Compressed Sensing for Loss-Tolerant Audio Transport

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 ) signal Sparse compressed 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 step 2. 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. 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 MOS score m as fraction of number of samples 6.829 Computer Networks

14. Quality of reconstruction m = 1233 d = 8821 Sum of squared differences between original and reconstructed signal 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* v r = 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 listed in Λ. A = Σ λ∈Λ φλ* sλ 6.829 Computer Networks

22. Iterative OMP • 1. Initializationr = v;s = 0d; • 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 6.829 Computer Networks

23. Iterative OMP -2 6.829 Computer Networks