sparse representation and compressive sensing
Post on 17-May-2015
5.604 Views
Preview:
TRANSCRIPT
Advanced Signal Processing
Sparse Representation and Compressive Sensing
Dr. M. Sabarimalai Manikandan
Assistant ProfessorCenter for Excellence in Computational Engineering and Networking
Amrita University, Coimbatore CampusE-mail: msm.sabari@gmail.com
September 16, 2011
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Why Signal Processing?
◮ Most natural signals are non-stationary and have highlycomplex 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 aretypically having different shapes, amplitudes, durations andfrequency content, which are not known in many differentapplications and systems
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Signal Representation Using Basis Functions
◮ A set {ψn}Nn=1 is called a orthonormal basis for RN if thevectors in the set span R
N and are linearly independent
◮ Let x ∈ RN×1 be the input signal that is spanned by N basis
functions {ψn}Nn=1. Then, a discrete-time signal x can berepresented as
x =
N∑
n=1
αnψn = Ψα (1)
where α = [α1, α2, α3, ......αN ] is the transform coefficientsvector that is computed as αn = 〈x ,ψn〉.
◮ For some transform matrix, the transform coefficients vector αhas a small number of large amplitude coefficients and a largenumber of small amplitude coefficients
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
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. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Limitations of Fixed Representation Matrix
◮ The Fourier transform is suitable for analysis of thesteady-state sinusoidal signals but it fails to capture the sharpchanges and discontinuities in the signals.
◮ In the STFT-based methods, the choices for widths of thetime-window affect the frequency and time resolution.
◮ The common problem in well-known wavelet transform-basedmethods is which mother wavelet function and characteristicscale provides the best time-frequency resolution for detectionof transients and non-transients.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Sparse Representation/Recovery
◮ Definition: The sparse representation theory has shown thatsparse signals can be exactly reconstructed from a smallnumber of elementary signals (or atoms).
◮ The sparse representation of natural signals can be achievedby exploiting its sparsity or compressibility.
◮ A natural signal is said to be sparse signal if that can becompactly expressed as a linear combination of a few smallnumber of basis vectors.
◮ Sparse representation has become an invaluable tool ascompared to direct time-domain and transform-domain signalprocessing methods.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
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
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Sparse Signal: Sparsity
◮ Definition: A signal can be sparse or compressible in sometransform matrix Ψ when the transform coefficients vector αhas a small number of large amplitude coefficients and a largenumber of small amplitude coefficients.
◮ Observations: Most of the energy is concentrated in a fewtransform coefficients in a vector α
◮ The other N − K coefficients have less contribution inrepresenting a signal vector x ∈ R
N×1.
◮ The insignificant coefficients are set to zero in coding scheme.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Sparse Signal: Sparsity
◮ The value of K is computed as K = ‖α‖0, where ‖.‖0 denotesthe ℓ0-norm which counts the number of non-zero entries inα.
◮ Concluding Remarks: A sparse signal x can be exactlyrepresented or approximated by the linear combination of Kbasis functions with shorter transform coefficients vector.
◮ In such a reconstruction process, the reconstruction error by aK -term representation decays exponentially as K increases.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Sparse Representation: Dictionary Learning
◮ Need for Dictionary Learning: In practice, a signal iscomposed of impulsive and oscillatory transients, spikes andlow-frequency components.
◮ Nature: The composite signal may not exhibit sparsity in onetransform basis matrix because some of its components aresparse in one domain while other components are sparse inanother domain.
◮ The signals may exhibit sparsity in either time-domain orfrequency-domain.
◮ For example, the 50 Hz powerline signal is sparse in thefrequency-domain and the impulse or spikes component issparse in the time-domain.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Sparse Representation: Dictionary Learning
◮ Problem with Fixed Basis: In practice, the composite signal(spike is superimposed on powerline signal) exhibits sparsity inneither time-domain nor frequency-domain.
◮ In such cases, a fixed orthogonal basis functions are notflexible enough to capture the complex local waves of a signal.
◮ For example, a fixed elementary cosine waveforms of discretecosine transform (DCT) matrix fails to capture transient partsof biosignals.
◮ Detection and suppression of impulsive noise in speechwaveform.
◮ Compression of slow varying signals with spikes.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Sparse Representation: Need for Best Basis Functions
◮ Remedies: To improve the sparsity of composite signals, onehas to construct a transform matrix with the best basisfunctions.
◮ One way to process such signal is to work with an largedictionary matrix.
◮ A best basis set from a dictionary matrix used to sparsify thedata may yield highly compact representations of manynatural signals.
◮
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Sparse Representation: What is Dictionary?
◮ Dictionary: A dictionary is a collection of elementarywaveforms or prototype atoms or basis functions.
◮ A dictionary matrix D of dimension N × L can be representedas D = {ψ1|ψ2|ψ3|..........|ψL}.
◮ The column vectors {ψl}Ll=1 of an dictionary D arediscrete-time elementary signals of length N × 1, calleddictionary atoms or basis functions.
◮ The atoms in the pre-defined dictionary may be pairwiseorthogonal, linear independent, linear dependent, or notorthogonal.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Sparse Representation: Classification of Dictionaries
◮ Based on the number of atoms L and the signal length N, thepre-defined dictionary, D ∈ R
N×M , could be classified intothree categories:
◮ (i) when L > N, D is called overcomplete, or redundantdictionary.
◮ (ii) when L < N, D is called undercomplete dictionary.
◮ (iii) D is said to be complete dictionary if L = N.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Some of Sparse Transform Matrices
◮ dirac and heaviside functions
◮ Fourier transform matrix and Fourier, short-time Fouriertransform (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. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Sparse Representation: Research Problems
◮ The dirac dictionary can be used to detect the spikes in asignal and the discrete cosines dictionary can providesinusoidal waveforms
◮ The SR from redundant dictionaries may provide better waysto reveal/capture the structures in nonstationaryenvironments
◮ The SR may offer better performance in signal modeling andclassification problems
◮ An efficient and flexible dictionary matrix has to be built forseparation of mixtures of events
◮ Many researchers have attempted to build dictionary forspecific signal processing tasks
◮ How to learn the dictionary from the training datasets
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
The CS Measurement System
◮ Performing reduction/compression when sensing analogsignals
◮ The CS is a new data acquisition theory
◮ The number of measurements is typically below the number ofsamples obtained from the Nyquist sampling theorem
◮ The nonadaptive linear measurements of the input signalvector are computed as
y = Φx (2)
where y is an M × 1 measurement vector, M ≪ N and Φ isan M × N measurement/sensing matrix.
◮ Measurements using a second basis matrix Φ ∈ RM×N that is
incoherent with the sparsity basis matrix Ψ ∈ RN×N
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
The CS System: Reduction and Information
◮ The measurement system actually performs dimensionalityreduction
◮ Measurements are able to completely capture the usefulinformation content embedded in a sparse signal
◮ Measurements are information of the signals and thus can beused as features for signal modeling
◮ If the Φ consists of elementary sinusoid waveforms, then α isa vector of Fourier coefficients.
◮ If the Φ consists of Dirac delta functions, then α is a vector ofsampled values of continuous time signal x(t).
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
The CS Recovery: Issues
◮ The y can be written as
y = ΦΨα (3)
◮ We define the matrix D = ΦΨ with a size of M × N.
◮ The major problem associated with CS concept is that wehave to solve an underdetermined system of equations torecover the original signal x from the measurement vector y .
◮ This system has infinitely many solutions since the number ofequations 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 recoveryto be efficient
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
CS Reconstruction: Incoherence
◮ the CS recovery relies on two basic principles [?]:
◮ (i) the row vectors of the measurement matrix Φ cannotsparsely represent the column vectors of the sparsity matrix Ψ,and vice versa
◮ (ii) the number of measurements M is greater thanO(cKlog(N
K))
◮ these conditions can ensure that it is possible to recover theset of nonzero elements of sparse vector α from measurementsy .
◮ the input signal x can be reconstructed by the lineartransformation of α: .
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
CS Reconstruction: Incoherence
◮ Sparsity basis matrix Ψ is orthonormal and the sensing matrixΦ consists of M row vectors drawn randomly from some basismatrix Φ̃ ∈ R
N×N
◮ The mutual coherence is computed as:
µ(Φ,Ψ) =√N max
1≤k,j≥N|〈Φk ,Ψj〉| (4)
◮ It measures the largest correlation between any two elementsof Φ and Ψ and will take a value between 1 and
√N .
◮ The value of coherence is large when the elements of Φ andΨ are highly correlated and thus CS system requires moremeasurements.
◮ The smaller value µ(Φ,Ψ) indicates maximally incoherentbases and hence, the number of measurements will be less
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
CS Recovery: How many measurements?
◮ The recovery performance is perfect and optimal when thebases are perfectly incoherent, and unavoidably decreaseswhen the mutual coherence µ increases.
◮ The number of measurements M required for perfect signalreconstruction can be computed as:
M ≥ c · K · µ2(Φ,Ψ) · log(N) (5)
where c is positive constant, µ is the mutual coherence, K isthe sparsity factor, and N is the length of the input vector.
◮ The value of coherence is large when the elements of Φ andΨ are highly correlated and thus CS system requires moremeasurements.
◮ The smaller value µ(Φ,Ψ) indicates maximally incoherentbases and hence, the number of measurements in (4) can bethe smallest
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
CS Recovery: How many measurements?
◮ Under very low mutual coherence value, k-sparse signal canbe reconstructed from k .log(N) measurements using basispursuit
◮ Examples of such pairs (maximal mutual incoherence)are: Φ is the spike basis and Ψ is the Fourier basis
◮ Φ is the noiselet basis and Ψ is the wavelet basis. Noiseletsare also maximally incoherent with spikes and incoherent withthe Fourier basis.
◮ Φ is a random matrix and Ψ is any fixed basis
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Compressive Sensing/Measurement Matrices
◮ The the entries of Φ are: (i) samples of independent andidentically 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 theset of vectors that are K -sparse, and a matrix D satisfies theK−restricted isometry property if there exists the smallestnumber, δs ∈ [0 1], such that(1− δs)‖α‖22 ≤ ‖Dα‖22 ≤ (1 + δs)‖α‖22.The constant δs depends on K , Φ, and α.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
CS Recovery by ℓ1-norm Optimization
◮ The goal of a sparse recovery algorithm is to obtain anestimate of α given only y and D = ΦΨ
◮ The recovery of the K -sparse signal x from the measurementsy is ill-posed since M < N
◮ The CS system of equations is underdetermined
◮ the sparest vector is computed by solving the well-knownunderdetermind problem with sparsity constraint,
α̂ = argminα
‖α‖0 subject to y = ΦΨα = Dα (6)
where ‖ • ‖ denotes ℓ0-norm that counts the number ofnonzero entries in a vector.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
CS Recovery by ℓ1-norm Optimization
◮ By allowing a certain degree of reconstruction error given bythe magnitude of the noise
◮ the optimization constraint is now relaxed:
α̂ = argminα
{‖α‖0 +λ
2‖y −ΦΨα‖22} (7)
where λ ∈ R+, which controls the relative importance applied
to the reconstruction error term and the sparseness term.
◮ the solution needs a combinatorial search among all possiblesparse α, which is infeasible for most problems of interest
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
CS Recovery by ℓ1-norm Optimization
◮ To overcome this problem, many nonlinear optimization-basedmethods have been proposed to obtain sparest vector α byconverting (6) into a convex problem which relaxes theℓ0-norm to an ℓ1-norm problem
α̂ = argminα
‖α‖1 subject to y = ΦΨα = Dα (8)
α̂ = argminα
{‖α‖1 +λ
2‖y −ΦΨα‖22} (9)
◮ which can be solved by linear programming such as BP, MPand OMP
◮ The solution to equation (8) is exact or optimal if the numberof measurements K is large enough compared to the sparsityfactor K , K < M < N and the measurements are chosenuniformly at random
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Analog to Information Converter
�
Figure: The block diagram of AIC
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
SR Applications: Transients Detection
0 20 40 60 80 100−2
−1
0
1
2
ampli
tude
0 20 40 60 80−1
0
1
−1
0
1
ampli
tude
0 20 40 60 80 100 120 140−2
−1
0
1
sample number
ampli
tude
(b)
(a)
(c)
Figure: Examples of measured transients with 50 Hz power supplywaveforms: (a) spike, (b) microinterruption, and (c) oscillatory transient.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
SR Applications: Transients Detection
◮ The over-complete dictionary Ψ with size of N × 2N isconstructed as
Ψ = [ I C] (10)
where I is the N × N identity (or spike-like) matrix, and C isthe N × N DCT matrix.
◮
Cij =
{
1√
M, i = 0, 0 ≤ j ≤ N − 1
√
2M
cos(π(2j+1)i
2N), 1 ≤ i ≤ N − 1, 0 ≤ j ≤ N − 1
(11)
and the spike like matrix is constructed as
Iij =
1 0 · · · 0 00 1 · · · 0 00 0 1 0 0...
......
. . . 00 0 0 · · · 1
N×N
(12)
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
SR Applications: Transients Detection
◮ the signal x can be written as
x ≈ Ψα̃ = [ I C]α̃ = Id+ Ca. (13)
and can be rewritten as
y =
N∑
n=1
dnIn +
N∑
n=1
anCn. (14)
◮ The common problem in well-known wavelet transform-basedmethods is which mother wavelet function and characteristicscale provides the best time-frequency resolution for detectionof transients and non-transients.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
SR Applications: Transients Detection
1. Input: N × 1 input signal vector y .2. Specify the value of regularization parameter λ.3. Read the N × 2N over-complete dictionary matrix Ψ.4. Solve the ℓ1-norm minimization problem:
α̃ = argminα{‖Ψα− y‖22 + λ‖α‖1}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. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
SR Applications: Impulsive Transients
−1
0
1Powerline with impulsive noise
orig
inal
−0.5
0
0.5
deta
ilco
mpo
nent
0 0.02 0.04 0.06 0.08 0.1 0.12
−0.5
0
0.5
1
Time (sec)
appr
oxim
atio
nco
mpo
nent
Figure: Illustrates the detail and approximation components extracted byusing the proposed method. The power supply waveform is corrupted byspikes.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
SR Applications: Transients Detection
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−1
0
150 Hz powerline with microinterruption
orig
inal
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0.2
0.4
0.6
0.8
deta
il co
mpo
nent
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−1
0
1
Time (sec)
appr
oxim
atio
n co
mpo
nent
Figure: Illustrates the detail and approximation components extracted byusing the proposed method. The The power supply waveform withmicrointerruption.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
SR Applications: Transients Detection
0 0.01 0.02 0.03 0.04 0.05 0.06
−1
0
1
orig
inal
sig
nal
0 0.01 0.02 0.03 0.04 0.05 0.06−1
−0.5
0
0.5
1
dete
cted
tran
sien
t(d
etai
l sig
nal e
xtra
cted
)
0 0.01 0.02 0.03 0.04 0.05 0.06
−1
0
1
Time (sec)
appr
oxim
atio
n s
igna
l ext
ract
ed
0 0.01 0.02 0.03 0.04 0.05 0.06
−1
0
1
orig
nal s
igna
l
0 0.01 0.02 0.03 0.04 0.05 0.06−1.5
−1
−0.5
0
0.5
1
dete
cted
tran
sien
t (d
etai
l sig
nal e
xtra
cted
)
0 0.01 0.02 0.03 0.04 0.05 0.06
−1
0
1
Time (sec)ap
prox
imat
ion
sig
nal e
xtra
cted
(a)
(f)
low−amplitude transients
(b)
(e)
(d)(c)
high−amplitude transients
Figure: Example of waveforms S1 and S2 illustrates signals corrupted bylow-amplitude transient S1 and high-amplitude transient S2 due tocapacitor switching, respectively. The detected transient events by usingour method.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
SR Applications: Transients Detection
0.01 0.02 0.03 0.04 0.05
−0.2
0
0.2
0.4
spik
e w
ith n
oise
0.01 0.02 0.03 0.04 0.050
0.1
0.2
Time (sec)
dete
cted
spi
ke
by o
ur m
etho
d
0 0.01 0.02 0.03 0.04 0.05
−0.2
0
0.2
0.4
wav
elet
met
hod
(Firs
t Det
ail)
0 0.01 0.02 0.03 0.04
−0.2
0
0.2
0.4
wav
elet
met
hod
(Sec
ond
Det
ail)
0 0.01 0.02 0.03 0.04 0.05−1
0
1
sign
al w
ithsp
ike
0 0.01 0.02 0.03 0.04 0.05−1
0
1
Time (sec)de
tect
ed s
pike
by o
ur m
etho
d
0 0.01 0.02 0.03 0.04 0.05
−1
0
1
wav
elet
met
hod
(Firs
t Det
ail)
0 0.01 0.02 0.03 0.04 0.05−1
0
1
wav
elet
met
hod
(Sec
ond
Det
ail)
(d1)
(b1)
(a1)
(c2)
(a2)
(d2)
(b2)
(c1)
Figure: Example of transient signals S3 and S4: (a1) the spike buried instrong noise with SNR value of -10 dB; (a2) the 50 Hz sinusoidal signalaffected by a superimposed spike; Plots are the outputs from thewavelet-based methods and the proposed method.
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
SR Applications: Removal of Powerline
200 400 600 800 1000 1200 1400 1600 1800 2000
-0.5
0
0.5
ampl
itude
Time (sec)
original ECG signal
200 400 600 800 1000 1200 1400 1600 1800 2000
-0.5
0
0.5
original ECG signal plus powerline (10 degree)
ampl
itude
Time (sec)
200 400 600 800 1000 1200 1400 1600 1800 2000
-0.5
0
0.5
Output of CS-based approach
ampl
itude
Time (sec)
200 400 600 800 1000 1200 1400 1600 1800 2000
-0.5
0
0.5
ampl
itude
Time (sec)
original ECG signal
200 400 600 800 1000 1200 1400 1600 1800 2000
-0.5
0
0.5
original ECG signal plus powerline (86 degree)
ampl
itude
Time (sec)
200 400 600 800 1000 1200 1400 1600 1800 2000
-0.5
0
0.5
Output of CS-based approach
ampl
itude
Time (sec)
Figure: Removal of Powerline from ECG Signal
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
SR Applications: Removal of Artifacts
500 1000 1500 2000 2500 3000-1
-0.5
0
0.5
ampli
tude
Time (sec)
original ECG signal
500 1000 1500 2000 2500 3000
-1
-0.5
0
0.5
original ECG signal plus powerline
ampli
tude
Time (sec)
500 1000 1500 2000 2500 3000
-0.5
0
0.5
Output of CS-based approach
ampli
tude
Time (sec)
500 1000 1500 2000 2500 3000
-0.4
-0.2
0
0.2
0.4
Output of CS-based baseline wander removal
ampli
tude
Time (sec)
Figure: Simultaneous removal of Powerline and LF artifact from ECGSignal
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
Advanced Signal Processing
Sparse Representation and Compressive Sensing
Thanks for your Attention!
Dr. M. Sabarimalai Manikandan Sparse Representation and Compressive Sensing
top related