Parallel MRI: Concepts and Methods - Advance Image Processing in Magnetic Resonance Imaging - page 71

x direction

( G g

y , p)

W k (m, p) φ

y(n)

x (m)

= X

Fourier-Transformed
Data S k (G y , m)

Column-by-Column
Reconstruction I (m, n)

Phase-Encoded Sensitivity
Profile Matrix

FIGURE 3.9 A full image reconstruction representation. Equation 3.17 is solved inde-
pendently for each column in the image.

accurate than estimates computed prior to the beginning of the dynamic acquisition,
leading to a minimization of artifacts. The inclusion of these center lines in the
reconstruction further contributes to a higher SNR in the image, due to the high
level of energy present at the center of k-space. Reconstructing the full image
amounts to a column-by-column reconstruction whereby each location along the
frequency-encoded direction can be computed independently; a schematic descrip-
tion of the full image reconstruction is shown in Figure 3.9.
Note that one can convert the given SPACE RIP linear system of equations to
the Generalized SMASH linear system of equations with an inverse- and forward-
unitary Fourier transform matrix between the system matrix in Equation 3.17 and
the reconstruction vector (12). This effectively converts the objective from a spatial
domain reconstruction to a k-space domain reconstruction, while maintaining the
basic structure if the linear system.

3.3.3.4.4.2.1 Conditioning of the Reconstruction Matrix
The reconstruction scheme outlined in the SPACE RIP technique is based on
matrix inversion. In order to ensure stable and robust reconstruction, the condition
number of the inverted matrices (defined as the ratio of the largest eigenvalue to
the lowest eigenvalue) should be minimized. Equation 3.17 shows that the con-
dition number is a function of the choice of the phase encodes F acquired per
coil; it is also a function of the sensitivity profile estimations of the receiver coil
array. Because the sensitivity profiles are coil dependent and generally fixed
during a dynamic acquisition, conditioning the reconstruction of the SPACE RIP
technique is practically performed by careful selection of the acquired k -space
lines as well as minimizing the noise in the sensitivity estimates. The SNR in the
resulting images is also a function of the condition number of the reconstruction
matrix shown in Equation 3.17. To avoid errors due to numerical propagation,
the pseudoinverse of each reconstruction matrix is restricted to those singular
values that are greater than a given threshold. This effectively removes any noise
amplification due to poor conditioning.