compressed sensing: a magnetic resonance imaging perspective d97945003 jia-shuo hsu 2009/12/10
TRANSCRIPT
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Compressed Sensing: A Magnetic Resonance Imaging Perspective
D97945003
Jia-Shuo Hsu
2009/12/10
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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
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Sampling Theorem bounds the number of
samples required for full signal recovery
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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
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1. Certain undersampling patterns “pack”
signals efficiently within given bandwidth
Two different 5-fold undersampling
Fourier Transform
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2. Time-varying Signals are Relatively Redundant in Time-Frequency Domain
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3.Non-Cartesian Sampling Distorts
Aliasing into Non-regular Pattern
Tsao.et al. Magnetic Resonance in Medicine 55:116–125 (2006)
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Compressible Signal Suggests
“Inhomogeneous” information distribution
Tutorial on Compressive Sensing, R. Baraniuk et al. (Feb 2008)
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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
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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
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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
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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
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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
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Medical images often demonstrate
inherent sparsities
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Incoherence
Sampling must generate noise-like aliasing in
image domain (more strictly, transform domain)
Very loosely speaking, patterns of sampling must
demonstrate enough randomness
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Random results in noise-like while regular
equally weights the artifacts
U. Gamper et al. Magnetic Resonance in Medicine 59:365–373 (2008)
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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
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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)
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Demo 1: Reconstructing Highly Undersampled Sparse Signal
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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
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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)
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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
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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
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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
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Choice of Sparsifying Transform is Essential to Performance
It’s all about finding the right
STFT, Gabor, WDF, S-Transform,
Wavelet Transform……
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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
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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)
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Demo 2: CS-reconstructed MR Image
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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)
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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?
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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
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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
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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?
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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
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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!!
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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/
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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