A Huber-derived Robust Multi-quadric Interpolation Method for DEM Construction

In this paper, we propose a robust multi-quadric method (MQ-H) based on Huber loss function to conduct interpolations of contaminated spatial points, especially those derived from remote-sensing techniques. The objective function of the MQ-H has two main parts ; an improved Huber loss function and a regularized penalty term used to improve robustness and avoid overfitting, respectively. A mathematical surface, subject to model error with different distributions, was employed to comparatively analyze the robustness of the MQ-H, the classical MQ, and a least absolute deviation based MQ (MQ-L). The results indicated that when sample errors follow a normal distribution or a Laplacian distribution, the performance of MQ-H is comparatively better than those of MQ, and more accurate than MQ-L. For sample errors with a contaminated normal distribution and Cauchy distribution, MQ-H is more robust than MQ-L and MQ. Moreover, MQ with the improved Huber loss function is superior to MQ with the classical Huber loss function. A real-world example of DEM construction with stereo-image-derived elevation points indicates that compared to the classical interpolation methods including IDW (inverse distance weighting), OK (ordinary Kriging) and ANUDEM (Australian National University DEM), MQ-H has a better ability to reduce the impact of outliers while maintaining subtle terrain features suitable for qualitative analysis.