Parallel MRI: Concepts and Methods - Advance Image Processing in Magnetic Resonance Imaging - page 61

Full k-space images
acquired in 4 coils

Full k-space body
coil image

Resulting Sensitivity Profile
in coil 1, using body coil

Coil #1:
I 1 (x, y)

Coil #2:
I 2 (x, y)

I (x, y) W 1 (x, y) = I 1 (x, y)/I (x, y)

Resulting Sensitivity Profile
in coil 1, using coil images

Sum of all coil images

Coil #3:
I 3 (x, y)

Coil #4:
I 4 (x, y)

Is = I 1 + I 2 + I 3 + I 4 ≈ I (x, y) W 1 (x, y) = I 1 (x, y)/Is (x, y)

FIGURE 3.1 Coil-sensitivity profile computation from full k -space data.

composite image in the least-squares sense. This was found to be the optimal
combination strategy by Roemer et al., in Reference 13.

3.3.2 S N R

AMPLING AT THE YQUIST ATE

E I

AND QUATION NDEPENDENCE

) described in Equation 3.1 is related by a Fourier transform
to the projection of the image onto the frequency-encoded direction. This can be
seen by taking the Fourier transform of Equation 3.2 along the

The time signal s ( t

k

x direction:

( ) ( ) =

∫

=

i

jGy γ ( τ )

FT s G t ⋅⋅

i

, S G x

i

, Ixy W xy ( , ) ⋅ ( , ) e dy (3.5)

1

k y k y k

In order to fully represent the time signal acquired, the sampling rate needs to
be at least equal to the Nyquist rate for one period of the signal. From a linear
algebraic standpoint, the samples acquired at the Nyquist rate provide an independent
and complete set of equations for the specified resolution. Sampling below the
Nyquist rate yields an underdetermined linear system, whereas sampling at a higher
rate will result in overdetermination for the chosen maximum resolution. Once the
FOV and the number of image pixels M along the frequency-encoded direction are
determined, image resolution in that direction is set. In practice, the sampling rate
of the signal is constant. The effect of varying the sampling rate is obtained by
varying the magnitude of the magnetic field gradient in the frequency-encoded