Parallel MRI: Concepts and Methods - Advance Image Processing in Magnetic Resonance Imaging - page 69

W 1 W 2

W 4 W 3

FIGURE 3.8 Local FOV coil-sensitivity encoding is used to untangle separate pixels in
aliased acquisitions.

thus, spatially aliased—image. The reconstruction equation is formed using esti-
mates of the coil sensitivity at the location of these aliased pixels.

vjx WjxIjx WjN xIjN x
1

(, ) = (, )(, ) + ( + /, )( 2 + /, ) 2 (3.13)

1 1

where v l ( r ) is the spatial-domain representation of acquired data for coil l . Note

that the coordinate frame r

′

is the aliased version of r . Ideally, each coil “sees”
a different portion of the FOV, as in Figure 3.8.
Collecting together the expressions for each coil yields the SENSE linear
system of equations

′

vjx
vjx

(, )
(, )

Wjx (, ) W ( jN x +

+

/ , 2

2

)

1 1 1

Wjx (, ) Wj N x ( /, ) Ijx

Ij N

(, )

2 2 2

= (3.14)

NNx ( + / , 2 x )

vjx (, ) Wjx (, ) Wj ( + /, ) 2

L L L

which is repeated for each pixel location in the acquired spatial-domain coordinate
system. Using an identical framework, this approach has also been used for variable
density subsampling patterns (17).

3.3.3.4.2 SPACE RIP
From the signal equation, SPACE RIP (9) takes the Fourier transform of Equation 3.2
along the x direction, when a phase-encoding gradient

G i is applied yields:

∫

i

i jGy γ ( τ )

SGx (, ) = IxyWxye (, )(, ) dy , (3.15)

k y k