Basic Physics of MR Signal and Image Generation - Advance Image Processing in Magnetic Resonance Imaging - page 31

where (see also Figure 1.15 ):

∆ k y are the frequency interval step sizes in the x and y directions,

respectively,
W kx and W ky are the maximum frequencies that contain the object infor-
mation

k x and ∆

∆ t is the readout sampling time interval and T is the readout gradient time

x

duration

y is the phase-encoding gradient step size
G y,max is the phase gradient maximum amplitude
Creating an MR image requires sampling the two-dimensional k-space with
sufficient density (

∆

k y ) over a specified extent (W kx , W ky ). The Nyquist
sampling theorem dictates the sampling spacing necessary to prevent spatial alias-
ing of the reconstructed object (parts of the object can alias to different locations).
The unaliased region is known as the field of view (FOV). The extent of the
acquisition in Fourier space dictates the high-spatial frequency content and, hence,
the spatial resolution.
For data acquired on a two-dimensional rectilinear grid in k-space and recon-
structed with a two-dimensional Fast Fourier transform (FFT), the spatial reso-
lution and FOV relationships are:

∆ k x , ∆

FOV = 1 / ∆ k FOV = 1 / ∆ k

x x y y

(1.47)

∆ xW = 1 / ∆ yW = 1 /

kx ky

Up to now, we have described the MR signal equations for MRI for a particular
pulse sequence (two-dimensional Fourier) and a homogeneous sample. In general,
the value of the MR parameters, (T 1, T 2, and T 2*) vary with position; this
generates the contrast between tissues. So, a more general form of Equation 1.45 is:

∫

st (( ) k = ζ (, ) r p e d r

− 2 π it kr () ⋅

sample

(1.48)

t

∫

k () t = γ G ( τ ) d τ

0

where ζ ( r , p ) is a function of position r , and p is a parameters vector p = ( ρ ( r ),

T1( r ), T 2( r ), … , TR, TE, α , … ) that describes the dependence on the tissue

, T1, T 2, etc.) and the scanner parameters (i.e., TR, TE, etc.). For
example, in the two-dimensional Fourier sequence, if there are no effects from
magnetic field inhomogeneities (i.e., T 2

parameters ( ρ

= T 2*):

ζ (, ) rp = ρ () r e

− TE T / 2 () r

( 1 − e

− TR T / 1 () r

) (1.49)

k (t), described in Equation 1.48, is the spatial frequency vector as a function of
time; the Fourier coefficients of the image s( k (t)) can be sampled along the pathway
defined by k (t).