1. Cs: compressed sensing
Jialin peng

2. Outline Introduction Exact/Stable Recovery Conditions
-norm based recovery
OMP based recovery
Some related recovery algorithms
Sparse Representation
Applications

3. Introduction high-density sensor high speed sampling ……
Data
Compression
Data Storage
high-density sensor
high speed sampling ……
A large amount of sampled data will be discarded
A certain minimum number
of samples is required in order to perfectly capture an arbitrary bandlimited signal
decompress
Receiving & Storage

4. Sparse Property
Important classes of signals have naturally sparse representations with respect to fixed bases (i.e., Fourier, Wavelet), or concatenations of such bases.
Audio, images …
Although the images (or their features) are naturally very high dimensional, in many applications images belonging to the same class exhibit degenerate structure.
Low dimensional subspaces, submanifolds
representative samples—sparse representation

5. Transform coding: JPEG, JPEG2000, MPEG, and MP3

6. The Goal Relying on structure in the input
Develop an end-to-end system
Sampling
processing
reconstruction
All operations are performed at a low rate: below the Nyquist-rate of the input (too costly, or even physically impossible)
Relying on structure in the input

7. Sparse: the simplest choice is the best one
Signals can often be well approximated as a linear combination of just a few elements from a known basis or dictionary.
When this representation is exact ,we say that the signal is sparse.
Remark:
In many cases these high-dimensional
signals contain relatively little information compared to their ambient dimension.

8. Introduction high-density sensor high speed sampling ……
Data
Compression
Data Storage
high-density sensor
high speed sampling ……
A large amount of sampled data will be discarded
A certain minimum number
of samples is required in order to perfectly capture an arbitrary bandlimited signal
decompress
Receiving & Storage

9. Introduction Alleviated sensor Reduced data …… Sparse priors of signal
Nonuniform sampling
Imaging algorithm: optimization
Alleviated sensor
Reduced data ……
Data Storage
modified sensor
Receiving & Storage
optimization

10. Introduction =
Sensing Matrix

11. compression Find the most concise representation:
Compressed sensing: sparse or compressible representation
A finite-dimensional signal having a sparse or compressible representation can be recovered from a small set of linear, nonadaptive measurements
how should we design the sensing matrix A to ensure that it preserves the information in the signal x?.
how can we recover the original signal x from measurements y?
Nonlinear:
Unknown nonzero locations results in a nonlinear model:
the choice of which dictionary elements are used can change from signal to signal .
2. Nonlinear recovering algorithms
the signal is well-approximated by a signal with only k nonzerocoefficients

12. Exact/Stable Recovery Condition
Introduction
Let be a matrix of size with
For a –sparse signal , let be the measurement vector.
Our goal is to exact/stable recovery the unknown signal from measurement.
The problem is under-determined.
Thanks for the sparsity, we can reconstruct the signal via
How can we recovery the unknown signal:
Exact/Stable Recovery Condition

13. Exact/stable recovery conditions
The spark of a given matrix A
Null space property (NSP) of order k
The restricted isometry property
Remark:
verifying that a general matrix A satisfies any of these properties has a combinatorial computational complexity

14. Exact/stable recovery conditions
The restricted isometry constant (RIC) is defined as the smallest constant which satisfy:
The restricted orthogonality condition (ROC) is the smallest number such that:
Restricted Isometry Property

15. Exact/stable recovery conditions
Solving minimization is NP-hard, we usually relax it to the or minimization.

16. Exact/stable recovery conditions
For the inaccurate measurement , the stable reconstruction model is

17. Exact/stable recovery conditions
Some other Exact/Stable Recovery Conditions:

18. Exact/stable recovery conditions
Braniuk et al. have proved that for some random matrices, such as
Gaussian,
Bernoulli, ……
we can exactly/stably reconstruct unknown signal with overwhelming high probability.

19. Exact/stable recovery conditions
cf: minimization 19

20. Exact/stable recovery conditions
Some evidences have indicated that with , can exactly/stably recovery signal with fewer measurements.

21. Quicklook Interpretation
Dimensionality-reducing projection.
Approximately isometric embeddings, i.e., pairwise Euclidean distances are nearly
preserved in the reduced space
RIP

22. Quicklook Interpretation

23. Quicklook Interpretation
the ℓ2 norm penalizes large coefficients heavily, therefore solutions tend to have many smaller coefficients.
In the ℓ1 norm, many small coefficients tend to carry a
larger penalty than a few large coefficients.

24. Algorithms L1 minimization algorithms iterative soft thresholding
iteratively reweighted least squares …
Greedy algorithms
Orthogonal Matching Pursuit
iterative thresholding
Combinatorial algorithms

25. CS builds upon the fundamental fact that
we can represent many signals using only a few non-zero coefficients in a suitable basis or dictionary.
Nonlinear optimization can then enable recovery of such signals from very few measurements.

26. Sparse property
The basis for representing the data
incoherent->task-specific (often overcomplete) dictionary or redundant one

27. MRI Reconstruction
MR images are usually sparse in certain transform domains, such as finite difference and wavelet.

28. Sparse Representation
Consider a family of images, representing natural and typical image content:
Such images are very diverse vectors in
They occupy the entire space?
Spatially smooth images occur much more often than highly non-smooth and disorganized images
L1-norm measure leads to an enforcement of sparsity of the signal/image derivatives.
Sparse representation

29. Matrix completion algorithms
Recovering a unknown (approximate) low-rank matrix from a sampling set of its entries.
NP-hard
Convex relaxation
Unconstraint