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A fully probe-corrected near-field far-field transformation employing plane wave expansion and diagonal translation operators has recently been presented, which is able to efficiently carry out transformations for arbitrary measurement point locations. Here, the algorithm is discussed in its multilevel version for the application to planar and quasi-planar measurement contours. The measurement points are grouped in a multilevel box structure and translations of the plane waves, representing the antenna under test, are only carried out to the box centres on the highest level. The plane waves are further processed through the different levels towards the measurement points with a decreasing sampling rate using a disaggregation and anterpolation procedure. This reduces the overall complexity to O(N q log N), N being the total number of measurement points, which is comparable to the classical fast Fourier transform-based algorithms for planar measurements. The proposed algorithm includes a full correction of arbitrary near-field probes and is able to cope with irregular measurement grids and is further optimised for highly directive antennas within this contribution.