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The decomposition of magnetic resonance imaging (MRI) data to generate water and fat images has several applications in medical imaging, including fat suppression and quantification of visceral fat. We introduce a novel algorithm to overcome some of the problems associated with current analytical and iterative decomposition schemes. In contrast to traditional analytical schemes, our approach is general enough to accommodate any uniform echo-shift pattern, any number of metabolites and signal samples. In contrast to region-growing method that use a smooth field-map initialization to resolve the ambiguities with the IDEAL algorithm, we propose to use an explicit smoothness constraint on the final fieldmap estimate. Towards this end, we estimate the number of feasible solutions at all the voxels, prior to the evaluation of the roots. This approach enables the algorithm to evaluate all the feasible roots, thus avoiding the convergence to the wrong solution. The estimation procedure is based on a modification of the harmonic retrieval (HR) framework to account for the chemical shift dependence in the frequencies. In contrast to the standard linear HR framework, we obtain the frequency shift as the common root of a set of quadratic equations. On most of the pixels with multiple feasible solutions, the correct solution can be identified by a simple sorting of the solutions. We use a region-merging algorithm to resolve the remaining ambiguity and phase-wrapping. Experimental results indicate that the proposed algebraic scheme eliminates most of the difficulties with the current schemes, without compromising the noise performance. Moreover, the proposed algorithm is also computationally more efficient.