引用本文: 陈传法, 刘凤英, 闫长青, 戴洪磊, 郭金运, 刘国林. DEM建模的多面函数Huber抗差算法[J]. 武汉大学学报 ( 信息科学版), 2016, 41(6): 803-809.
CHEN Chuanfa, LIU Fengying, YAN Changqing, DAI Honglei, GUO Jinyun, LIU Guolin. A Huber-derived Robust Multi-quadric Interpolation Method for DEM Construction[J]. Geomatics and Information Science of Wuhan University, 2016, 41(6): 803-809.
 Citation: CHEN Chuanfa, LIU Fengying, YAN Changqing, DAI Honglei, GUO Jinyun, LIU Guolin. A Huber-derived Robust Multi-quadric Interpolation Method for DEM Construction[J]. Geomatics and Information Science of Wuhan University, 2016, 41(6): 803-809.

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

• 摘要: 为了抑制采样点中粗差对数字高程模型(digital elevation model,DEM)建模的影响,以较高精度的多面函数(multi-quadric,MQ)为基函数,由改进Huber损失函数和权重惩罚项组成目标函数,发展了MQ抗差插值算法(MQ-H)。通过优化MQ-H目标函数,采样点权重计算最终转换为方程组求解。以数学曲面为研究对象,将MQ-H计算结果与传统MQ及最小绝对偏差MQ(MQ-L)进行比较,结果表明:当采样误差服从正态分布时,MQ-H计算精度与传统MQ相当,而远高于MQ-L;当采样误差服从拉普拉斯分布时,MQ-H计算精度略高于MQ-L及传统MQ;当采样点被粗差污染时,MQ-H计算精度远高于传统MQ及MQ-L。在实例分析中,以无人遥测飞艇立体像对获取的地面离散高程点为基础数据,基于MQ-H构建测区DEM,并将计算结果与传统插值算法,如反距离加权(inverse distance weighting,IDW)、普通克里金(ordinary Kriging,OK)和专业DEM插值软件ANUDEM(Australian National University DEM)进行比较,结果表明,传统插值方法在不同程度上受采样点中异常值或偶然误差影响,而MQ-H受异常值影响较小,且能准确捕捉到地形细节信息。

Abstract: 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.

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