Introduction to compressive sensing

1. Introduction to Compressive Sensing BY / ENG. AHMED NASSER AHMED DEMONSTRATOR AT FACULTY OF ENGINEERING SUEZ CANAL UNIVERSITY IntroductiontoCompressiveSensing 1

2.  Contents 1- What is compressive sensing (CS) •Sparsity •Incoherence Sampling 2- UNDERSAMPLING AND SPARSE SIGNAL RECOVERY 3- Robust Compressive Sensing •Restricted Isometry Property(RIP) •Random Sensing •General Data Recovery From Undersampling Data •Robust Signal Recovery From Noisy Data 4- Compressive Sensing applications Introduction to Compressive Sensing 2

3. Nyquist Rate Introduction to Compressive Sensing 3 Sampling rate > 2 * max frequency What about 4K HD videos ?! The solution is : Compressed Sensing

4. What is compressive sensing (CS)  compressive sensing (CS) theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use  CS relies on two principle 1. sparsity: which pertains to the signal of interest 2. In coherence : which pertains to the sensing modality Introduction to Compressive Sensing 4

5. Sparsity  Sparsity expresses the idea that the “information rate” of a continuous time signal may be much smaller than suggested by its bandwidth, or that a discrete-time signal depends on a number of degrees of freedom which is comparably much smaller than its (finite) length  CS exploits the fact that many natural signals are sparse or compressible in the sense that they have concise representations when expressed in the proper basis Ψ. Introduction to Compressive Sensing 5

6. Sparsity  The basis Ψ ⋴ R 𝑛 must be orthonormal basis (Orthogonal + Normalize)  2-D axis R2 1 0 0 1 2 3 = 2 3  3-D axis R3  1 0 0 0 1 0 0 0 1 2 4 5 = 2 4 5 Introduction to Compressive Sensing 6

7. Sparsity Introduction to Compressive Sensing 7

8. Sparsity  Many natural signals have concise representations when expressed in a convenient basis.  Consider, for example, the image in Figure 1(a) and its wavelet transform in (b). Although nearly all the image pixels have nonzero values, the wavelet coefficients offer a concise summary: most coefficients are small, and the relatively few large coefficients capture most of the information. Introduction to Compressive Sensing 8

9. Sparsity  If we have a vector which we expand in an orthonormal basis (such as a wavelet basis)  Where :  X : is the coefficient sequence of ƒ  Ψ : is the n x n matrix with Ψ1 , …….., Ψ 𝑛 as columns  S-Sparse signal: is the signal that has S nonzero entries Introduction to Compressive Sensing 9

10. Incoherence Sampling Introduction to Compressive Sensing 10

11. Incoherence Sampling  Suppose we are given a pair (Φ, Ψ) of orthobases of R 𝑛 .  The first basis Φ is used for sensing the object ƒ  The second Ψ is used to represent ƒ  the coherence measures the largest correlation between any two elements of Φ and Ψ. If Φ and Ψ contain correlated elements, the coherence is large. Otherwise, it is small  The smaller the coherence, the fewer sample are needed. Introduction to Compressive Sensing 11

12. Incoherence Sampling  random matrices are largely incoherent with any fixed basis Ψ .  Select an orthobasis Φ uniformly at random, which can be done by orthonormalizing n vectors sampled independently and uniformly on the unit sphere. Then with high probability, the coherence between Φ and Ψ is about .  By extension, random waveforms (ϕk (t )) with independent identically distributed (i.i.d.) entries, e.g., Gaussian or ±1 binary entries, will also exhibit a very low coherence with any fixed representation . Introduction to Compressive Sensing 12

14. UNDERSAMPLING AND SPARSE SIGNAL RECOVERY  We can use ℓ1- normalization to recover the sparse signal  if  the proposed reconstruction ƒ ∗ us given by ƒ ∗ = Ψ 𝑥∗  Where 𝑥∗ is the convex optimization program  Among all objects consistent with the data, we pick that whose coefficient sequence has minimalℓ1 norm . Introduction to Compressive Sensing 14

15. UNDERSAMPLING AND SPARSE SIGNAL RECOVERY  Result asserts that when f is sufficiently sparse, the recovery via ℓ1 minimization is provably exact  ℓ1-minimization is not the only way to recover sparse solutions; other methods, such as greedy algorithms, have also been proposed. Introduction to Compressive Sensing 15

16. UNDERSAMPLING AND SPARSE SIGNAL RECOVERY  Theorem 1 :  For ƒ ⋴ R 𝑛 , and the coefficient sequence x of f in the basis Ψ is S-sparse  Then if the Selected m measurements in the Φ domain uniformly at random equal :  the solution to (5) is exact with overwhelming probability.(the probability of success exceeds 1-δ if Introduction to Compressive Sensing 16

17. UNDERSAMPLING AND SPARSE SIGNAL RECOVERY  According to there are three comments: 1) the smaller the coherence, the fewer samples are needed, hence our emphasis on low coherence systems 2) One suffers no information loss by measuring just about any set of m coefficients which may be far less than the signal size apparently demands. If μ(Φ, Ψ) is equal or close to one, then on the order of Slog n samples suffice instead of n. 3) The signal f can be exactly recovered from our condensed data set by minimizing a convex functional which does not assume any knowledge about the number of nonzero coordinates of x, their locations, or their amplitudes which we assume are all completely unknown a priori. We just run the algorithm and if the signal happens to be sufficiently sparse, exact recovery occurs. Introduction to Compressive Sensing 17

19. Robust Compressive Sensing  in order to be really powerful, CS needs to be able to deal with both nearly sparse signals and with noise. So general objects of interest are not exactly sparse but approximately sparse. CS must deal with two issue: 1) First, is whether or not it is possible to obtain accurate reconstructions of such objects from highly undersampled measurements. 2) Second, in any real application measured data will invariably be corrupted by at least a small amount of noise as sensing devices do not have infinite precision  In general 𝑦 = 𝐴𝑥 + 𝑧 where  A=RΦΨ : is an m× n “sensing matrix” giving us information about x,  R is the m× n matrix extracting the sampled coordinates in M  z: is a stochastic or deterministic unknown error term Introduction to Compressive Sensing 19

20. Restricted Isometry Property(RIP)  Restricted Isometry Property(RIP) has proved to be very useful to study the general robustness of CS  Restricted Isometry Property(RIP) measure the orthogonality of all subsets of S columns taken from A which is very important for sparse signal recovery  For each integer S = 1, 2, . . . , define the Isometry constant δ 𝑠 of a matrix A as the smallest number such that  We will loosely say that a matrix A obeys the RIP of order S if δ 𝑠 is not too close to one.  When the RIP property holds this mean  Approximately preserves the Euclidean length of S-sparse signals, which in turn implies that S-sparse vectors cannot be in the null space of A.  Or , means that that all subsets of S columns taken from A are in fact nearly orthogonal Introduction to Compressive Sensing 20

21. Random Sensing  we would like to find sensing matrices with the property that column vectors taken from arbitrary subsets are nearly orthogonal  the following sensing matrices can be considered 1. form A by sampling n column vectors uniformly at random on the unit sphere of Rm; 2. form A by sampling i.i.d. entries from the normal distribution with mean 0 and variance 1/m; 3. form A by sampling a random projection P as in “Incoherent Sampling” and normalize: A = n/m P 4. form A by sampling i.i.d. entries from a symmetric Bernoulli distribution or other sub-gaussian distribution. 5. RIP can also hold for sensing matrices A = ΦΨ, where Ψ is an arbitrary orthobasis and Φ is an m× n measurement matrix drawn randomly from a suitable distribution  And so we can substitute m to Introduction to Compressive Sensing 21

22. General Data Recovery From Undersampling Data  THEOREM 2 :  If the RIP holds, then the following linear program gives an accurate reconstruction of the sparse signal: Where 𝑥∗ obey to And Introduction to Compressive Sensing 22

23. General Data Recovery From Undersampling Data  The conclusions of Theorem 2 are stronger than those of Theorem 1 as : 1. this new theorem deals with all signals. If x is not S-sparse, then theorem asserts that the quality of the recovered signal is as good as if one knew ahead of time the location of the S largest values of x and decided to measure those directly. 2. In other words, the reconstruction is nearly as good as that provided by an oracle which, with full and perfect knowledge about x, extracts the S most significant pieces of information for us. 3. Another striking difference with our earlier result is that it is deterministic; it involves no probability. If we are fortunate enough to hold a sensing matrix A obeying the hypothesis of the theorem, we may apply it, and we are then guaranteed to recover all sparse S-vectors exactly, and essentially the S largest entries of all vectors otherwise; i.e., there is no probability of failure. Introduction to Compressive Sensing 23

24. Robust Signal Recovery From Noisy Data  THEOREM 3:  If We are given noisy data as in and use 1 minimization with relaxed constraints for reconstruction:  Where 𝑥∗ obey to And , 𝐶1, 𝐶2 are typically Small Introduction to Compressive Sensing 24

25. Applications  Data Compression  Channel Coding  Inverse Problem  Data Acquisition  Wireless Channel Estimation Introduction to Compressive Sensing 25

26. References  [1] Candès, Emmanuel J., and Michael B. Wakin. "An introduction to compressive sampling." Signal Processing Magazine, IEEE 25.2 (2008): 21- 30.  [2] Baraniuk, Richard G. "Compressive sensing." IEEE signal processing magazine 24.4 (2007). Introduction to Compressive Sensing 26

27. Introduction to Compressive Sensing 27 Thank You Contact me: Web site: www.ahmed_nasser_eng.staff.scuegypt.edu.eg Email: ahmed.nasserahmed@gmail.com Ahmed.nasser@eng.suez.edu.eg

Introduction to compressive sensing

Ahmed Nasser Agag

Introduction to compressive sensing