Introduction to compressive sensing

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

 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

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

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
4

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.
5

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
6

Sparsity
7

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.
8

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
9

Incoherence Sampling
10

 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.
11

 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 .
12

 Contents
•Sparsity
•Random Sensing
13

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 .
14

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.
15

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
16

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.
17

 Contents
•Sparsity
•Random Sensing
18

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
19

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
20

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
21

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
22

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.
23

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
24

Applications
 Data Compression
 Channel Coding
 Inverse Problem
 Data Acquisition
 Wireless Channel Estimation
25

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).
26

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

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