Compressed Sensing: A Magnetic Resonance Imaging Perspective

D97945003

Jia-Shuo Hsu

2009/12/10

Magnetic Resonance Imaging (MRI) Acquires Data from Frequency Other than Image Domain

Characteristics:

Samples frequency domain

then retain image

Undersample shortens scan

time directly

Mostly Fourier Encoding

Wavelet domain

Imagedomain

Spatial freq Image

Sampling Theorem bounds the number of

samples required for full signal recovery

nN

jkN

n

enxkX21

0

][][

n

NjkN

n

enxkX2

212/

0

]2[][

V.S.

Techniques adopted to get around

1. Efficient Sampling Pattern Ex: Optimized Lattice Grid Sampling

2. Exploit spatio-temporal redundancy Ex: Short-Time FT to aperiodic signal

3. Alter characteristics of aliasing Ex: Various choice of time-frequency analysis that alters

the shape of spectrum

1. Certain undersampling patterns “pack”

signals efficiently within given bandwidth

Two different 5-fold undersampling

Fourier Transform

2. Time-varying Signals are Relatively Redundant in Time-Frequency Domain

3.Non-Cartesian Sampling Distorts

Aliasing into Non-regular Pattern

Tsao.et al. Magnetic Resonance in Medicine 55:116–125 (2006)

Compressible Signal Suggests

“Inhomogeneous” information distribution

Tutorial on Compressive Sensing, R. Baraniuk et al. (Feb 2008)

Possibility to fully recover highly undersampled signal ??

Emmanuel J. Candès, Justin Romberg, Member, IEEE, and Terence TaoIEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 2, FEBRUARY 2006

512*512 Shepp-Logan Undersampled by 22 radial lines

Normal Reconstruction

??????????????????

Introduce Compressed Sensing

Fulfilling certain criteria, 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

Sparsity

Number of significant(strictly speaking,

nonzero)components is relatively small

compared to signal length Ex: [1 0 10 0 0 0 0 0…….0 0]

Sparsity Representation: Lp-Norm: L0 norm counts the number of non-zero

components of x Ex: if x=[1, 100000, 2, 0], then L0-Norm=3

ppN

iip xx

1

1

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

Random results in noise-like while regular

equally weights the artifacts

U. Gamper et al. Magnetic Resonance in Medicine 59:365–373 (2008)

Non-linear Reconstruction

Lacks the linearity of FFT and iFFT

Does not have analytical solution as in STFT,

Gabor Transform, WDF….etc

Involves optimizations (often iterative) satisfying

certain boundary conditions

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)

Demo 1: Reconstructing Highly Undersampled Sparse Signal

Random sampling generates

noise-like artifacts

M.Lustig et.al Magnetic Resonance in Medicine 58:1182–1195 (2007)

(a) given that desired

signal is sparse

(b) different k-space

sampling pattern

(c) regular undersampling

begets regular aliasing

(d) random undersampling

begets noise-like

aliasing, preserving

most of the major

components

Signal satisfying CS1-3 are recovered

through CS

M.Lustig et.al Magnetic Resonance in Medicine 58:1182–1195 (2007)(e) detected strong

components above the

interference level

(f) obtain estimates by

thresholding

(g) convolve (f) with PSF,

obtain undersampled

version of the signal (f)

(h) subtract (g) from (e),

thus another major

component hindered by

noise reveals

M.Lustig et.al Magnetic Resonance in Medicine 58:1182–1195 (2007)

CS mathematically

Minimize such that , where

x stands for reconstructed signal

y’ stands for the estimated measurement

y stands for the initial measurement

ε serves as the boundary condition (usually noise level)

In other words, among all possible solutions of x, find one with

the smallest L0-norm(i.e. sparsest) whose estimated

measurement y’ remains consistent with the initial measurement

y with deviation less than ε

0x

2' yy

Many signals are not as sparse, strictly

limiting the application?

Sparsity (i.e. Compressibility) can be generated

through sparsifying transform

Signals that are compressible demonstrate

sparsities in their sparsifying transform domains

Revisit CS mathematically

Minimize such that , where

x stands for reconstructed signal

stands for sparsifying transform

y’ stands for the estimated measurement

y stands for the initial measurement

ε serves as the boundary condition (usually noise level)

Among all possible sparsified solutions, find one with the smallest L0-

norm(i.e. sparsest) whose estimated measurement y’ remains consistent

with initial measurement y with deviation less than ε

Most use L1-norm, i.e. minimize instead

0x

2' yy

1x

Choice of Sparsifying Transform is Essential to Performance

It’s all about finding the right

STFT, Gabor, WDF, S-Transform,

Wavelet Transform……

MRI suits CS in certain perspectives

Data is acquired in sampling space

Medical images posses sparsities

Achieved results in angiography, dynamic

imaging, MRSI and other potential applications

CS applies as long as CS1-3 holds in

sparsifying transform domain

uF 1uF

If image is already sparse

Non-linear reconstruction

1

M.Lustig et.al Magnetic Resonance in Medicine 58:1182–1195 (2007)

Demo 2: CS-reconstructed MR Image

Challenges and works to be done lie in every

aspects of CS procedure

uF 1uF

If image is already sparse

Non-linear reconstruction

1

M.Lustig et.al Magnetic Resonance in Medicine 58:1182–1195 (2007)

Criteria of CS remain to be further

customized to fit different application

Sparsity: Representation: What represents sparsity? Degree: How sparse is enough? Compressibility: Which sparsifying transform?

Incoherence Representation: What represents randomness? Degree: How random is enough?

Non-Linearity

Choice of method and complexity?

Representation of Sparsity is essential to

required sample number

L0-norm is ideal, yet intractable Needs only M=K+1 samples for K-sparse signals is an NP problem when p=0

L2-norm(i.e.) is well-known, yet inaccurate p=2 represents Least Mean Square

L1-norm requires more samples than L0, yet is most

feasible in its tractability and accuracy Needs approximately K log(N/K) samples, yet no longer NP L1-norm minimization is equivalent to a classical convex

optimization problem with many well-established approaches

ppN

iip xx

1

1

Ways to measure and achieve

incoherence remains to be developed

Approaches were taken,

yet reliabilities to be

verified

Inherent regularity of

Fourier basis limits

degree of randomness

Randomness doesn’t

guarantee performance

Non-Fourier

Fourier Basis

Reconstruction involves optimization with

unpredictable non-linearity

Complexity of the reconstruction is unpredictable

M.Lustig et.al Magnetic Resonance in Medicine 58:1182–1195 (2007)

How Long does the loop loops?

Summary

Theory of Compressed Sensing: From CS to MRI

Sparsity, incoherence, non-linear reconstruction

Sometimes requires transform (compression) to achieve sparsity

Random sampling of k-space generates noise-like aliasing artifacts

Non-linear reconstruction ties to some well-known optimization problem

Challenges and Focus

Acquisition mechanism of MRI is unfavorable to randomness

Prior knowledge of image on sparsity is required

Criteria of CS and their representations remain to be customized in MRI

Suitable applications are to be further explored

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!!

Appendix A: Online Resources

Open Source Softwares http://sparselab.stanford.edu/

A free matlab toolbox consists of CS algorithms

Collection of current works http://www.dsp.ece.rice.edu/cs/

MRI-specific of CS http://www.stanford.edu/~mlustig/

Appendix B: Recommended Literatures

Sparse MRI: The Application of Compressed

Sensing for Rapid MR Imaging An MRM publication with many results of CS in MRI

http://www.dsp.ece.rice.edu/cs/CS_notes.pdf A succinct note on theory of CS

http://www.dsp.ece.rice.edu/~richb/talks/cs-tutori

al-ITA-feb08-complete.pdf

A broad view of CS from theory to application