Advanced Image Reconstruction Methods in MRI - Advance Image Processing in Magnetic Resonance Imaging - page 51

(a) (b)

(c) (d)

ρ r

FIGURE 2.3 Regularized SENSE reconstructions with 4 coils, R = 4, = GS reconstruc-
tion, and λ (a) being a constant, (b) using GCV, (c) using L -curve, and (d) using the method
proposed in Reference 21.

As expected, the result in Equation 2.29 is a special case of Equation 2.36
when λ = 0. Comparing Equation 2.36 and Equation 2.29 yields

σ

max

kS () = , (2.38)

λ 2

σ +

λ
σ q

q

which can be regarded as the effective condition number of the regularized recon-
struction. Clearly, k ( S λ ) is reduced by increasing λ .

2.4.3 A PPLICATION E XAMPLE

An example is shown in Figure 2.4 , where the data were collected using three

coils in a dynamic contrast-enhanced MRI experiment. In addition to the usual
SENSE data, eight encodings were collected at the Nyquist rate in central k -space