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In this paper a new estimation scheme for multi-dimensional (MD) harmonic estimation is developed. Initiating from the MD rank reduction (RARE) estimator, from which independent sets of frequency estimates along the various axes are obtained, new algebraic properties are derived that allow to solve the parameter association problem efficiently. Existing closed-form algorithms partly exploit structural properties to develop parameter estimation schemes with low computational burden. While the well-known ESPRIT-type algorithms exploit shift-invariance between specific partitions of the signal matrix, the RARE algorithm exploits their internal Vandermonde structure. Estimation schemes which exploit both structural properties jointly show improved performance.