Terrain-aided navigation (TAN) offers autonomous positioning without reliance on external signals, complementing satellite navigation systems that remain susceptible to interference and spoofing. In three-dimensional terrain matching, the choice of feature descriptor governs how reliably one terrain patch can be distinguished from another. Among the global descriptors used for this task, 3D Zernike moments stand out for their noise resistance and rotational invariance, properties well suited to the noisy, orientation-uncertain conditions typical of terrain sensing. The standard 3D Zernike moment descriptor, however, embeds information unrelated to terrain elevation. This extraneous content dilutes discriminability and degrades matching accuracy. The present work identifies the structural source of this redundancy and constructs an optimized descriptor that removes it.

The redundancy was traced to the coordinate power structure inside the 3D Zernike moment computation. Geometric moments whose elevation exponent was zero (t = 0) encode no height information and contribute only noise to the descriptor. By expanding the Zernike basis functions and analyzing parity constraints on the constituent terms, we isolated the moment orders in which these elevation-independent contributions do not cancel out: even-order moments satisfying m=4a (a∈Z). This theoretical prediction was tested empirically by computing Zernike moments across 5,484 non-overlapping digital elevation model (DEM) maps, each measuring 80 × 80 grid cells and drawn from global terrain data. For each map, individual moment magnitudes were correlated against the local elevation standard deviation. An improved descriptor was then constructed by discarding all moments identified as redundant while retaining odd-order terms and the informative even-order terms with m=4a+2. Matching performance was evaluated on region p and region f (elevation standard deviations of 197.41 m and 57.86 m, respectively) under three levels of additive Gaussian noise (5%, 8%, and 10% of the elevation standard deviation) and four heading angles (0°, 5°, 45°, and 75°). Each condition was tested with 1,000 independent trials. The proposed descriptor, comprising 29 elements from the first 10 orders, was compared against the full descriptor (34 elements) and the descriptor that uses only odd-order moments (14 elements).

The correlation measurements confirm the theoretical analysis. Even-order moments with m=4a show negative correlation with elevation standard deviation, and their magnitudes converge to a non-zero constant as terrain variation approached zero. This behavior contradicts the expected physical relationship: a flat surface, devoid of elevation features, should produce near-zero descriptor values. All other moments exhibit positive correlation, consistent with their role in capturing terrain expressiveness. Removing the redundant moments raises matching success rates across all test conditions. On the region p at 10% noise and 45° rotation, the success rate increases from 73.7% (full descriptor) to 91.1% (improved descriptor); on the region f under the same conditions, it rose from 80.1% to 90.9%. The odd-order-only descriptor produces intermediate results of 83.9% and 80.8%, respectively. These are better than the full descriptor but below the proposed version. This gap shows that the retained even-order terms with m=4a+2 carry terrain-relevant information that the odd-order-only approach discards. Across all 24 combinations of terrain type, noise level, and rotation angle, the improved descriptor outperforms both baselines. Success rates remain above 90% even under the combined stress of 10% noise and large rotation angles, confirming robust anti-noise and anti-rotation performance.

The 3D Zernike moment descriptor contains structurally identifiable terms that carry no elevation information and weaken terrain discriminability. These redundant terms are confined to even-order moments with m=4a and can be removed without compromising rotational invariance. The resulting descriptor is more compact (29 versus 34 elements) and achieves higher matching reliability because it discards noise while retaining terrain-relevant signal. The noise resistance and rotational robustness demonstrated here illustrate a broader design principle: moment-based descriptors should be pruned to reflect the actual geometry of the data they represent. The same redundancy-aware selection strategy may extend to other applications of 3D Zernike moments in which a surface, rather than a full volumetric body, carries the meaningful signal.