Morgan UlloaMarch 20, 2008

Magnetic Resonance Imaging and the Fourier

Transform

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(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

dtetfF ti )()(

Inverse transform• Also an integral:

• 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

deFtf ti

)(21)(

Different functions and their Fourier transforms

Example:1) Define time-domain function 2) Compute our integral:

=

Improper integral

= =

Note: Has a real part and an imaginary part

0

)()( dtetfF ti

)4sin()( ttf

0)4sin( dtet ti

a ti

adtet

0)4sin(lim

a

adttitt

0))sin())(cos(4sin(lim

dtttidtttaa

a 00)sin()4sin()cos()4sin(lim

0)4(2

))4sin(()4(2))4sin((

0)4(2))4cos((

)4(2))4cos((lim

attiatt

a

from the definition of the integral of products of trigonometric functions

=

simplifying the fractions =

3) Understanding what this means- searching for a specific frequency (ω)

0)4)(4(2

))4cos(())4cos((4))4cos((4))4cos((2limatttt

a

0)4)(4(2

))4sin(())4sin((4))4sin((4))4sin((2atttti

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 TransformFinally 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 time-domain 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 frequency-domain 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