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Water–fat separation in magnetic resonance imaging (MRI) is of great clinical importance, and the key to uniform water–fat separation lies in field map estimation. This work deals with three-point field map estimation, in which water and fat are modelled as two single-peak spectral lines, and field inhomogeneities shift the spectrum by an unknown amount. Due to the simplified spectrum modelling, there exists inherent ambiguity in forming field maps from multiple locally feasible field map values at each pixel. To resolve such ambiguity, spatial smoothness of field maps has been incorporated as a constraint of an optimization problem. However, there are two issues: the optimization problem is computationally intractable and even when it is solved exactly, it does not always separate water and fat images. Hence, robust field map estimation remains challenging in many clinically important imaging scenarios. This paper proposes a novel field map estimation technique called JIGSAW. It extends a loopy belief propagation (BP) algorithm to obtain an approximate solution to the optimization problem. The solution produces locally smooth segments and avoids error propagation associated with greedy methods. The locally smooth segments are then assembled into a globally consistent field map by exploiting the periodicity of the feasible field map values. In vivo results demonstrate that JIGSAW outperforms existing techniques and produces correct water–fat separation in challenging imaging scenarios. [ABSTRACT FROM AUTHOR]