of Fig. 1. shows the Daubechies 4 coefficient vectors Dj , j = 1, . . . , 4, and A4 after circularly shifting them so that they are plotted against a physically meaningful time. The coarsest scale is L = 4. First, notice that the node a4,2 is above d4,2 , which in turn is above d3,2 and d3,3 . Continuing on down, d3,2 is above d2,2 and d2,3 , while d3,3 is

Proc. of SPIE Vol. 8365 83650E-3 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/26/2015 Terms of Use: http://spiedl.org/terms

ECG sianl

-100

-200 -300

-400 500

1000

1500

2000

2500

3000

3500

3000

3500

Wavelet coefficients

A4

D4

D3

D2

H-

Dl

SOC

lOOfl

1500

2000

2500

Figure 1. Daubechies 4 wavelet representation for ECG time 'I series. Record 117. L=4. *1. x Cter

above d2,4 and d2,5 and so forth. Therefore, there is a tree structure under each of the level L coefficients. Second, notice that the large R wave events line up well with spikes at different scales forming connected subtrees. Thus, the ECG signals have a well-defined tree structure in their wavelet coefficients that can be exploited to reconstruct the QRS complex with high quality. Moreover, any abnormality, such as arrhythmia, can be reconstructed given that singularities are represented as chains of large coefficients at different scales in the wavelet domain.

3.2 Joint sparsity of consecutive heartbeats In addition to the wavelet domain dependencies, there exists a high correlation between consecutive heartbeats due to the quasi-periodic structure of ECG signals. This correlation implies that a high fraction of common support can be exploited to further reduce the required number of samples for accurate reconstruction. Let {s1 , . . . , sJ } represents a sequence of the wavelet representation coefficients of J consecutive heartbeats and define the ground set [N ] = {1, 2, . . . , N }. Let supp(M(sj , K)) denotes the support of the best K-term

Proc. of SPIE Vol. 8365 83650E-4 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/26/2015 Terms of Use: http://spiedl.org/terms

model sparse approximation of sj using a connected tree model for the wavelet coefficients. The common support between J consecutive heartbeats is defined as ΩC = {z ∈ [N ] : z ∈ supp(M(sj , K)) for all j, 1 ≤ j ≤ J}.

(5)

Fig. 2 illustrates how the fraction of common support between J consecutive heartbeats, |ΩC |/K, varies as the number of heartbeats J increases. The experiment was carried out over a single-lead ECG extracted from the MIT-BIH Arrhythmia Database and averaged over records 100, 102, 107, 109, 115 and 117 . The decomposition level was set to L = 4 and the heartbeats were normalized to have a length N = 1024. The results of Fig. 2 suggest to model the support of each heartbeat as the sum of a common sparse component and a sparse innovation component, where the cardinality of the common sparse component depends on the number of consecutive heartbeats J. Fraction of common support

0.7 0.65 0.6 0.55 0.5 0

5

10 15 20 Number of heartbeats

25

30

Figure 2. Fraction of common support as a function of the number of consecutive heartbeats

4. COMPRESSIVE SENSING RECONSTRUCTION OF ECG SIGNALS The block diagram of the proposed compression scheme is presented in Fig. 3 and is composed by an encoder and a decoder stage. Beat information ntizati

Original _-_, ECG data

Beat Detection

Period Normalization

Sampling

-

Quantization

Store/transmit

Store/transmit

(a) Encoder

Beat irdormation

Compressed data

CS reconstruction algorithm

Period Recovery

Recovered ECG signal

(b) Decoder

Figure 3. Block diagram of the proposed compression scheme

Proc. of SPIE Vol. 8365 83650E-5 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/26/2015 Terms of Use: http://spiedl.org/terms

4.1 Encoder The encoder starts with the period normalization of the signals to properly exploit the interbeat dependencies. Each heartbeat is normalized to a fixed length N using cubic spline interpolation. The normalized heartbeats are organized column-wise in a matrix denoted by X while the original periods are sent to the decoder as side information since they are needed for the reconstruction stage. The information we gather about X can be described by Y = ΦX, where Φ is a m × N matrix, that needs to satisfy the restricted amplification property [4] in order to recover the K-term model sparse approximation of the original signal. After sampling, we encode each column of Y through scalar quantization.

4.2 Decoder The decoder contains the reconstruction algorithm and the period recovery stage to restore the signals to their original length. Let xj , j = 1, 2, . . . , J denote each column of the matrix X, and let sj denotes the corresponding wavelet representation of xj . For ECG signals, the support of sj , denoted as Ωj , is assumed to share a common component with the support of the other heartbeats, denoted as ΩC , and also contains a sparse innovation component Ωji : Ωj = ΩC + Ωji , j ∈ {1, 2, . . . , J}. The proposed algorithm for the reconstruction of heartbeats starts using the CoSOMP algorithm to find the common sparse component ΩC . Instead of using the best K-term approximation A(x, K) in the pruning steps as is done in the regular CoSOMP, we propose to use the best K-term model sparse approximation M(x, K) since we already know that the most significant wavelet coefficients of the hearbeats lie in a connected subtree structure. Next, ΩC is passed as input to the model-based version of the CoSaMP algorithm with partially know support10 . In this step, again, the pruning of the signal residual estimate and the pruning of the signal estimate are done according to the connected tree model. The entire algorithm is specified in Algorithm 1. Note that the individual reconstructions can be performed in parallel, thus representing an advantage in speed where multicore architectures are available.

5. EXPERIMENTAL RESULTS This section illustrates the effectiveness of the proposed algorithm by means of numerical experiments and their comparison with other CS reconstruction algorithms. The performance of the proposed compression scheme is also compared with a state-of-the-art algorithm for ECG compression. Experiments are carried out over a 10-min long single-ECG lead from the MIT-BIH Arrhythmia Database. Daubechies db4 wavelets are used as sparsifying transform with a decomposition level L = 4. The signals are measured using matrices that have i.i.d. entries drawn from a standard normal distribution with normalized columns. For our simulations, we average 20 repetitions of each experiment. Instead of recovering the ECG signal heartbeat by heartbeat, we incorporate a sliding window of fixed length in order to exploit the common support between adjacent heartbeats. However, since we are also interested in generating an approximate real time transmission of the recovered signal, the length of the window is set to only six heartbeats. The compression ratio (CR) and the percentage root-mean-square difference (PRD) are used as performance measures. The CR is computed as follows: CR =

11 × N (Bm × m) + (Bp × P )

where N is the length of the input signal, 11 is the number of bits to encode each sample, Bm are the bits for the measurements, m is the number of measurements, P is the number of ECG cycles and Bp are the bits for storage of each cycle. Let x and x ˆ be the N -dimensional vectors representing the original and reconstructed signals, respectively. The PRD is defined as PRD = (kx − x ˆk2 /kxk2 ) × 100.

Proc. of SPIE Vol. 8365 83650E-6 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/26/2015 Terms of Use: http://spiedl.org/terms

CoSOMP−I CoSOMP− Model CoSOMP CoSaMP CoSaMP− Model

20 0 −20 −40 −60 0.1

0.2

0.3 m/N

0.4

0.5

40 CoSOMP−I CoSOMP− Model CoSOMP CoSaMP CoSaMP− Model

20 0 −20 −40 −60 0.1

0.2

0.3 m/N

0.4

Reconstruction SNR

40

Reconstruction SNR

Reconstruction SNR

Algorithm 1 CS-based algorithm for the reconstruction of ECG signals Require: CS matrix Φ, measurements {yj }Jj=1 , sparsity level K, cardinality of the known support KC = |ΩC |. {Lines 1 to 13 correspond to the CoSOMP algorithm. We incorporate the wavelet tree model in lines 6 and 10.} 1: Initialize x ˆj,0 = 0, rj = yj , l = 0. 2: while halting criterion false do 3: l ← l + 1. 4: ej ← ΦT rj , j = 1, . . . , J PJ 5: e = j=1 (ej · ej ) 6: Λ = supp(M(e, 2KC )) 7: T ← Λ ∪ supp (ˆ xj,l−1 ) 8: bj |T ← Φ†T yj , bj |T C ← 0, j = 1, . . . , J P 9: b = Jj=1 (bj · bj ) 10: ΩC = supp(M(b, KC )) 11: x ˆj,l |ΩC ← bj |ΩC , xˆj,l |ΩCC ← 0, j = 1, . . . , J 12: rj ← yj − Φˆ xj,l , j = 1, . . . , J 13: end while {Lines 15 to 24 corresponds to the model-based version of CoSaMP with partially known support. We incorporate the wavelet tree model in lines 18 and 22.} 14: Ki = K − KC ; 15: while halting criterion false do 16: l ← l + 1. 17: ej ← ΦT rj , j = 1, . . . , J 18: Λj = supp(M(ej , 2Ki )), j = 1, . . . , J 19: Tj ← Λj ∪ supp (ˆ xj,l−1 ), j = 1, . . . , J 20: bj |Tj ← Φ†Tj yj , bj |TjC ← 0, j = 1, . . . , J 21: Aj |ΩCC ← bj |ΩCC , Aj |ΩC ← 0, j = 1, . . . , J 22: x ˆj,l ← bj |(ΩC ∪ supp(M(Aj ,Ki )) , j = 1, . . . , J 23: rj ← yj − Φˆ xj,l , j = 1, . . . , J 24: end while 25: return xj ← x ˆj,l , j = 1, . . . , J

0.5

40 CoSOMP−I CoSOMP− Model CoSOMP CoSaMP CoSaMP− Model

20 0 −20 −40 −60 0.1

0.2

0.3 m/N

0.4

0.5

Figure 4. Performance of the algorithms CoSOMP-I, CoSOMP, CoSOMP-Model, CoSaMP and CoSaMP-Model. The reconstruction SNR is calculated as a function of the number of measurements. L: Record 117, M: Record 119, R: Record 115.

5.1 Performance evaluation of the reconstruction algorithm The first experiment seeks to compare the performance of Algorithm 1 (denoted as CoSOMP-I) with CoSOMP, CoSOMP using the the best K-term model sparse approximation in the pruning steps (denoted as CoSOMPModel), CoSaMP and Model-based CoSaMP (denoted as CoSaMP-Model). From Fig. 1, we assume that an appropriate value for the percentage of common support between the six heartbeats in the sliding window is 58%. The simulation results are shown in Fig. 4 for the records 117, 119 and 115.

Proc. of SPIE Vol. 8365 83650E-7 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/26/2015 Terms of Use: http://spiedl.org/terms

The incorporation of a tree structure in the wavelet coefficients of the heartbeats results in a large improvement in terms of the number of measurements required for reconstruction as shown in Fig. 4. By comparing the plots of CoSOMP and CoSaMP, we can also assert that the common support model leads to an improvement in the performance of the algorithms since the number of measurements is significantly reduced by exploiting correlation between adjacent heartbeats. As expected, the performance of CoSaMP is inferior compared to the other algorithms since it exploits neither wavelet domain dependencies nor common support model. Although assuming a common support model between consecutive heartbeats improves the performance of the reconstruction algorithms, it is not the right model since consecutive heartbeats do not share the exact same support; there exists the contribution of an innovation sparse component. That is why Algorithm 1 outperforms CoSOMP and CoSOMP-Model for all the records.

5.2 Comparison of the proposed ECG compression method with SPIHT Fig. 5 shows the performance of the proposed compressor compared with SPIHT for ECG records 117, 115 and 119. The records 115 and 119 are included in the set to evaluate the performance of the algorithm in the case of irregular heartbeats. We use four bits for the quantization of the measurements, Bm = 4, and Bp = 4 bits for the quantization of the beat periods. Although SPIHT also exploits the underlying tree structure of the wavelet coefficients, the results in Fig. 5 indicate that the proposed algorithm has better performance for low PRDs, which is precisely the range of main interest, where the reconstructed signals have very good quality for medical diagnosis. For low PRDs, CoSOMPI outperforms SPIHT in the sense that it achieves higher compression ratios with better signal quality for the ECG records 117, 119 and 115. As expected, given that the heartbeats in the record 117 are very regular, the compression ratio achieved for this record is higher than for the others. 16 Proposed method SPIHT

14

5 2

3

12

PRD

4

5

6

CR

10

1

14

Proposed method SPIHT

12 CR

CR

15

10

10

8

8

6

6

4 1

2

3

4 PRD

5

6

7

Proposed method SPIHT

4 1

2

3

4 PRD

5

6

7

Figure 5. Comparison of the proposed ECG compression method with SPIHT. Compression ratio as a function of the PRD for different ECG records. L: Record 117, M: Record 119, R: Record 115.

6. CONCLUSION This paper proposed a CS-based ECG compression scheme aimed to demonstrate that there are significant performance gains to be made by exploiting the highly structured nature of the ECG signal, beyond the simplistic sparse model. Compared to SPIHT, we obtained superior results for low PRD values, indicating that the proposed algorithm is able to compress the ECG data, and still achieve good quality in the reconstructed signal for accurate medical diagnosis. We have provided an algorithm that exploits the high correlation between adjacent heartbeats, however, without ignoring the unique information provided by each individual heartbeat for medical diagnosis. Unlike state-of-the-art algorithms for ECG compression, our method does not need to encode the location of the largest coefficients in the wavelet domain and offers a low-complexity encoder.

Proc. of SPIE Vol. 8365 83650E-8 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/26/2015 Terms of Use: http://spiedl.org/terms

REFERENCES [1] Blanco-Velasco, M., Cruz-Rold´an, F., L´opez, F., Bravo, A., and Mart´ınez, D., “A low computational complexity algorithm for ECG signal compression,” Medical Eng. Phys. 26, 553 –568 (Sept. 2004). [2] Hilton, M., “Wavelet and wavelet packet compression of electrocardiograms,” IEEE T-BME 44, 394 –402 (May 1997). [3] Lu, Z., Kim, D. Y., and Pearlman, W. A., “Wavelet compression of ECG signals by the set partitioning in hierarchical trees algorithm,” IEEE T-BME 47, 849 –856 (July 2000). [4] Cand`es, E. and Wakin, M., “An introduction to compressive sampling,” IEEE Signal Processing Magazine 25, 21 –30 (Mar. 2008). [5] Polania, L. F., Carrillo, R. E., Blanco-Velasco, M., and Barner, K. E., “Compressed sensing based method for ECG compression,” in [Proc., IEEE ICASSP], (May 2011). [6] Kanoun, K., Mamaghanian, H., Khaled, N., and Atienza, D., “A real-time compressed sensing-based personal electrocardiogram monitoring system,” in [Design, Automation Test in Europe Conference Exhibition (DATE)], 1 –6 (March 2011). [7] Mamaghanian, H., Khaled, N., Atienza, D., and Vandergheynst, P., “Structured sparsity models for compressively sensed electrocardiogram signals: A comparative study,” in [IEEE Biomedical Circuits and Systems Conference (BioCAS)], 125 –128 (Nov. 2011). [8] Tropp, J., Gilbert, A., and Strauss, M., “Simultaneous sparse approximation via greedy pursuit,” in [Proc., IEEE ICASSP], 5, v/721 – v/724 Vol. 5 (Mar. 2005). [9] Duarte, M., Cevher, V., and Baraniuk, R., “Model-based compressive sensing for signal ensembles,” in [47th Annual Allerton Conference on Communication, Control, and Computing], 244 –250 (Oct 2009). [10] Carrillo, R. E., Polania, L. F., and Barner, K. E., “Iterative algorithms for compressed sensing with partially known support,” in [Proc., IEEE ICASSP], 3654 –3657 (Mar. 2010). [11] Needell, D. and Tropp, J. A., “Cosamp: Iterative signal recovery from incomplete and inaccurate samples,” Applied and Computational Harmonic Analysis 26, 301–321 (Apr. 2008). [12] Baraniuk, R., Cevher, V., Duarte, M., and Hegde, C., “Model-based compressive sensing,” IEEE Transactions on Information Theory 56, 1982 –2001 (April 2010). [13] Zigel, Y., Cohen, A., and Katz, A., “The weighted diagnostic distortion (wdd) measure for ECG signal compression,” IEEE Transactions on Biomedical Engineering 47, 1422 –1430 (Nov. 2000).

Proc. of SPIE Vol. 8365 83650E-9 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/26/2015 Terms of Use: http://spiedl.org/terms