10/06/20041 resolution enhancement in mri by: eyal carmi joint work with: siyuan liu, noga alon,...
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10/06/2004 1
Resolution Enhancementin MRI
By: Eyal Carmi
Joint work with:
Siyuan Liu, Noga Alon,
Amos Fiat & Daniel Fiat
2
Lecture Outline
Introduction to MRIThe SRR problem (Camera & MRI)Our Resolution Enhancement AlgorithmResultsOpen Problems
3
Introduction to MRI
Magnetic resonance imaging (MRI) is an imaging technique used primarily in medical settings to produce high quality images of the inside of the human body.
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Introduction to MRI
The nucleus of an atom spins, or precesses, on an axis.
Hydrogen atoms – has a single proton and a large magnetic moment.
5
Magnetic Resonance Imaging
Uniform Static Magnetic Field –
Atoms will line up with the direction of the magnetic field.
0 0
0
0
w B
B MagneticField
GyromagneticRatio
w LarmorFrequency
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Magnetic Resonance Imaging
Resonance – A state of phase coherence among the spins.
Applying RF pulse at
Larmor frequency When the RF is turned off
the excess energy is released
and picked up.
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Magnetic Resonance Imaging
Gradient Magnetic Fields –
Time varying magnetic fields
(Used for signal localization)
x
y
z
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Magnetic Resonance Imaging
Gradient Magnetic Fields: 1-D
X
Y
B0
B1
B2
B3
B4
0 0
0
0
w B
B MagneticField
GyromagneticRatio
w LarmorFrequency
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Signal Localization slice selection
Gradient Magnetic Fields for slice selection
z
ω
z1 z2 z3
0B
Gz,1
Gz,2
z4
ω1
ω2FT
B1(t)
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Signal Localizationfrequency encoding
Gradient Magnetic Fields for in-plane encoding
t
x
B0B=B0+Gx(t)x
t
Gx(t)
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Signal Localizationphase encoding
Gradient Magnetic Fields for in-plane encoding
t
x
B0B=B0+Gx(t)x
t
Gx(t)
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The Super Resolution Problem
Definition:
SRR (Super Resolution Reconstruction): The process of combining several low resolution images to create a high-resolution image.
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SRR – Imagery Model
The imagery process model:
Yk – K-th low resolution input image.
Gk – Geometric trans. operator for the k-th image.
Bk – Blur operator of the k-th image.
Dk – Decimation operator for the k-th image.
Ek – White Additive Noise.
NkEXGBDY kkkkk 1
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SRR – Main Approaches
Frequency Domain techniques
Tsai & Huang [1984]
Kim [1990]
Frequency Domain
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SRR – Main Approaches
Iterative Algorithms
Irani & Peleg [1993] : Iterative Back Projection
Current HR Best Guess
Back Projection
Back Projected LR images
Original LR images
Iterative Refinement
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SRR – Main Approaches
Patti, Sezan & Teklap [1994]
POCS:
Elad & Feuer [1996 & 1997]
ML:
MAP:
POCS & ML
EXHY
2XHY
XYPXmaxarg
XPXYPX
maxarg
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SRR – In MRI
Peled & Yeshurun [2000]
2-D SRR, IBP, single FOV,
problems with sub-pixel shifts.
Greenspan, Oz, Kiryati and Peled [2002] 3-D SRR (slice-select direction), IBP.
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Resolution Enhancement Alg.
A Model for the problemReconstruction using boundary values1-D Algorithm2-D Algorithm
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Modeling The Problem
True Image : A matrix of real values associated with a rectilinearly aligned grid of arbitrary high resolution.
23895
96644
53492
61476
88135(0,0)
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Modeling The Problem
A Scan of the image:
Pixel resolution –
Offset –
),,(),( yxmFjiS mm
True Image
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1632
1122
1122
4433
4433),( yx
),( yx
Image Scan, m=2
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Modeling The Problem
Definitions:
Maximal resolution –
Pixel resolution =
Maximal Offset resolution – We can perform scans at offsets where
with pixel resolution
n1
n mmm ,
n
),( yx
3,2,1 ,, kkyx
Zccm ,1
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Modeling The Problem
Goal: Compute an image of the subject area with pixel resolution while the maximal measured pixel resolution is
11 δn, nn 1
Errors:
1. Errors ~ Pixel Size & Coefficients
2. Immune to Local Errors =>
localized errors should have localized effect.
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Using boundary value conditions Assumption: or
Reconstruct using multiple scans with the same pixel resolution. Introduce a variable for each HR pixel of physical dimension . Algorithm: Perform c2 scans at all offsets & Solve linear equations (Gaussian
elimination).
mm
0000
000
CCC
CCC
0 C
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Problems using boundary valuesExample: 4 Scans, PD=2x2
Add more information
•Solve using LS
•Propagation
problem
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Demands On the algorithm No assumptions on the values
of the true image.
Over determined set of equations Use LS to reduce errors:
Error propagation will be localized.
????
????
???
???
2min bAx
lkl bklA &, where
A x b
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The One dimensional algorithm Input: Pixel of dimension Notation:
gcd(x,y) – greatest common divisor of x & y.
(Extended Euclidean Algorithm)
, . gcd( , )a b ax by x y
Nyxyx , 1 & 1
ixiixv offset at pixel 1 theof value- ),,(
,a y b x
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The One dimensional algorithm
(Extended Euclidean Algorithm)
, . gcd( , )a b ax by x y
Given all ( , ) & ( , )
We can compute all (gcd( , ), )
v x i v y i
v x y i
gcd( , ) 1 (1, )x y v i
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The One dimensional algorithmAlgorithm: Given
w.l.g, let: a>0 & b<0
To compute , compute:
( , ),0
( , ),0
v x j j m x
v y i j m x
(gcd( , ), )v x y i
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0 0
, ,ba
j j
v x i xj v y i yj
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The One dimensional algorithmLocalized Reconstruction
1, 0, 0ax by a b
1 ( mod )ax y
Localized ReconstructionEffective Area: x+y high-resolution pixels
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Two and More Dimensions Given pixels of size:
Where x,y & z are relatively prime.Reconstruct 1x1 pixels.
Error Propagation is limited to an area of O(xyz) HR pixels.
, & x x y y z z
x x
y y
xy x
xy y
1xy
1-D
AlgorithmStack
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Two and More Dimensions1
1
xy
xz
1x
1-D
Algorithm
gcd(xy,xz)=x
1
1
xy
zy
1y
gcd(xy,zy)=y
1
1
x
y
1 1
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Experiment
GE clinical 1.5T MRI
scanner was used. Phantom:
- plastic frames
- filled with water Three FOV: 230.4,
307.2 & 384 mm.
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Problems
Homogeneity of the phantomPhantom OrientationRectangular Blur Vs. Gauss-like Blur
Truncated
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Problems
Homogeneity of the phantomPhantom OrientationRectangular Blur Vs. Gauss-like Blur k-space and Fourier based MRI
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k-space and Fourier based MRIy
x
Wx
Wy
1-D path
k-space
Δkx
Sampling Points
DFT
2i kr
Object
S k r e dr
Δkx
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Problems
# SamplesNoiseProblem
InfiniteNoOne scan is enough
InfiniteYesSNR too low
FiniteYesNo perfect reconstruction
Apply manual shifts → Different experiment
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Open Problems
Optimization problems: “What is the smallest number of scans we can do to reconstruct the high resolution image?“
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Open Problems
Optimization problems: “What is the smallest number of scans we can do to reconstruct the high resolution image?“
Decision/Optimization problem: Given a set of scans, what can we reconstruct?
Design problem: Plan a set of scans for “good” error localization.