Complex Contour Reconstruction of Geological Bodies Based on Fuzzy Matching and Multi-feature Constrained Interpolation
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摘要:
针对地质体相邻轮廓线重建中因轮廓特征差异显著导致的匹配准确率低下、拓扑关系失真及几何形态畸变问题，提出了一种基于模糊匹配与多特征约束插值的复杂轮廓重建算法。该方法首先融合顶点的空间位置、局部邻接关系和整体轮廓特征，建立模糊域匹配策略，评估相邻轮廓的顶点相似度并在相似顶点间建立单映射匹配。基于匹配结果生成源轮廓与目标轮廓的最大接近轮廓后，采用线性插值和离散多边形演化分别处理最大接近轮廓之间以及原始轮廓与最大接近轮廓之间的过渡形态。最后，结合包围盒约束的几何变换校正，基于匹配结果对插值后的轮廓序列进行三维重建。采用3组典型地质特征的勘探线剖面数据进行重建测试。在方法对比方面，选取GOCAD标准重建算法作为基准，同时引入局部优化约束和全局优化约束的改进算法进行系统比较。结果表明，所提方法有效解决了传统重建方法中存在的轮廓自交和拓扑结构紊乱问题。基于三角形相似度、跨距长度和空间夹角构建的几何评价体系验证表明，该方法重建的TIN模型在几何精度和拓扑一致性方面均表现出显著优势。本研究提出的方法降低了轮廓插值对匹配结果的依赖，为轮廓线重建中的对应问题和插值问题提供了算法创新与理论参考。
Abstract:ObjectiveTo address feature differences in adjacent geological body contours and the limitations of full-mapping matching, this study proposes a complex contour reconstruction algorithm based on fuzzy matching and multi-feature constrained interpolation.
MethodsTo tackle the challenges of low matching accuracy, topological distortion, and geometric deformation caused by significant contour feature differences, we developed an algorithm that combines fuzzy matching with multi-feature constraints. Initially, the method integrates vertex spatial positions, local adjacency relationships, and global contour features to establish a fuzzy-domain matching strategy, evaluating vertex similarity and constructing one-to-one mappings between adjacent contours. Based on these matching results, maximum proximity contours between source and target contours are generated. Subsequently, linear interpolation and discrete polygon evolution methods are applied to generate transitions between maximum proximity contours and between original and maximum proximity contours. Finally, three-dimensional reconstructions are performed using bounding-box-constrained geometric transformation corrections.
ResultsTests were conducted using three sets of typical geological exploration-line profile data, with the standard GOCAD reconstruction algorithm serving as the baseline for comparison, alongside its locally and globally optimized variants. The proposed method effectively resolved issues of contour self-intersection and topological disorder observed in conventional approaches. Evaluations using triangle similarity, span length, and spatial angles demonstrated significant improvements in geometric accuracy and topological consistency within the reconstructed TIN models.
ConclusionThis approach reduces dependency on matching results during interpolation and provides both algorithmic and theoretical references for addressing contour reconstruction challenges.
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图 1 几种插值结果的比较
a. 基于顶点模糊域的匹配结果；b. 单映射的顶点匹配；c. 重顶点的匹配；d. 基于离散多边形演化的插值结果；e. 基于单映射匹配的插值结果；f. 基于重顶点匹配的插值结果。$ \left\{{p}_{{1}},\; {p}_{{2}},\; {p}_{{3}}\right\} $，$ \left\{{q}_{{1}},\; {q}_{{2}},\; {q}_{{3}},\; {q}_{{4}}\right\} $均为点集；$ \left\{p_{{0}}^{i},\; p_{{1}}^{i}\right\} $，$ \left\{q_{{0}}^{i}\right\} $均为角度插值的插入点集；蓝色点和红色点分别为2组待匹配轮廓的特征点
Figure 1. Comparison of several interpolation results
图 2 模糊匹配与最大接近轮廓
a. 源轮廓与目标轮廓；b. 源轮廓的U区域；c. 基于顶点模糊域的匹配；d. 源轮廓与目标轮廓的最大接近轮廓；e. 源轮廓到其最大接近轮廓的变形；f. 目标轮廓到其最大接近轮廓的变形。$ \{{p}_{1},\;{p}_{3},\;{p}_{5}\} $为孤立点；$ \{{p}_{0},\;{p}_{2},\;{p}_{4},\;{p}_{6},\;{p}_{7},\;{p}_{8},\;{p}_{9},\;{p}_{10},\;{p}_{11},\;{p}_{12},\;{p}_{13}\}$为匹配点；蓝色圆点和红色圆点分别为2组待匹配轮廓的特征点
Figure 2. Fuzzy Matching and maximum proximity contour
图 3 插值路径的自交与规避机制
a. 离散演化方法存在的路径自交；b. 匹配点的可能区域；c. 一种匹配点为凹顶点的情形；d. 匹配点位于邻接三角形内；e. 产生自交的插值路径；f. 合理的插值路径。A，B，C. 存在邻接关系的3个点（蓝色圆点）；D，D'，D''. 匹配点（红色圆点）；Ⅰ，Ⅱ，Ⅲ，Ⅳ. 匹配点的可能区域编号
Figure 3. Self-intersection and avoidance mechanisms in interpolation paths
图 4 基于离散多边形演化的特征过渡
a. 凹顶点的首轮识别与演化；b. 凹顶点的次轮识别与演化；c. 凹顶点的第三轮识别与演化；d. 形状贡献度排序与凸顶点的首轮演化；e. 形状贡献度排序与凸顶点的次轮演化；f. 凸顶点的第三轮演化。$ \{{{p}}_{{1}},\; {{p}}_{{2}},\; \cdots,\; p_{{17}}\} $为孤立点，其中蓝色圆点为不需要被处理的点，红色圆点为需要被处理的点
Figure 4. Feature transition based on discrete polygon evolution
图 5 基于包围盒约束（a～d）与几何变换（e～f）的轮廓变形
a. 原始轮廓的直接匹配结果；b. 基于包围盒约束的轮廓比例修正；c. 基于包围盒中心重合约束的轮廓平移；d. 基于包围盒重合约束的轮廓缩放；e. 变比例缩放；f. 等比例缩放；g. 平移变换。蓝色圆点和红色圆点分别为2组待匹配轮廓的特征点；(x, y)为变换前的原始坐标；(x', y')为变换后的坐标；$ {S}_{x} $和$ {S}_{y} $分别为沿着x方向和y方向的缩放比例；S为缩放比例；($ { \Delta }x $，$ { \Delta }y $)分别为原始坐标(x, y)的增量
Figure 5. Contour deformation based on bounding box constraints and geometric transformation
图 6 狭窄勘探线剖面轮廓的插值与重建测试
a. 原始勘探线剖面轮廓；b. 原始轮廓的GOCAD命令重建；c. 原始轮廓的模糊匹配重建；d. 轮廓插值结果（多特征约束插值）；e. 轮廓插值结果1（反距离定权插值）；f. 轮廓插值结果2（反距离定权插值）；g. 插值结果e的重建渲染模型；h. 插值结果d的重建TIN模型；i. 插值结果d的重建渲染模型。红色线圈内为轮廓自交
Figure 6. Interpolation and reconstruction testing of narrow exploration line profiles
图 7 宽体勘探线剖面轮廓的插值与重建测试
a. 原始勘探线剖面轮廓；b. 原始轮廓的GOCAD命令重建；c. 原始轮廓的模糊匹配重建；d. 轮廓插值结果（多特征约束插值）；e. 轮廓插值结果（反距离定权插值）；f. 插值结果e的重建渲染模型；g. 插值结果d的重建渲染模型（模糊匹配）；h. 插值结果d的重建渲染模型（全局最优匹配）；i. 插值结果d的重建渲染模型（局部最优匹配）。红色线圈内为轮廓自交
Figure 7. Interpolation and reconstruction testing of wide exploration line profiles
图 8 复杂勘探线剖面轮廓的插值与重建测试
a. 原始勘探线剖面轮廓；b. 多特征约束方法的插值结果；c. 反距离定权方法的插值结果；d. 轮廓重建TIN模型(插值结果c）；e. 轮廓重建 TIN模型（视角1）；f. 轮廓重建TIN模型（视角2）；g. 轮廓重建 TIN模型（视角3）；h. 轮廓重建TIN模型（视角4）
Figure 8. Interpolation and reconstruction testing of complex exploration line profiles
图 9 复杂勘探线剖面轮廓的重建对比
a. 原始轮廓全局最优约束重建；b. 原始轮廓的局部最优约束重建；c. 插值后轮廓的全局最优约束重建1；d. 插值后轮廓的全局最优约束重建2；e. 插值后轮廓的局部最优约束重建；f. 插值后轮廓的模糊匹配方法重建
Figure 9. Comparison of complex exploration line profiles
图 10 基于特征量度的TIN模型的质量评估
a. 基于距离的相似度（以等边三角形为基准）；b. 基于角度的相似度（以等边三角形为基准）；c. 基于距离的相似度（以等腰三角形为基准）；d. 基于角度的相似度（以等腰三角形为基准）；e. 邻接三角形的距离相似度；f. 邻接三角形的角度相似度；g. 邻接跨距的夹角（弧度制）；h. TIN（不规则三角网）的跨距长度；i. 基于法向粗糙度的邻接三角形相似度
Figure 10. Quality assessment of TIN models based on feature metrics
表 1 基于顶点模糊域的匹配算法
Table 1. Matching algorithm based on vertex fuzzy domain
输入：轮廓P与Q的点集$ V(P) $与$V{(Q)} $ 输出：匹配完成的点对$ {(}p{,q)} $ ①计算轮廓P与Q的U区域半径$ {r}_{P} $与$ {r}_{Q} $，使用$ \min({r}_{P},\; {r}_{Q}) $作为U区域的半径 ②比较$ {(V(P).{\mathrm{Count}},V(Q).{\mathrm{Count}})} $，对顶点数量少的轮廓，使用$ \min({r}_{P},\; {r}_{Q}) $建立U区域 ③根据定义4判断点对是否接近，对接近的点对建立匹配 ④重复步骤③，遍历U区域，算法结束 
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表 2 基于离散演化的轮廓插值算法
Table 2. Contour interpolation algorithm based on discrete evolution
输入：轮廓P的点集$ {V}{(P)} $，轮廓P的最大接近轮廓的点集$ {V}{(}{{P}}^{\max }{)} $ 输出：插值得到的过渡轮廓序列$ \{{P}_{1},\; {P}_{2},\; \cdots ,\;{P}_{{n}}\} $ ①依据旋向判断顶点的凹凸性 ②若顶点$ {{p}}_{{k}} $为凹顶点，则建立区域$ {U}{( \Delta }{{p}}_{{k}{-1}}{{p}}_{{k}}{{p}}_{{k}{+1}}{)} $，顶点$ {{p}}_{{k}} $的演化过程为$ {{ \theta p}}_{{k}}{{p}}_{{1+k}}{{p}}_{{2+k}} $的角平分线的反向延长线 ③遍历所有凹顶点，重复步骤②，完成凹区域的演化 ④计算顶点的形状贡献程度$ {C}=\left\{{{c}}_{{0}},\; {{c}}_{{1}}{ \cdots }{{c}}_{{m}}\right\} $，搜寻形状贡献程度最小的点$ {{p}}_{{i}},\; {{c}}_{{i}}=\min {(C)} $，依据定义9，建立区域$ {U}{( \Delta }{{p}}_{{i}{-1}}{{p}}_{{i}}{{p}}_{{i}{+1}}{)} $ ⑤搜寻点$ {{p}}_{{i}} $的次邻接点，建立区域$ {U}{( \Delta }{{p}}_{{i}{+1}}{{p}}_{{i}{+2}}{{p}}_{{i}{+3}}{)} $和$ {U}{( \Delta }{{p}}_{{i}{-3}}{{p}}_{{i}{-2}}{{p}}_{{i}{-1}}{)} $ ⑥用$ {{p}}_{{i}} $的次邻接点取代$ {{p}}_{{i}} $，遍历剩余顶点，重复步骤⑤，完成凸区域的一轮演化 ⑦删除步骤⑥中需要演化的顶点，执行步骤④⑤⑥，完成凸区域演化 ⑧遍历剩余顶点，重复步骤④⑤⑥⑦，至点集为空，算法结束
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