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Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Center for Excellence in Computational Engineering and Networking Amrita University, Coimbatore Campus E-mail: [email protected]

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Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Elementary 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 n

Complex 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 Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Signal 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 Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Why 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 Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Signal Representation Using Basis Functions

A 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 as

N

x=

n=1

n 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 coefficients

Sparse Signal Representation and Applications

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Some 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 on

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Limitations of Fixed Representation Matrix

The 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 Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Sparse Representation/Recovery

Definition: 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 Applications

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Representation: Applications

audio/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 technologies

Sparse Signal Representation and Applications

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Signal: Sparsity

Definition: 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 Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Example for non-sparse signal

Random Noise Sequence 2 0 -2 50 100 150 200 250 300 350 400 450 500

Original Sequence: Histogram 15 10 5 0 -3 -2 -1 0 DFT: Histogram 20 10 1 2

DFT coefficients 0.8 0.6 0.4 0.2 50 100 150 200 250 300 350 400 450 500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

DWT coefficients 0.5 0 -0.5 -1 50 100 150 200 250 300 350 400 450 500 550

DWT: Histogram 20 10 0 -1

amplitude

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

DCT coefficients 0.5 0 -0.5 -1 50 100 150 200 250 300 sample number 350 400 450 500

DCT: Histogram 15 10 5 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Example for sparsity in time-domain

Random Noise Sequence 0.5 0 -0.5 -1 -1.5 50 100 150 200 250 300 350 400 450 500 DFT coefficients

8 400 200 0 Original Sequence: Histogram

-1.5

-1

-0.5 DFT: Histogram

0

0.5

0.8 0.6 0.4 0.2 50 1

amplitude

6 4 2

100

150

200

250

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

DWT coefficients

400

DWT: Histogram

0.5 0

200

0

50 1 0.5 0 -0.5

100

150

200

250

300

350

400

450

500

550

15 10 5 0

-0.2

0

0.2

0.4

0.6

0.8

1

DCT coefficients

DCT: Histogram

50

100

150

200 250 300 sample number

350

400

450

500

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Spectrogram of a Speech Signal

Utterance of the word

Amplitude

0.2 0 -0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

Time (sec)

8000

Frequency (Hz)

6000 4000 2000 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Time (sec)

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Spectrum of a Voiced and Unvoiced Sounds

Voiced Sounds 1

Amplitude

Voiced Sounds 1 0.5 0 -0.5 -1

Amplitude

0.5 0 -0.5 -1

200

800 1000 1200 1400 1600 Time (seconds) Single-Sided Amplitude Spectrum of Voiced Sounds

400

600

200 400 600 800 1000 1200 1400 1600 1800 2000 Time (seconds) Single-Sided Amplitude Spectrum of Unvoiced Sounds

300

30

|Y(f)|

|Y(f)|

200

20

10

100

0 0 1000 2000 3000 4000 5000 Frequency (Hz) 6000 7000 8000

0

0

1000

2000

3000 4000 5000 Frequency (Hz)

6000

7000

8000

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Multiresolution signal decomposition

P R T RS ECG beat ECG+noise ECG+baseline artifact

2 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 1000

x[n] D1[n]

D2[n] D [n]

3

amplitude

D [n] 4 D5[n] A5[n]

sample number

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Sparse Signal: Sparsity

The 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 Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Sparse Representation: Dictionary Learning

Need 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 Applications

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Representation: Dictionary Learning

Problem 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 Applications

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Representation: Need for Best Basis Functions

Remedies: 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 Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Sparse 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 Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Sparse Representation: Classification of Dictionaries

Based 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 > N, D is called overcomplete, or redundant dictionary. (ii ) when L < 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 Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Some of Sparse Transform Matrices

dirac 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 Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Sparse Representation: Research Problems

The 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 datasets

Sparse Signal Representation and Applications

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

The CS Measurement System

Performing 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×N

Sparse Signal Representation and Applications

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

The CS System: Reduction and Information

The 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 Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

The CS Recovery: Issues

The 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 "sparsity" and "incoherence" that are introduced for for this signal recovery to be efficient

Sparse Signal Representation and Applications

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

CS 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 Applications

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

CS Reconstruction: Incoherence

Sparsity 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 less

Sparse Signal Representation and Applications

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

CS 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 smallest

Sparse Signal Representation and Applications

M c · K · µ2 (, ) · log(N)

(6)

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

CS 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 basis

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Compressive Sensing/Measurement Matrices

The 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 Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

CS Recovery by 1 -norm Optimization

The 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 < 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 to

y = = 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 Applications

CS Recovery by 1 -norm Optimization

By 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 interest

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

CS Recovery by 1 -norm Optimization

To 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 1

subject to +

y = = D (9) (10)

= arg min{ ^

y - 2 } 2 2

which 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 < M < N and the measurements are chosen uniformly at random

Sparse Signal Representation and Applications

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Analog to Information Converter

Figure: The block diagram of AIC

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

SR Applications: Transients Detection

2 1 0 -1 -2

amplitude

0

20

40

60

80

100

(a)

1 0

amplitude

-1 1 0 -1 0 20 40 60 80

(b)

1

amplitude

0 -1 -2

0

20

40

60

(c)

80 100 120 sample number

140

Figure: 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 Applications

SR Applications: Transients Detection

The 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 ), M

i = 0, 1 i N - 1,

0 j N -1 0 j N -1

(12)

and the spike like matrix 1 0 Iij = 0 . . . 0

is constructed as 0 ··· 0 0 1 ··· 0 0 0 1 0 0 . . .. . . . 0 . . 0 0 ··· 1

(13)

N×N

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

SR Applications: Transients Detection

the signal x can be written as x = [ I ~ and can be rewritten as

N N

C]~ = Id + Ca.

(14)

y=

n=1

dn I n +

n=1

an 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 Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

SR Applications: Transients Detection

1. 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 portions

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

SR Applications: Impulsive Transients

Powerline with impulsive noise

1

original detail component approximation component

0 -1 0.5 0

-0.5 1 0.5 0 -0.5 0 0.02 0.04 0.06 0.08 0.1 0.12

Time (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 Applications

SR Applications: Transients Detection

50 Hz powerline with microinterruption

1

original

0

-1 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

detail component

0.8 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 0.1

approximation component

1

0

-1 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time (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 Applications

SR Applications: Transients Detection

original signal orignal signal

1 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 0.06

detected transient (detail signal extracted)

1 0.5 0 -0.5 -1 0 0.01 0.02 0.03 0.04 0.05 0.06

detected 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 extracted

1 0 -1 0 0.01 0.02 0.03 0.04 0.05 0.06

approximation signal extracted

1 0 -1 0 0.01 0.02 0.03 0.04 0.05 0.06

Time (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 Applications

SR Applications: Transients Detection

0.4

spike with noise

0.2 0 -0.2 0.01 0.02

signal with spike

0.03 0.04 0.05

1 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 0.02

wavelet method (First Detail)

1 0 -1 0 0.01 0.02

(b1)

0.03

0.04

0.05

(b2)

0.03

0.04

0.05

wavelet method (Second Detail)

0.4 0.2 0 -0.2 0 0.01 0.02

wavelet method (Second Detail)

1 0 -1 0

(c1)

0.03

0.04

0.01

0.02

(c2)

0.03

0.04

0.05

detected spike by our method

detected spike by our method

0.01 0.02 0.03 Time (sec) 0.04 0.05

0.2 0.1 0

1 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 Applications

SR Applications: Removal of Powerline

original ECG signal

0.5

0.5

original ECG signal

a m p litu d e

0 -0.5 200 400 600 800 1000 1200 1400 1600 1800 2000

am plitude

0 -0.5 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (sec)

Time (sec)

original ECG signal plus powerline (10 degree)

original ECG signal plus powerline (86 degree)

a m p litu d e

am plitude

0.5 0 -0.5 200 400 600 800 1000 1200 1400 1600 1800 2000

0.5 0 -0.5 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (sec)

Time (sec)

Output of CS-based approach

Output of CS-based approach

0.5

a m p litu d e

am plitude

0.5 0 -0.5

0 -0.5 200 400 600 800 1000 1200 1400 1600 1800 2000

200

400

600

800

1000

1200

1400

1600

1800

2000

Time (sec)

Time (sec)

Figure: Removal of Powerline from ECG Signal

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

SR Applications: Removal of Artifacts

original ECG signal

amplitude

0.5 0 -0.5 -1 500 1000 1500 2000 2500 3000

T ime (sec)

original ECG signal plus powerline

amplitude

0.5 0 -0.5 -1 500 1000 1500 2000 2500 3000

T ime (sec)

Output of CS-based approach

0.5

amplitude

0 -0.5 500 1000 1500 2000 2500 3000

T ime (sec)

Output of CS-based baseline wander removal

amplitude

0.4 0.2 0 -0.2 -0.4 500 1000 1500 2000 2500 3000

T ime (sec)

Figure: Simultaneous removal of Powerline and LF artifact from ECG Signal

Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan Sparse Signal Representation and Applications

Sparse Signal Representation and Applications

Sparse Representation and Compressive Sensing

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Dr. K. P. Soman and Dr. M. Sabarimalai Manikandan

Sparse Signal Representation and Applications

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Sparse Signal Representation and Applications

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Sparse Signal Representation and Applications