Medical Imaging - Introduction to Biomedical Engineering - page 1095

in which the s is a function of t and G y ¼ 0, S 0 is a constant, and the Larmor frequency asso-

ciated with location

x n is

o n ¼ g G x x n ð 16 : 83 Þ

after mixing removes o 0 at the isocenter.

The one-dimensional Fourier transform of Eq. (16.82) is

1 1

h i h i

Þ e i 2p ft dt ¼ e j t T E j= T 2 n e i 2p f n T E e 2pð f f n Þ t

S n ð f ,0 Þ¼ s ð t ,0 S dt ð 16 : 84a Þ

0

1 1

The form of the complex exponents is recognizable from Eq. (16.77d) as the transform of an
impulse function centered on

. This relation implies that the Fourier transform of the func-
tion in the left brackets is centered on

f n

. To show this result, it is necessary to determine if
the Fourier transform of the first term in brackets is of the form of Eqs. (16.71a) and (16.71d)
with

f n

S 0 as the constant,

1 1

h i

e i 2p ft dt ¼

S

0

Þ e i 2p ft dt ¼ S 0 e j t T E j T 2 n W n ð f ,0 Þ¼ w n ð t ,0 ð 16 : 84b Þ

2

1 þð 2 p fT

n Þ

2

1 1

then S n ( f , 0) can be expressed as

Þ¼ e i 2p f n T E W n ð f f n Þ

S n ð f ,0 ð 16 : 84c Þ

From the relation for the Larmor frequency, this equation can be rewritten in terms of a
scaled variable for frequency from Eq. (16.83) as

g G x
2

Þ¼ e i g G x x n T E W n S n ð x ,0

p ð x x n Þ

ð 16 : 84d Þ

The derivation of this key equation reveals a second important principle in magnetic reso-
nance imaging. First, a locally excited resonance encodes a position into a frequency

as
part of the phase of a time waveform. Second, this phase, in terms of Fourier transforms,
ensures that the signal function has a spectrum centered on the frequency associated with
the location of the net magnetization density. Third, through simple scaling, the scaled
spectra are translated into spatial locations along the

f n

-axis (Figure 16.48b). This process
maps scaled spectral magnitudes into their spatial locations along the

x

-axis.
To advance to two dimensions, the one-dimensional inverse Fourier transforms are com-
bined in a two-dimensional relation:

x

ðð

G y Þ¼ `` 1

u Þ e i 2p ft e i 2p uG y dfdu

s ð t

, ½ I ð f , u Þ ¼ I ð f , ð 16 : 85 Þ

To find the object distribution, I , from the measured set of signals from a location ( x n , y m ),

a forward 2D Fourier transform is

ðð

G y Þ e i 2p ft e i 2p uG y dtdG y I ð f , u Þ¼ S mn ð f , u Þ¼ `` ½ s mn ð t , G y Þ ¼ s mn ð t , ð 16 : 86 Þ