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 )
signal
Sparse
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compressed
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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
1.
2.
3.
4.
Use a modified version of the classical
Orthogonal Matching Pursuit
optimized the main iterative step
dealt with MATLAB memory overflow for
matrix storage
split original large data samples into
smaller frames and combine at the end
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)
• Input
Φ: N x d measurement matrix
v: N-dimensional data vector
m: data sparsity
• Output
s: estimated signal in Rd
v= Φ * s
• Procedure
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OMP Procedure
Determine which columns of Φ participate in the
measurement vector v, in greedy fashion.
1. Initialization
2. Iteration
In 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. Reconstruction
Use 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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MOS score
Sparsity and MOS
m as fraction of number of samples
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Quality of reconstruction
m = 1233
d = 8821
Sum of squared
differences
between
original and
reconstructed
signal
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* v
r=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 listed in Λ.
A = Σ λ∈Λ φλ* sλ
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Iterative OMP
• 1. Initialization
r = v;
s = 0 d;
• 2. Iteration
For 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
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