23
6.829 Computer Networks 1 Compressed Sensing for Loss-Tolerant Audio Transport Clay, Elena, Hui
6.829 Computer Networks 1
Compressed Sensing for Loss-Tolerant Audio Transport
Clay, Elena, Hui
6.829 Computer Networks 2
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
6.829 Computer Networks 3
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?
6.829 Computer Networks 4
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
6.829 Computer Networks 5
Motivation
Direct applicability to low-power sensor networks (data is sparse)
Applications to medical imaging
How does CS apply to audio signals?
6.829 Computer Networks 6
CS and sound reconstruction
Compressed Sensing is:
loss-tolerant
universal
But:
is it practical? Particularly for audio?
how about quality of reconstructed sound?
6.829 Computer Networks 7
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.
6.829 Computer Networks 8
Parameters
• m: sparsity level of original data
• d: data space dimension
• N: # of measurements
N= c m ln(d)
6.829 Computer Networks 9
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
6.829 Computer Networks 10
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.
6.829 Computer Networks 11
I-OMP on Audio Signal Recovery
• Original sound signal (Source: s4d.wav)
• Reconstructed by setting m = 256 and 500
6.829 Computer Networks 12
Tests
Test the impact of the parameters m, c on the quality of the reconstruction
Method: MOS (Mean Opinion Score)
6.829 Computer Networks 13
Sparsity and MOS
m as fraction of number of samples
MO
S
scor
e
6.829 Computer Networks 14
Quality of reconstruction
Sum of squared differences between original and reconstructed signal
m = 1233
d = 8821
c
6.829 Computer Networks 15
Piecewise Compression
• Original:
• Recovered:
• MOS = 2.8
6.829 Computer Networks 16
I-OMP on image recovery
Different m = 256, 512, 1024
Source: moon.bmp
6.829 Computer Networks 17
I-OMP on Image Recovery
• Different value of parameter c
• Original, c=2,4,20
6.829 Computer Networks 18
The End
6.829 Computer Networks 19
• 1. Initialization residual r = v; Index set Λ = empty;
• 2. Iteration
• 3. Reconstruction
6.829 Computer Networks 20
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
6.829 Computer Networks 21
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
6.829 Computer Networks 22
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
6.829 Computer Networks 23
Iterative OMP -2
