An efficient probe-compensated near-field-far-field (NF-FF) transformation with a noncoventional scanning optimized for flat antennas under test (AUTs) is here developed. It relies on the nonredundant sampling representations of the electromagnetic fields and on the optimal sampling interpolation expansions, and considers a flat AUT as enclosed in a dish having diameter equal to its maximum dimension. Since the number of needed NF samples is related to the area of surface enclosing the AUT, it results significantly reduced with respect to the use of the other AUT modelings available for quasi-planar AUTs and based on the oblate spheroid and the double bowl. Numerical results assessing the effectiveness of the proposed NF-FF transformation a...