#### Read Sparse Signal Representation and Applications text version

`Sparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsDr. K. P. Soman and Dr. M. Sabarimalai ManikandanCenter for Excellence in Computational Engineering and Networking Amrita University, Coimbatore Campus E-mail: [email protected].Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsElementary signals: what and why they are?      Unit sample sequence [n] Unit step sequence u[n]k Rectangular pulse rect( 2N+1 )Signum sgn[n] Sinc sinc(o n) =sin(o n) o nComplex exponentials x[n] = e (+jo )n Sinusoidal signal x[n] = Acos(o n + )Elementary signals are used to represent more complicated signals. Representation simplifies the analysis and design of systems.Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSignal Operations: why do we perform?   Time shifting x[n + K ] Time Scaling x[Cn]] Time Inversion x[-n] Combined Operations x[Cn + K ]Any arbitrary signal x[n] can be represented as a linear combination of time-shifted impulse functions:x[n] ==-x[k][n - k](1)Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsWhy Signal Processing?Most natural signals are non-stationary and have highly complex time-varying spectro-temporal characteristics. Mixture of many sources Composition of mixed events Various kinds of noise and artifacts The SP is challenging task because the natural signals are typically having different shapes, amplitudes, durations and frequency content, which are not known in many different applications and systems   Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSignal Representation Using Basis FunctionsA set {n }N is called a orthonormal basis for RN if the n=1 vectors in the set span RN and are linearly independent Let x  RN×1 be the input signal that is spanned by N basis functions {n }N . Then, a discrete-time signal x can be n=1 represented asNx=n=1n  n = (2)where  = [1 , 2 , 3 , ......N ] is the transform coefficients vector that is computed as n = x,  n .For some transform matrix, the transform coefficients vector  has a small number of large amplitude coefficients and a large number of small amplitude coefficientsSparse Signal Representation and ApplicationsDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSome of Representation or Transform Matrices       Fourier transform matrix discrete cosines (DCT matrix) and discrete sines (DST matrix) Haar transform matrix wavelet and wavelet packets matrices Gabor filters curvelets, ridgelets, contourlets, bandelets, shearlets directionlets, grouplets, chirplets Hermite polynomials, and so onDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsLimitations of Fixed Representation MatrixThe Fourier transform is suitable for analysis of the steady-state sinusoidal signals but it fails to capture the sharp changes and discontinuities in the signals. In the STFT-based methods, the choices for widths of the time-window affect the frequency and time resolution. The common problem in well-known wavelet transform-based methods is which mother wavelet function and characteristic scale provides the best time-frequency resolution for detection of transients and non-transients.Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSparse Representation/RecoveryDefinition: The sparse representation theory has shown that sparse signals can be exactly reconstructed from a small number of elementary signals (or atoms). The sparse representation of natural signals can be achieved by exploiting its sparsity or compressibility. A natural signal is said to be sparse signal if that can be compactly expressed as a linear combination of a few small number of basis vectors. Sparse representation has become an invaluable tool as compared to direct time-domain and transform-domain signal processing methods.Sparse Signal Representation and ApplicationsDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Representation: Applicationsaudio/image/video processing tasks (compression, denoising, deblurring, inpainting, and superresolution) speech enhancement and recognition signal detection and classification face recognition, array processing, blind source separation sensor networks and cognitive radios power quality disturbances underwater acoustic communications data acquisition and imaging technologiesSparse Signal Representation and Applications      Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal: SparsityDefinition: A signal can be sparse or compressible in some transform matrix  when the transform coefficients vector  has a small number of large amplitude coefficients and a large number of small amplitude coefficients. Observations: Most of the energy is concentrated in a few transform coefficients in a vector  The other N - K coefficients have less contribution in representing a signal vector x  RN×1 . The insignificant coefficients are set to zero in coding scheme.Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsExample for non-sparse signalRandom Noise Sequence 2 0 -2 50 100 150 200 250 300 350 400 450 500Original Sequence: Histogram 15 10 5 0 -3 -2 -1 0 DFT: Histogram 20 10 1 2DFT coefficients 0.8 0.6 0.4 0.2 50 100 150 200 250 300 350 400 450 50000.10.20.30.40.50.60.70.80.91DWT coefficients 0.5 0 -0.5 -1 50 100 150 200 250 300 350 400 450 500 550DWT: Histogram 20 10 0 -1amplitude-0.8-0.6-0.4-0.200.20.40.60.8DCT coefficients 0.5 0 -0.5 -1 50 100 150 200 250 300 sample number 350 400 450 500DCT: Histogram 15 10 5 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsExample for sparsity in time-domainRandom Noise Sequence 0.5 0 -0.5 -1 -1.5 50 100 150 200 250 300 350 400 450 500 DFT coefficients8 400 200 0 Original Sequence: Histogram-1.5-1-0.5 DFT: Histogram00.50.8 0.6 0.4 0.2 50 1amplitude6 4 210015020025000.10.20.30.40.50.60.70.80.9DWT coefficients400DWT: Histogram0.5 0200050 1 0.5 0 -0.510015020025030035040045050055015 10 5 0-0.200.20.40.60.81DCT coefficientsDCT: Histogram50100150200 250 300 sample number350400450500-0.8-0.6-0.4-0.200.20.40.60.8Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSpectrogram of a Speech SignalUtterance of the wordAmplitude0.2 0 -0.200.10.20.30.40.50.6Time (sec)8000Frequency (Hz)6000 4000 2000 00.050.10.150.20.250.30.350.40.450.50.55Time (sec)Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSpectrum of a Voiced and Unvoiced SoundsVoiced Sounds 1AmplitudeVoiced Sounds 1 0.5 0 -0.5 -1Amplitude0.5 0 -0.5 -1200800 1000 1200 1400 1600 Time (seconds) Single-Sided Amplitude Spectrum of Voiced Sounds400600200 400 600 800 1000 1200 1400 1600 1800 2000 Time (seconds) Single-Sided Amplitude Spectrum of Unvoiced Sounds30030|Y(f)||Y(f)|20020101000 0 1000 2000 3000 4000 5000 Frequency (Hz) 6000 7000 800000100020003000 4000 5000 Frequency (Hz)600070008000Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsMultiresolution signal decompositionP R T RS ECG beat ECG+noise ECG+baseline artifact2 0 -2 0.05 0 -0.05 -0.1 0.2 0 -0.2 0.5 0 -0.5 0.5 0 -0.5 0.5 0 -0.5 1 0 -1 500 1000 1500 2000 500 1000 1500 2000 500 1000x[n] D1[n]D2[n] D [n]3amplitudeD [n] 4 D5[n] A5[n]sample numberDr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSparse Signal: SparsityThe value of K is computed as K =  0 , where . 0 denotes the 0 -norm which counts the number of non-zero entries in . Concluding Remarks: A sparse signal x can be exactly represented or approximated by the linear combination of K basis functions with shorter transform coefficients vector. In such a reconstruction process, the reconstruction error by a K -term representation decays exponentially as K increases.Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSparse Representation: Dictionary LearningNeed for Dictionary Learning: In practice, a signal is composed of impulsive and oscillatory transients, spikes and low-frequency components. Nature: The composite signal may not exhibit sparsity in one transform basis matrix because some of its components are sparse in one domain while other components are sparse in another domain. The signals may exhibit sparsity in either time-domain or frequency-domain. For example, the 50 Hz powerline signal is sparse in the frequency-domain and the impulse or spikes component is sparse in the time-domain.Sparse Signal Representation and ApplicationsDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Representation: Dictionary LearningProblem with Fixed Basis: In practice, the composite signal (spike is superimposed on powerline signal) exhibits sparsity in neither time-domain nor frequency-domain. In such cases, a fixed orthogonal basis functions are not flexible enough to capture the complex local waves of a signal. For example, a fixed elementary cosine waveforms of discrete cosine transform (DCT) matrix fails to capture transient parts of biosignals. Detection and suppression of impulsive noise in speech waveform. Compression of slow varying signals with spikes.Sparse Signal Representation and ApplicationsDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Representation: Need for Best Basis FunctionsRemedies: To improve the sparsity of composite signals, one has to construct a transform matrix with the best basis functions. One way to process such signal is to work with an large dictionary matrix. A best basis set from a dictionary matrix used to sparsify the data may yield highly compact representations of many natural signals.Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSparse Representation: What is Dictionary?Dictionary: A dictionary is a collection of elementary waveforms or prototype atoms or basis functions. A dictionary matrix D of dimension N × L can be represented as D = { 1 | 2 | 3 |..........| L }. The column vectors { l }L of an dictionary D are l=1 discrete-time elementary signals of length N × 1, called dictionary atoms or basis functions. The atoms in the pre-defined dictionary may be pairwise orthogonal, linear independent, linear dependent, or not orthogonal.Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSparse Representation: Classification of DictionariesBased on the number of atoms L and the signal length N, the pre-defined dictionary, D  RN×M , could be classified into three categories: (i ) when L &gt; N, D is called overcomplete, or redundant dictionary. (ii ) when L &lt; N, D is called undercomplete dictionary. (iii ) D is said to be complete dictionary if L = N. Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSome of Sparse Transform Matricesdirac and heaviside functions Fourier transform matrix and Fourier, short-time Fourier transform (STFT) discrete cosines (DCT matrix) and discrete sines (DST matrix) Haar transform matrix wavelet and wavelet packets matrices Gabor filters curvelets, ridgelets, contourlets, bandelets, shearlets directionlets, grouplets, chirplets Hermite polynomials, and so on       Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSparse Representation: Research ProblemsThe dirac dictionary can be used to detect the spikes in a signal and the discrete cosines dictionary can provide sinusoidal waveforms The SR from redundant dictionaries may provide better ways to reveal/capture the structures in nonstationary environments The SR may offer better performance in signal modeling and classification problems An efficient and flexible dictionary matrix has to be built for separation of mixtures of events Many researchers have attempted to build dictionary for specific signal processing tasks How to learn the dictionary from the training datasetsSparse Signal Representation and ApplicationsDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsThe CS Measurement SystemPerforming reduction/compression when sensing analog signals The CS is a new data acquisition theory The number of measurements is typically below the number of samples obtained from the Nyquist sampling theorem The nonadaptive linear measurements of the input signal vector are computed as y = x (3) where y is an M × 1 measurement vector, M  N and  is an M × N measurement/sensing matrix. Measurements using a second basis matrix   RM×N that is incoherent with the sparsity basis matrix   RN×NSparse Signal Representation and ApplicationsDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsThe CS System: Reduction and InformationThe measurement system actually performs dimensionality reduction Measurements are able to completely capture the useful information content embedded in a sparse signal Measurements are information of the signals and thus can be used as features for signal modeling If the  consists of elementary sinusoid waveforms, then  is a vector of Fourier coefficients. If the  consists of Dirac delta functions, then  is a vector of sampled values of continuous time signal x(t).Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsThe CS Recovery: IssuesThe y can be written as y =  (4) We define the matrix D =  with a size of M × N. The major problem associated with CS concept is that we have to solve an underdetermined system of equations to recover the original signal x from the measurement vector y . This system has infinitely many solutions since the number of equations is less than the number of unknowns It is necessary to impose constraints such as &quot;sparsity&quot; and &quot;incoherence&quot; that are introduced for for this signal recovery to be efficientSparse Signal Representation and ApplicationsDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsCS Reconstruction: Incoherence the CS recovery relies on two basic principles [?]: (i ) the row vectors of the measurement matrix  cannot sparsely represent the column vectors of the sparsity matrix , and vice versa (ii ) the number of measurements M is greater than N O(cKlog ( K )) these conditions can ensure that it is possible to recover the set of nonzero elements of sparse vector  from measurements y. the input signal x can be reconstructed by the linear transformation of : .Sparse Signal Representation and ApplicationsDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsCS Reconstruction: IncoherenceSparsity basis matrix  is orthonormal and the sensing matrix  consists of M row vectors drawn randomly from some basis ~ matrix   RN×N The mutual coherence is computed as:  µ(, ) = N max | k , j |1k,jN(5)It measures the largest correlation between any two elements  of  and  and will take a value between 1 and N. The value of coherence is large when the elements of  and  are highly correlated and thus CS system requires more measurements. The smaller value µ(, ) indicates maximally incoherent bases and hence, the number of measurements will be lessSparse Signal Representation and ApplicationsDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsCS Recovery: How many measurements?The recovery performance is perfect and optimal when the bases are perfectly incoherent, and unavoidably decreases when the mutual coherence µ increases. The number of measurements M required for perfect signal reconstruction can be computed as: where c is positive constant, µ is the mutual coherence, K is the sparsity factor, and N is the length of the input vector. The value of coherence is large when the elements of  and  are highly correlated and thus CS system requires more measurements. The smaller value µ(, ) indicates maximally incoherent bases and hence, the number of measurements in (5) can be the smallestSparse Signal Representation and ApplicationsM  c · K · µ2 (, ) · log(N)(6)Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsCS Recovery: How many measurements?Under very low mutual coherence value, k-sparse signal can be reconstructed from k.log (N) measurements using basis pursuit Examples of such pairs (maximal mutual incoherence) are:  is the spike basis and  is the Fourier basis  is the noiselet basis and  is the wavelet basis. Noiselets are also maximally incoherent with spikes and incoherent with the Fourier basis.  is a random matrix and  is any fixed basisDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsCompressive Sensing/Measurement MatricesThe the entries of  are: (i ) samples of independent and identically distributed (iid) Gaussian or Bernoulli entries (ii ) randomly selected rows of an orthogonal N × N matrix The RIP says that D acts as an approximate isometry on the set of vectors that are K -sparse, and a matrix D satisfies the K -restricted isometry property if there exists the smallest number, s  [0 1], such that (1 - s )  2  D 2  (1 + s )  2 . 2 2 2 The constant s depends on K , , and . Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsCS Recovery by 1 -norm OptimizationThe goal of a sparse recovery algorithm is to obtain an estimate of  given only y and D =  The recovery of the K -sparse signal x from the measurements y is ill-posed since M &lt; N The CS system of equations is underdetermined the sparest vector is computed by solving the well-known underdetermind problem with sparsity constraint,  = arg min  ^ 0 subject toy =  = D(7)where · denotes 0 -norm that counts the number of nonzero entries in a vector.Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsCS Recovery by 1 -norm OptimizationBy allowing a certain degree of reconstruction error given by the magnitude of the noise the optimization constraint is now relaxed:  = arg min{  ^ 0+ y -  2 } 2 2(8)where   R+ , which controls the relative importance applied to the reconstruction error term and the sparseness term.the solution needs a combinatorial search among all possible sparse , which is infeasible for most problems of interestDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsCS Recovery by 1 -norm OptimizationTo overcome this problem, many nonlinear optimization-based methods have been proposed to obtain sparest vector  by converting (7) into a convex problem which relaxes the 0 -norm to an 1 -norm problem  = arg min  ^ 1 1subject to +y =  = D (9) (10) = arg min{  ^  y -  2 } 2 2which can be solved by linear programming such as BP, MP and OMP The solution to equation (9) is exact or optimal if the number of measurements K is large enough compared to the sparsity factor K , K &lt; M &lt; N and the measurements are chosen uniformly at randomSparse Signal Representation and ApplicationsDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsAnalog to Information ConverterFigure: The block diagram of AICDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSR Applications: Transients Detection2 1 0 -1 -2amplitude020406080100(a)1 0amplitude-1 1 0 -1 0 20 40 60 80(b)1amplitude0 -1 -20204060(c)80 100 120 sample number140Figure: Examples of measured transients with 50 Hz power supply waveforms: (a) spike, (b) microinterruption, and (c) oscillatory transient.Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSR Applications: Transients DetectionThe over-complete dictionary  with size of N × 2N is constructed as  = [ I C](11)where I is the N × N identity (or spike-like) matrix, and C is the N × N DCT matrix. Cij =1  , M (2j+1)i 2 cos( 2N ), Mi = 0, 1  i  N - 1,0  j  N -1 0  j  N -1(12)and the spike like matrix  1 0   Iij = 0 . . . 0is constructed as  0 ··· 0 0 1 ··· 0 0  0 1 0 0   . . .. . . . 0 . . 0 0 ··· 1(13)N×NDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSR Applications: Transients Detectionthe signal x can be written as x   = [ I ~ and can be rewritten asN NC]~ = Id + Ca. (14)y=n=1 dn I n +n=1an Cn .(15)The common problem in well-known wavelet transform-based methods is which mother wavelet function and characteristic scale provides the best time-frequency resolution for detection of transients and non-transients.Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSR Applications: Transients Detection1. 2. 3. 4.Input: N × 1 input signal vector y . Specify the value of regularization parameter . Read the N × 2N over-complete dictionary matrix . Solve the 1 -norm minimization problem:  = arg min {  - y 2 +   1 } ~ 2 5. Obtain the detail and approximation coefficient vectors. 6. Process detail vector for detecting boundaries of transient event. 7. Output: time-instants and transient portionsDr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSR Applications: Impulsive TransientsPowerline with impulsive noise1original detail component approximation component0 -1 0.5 0-0.5 1 0.5 0 -0.5 0 0.02 0.04 0.06 0.08 0.1 0.12Time (sec)Figure: Illustrates the detail and approximation components extracted by using the proposed method. The power supply waveform is corrupted by spikes.Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSR Applications: Transients Detection50 Hz powerline with microinterruption1original0-1 00.010.020.030.040.050.060.070.080.090.1detail component0.8 0.6 0.4 0.2 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1approximation component10-1 00.010.020.030.040.050.060.070.080.090.1Time (sec)Figure: Illustrates the detail and approximation components extracted by using the proposed method. The The power supply waveform with microinterruption.Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSR Applications: Transients Detectionoriginal signal orignal signal1 0 -1 0 0.01 0.02 0.03 0.04 0.05 0.06 low-amplitude transients high-amplitude transients 1 0 -1 0 0.01 0.02 0.03 0.04 0.05 0.06detected transient (detail signal extracted)1 0.5 0 -0.5 -1 0 0.01 0.02 0.03 0.04 0.05 0.06detected transient (detail signal extracted)(a)(b)1 0.5 0 -0.5 -1 -1.5 0 0.01 0.02 0.03 0.04 0.05 0.06(c)(d)approximation signal extracted1 0 -1 0 0.01 0.02 0.03 0.04 0.05 0.06approximation signal extracted1 0 -1 0 0.01 0.02 0.03 0.04 0.05 0.06Time (sec) (e)Time (sec) (f)Figure: Example of waveforms S1 and S2 illustrates signals corrupted by low-amplitude transient S1 and high-amplitude transient S2 due to capacitor switching, respectively. The detected transient events by using our method.Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSR Applications: Transients Detection0.4spike with noise0.2 0 -0.2 0.01 0.02signal with spike0.03 0.04 0.051 0 -1 0 0.01 0.02 0.03 0.04 0.05(a1)(a2)wavelet method (First Detail)0.4 0.2 0 -0.2 0 0.01 0.02wavelet method (First Detail)1 0 -1 0 0.01 0.02(b1)0.030.040.05(b2)0.030.040.05wavelet method (Second Detail)0.4 0.2 0 -0.2 0 0.01 0.02wavelet method (Second Detail)1 0 -1 0(c1)0.030.040.010.02(c2)0.030.040.05detected spike by our methoddetected spike by our method0.01 0.02 0.03 Time (sec) 0.04 0.050.2 0.1 01 0 -1 0 0.01 0.02 0.03 Time (sec) 0.04 0.05(d1)(d2)Figure: Example of transient signals S3 and S4 : (a1) the spike buried in strong noise with SNR value of -10 dB; (a2) the 50 Hz sinusoidal signal affected by a superimposed spike; Plots are the outputs from the wavelet-based methods and the proposed method.Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSR Applications: Removal of Powerlineoriginal ECG signal0.50.5original ECG signala m p litu d e0 -0.5 200 400 600 800 1000 1200 1400 1600 1800 2000am plitude0 -0.5 200 400 600 800 1000 1200 1400 1600 1800 2000Time (sec)Time (sec)original ECG signal plus powerline (10 degree)original ECG signal plus powerline (86 degree)a m p litu d eam plitude0.5 0 -0.5 200 400 600 800 1000 1200 1400 1600 1800 20000.5 0 -0.5 200 400 600 800 1000 1200 1400 1600 1800 2000Time (sec)Time (sec)Output of CS-based approachOutput of CS-based approach0.5a m p litu d eam plitude0.5 0 -0.50 -0.5 200 400 600 800 1000 1200 1400 1600 1800 2000200400600800100012001400160018002000Time (sec)Time (sec)Figure: Removal of Powerline from ECG SignalDr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSR Applications: Removal of Artifactsoriginal ECG signalamplitude0.5 0 -0.5 -1 500 1000 1500 2000 2500 3000T ime (sec)original ECG signal plus powerlineamplitude0.5 0 -0.5 -1 500 1000 1500 2000 2500 3000T ime (sec)Output of CS-based approach0.5amplitude0 -0.5 500 1000 1500 2000 2500 3000T ime (sec)Output of CS-based baseline wander removalamplitude0.4 0.2 0 -0.2 -0.4 500 1000 1500 2000 2500 3000T ime (sec)Figure: Simultaneous removal of Powerline and LF artifact from ECG SignalDr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and ApplicationsSparse Signal Representation and ApplicationsSparse Representation and Compressive SensingThanks for your Attention!Dr. K. P. Soman and Dr. M. Sabarimalai ManikandanSparse Signal Representation and Applications`

#### Information

##### Sparse Signal Representation and Applications

46 pages

Find more like this

#### Report File (DMCA)

Our content is added by our users. We aim to remove reported files within 1 working day. Please use this link to notify us:

Report this file as copyright or inappropriate

323381

### You might also be interested in

BETA
Sparse Signal Representation and Applications