Cramér–Rao bounds for three‐point decomposition of water and fat

The noise analysis for three‐point decomposition of water and fat was extended to account for the uncertainty in the field map. This generalization leads to a nonlinear estimation problem. The Crámer–Rao bound (CRB) was used to study the variance of the estimates of the magnitude, phase, and field map by computing the maximum effective number of signals averaged (NSA) for any choice of echo time shifts. The analysis shows that the noise properties of the reconstructed magnitude, phase, and field map depend not only on the choice of echo time shifts but also on the amount of fat and water in each voxel and their alignment at the echo. The choice of echo time shifts for spin‐echo, spoiled gradient echo, and steady‐state free precession imaging techniques were optimized using the CRB. The noise analysis for the magnitude explains rough interfaces seen clinically in the boundary of fat and water with source images obtained symmetrically about the spin‐echo. It also provides a solution by choosing appropriate echo time shifts (−π/6 + πk, π/2 + πk, 7π/6 + πk), with k an integer. With this choice of echo time shifts it is possible to achieve the maximum NSA uniformly across all fat:water ratios. The optimization is also carried out for the estimation of phase and field map. These theoretical results were verified using Monte Carlo simulations with a newly developed nonlinear least‐squares reconstruction algorithm that achieves the CRB. Magn Reson Med, 2005. © 2005 Wiley‐Liss, Inc.