compressed sensing for loss-tolerant audio transport
DESCRIPTION
Compressed Sensing for Loss-Tolerant Audio Transport. Clay, Elena, Hui. Introduction to CS. Basic idea: Given a signal S of length d (large) S can be recovered from a much smaller measurement vector v ! ( if S is sparse ). signal. Sparse. compressed. Introduction to CS. - PowerPoint PPT PresentationTRANSCRIPT
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Compressed Sensing for Loss-Tolerant Audio Transport
Clay, Elena, Hui
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Introduction to CS
Basic idea:
Given a signal S of length d (large)
S can be recovered from a much smaller measurement vector v ! ( if S is sparse )
Sparse compressedsignal
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Introduction to CS
signal: s= (0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1)
measurements: projections of s onto some small number of basis vectors
Questions: 1. what basis vectors?2. how many measurements are enough?
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Intro to CS
Sometimes imperfection is OK! We only want to have to transmit enough for a “reasonable” reconstruction.
Reduce the number of bits used to transmit a signal
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Motivation
Direct applicability to low-power sensor networks (data is sparse)
Applications to medical imaging
How does CS apply to audio signals?
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CS and sound reconstruction
Compressed Sensing is:
loss-tolerant
universal
But:
is it practical? Particularly for audio?
how about quality of reconstructed sound?
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Approach/contributions
- Use a modified version of the classical Orthogonal Matching Pursuit
1. optimized the main iterative step2. dealt with MATLAB memory overflow for
matrix storage3. split original large data samples into
smaller frames and combine at the end4. Quantify relationship between quality and
compression parameters m, c.
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Parameters
• m: sparsity level of original data
• d: data space dimension
• N: # of measurements
N= c m ln(d)
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OMP(Orthogonal Matching Pursuit)
• InputΦ: N x d measurement matrixv: N-dimensional data vectorm: data sparsity
• Outputs: estimated signal in Rd
• Procedure
v= Φ * s
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OMP Procedure
Determine which columns of Φ participate in the measurement vector v, in greedy fashion.1. Initialization 2. IterationIn each iteration, choose one column Φ that is most strongly correlated with the remaining part of v. Then we subtract off its contribution to v and iterate on the residual.3. ReconstructionUse the chosen columns of Φ and approximation to reconstruct the signal.
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I-OMP on Audio Signal Recovery
• Original sound signal (Source: s4d.wav)
• Reconstructed by setting m = 256 and 500
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Tests
Test the impact of the parameters m, c on the quality of the reconstruction
Method: MOS (Mean Opinion Score)
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Sparsity and MOS
m as fraction of number of samples
MO
S
scor
e
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Quality of reconstruction
Sum of squared differences between original and reconstructed signal
m = 1233
d = 8821
c
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Piecewise Compression
• Original:
• Recovered:
• MOS = 2.8
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I-OMP on image recovery
Different m = 256, 512, 1024
Source: moon.bmp
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I-OMP on Image Recovery
• Different value of parameter c
• Original, c=2,4,20
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The End
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• 1. Initialization residual r = v; Index set Λ = empty;
• 2. Iteration
• 3. Reconstruction
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OMP Procedure• 1. Initialization• 2. Iteration
For t=0: m-1• Find the index λ that solves• λ= arg max j=1,…,d |<r,φj>|• Λ = Λ U {λ}• Re-compute projection P on φΛ.
A = P* vr = v - A
• 3. Reconstruction
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OMP Procedure
• 1. Initialization
• 2. Iteration
• 3. Reconstruction
The estimate s for the ideal signal has non-zero coefficients sλ at the components li
sted in Λ.
A = Σ λ∈Λ φλ* sλ
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Iterative OMP
• 1. Initializationr = v;s = 0d;
• 2. IterationFor t=0: m-1
• Find the index λ that solves λ= arg max j=1,…,d |<r,φj>|
• sλ = <r, φλ >/ || φλ ||2
• r = r - sλ * φλ
• A= A + sλ * φλ
• 3. Reconstruction
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Iterative OMP -2