1-Compressed sensing. 2-Partial Fourier. 3-My thesis.

Introduce Compressed sensing it is possible to fully recover a signal from sampling points much fewer than that defined by Shannon's sampling theorem

Compressed Sensing Given x of length N, only M measurements (M<N) is required to fully recover x when x is K-sparse (K<M<N) However, three conditions named CS1-3 are to be satisfied for the above statement to be true

Three essential criteria Sparsity: The desired signal has a sparse representation in a known transform domain Incoherence Undersampled sampling space must generate noise-like aliasing in that transform domain Non-linear Reconstruction Requires a non-linear reconstruction to exploit sparsity while maintaining consistency with acquired data

Medical images often demonstrate inherent sparsities

Incoherence Sampling must generate noise-like aliasing in image domain (more strictly, transform domain) Very loosely speaking, patterns of sampling must demonstrate enough randomness

Conjugate Gradient: non-linear recon with iterative optimization A multi-dimensional optimization method suitable for non-cartesian sampled images M.S. Hansen et.al Magnetic Resonance in Medicine 55:85 – 91 (2006)

Partial k-space acquisition Partial acquisition in phase encode (PE) axis to reduce scan time Partial acquisition in frequency encode (FE) axis to reduce echo time

What is Half Fourier/Partial Fourier Reconstruction? Constrained reconstruction exploiting symmetry properties of Fourier transform For any real x[n] (Conjugate Symmetry)

What I want to is >>>> *Using Partial Fourier and compressed sensing to reduce number of samples as much as possible

Wavelets are no longer the central topic, despite the previous edition’s original title. It is just an important tool, as the Fourier transform is. Sparse representation and processing are now at the core - S. Mallat, 2009 Thanks for Your Attention!!