Fully‐Constrained Variable Projection for Water‐Fat Models

Purpose 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. Theory and Methods The linear coefficients are redefined according to the physical constraints, thereby introducing the fat fraction 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 can be expressed as a closed‐form function of local off‐resonance frequency and relaxation rate . Results For challenging data, estimation of , and becomes more stable using the proposed fully‐constrained VARPRO method. Conclusion Fully‐constrained VARPRO enables more reliable local parameter estimation, while preserving the efficiency of existing approaches.