This paper describes an interdisciplinary approach to geometry modeling of geospatial boundaries. The objective is to extract surfaces from irregular spatial patterns using differential geometry and obtain coherent directional predictions along the boundary of extracted surfaces to enable more targeted sampling and exploration. Specific difficulties of the data include sparsity, incompleteness, causality and resolution disparity. Surface slopes are estimated using only sparse samples from cross-sections within a geological domain with no other information at intermediate depths. From boundary detection to subsurface reconstruction, processes are automated in between. The key problems to be solved are boundary extraction, region correspondence and propagation of the boundaries via contour morphing. Established techniques from computational physics, computer vision and signal processing are used with appropriate modifications to address challenges in each area. To facilitate boundary extraction, an edge map synthesis procedure is presented. This works with connected component analysis, anisotropic diffusion and active contours to convert unordered points into regularized boundaries. For region correspondence, component relationships are handled via graphical decomposition. FFT-based spatial alignment strategies are used in region merging and splitting scenarios. Shape changes between aligned regions are described by contour metamorphosis. Specifically, local spatial deformation is modeled by PDE and computed using level-set methods. Directional predictions are obtained using particle trajectories by following the evolving boundary. However, when a branching point is encountered, particles may lose track of the wavefront. To overcome this, a curvelet backtracking algorithm has been proposed to recover information for boundary segments without particle coverage to minimize shape distortion.

中文翻译：

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地下边界几何建模：将计算物理、计算机视觉和信号处理技术应用于地球科学

本文描述了一种跨学科的地理空间边界几何建模方法。目标是使用微分几何从不规则的空间模式中提取表面，并沿着提取表面的边界获得连贯的方向预测，以实现更有针对性的采样和探索。数据的具体困难包括稀疏性、不完整性、因果关系和分辨率差异。仅使用来自地质域内横截面的稀疏样本估计地表坡度，中间深度没有其他信息。从边界检测到地下重建，两者之间的过程是自动化的。要解决的关键问题是边界提取、区域对应和通过轮廓变形的边界传播。来自计算物理学的成熟技术，使用计算机视觉和信号处理并进行适当修改以应对每个领域的挑战。为了便于边界提取，提出了边缘图合成程序。这适用于连通分量分析、各向异性扩散和活动轮廓，将无序点转换为正则化边界。对于区域对应，组件关系通过图形分解处理。基于 FFT 的空间对齐策略用于区域合并和分割场景。对齐区域之间的形状变化由轮廓变形来描述。具体而言，局部空间变形由 PDE 建模并使用水平集方法计算。方向预测是通过遵循演化边界使用粒子轨迹获得的。但是，当遇到分支点时，粒子可能会失去对波前的跟踪。为了克服这个问题，已经提出了一种曲波回溯算法来恢复没有粒子覆盖的边界段的信息，以最大限度地减少形状失真。