10/06/2004 1

Resolution Enhancementin MRI

By: Eyal Carmi

Joint work with:

Siyuan Liu, Noga Alon,

Amos Fiat & Daniel Fiat

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Lecture Outline

Introduction to MRIThe SRR problem (Camera & MRI)Our Resolution Enhancement AlgorithmResultsOpen Problems

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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.

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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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k-space interpretationy

x

Wx

Wy

1-D path

k-space

Δkx

Sampling Points

DFT

2i kr

Object

S k r e dr

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Magnetic Resonance Imaging

Collected

Data

(k-space)

2-D DFT

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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

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.

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

48

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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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 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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Using boundary value conditionsExample: 4 Scans, PD=2x2

Second sample

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Using boundary value conditionsExample: 4 Scans, PD=2x2

000

0

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Using boundary value conditionsExample: 4 Scans, PD=2x2

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Problems using boundary valuesExample: 4 Scans, PD=2x2

)0,0(

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Problems using boundary valuesExample: 4 Scans, PD=2x2

-)1-,1(

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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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One dimensional reconstruction

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One dimensional reconstruction

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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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One dimensional reconstruction

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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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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

Model ResultsExperiment ResultsProblems…

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Model Design

HR Scene Blur Sampling

LRImage

Noise

LRImage

LRImage

SRR

AlgorithmHR

Image

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Results

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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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Experiment

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Experiment Results

Modeled DataMRI Data

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Experiment Results

Modeled DataMRI Data

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Problems

Homogeneity of the phantomPhantom Orientation

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Problems

Homogeneity of the phantomPhantom OrientationRectangular Blur Vs. Gaussian-like Blur

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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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Scan Selection Problem

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Scan Selection ProblemS5x and…

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Scan Selection ProblemS3x and…

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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.

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Questions