Transcript Resolution Enhancement in MRI By: Eyal Carmi Joint work with: Siyuan Liu, Noga Alon, Amos Fiat & Daniel Fiat 10/06/2004

Resolution Enhancement
in MRI
By: Eyal Carmi
Joint work with:
Siyuan Liu, Noga Alon,
Amos Fiat & Daniel Fiat
10/06/2004
1
Lecture Outline

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
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
Introduction to MRI
The SRR problem (Camera & MRI)
Our Resolution Enhancement Algorithm
Results
Open Problems
2
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.
3
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.
4
Magnetic Resonance Imaging

Uniform Static
Magnetic Field –
Atoms will line up
with the direction of
the magnetic field.
w0   B0
B0  MagneticField
  GyromagneticRatio
w0  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)
z
x
y
7
Magnetic Resonance Imaging

Gradient Magnetic Fields: 1-D
Y
w0   B0
B0  MagneticField
B0
B1
  GyromagneticRatio
B2
B3
B4
w0  LarmorFrequency
X
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Signal Localization
slice selection

Gradient Magnetic Fields for slice selection
ω
Gz,1
B1(t)
FT
Gz,2
ω2
ω1
B0
z1
z2
z
z3
z4
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Signal Localization
frequency encoding

Gradient Magnetic Fields for in-plane encoding
x
B=B0+Gx(t)x
B0
t
Gx(t)
t
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Signal Localization
phase encoding

Gradient Magnetic Fields for in-plane encoding
x
B=B0+Gx(t)x
B0
t
Gx(t)
t
11
k-space interpretation
y
1-D path
Wy
k-space
x
DFT
Δkx
Sampling Points
Wx
S k  

  r e i 2 kr dr
Object
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Magnetic Resonance Imaging
Collected
Data
(k-space)
2-D DFT
13
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  Dk Bk Gk X  Ek

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k  1 N
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.
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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
Back
Projection
Current HR
Best Guess
Back Projected LR
images
Original LR images
Iterative Refinement
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SRR – Main Approaches
Y  HX E

Patti, Sezan & Teklap [1994]
POCS: Y  H  X

2

Elad & Feuer [1996 & 1997]
ML:
arg maxP Y X 
X
MAP: arg maxPY X   P X 
X
POCS & ML
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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.
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A Model for the problem
Reconstruction using boundary values
1-D Algorithm
2-D Algorithm
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Modeling The Problem

Subject Area: 1x1 rectilinearly aligned square
grid.
(0,0)
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Modeling The Problem

True Image : A matrix of real values associated
with a rectilinearly aligned grid of arbitrary high
resolution.
(0,0)
2
9
5
6
8
3
6
3
1
8
8
6
4
4
1
9
4
9
7
3
5
4
2
6
5
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Modeling The Problem
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A Scan of the image: S (i, j )  F (m,  x ,  y )
Pixel resolution – m m
Offset – ( x ,  y )
True Image
( x ,  y )
1
1
4
4
1
1
4
4
2
2
3
3
2
2
3
3
Image Scan, m=2
4
8
16 32
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Modeling The Problem
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Definitions:
Maximal resolution – n
Pixel resolution = m  m, m  n
Maximal Offset resolution – 1   n
We can perform scans at offsets ( x ,  y ) where
 x ,  y  k   , k  1,2,3
with pixel resolution m  1 c , c  Z 
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Modeling The Problem

Goal: Compute an image of the subject area
with pixel resolution 1  1  while the maximal
measured pixel resolution is n  n, n 1 δ

Errors:
1. Errors ~ Pixel Size & Coefficients
2. Immune to Local Errors =>
localized errors should have localized effect.
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Multiple offsets of a single
resolution scan?
2x2
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Multiple offsets of a single
resolution scan?
3x3
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Multiple offsets of a single
resolution scan?
4x2
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Using boundary value conditions
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Assumption:
0
0
0
0 0000
or
C
C
C
C CCC
Reconstruct using multiple scans with the same
m m pixel resolution.
Introduce a variable for each HR pixel of
physical dimension    .
Algorithm: Perform c2 scans at all offsets &
Solve linear equations (Gaussian elimination).
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Using boundary value conditions
Example: 4 Scans, PD=2x2
Second sample
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Using boundary value conditions
Example: 4 Scans, PD=2x2
0
0
0 0
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Using boundary value conditions
Example: 4 Scans, PD=2x2
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Problems using boundary values
Example: 4 Scans, PD=2x2
(0,0)
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Problems using boundary values
Example: 4 Scans, PD=2x2
(-1,-1)
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Problems using boundary values
Example: 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.
?
?
?
?
?
?
????
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Over determined set of equations 
Use LS to reduce errors: min Ax  b
l k
where A   , l  k & b  
A

l
x
2
b
Error propagation will be localized.
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The One dimensional algorithm
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Input: Pixel of dimension 1 x & 1 y x, y  N
Notation:
v( x, i), i   - valueof the1 x pixelat offseti
gcd(x,y) – greatest common divisor of x & y.
a, b. ax  by  gcd( x, y)
(Extended Euclidean Algorithm)
a  y, b  x
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The One dimensional algorithm
(Extended Euclidean Algorithm)
a, b. ax  by  gcd( x, y)
Given all v( x, i) & v( y, i)
We can compute all v(gcd( x, y), i)
gcd( x, y)  1  v(1, i)
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One dimensional reconstruction
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One dimensional reconstruction
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The One dimensional algorithm
Algorithm: Given v( x, j ),0  j  m  x

v( y, i),0  j  m  x
w.l.g, let: a>0 & b<0
To compute v(gcd( x, y), i) , compute:
a 1
b 1
j 0
j 0
 v  x, i  xj   v  y, i  yj 
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The One dimensional algorithm
Localized Reconstruction
ax  by  1, a  0, b  0
ax  1 ( mod y)
Localized Reconstruction
Effective Area: x+y high-resolution pixels
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One dimensional reconstruction
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Two and More Dimensions


Given pixels of size: x  x, y  y & z  z
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
 xy  x

 xy  y
xy  1
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Two and More Dimensions
 xy 1

 xz 1
x 1
 x 1

 y 1
 xy 1

 zy 1
y 1
1 1
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Example
Two dimensional reconstruction
PD=5x5
PD=3x3
PD=15x1
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Example
Two dimensional reconstruction
PD=4x4
PD=3x3
PD=12x1
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Example
Two dimensional reconstruction
PD=5x5
PD=4x4
PD=20x1
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Example
Two dimensional reconstruction
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Larger Dimensions
Generalize to Dimension k
Using
k+1 relatively prime
Low-Resolution pixels
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Results
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Model Results
Experiment Results
Problems…
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Model Design
Noise
HR
Scene
Blur
HR
Image
Sampling
LR
LR
LR
Image
Image
Image
SRR
Algorithm
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Results
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Experiment
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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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Experiment
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Experiment Results
MRI Data
Modeled Data
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Experiment Results
MRI Data
Modeled Data
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Problems
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Homogeneity of the phantom
Phantom Orientation
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Problems
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Homogeneity of the phantom
Phantom Orientation
Rectangular Blur Vs. Gaussian-like Blur
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Problems
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Homogeneity of the phantom
Phantom Orientation
Rectangular Blur Vs. Gauss-like Blur
Truncated
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Problems
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Homogeneity of the phantom
Phantom Orientation
Rectangular Blur Vs. Gauss-like Blur
k-space and Fourier based MRI
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k-space and Fourier based MRI
y
1-D path
Wy
k-space
x
DFT
Δk
Δkx x
Sampling Points
Wx
S k  

  r e i 2 kr dr
Object
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Problems
# Samples
Infinite
Noise
No
Problem
One scan is enough
Infinite
Yes
SNR too low
Finite
Yes
No 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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Scan Selection Problem
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Scan Selection Problem
S5x and…

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Scan Selection Problem
S3x and…


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Open Problems
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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.
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Questions
69