1. 6.829 Computer Networks1 Compressed Sensing for Loss-Tolerant Audio Transport Clay, Elena, Hui

2. 6.829 Computer Networks2 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 compressed signal

3. 6.829 Computer Networks3 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?

4. 6.829 Computer Networks4 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

5. 6.829 Computer Networks5 Motivation Direct applicability to low-power sensor networks (data is sparse) Applications to medical imaging How does CS apply to audio signals?

6. 6.829 Computer Networks6 CS and sound reconstruction Compressed Sensing is: loss-tolerant universal But: is it practical? Particularly for audio? how about quality of reconstructed sound?

7. 6.829 Computer Networks7 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 3.split original large data samples into smaller frames and combine at the end 4.Quantify relationship between quality and compression parameters m, c.

8. 6.829 Computer Networks8 Parameters m: sparsity level of original data d: data space dimension N: # of measurements N= c m ln(d)

9. 6.829 Computer Networks9 OMP(Orthogonal Matching Pursuit) Input Φ: N x d measurement matrix v: N-dimensional data vector m: data sparsity Output s: estimated signal in R d Procedure v= Φ * s

10. 6.829 Computer Networks10 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.

11. 6.829 Computer Networks11 I-OMP on Audio Signal Recovery Original sound signal (Source: s4d.wav) Reconstructed by setting m = 256 and 500

12. 6.829 Computer Networks12 Tests Test the impact of the parameters m, c on the quality of the reconstruction Method: MOS (Mean Opinion Score)

13. 6.829 Computer Networks13 Sparsity and MOS m as fraction of number of samples MOS score

14. 6.829 Computer Networks14 Quality of reconstruction Sum of squared differences between original and reconstructed signal m = 1233 d = 8821 c

15. 6.829 Computer Networks15 Piecewise Compression Original: Recovered: MOS = 2.8

16. 6.829 Computer Networks16 I-OMP on image recovery Different m = 256, 512, 1024 Source: moon.bmp

17. 6.829 Computer Networks17 I-OMP on Image Recovery Different value of parameter c Original, c=2,4,20

18. 6.829 Computer Networks18 The End

19. 6.829 Computer Networks19 1. Initialization residual r = v; Index set Λ = empty; 2. Iteration 3. Reconstruction

20. 6.829 Computer Networks20 OMP Procedure 1. Initialization 2. Iteration For t=0: m-1 Find the index λ that solves λ= arg max j=1,…,d | | Λ = Λ U {λ} Re-compute projection P on φ Λ. A = P* v r = v - A 3. Reconstruction

21. 6.829 Computer Networks21 OMP Procedure 1. Initialization 2. Iteration 3. Reconstruction The estimate s for the ideal signal has non-zero coefficients s λ at the components listed in Λ. A = Σ λ ∈ Λ φ λ * s λ

22. 6.829 Computer Networks22 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 | | s λ = / || φ λ || 2 r = r - s λ * φ λ A= A + s λ * φ λ 3. Reconstruction

23. 6.829 Computer Networks23 Iterative OMP -2