Holger Eggers^{1}

Fat is shifted relative to water in opposite directions in the odd and even single-echo images obtained with bipolar multi-echo sequences. Considering this effect in a water-fat separation permits suppressing associated artifacts, but has required regularization so far to cope with a locally ill-conditioned inverse problem in k-space. In this work, an alternative approach to limiting noise amplification in the water-fat separation is proposed and explored, which promises to avoid the loss of image sharpness observed with regularization.

For the sake of
simplicity, dual-echo Dixon imaging is considered in the following only. After estimation and demodulation of the main field inhomogeneity-induced phase from the single-echo images, the composite signal *S* in k-space is modelled by

$$\begin{bmatrix}S_{1k}\\S_{2k}\end{bmatrix} = \begin{bmatrix}1&c_{1k}\\1&c_{2k}\end{bmatrix} \begin{bmatrix}W_k\\F_k\end{bmatrix},$$

with

$$c_k=\sum_{n}w_n e^{2\pi i\Delta f_n t_k},$$

where *W* and *F* denote the water and fat signal in k-space, *t* the acquisition time,
and *w* and Δ*f* the relative amplitude and frequency of the peaks of the employed
spectral model of fat. A solution for *W* and *F* is given by

$$\begin{bmatrix} W_{k}\\F_{k}\end{bmatrix} = (C_k^H C_k+\lambda I)^{-1} C_k^H\begin{bmatrix}S_{1k}\\S_{2k}\end{bmatrix},$$

with

$$C_k=\begin{bmatrix}1&c_{1k}\\1&c_{2k}\end{bmatrix},$$

where *λ* denotes a regularization parameter. As
demonstrated in Fig. 2, a suitable choice of* λ* allows to limit the noise amplification factor to a desired
value. However, the diagonal loading of the matrix *C*^{H}*C* introduces
a bias in the water-fat separation. To avoid this, it is proposed to constrain
the initial phase of the water and fat signal in image space to be equal
whenever the condition of the matrix *C*^{H}*C* is deemed too poor.^{2 }The water and fat signal are then assumed to
be real in image space, and thus to be Hermitian in k-space. Considering one echo only, *S* is described by

$$\begin{bmatrix}S_{k_1 R}\\S_{k_1 I}\\S_{k_2 R}\\S_{k_2 I}\end{bmatrix}=\begin{bmatrix}1&0&c_{k_1 R}&-c_{k_1 I}\\0&1&c_{k_1 I}&c_{k_1 R}\\1&0&c_{k_2 R}&c_{k_2 I}\\0&-1&c_{k_2 I}&-c_{k_2 R}\end{bmatrix}\begin{bmatrix}W_{k_1 R}\\W_{k_1 I}\\F_{k_1 R}\\F_{k_1 I}\end{bmatrix},$$

with

$$k_2=-k_1,$$

where _{R} and _{I} denote the real and imaginary
part. Including the other echo as well leads to an overdetermined linear system
of eight equations. *W* and* F* are, in both cases, derived as above,
but without regularization.

Images of the pelvis of volunteers were acquired with a 3D T1-weighted spoiled dual-gradient-echo sequence on a 3 T Ingenia scanner
(Philips Healthcare, Best, The Netherlands) using anterior and posterior coil
arrays. A FOV of 350 (AP) x 350 (RL) x 200 (FH) mm^{3}, a resolution of 0.7 x 0.7 x
2.5 mm^{3}, a TE_{1}/TE_{2}/TR of 2.0/3.7/5.5 ms, and a
flip angle of 10° were chosen. The pixel bandwidth amounted to 650 Hz, corresponding
to a shift of the signal from the dominant peak of the fat spectrum of 0.65
pixel in the single-echo images.

The main field inhomogeneity-induced
phase and the initial phase of the water and fat signal were first determined and
eliminated from the single-echo images, ignoring any misregistration.^{3} The original complex two-point
separation, without and with regularization, as well as the proposed
hybrid complex-real two-point separation without regularization were then applied in k-space.

Noise amplification factor (NAF) as function of *k*_{x}
of the original complex two-point separation (red), without (dashed) and with (solid)
regularization, and of a real, or phase-constrained, one-point separation
(green) without regularization.

Water images reconstructed with the original complex two-point
separation, without (left) and with (right) regularization.

Water image reconstructed with the proposed
hybrid complex-real two-point separation without regularization.