Top Banner
MRI: from Physics to Signal Processing Matthieu Guerquin-Kern Biomedical Imaging Group Institute of Micro-engineering EPFL, Lausanne, Switzerland Internal presentation, June 2008
14

MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

May 09, 2018

Download

Documents

buidat
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

MRI: from Physics to Signal Processing

Matthieu Guerquin-Kern

Biomedical Imaging Group

Institute of Micro-engineeringEPFL, Lausanne, Switzerland

Internal presentation, June 2008

Page 2: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

TABLE OF CONTENT

2

Physics Excitation & Relaxation Detection

Imaging Principle in 1D Imaging with gradients of field

Parallel MRI Signal Processing point of view

Scalar products Linear problem formalism First step to reconstruction

Page 3: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

Physics

3

Resulting magnetization

Spinʼs motion

B0

M

dMdt

= !M!B

Bloch equation

!L = "B0

Page 4: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

B1

B0

! ! !L

Physics: Excitation

4

Resulting magnetization

dMdt

= !M!B

Bloch equation

Transverse RF pulse

Page 5: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X−component of the magnetization

Y−co

mpo

nent

of t

he m

agne

tizat

ion

100 200 300 400 500 600 700−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time in ps

X−co

mpo

nent

of t

he m

agne

tizat

ion

T2*

63%

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time in ps

Z−co

mpo

nent

of t

he m

agne

tizat

ion

T1

63%

Free induction decay

Physics: Relaxation

5

B0

Page 6: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

MR detection

6

Some formulae from electro-magnetism:Equivalent current distribution: JM(r, t) =!!!"M(r, t).

Magnetic vector potential A(r, t) = µ04!

!R3

JM(r!,t)!r"r!! d3r#.

Magnetic field B(r, t) =!!!"A(r, t)

Flux !"(t) =!C!

B(r, t) · d2S ="

#C!A(r, t) · dr,

!"(t) =#

R3M(r#, t) · Bu

"(r#) d3r#,

where we define

Bu"(r#) =

µ0

4!

$

#C!

dr" (r# r#)$r# r#$3

,

Page 7: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

7

MR detection By

Bx

B

I

!!(t) =!

R3M(r!, t) · Bu

!(r!) d3r!Flux

Synchronous demodulation

m!(t) ! !0

!

R3M(r, 0)e!j

R t0 !"(r,#) d#S!(r) d3r

Complex notations:M(r, t) = Mx(r, t) + jMy(r, t) = M(r, 0)e!j

R t0 !(r,") d"

S!(r) = Bu!,x(r)! jBu

!,y(r)

e!(t) = ! d!!

dt(t)

" ! Im!

!0

"

R3M(r, 0)e!j

R t0 "(r,#) d#S!(r) d3r

#Emf

Page 8: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

Imaging: principle in 1D

8Space

B = B0 + Gx

x

! = "BV (t) =

!!(x)ej!t dx

= ej"B0t

!!(x)ej"Gtx dx

" #$ %m(t)

Frequency

t

m(t) = !!(!"Gt)

!(x)

!(x1) < !(x2) < !(x3)

Page 9: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

Imaging: apply gradients!

9

z

Gz

!exc

!!

!z

zexc

•Slice selection

•Phase encoding m!kx(t), ky(t)

"=

##!(x, y)ej2!(kx(t)x+ky(t)y) dx dy

with kx(t) =!

2"

! t

0Gx(#) d# and ky(t) =

!

2"

! t

0Gy(#) d#

−8 −6 −4 −2 0 2 4 6 8

−8

−6

−4

−2

0

2

4

6

8 k−space trajectory

kx

k y

Page 10: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

10

Array of non-homogeneous coils

Non-cartesian and undersampled trajectory

SENSitivity Encoding[Pruessmann, 1999]

Phase encoding

−60 −40 −20 0 20 40 60−60

−40

−20

0

20

40

60

SenseSpiral k−space trajectory

kx

k y

500 1000 1500 2000 2500 3000 3500 4000 1

2

3

4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x 105

index of the coil

modulus of the measurements

index of the k−space sample

Parallel MRI: generalization

Page 11: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

SP point of view: projection

11

with !!,k(r) = S!!(r)e"2j"k·rm!(k) = !!!,k , ""

Page 12: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

Linear problem formalism

12

Continuous forward model:

Discretization: k(t) !" k[n]

S!(r)!(r) !!

p!Cs

s! [p]c[p]"(r" p)

Discretized model:

m!(k) =!

R2S!(r)!(r)e2j"!k , r" dr + b!(k)

m! [n] = !!(!2"kn)"

p!Cs

s! [p]c[p]e2j""kn , p# + b! [n]

Vector representation: m = Ec + b

measurements original signal

error to the model(noise)

Page 13: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

First step to reconstruction

13

Reconstruction often requires the adjoint operator to be computed...

MRI adjoint: MRIH {m!}(r) = S!!(r)

!

Rm!(t)e2j""k(t) , r# dt

MRI measurement followed by adjoint:MRIH MRI{!}(r) = S!

!(r) (g ! S!!) (r)

with g(r) =!

Re2j!!k(t) , r" dt

and "g(!!!) =!

R"(!!! + 2#k(t)) dt

Page 14: MRI: from Physics to Signal Processingbig · TABLE OF CONTENT 2 Physics Excitation & Relaxation Detection Imaging Principle in 1D Imaging with gradients of field Parallel MRI Signal

Questions?

14

Thanks for your attention!

Any question?