投审稿入口
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进行校正，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，该方法重建的不规则三角网（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 the problems of low matching accuracy, topological distortion, and geometric deformation caused by significant contour feature differences in adjacent geological body contour reconstruction, and to overcome the limitations of full-mapping matching, this study proposes a complex contour reconstruction algorithm based on fuzzy matching and multi-feature constrained interpolation.
MethodsInitially, the method integrated vertex spatial positions, local adjacency relationships, and global contour features to establish a fuzzy-domain matching strategy. Vertex similarity between adjacent contours was evaluated, and one-to-one mappings between similar vertices were constructed. Based on the matching results, maximum proximity contours between the source and target contours were generated. Subsequently, linear interpolation and discrete polygon evolution were applied to handle transitional shapes between maximum proximity contours and between original and maximum proximity contours, respectively. Finally, three-dimensional reconstruction was performed on the interpolated contour sequence based on the matching results, using bounding-box-constrained geometric transformation correction. Reconstruction tests were conducted using three sets of typical geological exploration-line profile data. The standard GOCAD reconstruction algorithm was selected as the baseline, and improved algorithms with local and global optimization constraints were introduced for systematic comparison.
ResultsThe results showed that the proposed method effectively resolved the problems of contour self-intersection and topological disorder in conventional reconstruction approaches. Evaluations using a geometric assessment system-comprising triangle similarity, span length, and spatial angles-demonstrated that the reconstructed triangulated irregular network (TIN) models exhibited significant advantages in geometric accuracy and topological consistency.
ConclusionThe proposed approach reduces the dependency of contour interpolation on matching results and provides both algorithmic innovation and theoretical references for addressing correspondence and interpolation problems in contour reconstruction.
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图 1 几种插值结果的比较
$ \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；蓝色圆点和红色圆点分别为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
$ \{{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}\}$为匹配点；蓝色圆点和红色圆点分别为2组待匹配轮廓的特征点
Figure 2. Fuzzy matching and maximum proximity contour
图 3 插值路径的自交与规避机制
A，B，C. 存在邻接关系的3个点（蓝色圆点）；D，D'，D''. 匹配点（红色圆点）；Ⅰ，Ⅱ，Ⅲ，Ⅳ. 匹配点的可能区域编号
Figure 3. Self-intersection and avoidance mechanisms in interpolation paths
图 4 基于离散多边形演化的特征过渡
a. 凹顶点的首轮识别与演化；b. 凹顶点的次轮识别与演化；c. 凹顶点的第三轮识别与演化；d. 形状贡献度排序与凸顶点的首轮演化；e. 形状贡献度排序与凸顶点的次轮演化；f. 凸顶点的第三轮演化。$ \{{{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} $，$ {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 $)分别为原始坐标(x, y)的增量；$ Side_{0}^{Q} $为轮廓Q在x方向的包围盒边长；$ Side_{1}^{Q} $为轮廓Q在y方向的包围盒边长；$ Side_{0}^{P} $为轮廓P在x方向的包围盒边长；$ Side_{1}^{P} $为轮廓P在y方向的包围盒边长
Figure 5. Contour deformation based on bounding-box constraints (a-d) and geometric transformation (e, f)
图 6 狭窄勘探线剖面轮廓的插值与重建测试
a. 原始勘探线剖面轮廓；b. 原始轮廓的GOCAD命令重建；c. 原始轮廓的模糊匹配重建；d. 轮廓插值结果（多特征约束插值）；e. 轮廓插值结果1（反距离定权插值）；f. 轮廓插值结果2（反距离定权插值）；g. 轮廓插值结果1（图e）的重建渲染模型；h. 轮廓插值结果（图d）的重建TIN模型；i. 轮廓插值结果（图d）的重建渲染模型。红色线圈内为轮廓自交
Figure 6. Contour interpolation and reconstruction tests of narrow exploration-line profiles
图 7 宽体勘探线剖面轮廓的插值与重建测试
a. 原始勘探线剖面轮廓；b. 原始轮廓的GOCAD命令重建；c. 原始轮廓的模糊匹配重建；d. 轮廓插值结果（多特征约束插值）；e. 轮廓插值结果（反距离定权插值）；f. 轮廓插值结果（图e）的重建渲染模型；g. 轮廓插值结果（图d）的重建渲染模型（模糊匹配）；h. 轮廓插值结果（图d）的重建渲染模型（全局最优匹配）；i. 轮廓插值结果（图d）的重建渲染模型（局部最优匹配）。红色线圈内为轮廓自交
Figure 7. Contour interpolation and reconstruction tests of wide exploration-line profiles
图 8 复杂勘探线剖面轮廓的插值与重建测试
a. 原始勘探线剖面轮廓；b. 多特征约束方法的插值结果；c. 反距离定权方法的插值结果；d. 轮廓插值结果（图c）的重建TIN模型；e. 轮廓插值结果（图b）的重建 TIN模型（视角1）；f. 轮廓插值结果（图b）的重建TIN模型（视角2）；g. 轮廓插值结果（图b）的重建 TIN模型（视角3）；h. 轮廓插值结果（图b）的重建TIN模型（视角4）
Figure 8. Contour interpolation and reconstruction tests of complex exploration-line profiles
图 9 复杂勘探线剖面轮廓的重建对比
a. 原始轮廓全局最优约束重建；b. 原始轮廓的局部最优约束重建；c. 插值后轮廓的全局最优约束重建1；d. 插值后轮廓的全局最优约束重建2；e. 插值后轮廓的局部最优约束重建；f. 插值后轮廓的模糊匹配方法重建
Figure 9. Comparison of contour reconstruction for complex exploration-line profiles
图 10 基于特征量度的TIN模型的质量评估
Example1～Example3 分别对应狭窄轮廓测试（实验一）、宽体轮廓测试（实验二）和复杂轮廓测试（实验三）的结果
Figure 10. Quality assessment of triangulated irregular network (TIN) models based on feature metrics
算法1 基于顶点模糊域的匹配算法 输入：轮廓P、Q的点集$ V(P) $、$V{(Q)} $ 算法步骤： ①计算轮廓P、Q的区域U半径$ {r}_{P} $、$ {r}_{Q} $，使用$ \min({r}_{P},\; {r}_{Q}) $作为区域U的半径 ②比较集合V(P)与V(Q)的元素数量$ {(V(P).{\mathrm{Count}},V(Q).{\mathrm{Count}})} $，对顶点数量少的轮廓，使用$ \min({r}_{P},\; {r}_{Q}) $建立区域U ③根据定义4判断点对是否接近，对接近的点对建立匹配 ④重复步骤③，遍历区域U，算法结束 输出：匹配完成的点对$ {(}p{,q)} $ 
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算法2 基于离散演化的轮廓插值算法 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 }{)} $ 算法步骤： 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}} $，遍历剩余顶点，重复步骤⑤，完成凸区域的一轮演化 ⑦删除步骤⑥中需要演化的顶点，执行步骤④⑤⑥，完成凸区域演化 ⑧遍历剩余顶点，重复步骤④⑤⑥⑦，至点集为空，算法结束 输出：插值得到的过渡轮廓序列$ \{{p}_{1},\; {p}_{2},\; \cdots ,\;{p}_{{n}}\} $
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