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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