Fully-Constrained Variable Projection for Water-Fat Models

Variable projection (VARPRO) is an effective technique to reduce the dimensionality of least squares optimization problems through the elimination of linear parameters. For water-fat models, constrained VARPRO variants have been proposed in the literature to ensure that water and fat components are aligned at TE = 0, thereby preventing overfitting of data. These models, however, still allow for anti-parallel orientation of water and fat and therefore remain potentially vulnerable. An alternative, fully-constrained VARPRO formulation is derived, which resolves these limitations without compromising dimensionality reduction and/or numerical performance.

The linear coefficients are redefined according to the physical constraints, thereby introducing the fat fraction  $f$ as an additional nonlinear parameter. To remove the latter and regain the original VARPRO dimensionality, it is shown that the maximum likelihood (ML) estimate of $f$ can be expressed as a closed-form function of local off-resonance frequency $\omega$ and relaxation rate $R_2^{\ast }$ .

For challenging data, estimation of $\omega$ , $R_2^{\ast }$ and $f$ becomes more stable using the proposed fully-constrained VARPRO method.

Fully-constrained VARPRO enables more reliable local parameter estimation, while preserving the efficiency of existing approaches.