Master's Thesis - Studierstube Augmented Reality Project - Graz ...

3.3 Signal localization
in one direction of the responding slice after the selective excitation of a segment. Then
the frequencies of the spins will be modulated with
ω(x) = γ(B 0 + G x x) (3.26)
for a chosen gradient in x-direction. When neglecting the proportionality relationship
to the flip angle and a notional spin density distribution ρ(x) the received signal of a
sample with simultaneously activated frequency encoding gradient G x follows from
S(t) =
∫ ∞
−∞
ρ(x)e −iγ(B 0+G xx)t dx =
∫ ∞
−∞
ρ(x)e −iγGxxt dxe −iω 0t
and after the removal of the carrier signal e −iω 0t (demodulation), a signal
S(t) =
results.
∫ ∞
−∞
(3.27)
ρ(x)e −iγGxxt dx (3.28)
Encoding the phase angle of a collection of spins can be performed similarly. Applying
a gradient G 1 y with a certain strength between the RF-pulse excitation and the
former described gradient will provoke a preparatory frequency encoding depending on
the time T the gradient is applied. In combination with the following frequency encoding
gradient 2 each row of the responding segment will accumulate different phase-angles
φ(y) = −γG y yT, G y = const. (3.29)
which leads to the term phase-encoding gradient for this special type of frequencyencoding
gradient. Figure 3.6 shows the resulting pulse sequence diagram to obtain a
sampled signal spectrum of one arbitrary line inside the activated slice. Such a sequence
has to be repeated to get all voxel signals of a slice.
3.3.3 Image Formation
With the substitutions
k x = −γG x t, (3.30)
k y = −γG y T (3.31)
1 G y is not necessarily different from G x. The notation G y is only used to distinguish from frequency
and phase encoding gradients.
2 This frequency encoding gradient is also called read-out gradient and notated as G x
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