An Introduction To Compressive Sampling

1. An Introduction To Compressive Sampling Advisor: 王聖智 教授 Student: 林瑋國

2. outline • Introduction • Sparse signal recovery • Theorem 1 • Theorem 2 • Theorem 3 • Summary

3. outline • Introduction • Sparse signal recovery • Theorem 1 • Theorem 2 • Theorem 3 • Summary

4. Introduction(1) • The signal can be present by the basis • Sometimes we can ignore the smaller coefficients and also can maintain the main information

5. Introduction(2) • Example:

6. Introduction(3) • We have the value “zero” and maybe we don’t send the value “zero” • But the received doesn’t know the position of zero • Someone thinks maybe we can compress the data

7. Introduction(4) • We use compress the data to Y • The received have the Y and use to recover the data X

8. Introduction(5) • There are many solutions of (1)

9. outline • Introduction • Sparse signal recovery • Theorem 1 • Theorem 2 • Theorem 3 • Summary

10. Theorem 1 • Theorem 1 • Selecting A uniformly from m by n matrix If for some positive constant C , the solution of (2) is exact with overwhelming probability

11. Theorem 2(1) • If the n by n matrix M satisfys the matrix has the “isometry” property

12. Theorem 2(2) • Restricted Isometry Property (RIP) s s s

13. Theorem 2(3) • If the RIP holds, then the following linear program gives an accurate reconstruction • Theorem 2 Assume then the solution of (3) obeys s

14. Theorem 3(1) • Sometimes the sparse recovery has to deal with noise • Thus we have to solve the problem

15. Theorem 3(2) • If the RIP holds, then the following linear program gives an accurate reconstruction • Theorem 3 Assume then the solution of (4) obeys

16. outline • Introduction • Sparse signal recovery • Theorem 1 • Theorem 2 • Theorem 3 • Summary

17. Summary • The Theorem1-3 has to use on the S-sparse vector • But the signal sometimes isn’t sparse vector and we have to represent by S-sparse vector • Then

18. Example