Transcript Magnetic Resonance Imaging and the Fourier Transform

Magnetic Resonance
Imaging and the Fourier
Transform
Morgan Ulloa
March 20, 2008
Outline
• The Fourier transform and the inverse Fourier transform
• Fourier transform example
• 2-D Fourier transform for 2-D images
• Magnetic Resonance Imaging (MRI)
• Basic MRI physics
• K-Space and the inverse Fourier transform
•
Overview of how an MRI machine works
• 3-D MRI
What is the Fourier transform
• A component of Fourier Analysis named for French
mathematician Joseph Fourier
(1768-1830)
• The Fourier transform in an operator that inputs a function and
outputs a function
• Inputs a function in the time-domain and outputs a function in
the frequency-domain
• the Fourier transforms is used in many different ways
• Continuous Fourier transform is used for MRI
What the Fourier transform looks like
• Written as an integral:
F ( )  


f (t )e
it
dt
• f(t) is a function in the time-domain
• ω=2πf known as the angular frequency
• i = square root of -1 (imaginary number)
• t is the variable time
• The Fourier transform takes a function in the time-domain
into the frequency-domain
Inverse transform
• Also an integral:
1
f (t ) 
2



F ( )e
it
d
• F(ω) is our Fourier transform in the frequency-domain
• ω=2πf known as the angular frequency
• i is the square root of -1
• t is the variable time
• 1/(2 π) is a conversion factor
• The inverse Fourier transform takes a function in the frequency-domain
back into the time-domain
Different functions and their Fourier transforms
Example:
1) Define time-domain function
2) Compute our integral:
F ( )  

0
it
f (t )e dt =


0
f (t )  sin( 4t )
it
sin( 4t )e dt
 Improper integral
a
lim
sin(
4
t
)(cos(

t
)

i
sin(

t
))
dt
sin(
4
t
)
e
dt
=
=


0
a


0
a  
lim
a
it
a
a

lim  sin( 4t ) cos(t )dt i  sin( 4t ) sin( t )dt 

0
a  
0
Note: Has a real part and an imaginary part
 from the definition of the integral of products of
trigonometric functions 
= lim  cos(( 4   )t )  cos(( 4   )t )  a  i  sin(( 4   )t )  sin(( 4   )t )  a 
a  
 
2(4   )
2(4   )
0

2(4   )
2(4   )  0 
simplifying the fractions
   cos(( 4   )t )  4 cos(( 4   )t )  4 cos(( 4   )t )   cos(( 4   )t )  a
= alim

2


2(4   )( 4   )
0
 
 sin(( 4   )t )  4 sin(( 4   )t )  4 sin(( 4   )t )   sin(( 4   )t )  a 
2i 
 0
2
(
4


)(
4


)

 
3) Understanding what this means
- searching for a specific frequency (ω)
The 2-D Fourier transform
• Begin with a 2-D
array of data: t’ by t’’
• Since this data is two
dimensional we say
that the data is in the
spatial domain
1st Fourier Transform
First Fourier transform in one direction: t’
2nd Fourier Transform
Finally in the t’’ direction
The spike corresponds to the intensity and location of
the frequency within the 2-D frequency domain
What is MRI?
• Originally called nuclear magnetic resonance (NMR) but now
it is called MRI in the medical field because of negative
associations with the word nuclear
– Thus, on the atomic level MRI utilizes properties of the nucleus,
specifically the protons
• It takes tomographic images of structures inside the human
body similar to an x-ray machine
– Unlike the x-ray this imaging technique is non-ionizing
Tomography
• Tomography: slice selection, with insignificant
thickness, of a 3-D object
Spin and spin states
• Certain atoms have protons that create tiny magnetic fields in
one of two directions: this is called spin
– some common and important types of atoms with this property are 1H,
13C, 19F, 31P
– Hydrogen protons are used for MRI in the human body
• Usually the spins of protons in a compound are oriented
randomly but if they are exposed to a magnetic field they either
align themselves with it, called parallel alignment, or against
it, called antiparallel alignment.
• The parallel and antiparallel alignments are called spin states
• The energy difference between the spin states lies within the
radio frequency spectrum
Resonance
• When a magnetic field is applied the protons oscillate between
their two spin states: between antiparallel and parallel
alignment
• A radio frequency is applied by the MRI machine and varied
until it matches the frequency of the oscillation: this is called
resonance
• Whilst in this state of resonance the protons will absorb and
release energy
– This is then measured by a radio frequency receiver on the
MRI machine
• The energy difference of the two spin states depends on the
response of protons to the magnetic field in which it lies and,
since the magnetic field is affected by the electrons of nearby
atoms, each MRI scan produces a spectrum that is unique to
the compounds present in the tomographic slice
Radio Frequency Spectrum
• Spectrum: the distribution of energy emitted
by a radiant source
Axes of a 2-D slice
• x-axis called the
phase-encoding
axis
• y-axis called the
frequency encoding
axis
Regions of spin
• A region of spin is simply a
location within the 2-D slice
plane where protons are
expressing their unique
characteristic: spin
• For the purposes of
explaining the 2-D Fourier
transform we will use 2
regions of spin with similar
material compositions but
different locations: labeled 1
and 2
Imagining these regions of spin
• These regions of spin can be depicted visually
by vectors (magnetization vectors) rotating
around the origin at a frequency corresponding
to their rates of oscillation
x-axis: Phase Encoding
• Magnetic field gradient is applied to the slice along the x-axis to
both regions of spin
• The radio frequency bursts are applied and both regions of spin
resonate at different frequencies because they have different
positions
– when the gradient is turned off they will have a different phase angle, φ
• Phase angle: the angle between the reference axis (y) and the
magnetization vectors
Phase encoding: Cuts the slice into rows
y-axis: Frequency encoding
• The magnetic field gradient is then applied along the
y-axis
• This results in the two spin vectors rotating at unique
frequencies about the origin
• Thus each region of spin now has a unique rotational
frequency and a unique phase angle
Frequency encoding: cuts the slice into
columns
Using the Fourier transform with this data
• Mapping these rotations about the origin as functions of time
we get two unique time-domain signals each with a unique
phase and rotational frequency and we can create a 2-D array
of data with our rows and columns
• To these we can apply the Fourier transform as we did before
in the t’ and t’’ directions but this time in the frequency
encoding direction and the phase encoding direction
– therefore our data is in the spatial domain, not the time domain as with
1-D Fourier transform
• This process identifies the the position and the intensity of the
spin within the 2-D slice plane
A visual of how this works…
• We have our two unique
signals plotted as a 2-D
array of data
…
• First Fourier transform in the frequency
encoding direction
…
• Then Fourier transform
in the phase encoding
direction
• The spikes indicate
frequency intensity and
location of our regions
of spin
What is the K-space?
• A matrix known as a temporary image space which holds the
spatial frequency data from a 2-D Fourier transform
• The number of entries in each row and column correspond to the
number of regions of spin within the slice plane and their location
• Each matrix entry is given a pixel intensity and thus each entry
contains both frequency and spatial data
• These entries form a grayscale image
– Whiter entries correspond to high intensity signals
– Darker entries correspond to low intensity signals
K-space for the two regions of spin
• The k-space for our two
regions of spin is the
following matrix which
clearly demonstrates the
position and intensity
(here the difference in color)
of each region
K-space and the Image
From the K-space to a recognizable image
• Inverse transforming the K-space yields a new grayscale
image that corresponds to the physical slice plane, thus
creating an accurate image representation of the slice in vivo
MRI imaging process
• A slice is selected from the body
• Magnetic Field gradient is applied to the slice in the
Phase encoding and Frequency encoding directions
– only the protons within the slice to oscillate between their
two energy states (spin states)
• At the same time radio frequency pulses are applied
to the slice with a bandwidth capable of exciting all
resonances simultaneously
• The emitted energy is measured by a radio frequency
receiver and converted into a spectrum in the timedomain in the x-direction and y-direction thereby
creating a 2-D array of data in the spatial domain
…MRI process continued
• The two time-domain functions of the spatial-domain
is then Fourier transformed into a 2-D frequencydomain function with information about the position
and frequency intensity of the spin regions
• This data is entered into the K-space and then inverse
Fourier transformed creating a corresponding
accurate image of the physical slice
Stacking Slices: the 3-D image
References:
•
Campbell, Iain D., and Raymond A. Dwek. Biological Spectroscopy. Menlo Park,
Ca: Benjamin/Cummings Company, 1984.
•
Gadian, David G. Nuclear Magnetic Resonance and Its Applications to Living
Systems. New York: Oxford UP, 1982.
•
Hornak, Joseph P. "The Basics of MRI." 1996. Rochester Institute of Technology.
30 Sept. 2007 <http://www.cis.rit.edu/htbooks/mri/>.
•
Hsu, Hwei P. Applied Fourier Analysis. Orlando: Harcourt Brace Jovanovich, 1984.
•
Knowles, P. F., D. Marsh, and H.W.E. Rattle. Magnetic Resonance of
Biomolecules. John Wiley & Sons, 1976.
•
Mansfield, P., and P. G. Morris. NMR Imaging in Biomedicine. New York:
Academic P, 1982.
•
Swartz, Harold M., James R. Bolton, and Donald C. Borg. Biological Applications
of Electron Spin Resonance. John Wiley & Sons, 1972.
Many thanks to
Professor Ron Buckmire
&
the Occidental Mathematics Department