PhD Thesis: Pratical Compressive Sensing by Becker, Stephen R.

First paper on Basis Pursuit??? Excellent discussion on the difference between basis versus matching pursuit by S Chen and D Donoho.

Terms and jargon commonly used:

A

B

Bandlimited: A function whose Fourier Transform has a bounded support.

C

Concentration of measure:

• Terry Tao's explanation

D

Dictionary

• A set of vectors (not-nessiceraly the minimum amount of vectors) that can be used to represent a signal. An orthogonal basis is a dictionary of minimum size.

Redundant Dictionaries:

• Easy to guess a dictionary with more vectors than nessecary to represent a single. The utility of redundant dictionaries come from the fact that signals become even more sparse if the redundant vectors are chosen correctly.
• For example "In natural languages, a richer dictionary helps to build shorter and more precise sentences. Similarly, dictionaries of vectors that are larger than bases are needed to build sparse representations of complex signals." - Stephane Mallat

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F

G

The Golden Rule For Solving Inverse Problems

• Search for approximate solutions satisfying additional constraints coming from the physics of the problem. Additional information is needed to compensate for the loss of information due to the mapping (aka imaging process).

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I

Ill-Posed Problem:

• A problem whose solution is not unique.

Ill-Condition Systems:

• A system in which the solution exists an is unique but is completely corrupted by a small error on the data

IRS: Iterative Reweighted Shrinkage

• IRS algorithms are faster than Ierative Shrinkage\Thresholding (IST) when the operator is very ill-conditioned.

IST: Iterative Shrinkage\Thresholding Algorithms

• IST alogirthms are used for a variety of optimization problems. Notably they have been used for the LIP. IST algorithms belong to a class of so-called forward-backward algorithms [1]. IST algorithms are said to be faster in terms of convergence than iterative reweight shrinkage (IRS) when the operators are mildly ill-conditions and there is strong noise. [2]

Invariant Subspace:

• A subspace U, whose vectors are closed under the linear operator T, in other words it maps onto itself whenver T to any of its vectors.

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K

L

Linear inverse problem (LIP):

• Estimate an unknown vector \textbf{x}

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N

O

Operator

• A linear amp from a vector space to itself is called an operator.

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Q

R

Regularization:

• http://en.wikipedia.org/wiki/Regularization_(mathematics)

S

Semi-Continuity:

• http://en.wikipedia.org/wiki/Semi-continuity

Space Limited: A functions whose support is bounded.

T

TwIST(2IST):

• Two step Iterative Shrinkage\Thresholding. Suppossed to be faster convergence with severely ill-conditioned operators. The update equation uses the two previous estimates, rather than only the previous one.

U
V

W

X

Y

Z

Overcomplete Signal Dictionary

References:

[1] P. Combetter and V. Wajs, "Signal recovery by proximal forward-backward splitting"

[2] J. Bioucas-Dias and M.A.T. Figueiredo, "Two-step algorithms for linear inverse problems with non-quadratic regularization"