ompressive Sensing
• But there are some flaws of this method: • 1) Considering the Shannon sampling theorem，the sampling interval will be very narrow to gain better signal resolution，which will make the original signal very long, so the processing of transformation costs lots of time. • 2) The positions of K components required to remain vary while the signal changes. Therefore, this strategy is self-adaptive, and we need to allocate more space to store these positions. • 3) Poor anti-interference. Once one of the K components lost in transmission, the output will be changed greatly.
An Overview of Compressive Sensing
• The definition of norm: • For a vector x, if there is a corresponded real function ||x||, which fits such conditions: 1) ||x||≥0, only if x=0, ||x||=0; 2) for any number a, ||ax||=|a|||x||; 3) for any vector x and y, ||x+y||≤||x||+||y||; • Then we call ||x|| the norm of x. • RIP: for δK∈(0,1) (1-δK)||x||22≤||Φx||22 ≤ (1+δK)||x||22
Coding Signal x Sampling Transformation y
Decoding Received data y
Inverse transformation
Reconstructed signal x*
coding：make orthogonal matrix Ψ, transformation y= Ψx, remain the most important K components of y and the corresponding positions. decoding：put K components back to the corresponding positions, let other positions be zero, make ΨH，inverse transformation x*=ΨHy*.
An Overview of Compressive Sensing
• Suppose x (n) is a digital signal, if it’s a K-sparse (has K non-zero values) or compressible signal, then we can estimate it with few coefficients by linear transformation. By compressive sensing we get the signal y (m) (m<n), • y=Φx • Φ is called sensing matrix with m×n dimension. • The dimension of y is much less than that of x, so the equation has infinitive solutions, which makes it difficult to rebuild original signal. Since x is Ksparse, we can rebuild x from y by solving the optimal problem below: • x*=min||x||0 s.t. y= Φx • Candes indicates that when m>Klog(n) and Φ has restricted isometry property (RIP), x (n) can be rebuilt.
An Overview of Compressive Sensing
• But few of natural signal is sparse. • According to compressive sensing theory, signal x can be sparse by some reversible transformation Ψ, that is x= Ψs, so we have • y=Φx= ΦΨs • Baraniuk indicates that the equivalent condition of RIP is that the measurement matrix Φ and the sparse base Ψ is irrelevant. It’s confirmed that when Φ is Guass random matrix, the condition is well fitted.
OMP Algorithm
• In some circumstance, we can replace l0 norm with l1 norm, that is • x*=min||x||1 s.t. y= Φx • The problem above can be solved by greed iterative algorithm, one of the most commonly used algorithm is the orthogonal matching pursuit (OMP) method. • The main idea of the OMP algorithm: choose the column of Φ by greed iterative method, which makes the chosen column and the present redundant vector related to the greatest extent, we subtract the related part from measurement vector, repeat the procedure above until the number of iterations up to K.
OMP Algorithm
• Input: sensing matrix Φ, sampling vector y, sparse degree K; • Output: the K-sparse approximation x* of x; • Initialization: the residual r0=y, index set Λ0=∅, t=1;
The Compressive Sensing Theory
And Practice of OMP Algorithm
2011-01-25
An Overview of Compressive Sensing
• For 1-D signal X∈RN×1, mostly，the information is redundant. 。 • We can compress it by orthogonal transformation.
An Overview of Compressive Sensing
• The advantages of compressive sensing: • 1) Non-adaptive, break through the limitation of Shannon sampling theorem. • 2) Strong Anti-interference ability, every component of the measurement is important, or unimportant. It can still be reconstructed while some components are lost. • The application prospect of compressive sensing is broad: • digital camera and audio acquisition device with low cost; astronomy (stars are sparse); network; military.
Coding Sparse signal x Received signal y Measurement, coding Decoding, reconstruction y
Decoding
Constructed signal x*
The measured value is not the signal itself, but the projective value from higher dimension to lower dimension.
OMP Algorithm
• Execute steps 1 to 5 circularly: • Step 1: find the maximum value of the inner product of residual r and the column of sensing matrix φj, the corresponding foot mark is λ; • Step 2: renew the index set Λt=Λt-1∪{λ} , the sensing matrix Φt=[Φt-1, φλ]; • Step 3: solve x*t=min||y- Φtx*||2 by least-square method; • Step 4: renew the residual rt=y-Φtx*t, t=t+1; • Step 5: if t>K, stop the iteration, else do step 1.