Basic Physics of MR Signal and Image Generation - Advance Image Processing in Magnetic Resonance Imaging - page 29

Generally, a sequence diagram can be split into three distinct sections, namely,
slice selection, phase encoding, and readout, according to the previous descrip-

tions. In Figure 1.14 such sections are separated in time (three time intervals)

but, especially in more recently developed sequences, some of them overlap.
The first event to occur in the imaging sequence represented in Figure 1.14
is to turn on the slice selection gradient, together with the RF pulse. As previously
described, the slice selective RF pulse should be a shaped pulse. Once the RF
pulse is complete, the slice selection gradient is turned off, and a phase-encoding
gradient is turned on. In order to obtain an MR echo signal, a negative read
gradient is switched on. Once the phase-encoding gradient has been turned off,
a positive frequency-encoding gradient is turned on and an echo signal is recorded.
This sequence of pulses is usually repeated m times, and each time the
sequence is repeated, the magnitude of the phase-encoding gradient is changed
according to Equation 1.41. The time between the repetitions of the sequence
is called the repetition time, TR.

1.10 ACQUIRING MR SIGNALS IN THE K-SPACE

According to Equation 1.36 and Equation 1.41 we can describe the signals result-
ing from a two-dimensional Fourier transform sequence as a function of both the
phase-encoding step and the time during the readout period. When M frequency-
encoded FIDs are obtained, each one experiences a different value of the phase-
encoding gradient amplitude; usually both positive and negative amplitudes are
applied:

&

∞ ∞

)
+

∫ ∫

st c

m ()

= ρ ( , ) xye

− ixGtymGt

x

γ ( + )

dxdy e

− i ω 0 t

(1.42)

y G

(

−∞ −∞

In order to obtain digital MR signals, data acquired during frequency gradient
activation are sampled. So that, k-space data are sampled data, memorized in a
matrix of N

M points, if N is the number of samples along reading gradient
and M the number of times the phase gradient is activated.
Then, if each FID is sampled and

×

t is the sampling time interval, and we
consider the demodulated signal, from Equation 1.42 we obtain:

∆

∞ ∞

∫ ∫ ρ

− ixGntymGt

x

γ ( ∆ + )

s n,m ( ) = c ( x,y e ) dxdy

y G

−∞ −∞

≤≤
−+≤≤

0
21

nN (1.43)

M/ m M/ 2

The formula in Equation 1.43 is usually referred to as the imaging equation .
So that, if M = 256 and the FID sampling points N are 256, then a 256 × 256

data matrix of complex numbers is the result.