{"id":4624,"date":"2025-11-21T11:37:01","date_gmt":"2025-11-21T03:37:01","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=4624"},"modified":"2026-01-30T11:47:28","modified_gmt":"2026-01-30T03:47:28","slug":"dual_marching_cubes_primal_contouring_dual_grids_paper_interpretation","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2025\/11\/21\/dual_marching_cubes_primal_contouring_dual_grids_paper_interpretation\/","title":{"rendered":"\u300aDual Marching Cubes: Primal Contouring of Dual Grids\u300b\u8bba\u6587\u89e3\u8bfb"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u4e00\u5468\u524d\u7ec8\u4e8e\u7ed3\u675f\u4e86\u957f\u8fbe3\u4e2a\u6708\u76842025\u4e0b\u534a\u5e74\u7cfb\u7edf\u67b6\u6784\u8bbe\u8ba1\u5e08\u8f6f\u8003\u5907\u8003\u5386\u7a0b, \u81ea\u5df1\u5bf9\u4e8eUE\u51e0\u4f55\u90e8\u5206\u7684\u5b66\u4e60\u8282\u594f\u4e5f\u65ad\u4e86\u597d\u957f\u4e00\u9635\u5b50. \u76ee\u524d\u6bd4\u8f83\u611f\u5174\u8da3\u7684\u4e14\u80fd\u5bf9\u81ea\u5df1\u7684\u51e0\u4f55\u529f\u529b\u6709\u6240\u63d0\u5347\u7684\u4fbf\u662f\u7f51\u683c\u751f\u6210\u4e0e\u66f2\u9762\u7ec6\u5206, \u800c\u8fd9\u4e24\u90e8\u5206\u6070\u597d\u662f\u300aGPU Gems 3\u300b \u4e2d\u51e0\u4f55\u4e00\u7ae0\u7684\u4e24\u5c0f\u8282\u5185\u5bb9. \u5982\u6b64\u4e00\u6765, \u4eff\u4f5b\u4e00\u4e0b\u5b50\u627e\u5230\u4e86\u6307\u8def\u660e\u706f, \u63a5\u4e0b\u6765\u4e00\u6bb5\u65f6\u95f4\u4e5f\u4f1a\u4ee5\u6b64\u4f5c\u4e3a\u51e0\u4f55\u90e8\u5206\u7684\u5b66\u4e60\u7eb2\u9886. \u5148\u4ece\u7f51\u683c\u751f\u6210\u90e8\u5206\u5f00\u59cb, \u4ee5\u672c\u6587\u89e3\u8bfb\u4e00\u7bc7\u5728\u7f51\u683c\u751f\u6210\u9886\u57df\u5341\u5206\u7ecf\u5178\u7684\u8bba\u6587\u300aDual Marching Cubes: Primal Contouring of Dual Grids\u300b.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. Schaefer S, Warren J. Dual marching cubes: Primal contouring of dual grids[C]\/\/12th Pacific Conference on Computer Graphics and Applications, 2004. PG 2004. Proceedings. IEEE, 2004: 70-76.<\/p>\n<p><strong>1. \u7279\u5f81\u9694\u79bb<\/strong><\/p>\n<p>\u7ed9\u5b9a\u4e00\u4e2a\u9690\u51fd\u6570$f(x, y, z)$\u548c\u4e00\u4e2a\u516b\u53c9\u6811, \u4f5c\u8005\u7684\u76ee\u6807\u662f\u5728\u7279\u5f81\u9694\u79bb\u4e2d\u4e3a\u516b\u53c9\u6811\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u6784\u9020\u4e00\u4e2a\u9876\u70b9, \u8be5\u9876\u70b9\u5c06\u6210\u4e3a\u5bf9\u5076\u7f51\u683c\u7684\u9876\u70b9. \u4e0e\u5bf9\u5076\u8f6e\u5ed3\u6cd5(Dual Contouring) \u7b49\u65b9\u6cd5\u4e0d\u540c, \u6b64\u9876\u70b9\u5e76\u975e\u4e0e\u8868\u9762\u7684\u51e0\u4f55\u7279\u5f81\u5bf9\u9f50, \u800c\u662f\u4e0e\u9690\u51fd\u6570\u672c\u8eab\u7684\u7279\u5f81\u5bf9\u9f50. \u56e0\u6b64, \u6bcf\u4e2a\u9876\u70b9\u4e0d\u4ec5\u5305\u542b\u4f4d\u7f6e\u5750\u6807$(x, y, z)$, \u8fd8\u5305\u542b\u4e00\u4e2a\u6807\u91cf\u503c$w$, \u8868\u793a\u8be5\u9876\u70b9\u5904\u9690\u51fd\u6570\u503c$f(x, y, z)$\u7684\u4f30\u8ba1\u7ed3\u679c.<br \/>\n$\\\\$ \u4f5c\u8005\u4ece\u9690\u51fd\u6570\u4e2d\u68c0\u6d4b\u7279\u5f81\u7684\u65b9\u6cd5\u4e0e\u8bba\u6587M. Garland and P. S. Heckbert. Surface simplification using quadric error metrics. In Proceedings of SIGGRAPH 97, Computer Graphics Proceedings, Annual Conference Series, pages 209\u2013216, Los Angeles, California, August 1997. ACM SIGGRAPH \/ Addison Wesley.\u63d0\u51fa\u7684\u8868\u9762\u7279\u5f81\u68c0\u6d4b\u6280\u672f\u7c7b\u4f3c. \u4e3a\u4e86\u786e\u5b9a\u8fd1\u4f3c\u5355\u5143\u683c\u5185\u9690\u51fd\u6570\u7279\u5f81\u7684\u9876\u70b9, \u4f5c\u8005\u4f7f\u7528\u4e8c\u6b21\u8bef\u5dee\u51fd\u6570(QEFs)(\u5982\u6587\u732eM. Garland and P. S. Heckbert. Surface simplification using quadric error metrics. In Proceedings of SIGGRAPH 97, Computer Graphics Proceedings, Annual Conference Series, pages 209\u2013216, Los Angeles, California, August 1997. ACM SIGGRAPH \/ Addison Wesley\u6240\u8ff0). \u751f\u6210\u8fd9\u4e9bQEF\u65f6, \u4f5c\u8005\u5728\u5355\u5143\u683c$c$\u4e0a\u91c7\u6837\u7684\u7f51\u683c\u70b9\u5904\u8ba1\u7b97\u5173\u4e8e\u9690\u51fd\u6570$f(x, y, z)$\u56fe\u50cf\u7684\u5207\u5e73\u9762. \u5728\u91c7\u6837\u70b9$(x_i, y_i, z_i)$\u5904, \u9690\u51fd\u6570$w = f(x, y, $$ z)$\u56fe\u50cf\u7684\u5207\u5e73\u9762\u65b9\u7a0b\u4e3a$w = T_i(x, y, z)$, \u5176\u4e2d$$T_i(x, y, z) = \\nabla f(x_i, y_i, z_i) \\cdot ((x, y, z) &#8211; (x_i, y_i, z_i)),$$$\\nabla f(x_i, y_i, z_i)$\u662f\u51fd\u6570$f(x_i, y_i, z_i)$\u7684\u68af\u5ea6. \u4e3a\u4e86\u6784\u5efaQEF, \u4f5c\u8005\u5bf9\u8fd9\u4e00\u65b9\u7a0b\u8fdb\u884c\u5e73\u65b9, \u5e76\u5bf9\u6240\u6709\u91c7\u6837\u70b9\u6c42\u548c, \u4ece\u800c\u5f97\u5230:$\\begin{align}E(w, x, y, z) = \\sum_{i} \\frac{(w &#8211; T_i(x, y, z))^2}{1 + \\left | \\nabla f(x_i, y_i, z_i) \\right | } . \\label{QEF} \\end{align}$\u8be5\u8868\u8fbe\u5f0f\u7684\u5206\u6bcd\u5bf9\u6bcf\u4e2a\u5207\u5e73\u9762\u7684\u8d21\u732e\u8fdb\u884c\u5f52\u4e00\u5316, \u4ee5\u4f7f\u5176\u5177\u6709\u76f8\u7b49\u7684\u6743\u91cd.<br \/>\n$\\\\$ \u63a5\u4e0b\u6765, \u4f5c\u8005\u5728\u5355\u5143\u683c$c$\u5185\u6700\u5c0f\u5316\u8be5\u4e8c\u6b21\u51fd\u6570\u4ee5\u627e\u5230\u5bf9\u5076\u7f51\u683c$(w, x, y, z $$ )$\u7684\u9876\u70b9. \u82e5\u6700\u5c0f\u5316\u6c42\u89e3\u5668\u5b58\u5728\u6b20\u5b9a\u6027(\u5373\u89e3\u4e0d\u552f\u4e00), \u4f5c\u8005\u91c7\u7528P. Lindstrom. Out-of-core simplification of large polygonal models. In Proceedings of SIGGRAPH 2000, Computer Graphics Proceedings, Annual Conference Series, pages 259\u2013262. ACM Press \/ ACM SIGGRAPH \/ Addison Wesley Longman, July 2000.\u7684\u65b9\u6cd5\uff0c\u901a\u8fc7\u4f2a\u9006\u77e9\u9635\u5c06\u6700\u5c0f\u503c\u70b9\u5b9a\u4f4d\u5230\u5c3d\u53ef\u80fd\u9760\u8fd1\u5355\u5143\u683c$c$\u4e2d\u5fc3\u7684\u4f4d\u7f6e.<br \/>\n$\\\\$ \u9700\u8981\u6ce8\u610f\u7684\u662f, \u4f5c\u8005\u5bf9\u7279\u5f81\u9694\u79bb\u9636\u6bb5\u7684\u63cf\u8ff0\u5e76\u672a\u9650\u5b9a\u9690\u51fd\u6570$f(x, y, z)$\u7684\u5177\u4f53\u8f93\u5165\u683c\u5f0f. \u5176\u63cf\u8ff0\u5141\u8bb8\u591a\u79cd\u53ef\u80fd\u7684\u8f93\u5165\u5f62\u5f0f. \u4f8b\u5982, Marching Cubes\u4e2d\u5e38\u7528\u7684\u5747\u5300\u6807\u91cf\u7f51\u683c\u53ef\u88ab\u91c7\u7528, \u5176\u4e2d\u5c06\u7f51\u683c\u4e2d\u7684\u6bcf\u4e2a\u7acb\u65b9\u4f53\u89c6\u4e3a\u4e09\u7ebf\u6027\u51fd\u6570, \u4ece\u800c\u4f7f\u5f97$f(x, $$ y, z)$\u548c\u5176\u68af\u5ea6$\\nabla f(x, y, z)$\u5728\u5355\u5143\u5185\u6709\u660e\u786e\u5b9a\u4e49. \u6b64\u5916, \u4e5f\u53ef\u4ee5\u5c06\u4e0a\u8ff0\u91c7\u6837\u9650\u5236\u5728\u5747\u5300\u7f51\u683c\u7684\u9876\u70b9\u4e0a, \u5e76\u901a\u8fc7\u5dee\u5206\u6cd5\u5728\u8fd9\u4e9b\u9876\u70b9\u5904\u4f30\u8ba1$\\nabla f(x, y, z)$. \u540e\u8005\u7684\u4f18\u52bf\u5728\u4e8e\u80fd\u591f\u5728\u7f51\u683c\u9876\u70b9\u5904\u5b9a\u4e49\u5355\u4e00\u7684\u6cd5\u7ebf$\\nabla f(x, y, z)$, \u800c\u975e\u524d\u8005\u4ea7\u751f\u7684\u591a\u4e2a\u4e0d\u8fde\u7eed\u6cd5\u7ebf. \u6709\u5411\u8ddd\u79bb\u573a\u5219\u63d0\u4f9b\u4e86\u53e6\u4e00\u79cd\u66ff\u4ee3\u65b9\u6848, \u5176\u901a\u8fc7\u975e\u5747\u5300\u6807\u91cf\u573a\u53ef\u63d0\u4f9b\u66f4\u7cbe\u786e\u7684\u68af\u5ea6\u4fe1\u606f, \u4ece\u800c\u66f4\u51c6\u786e\u5730\u91cd\u6784\u5c16\u9510\u7279\u5f81.<br \/>\n$\\\\$ CSG\u6811(Constructive Solid Geometry Trees) \u662f\u672c\u7b97\u6cd5\u7684\u53e6\u4e00\u79cd\u6f5c\u5728\u8f93\u5165\u5f62\u5f0f. CSG\u901a\u8fc7\u96c6\u5408\u64cd\u4f5c(\u5982\u5e76\u96c6, \u4ea4\u96c6) \u7ec4\u5408\u51e0\u4f55\u4f53, \u5176\u4e2dCSG\u6811\u7684\u53f6\u8282\u70b9\u8868\u793a\u57fa\u672c\u51e0\u4f55\u4f53(\u5982\u5e73\u9762, \u5706\u67f1\u4f53\u7b49), \u5185\u90e8\u8282\u70b9\u5219\u8868\u793a\u96c6\u5408\u64cd\u4f5c. \u5728\u4f5c\u8005\u7684\u7b97\u6cd5\u4e2d\u4f7f\u7528\u6b64\u7c7bCSG\u6811\u65f6, \u5fc5\u987b\u5c06\u5176\u8f6c\u6362\u4e3a\u5177\u6709\u660e\u786e\u5b9a\u4e49\u68af\u5ea6$\\nabla f(x $$ , y, z)$\u7684\u6807\u91cf\u503c\u51fd\u6570$f(x, y, z)$. \u5bf9\u4e8eCSG\u6811\u7684\u53f6\u8282\u70b9, \u4f5c\u8005\u5c06\u5176\u51e0\u4f55\u4f53\u66ff\u6362\u4e3a\u5bf9\u5e94\u7684\u57fa\u672c\u4f53\u5e26\u7b26\u53f7\u6b27\u51e0\u91cc\u5f97\u8ddd\u79bb\u51fd\u6570(\u5982\u5e73\u9762\u3001\u5706\u67f1\u4f53\u7b49). \u5bf9\u4e8e\u5185\u90e8\u8282\u70b9, \u5219\u5c06\u5e76\u96c6\u548c\u4ea4\u96c6\u64cd\u4f5c\u5206\u522b\u66ff\u6362\u4e3a\u6700\u5c0f\u503c\u548c\u6700\u5927\u503c\u64cd\u4f5c. \u8ba1\u7b97CSG\u6811\u5728\u67d0\u70b9$(x, y, z)$\u5904\u7684\u503c\u65f6, \u53ea\u9700\u8ba1\u7b97\u8be5\u70b9\u5bf9\u6240\u6709\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\u51fd\u6570, \u5e76\u5728CSG\u6811\u7684\u5185\u90e8\u8282\u70b9\u4e0a\u6267\u884c\u5bf9\u5e94\u7684Min\/Max\u64cd\u4f5c. \u8ba1\u7b97\u68af\u5ea6$\\nabla  $$ f(x, y, z)$\u65f6, \u9700\u5728CSG\u6811\u7684\u6bcf\u4e2a\u53f6\u8282\u70b9\u5904\u8bc4\u4f30\u5bf9\u5e94\u7684\u68af\u5ea6, \u5e76\u5728Min\/Max\u64cd\u4f5c\u4e2d\u968f\u51fd\u6570\u503c\u4e00\u540c\u5411\u4e0a\u4f20\u9012. \u56fe1\u4e0e\u56fe2\u5747\u901a\u8fc7\u8ba1\u7b97CSG\u6811\u751f\u6210.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/thin_walled_room_defined_csg.png\" alt=\"\" width=\"1050\" height=\"1050\" class=\"aligncenter size-full wp-image-4632\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/thin_walled_room_defined_csg.png 1050w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/thin_walled_room_defined_csg-300x300.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/thin_walled_room_defined_csg-150x150.png 150w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/thin_walled_room_defined_csg-768x768.png 768w\" sizes=\"(max-width: 1050px) 100vw, 1050px\" \/><center><em><strong>\u56fe1:<\/strong> \u901a\u8fc7CSG\u5b9a\u4e49\u7684\u8584\u58c1\u623f\u95f4(\u5de6\u4e0a). \u591a\u8fb9\u5f62\u8fd1\u4f3c\u7531\u4ee5\u4e0b\u65b9\u6cd5\u751f\u6210:<br \/>\n$\\cdot$ Marching Cubes(\u5de6\u4e0b, 67,000\u9762)<br \/>\n$\\cdot$ \u5bf9\u5076\u8f6e\u5ed3\u6cd5(\u53f3\u4e0b, 17,000\u9762)<br \/>\n$\\cdot$ \u5bf9\u5076Marching Cubes(\u53f3\u4e0a, 440\u9762)<br \/>\n\u4f7f\u7528\u5bf9\u5076Marching Cubes\u65f6, \u7b49\u503c\u9762\u7f51\u683c\u7684\u89c4\u6a21\u5bf9\u5899\u58c1\u539a\u5ea6\u4e0d\u654f\u611f.<\/em><\/center><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/csg_model_rocket_models_approximating_shape_using_same_number_polygons.png\" alt=\"\" width=\"1050\" height=\"1357\" class=\"aligncenter size-full wp-image-4637\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/csg_model_rocket_models_approximating_shape_using_same_number_polygons.png 1050w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/csg_model_rocket_models_approximating_shape_using_same_number_polygons-232x300.png 232w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/csg_model_rocket_models_approximating_shape_using_same_number_polygons-768x993.png 768w\" sizes=\"(max-width: 1050px) 100vw, 1050px\" \/><center><em><strong>\u56fe2:<\/strong> \u901a\u8fc7CSG\u5b9a\u4e49\u7684\u706b\u7bad\u6a21\u578b(\u5de6\u4e0a), \u4ee5\u53ca\u4f7f\u7528\u76f8\u540c\u591a\u8fb9\u5f62\u6570\u91cf\u7684\u8fd1\u4f3c\u6a21\u578b:<br \/>\n$\\cdot$ \u5bf9\u5076Marching Cubes(\u53f3\u4e0a)<br \/>\n$\\cdot$ Marching Cubes(\u5de6\u4e0b)<br \/>\n$\\cdot$ \u5bf9\u5076\u8f6e\u5ed3\u6cd5(\u53f3\u4e0b)<\/em><\/center><\/p>\n<p><strong>1.1 \u516b\u53c9\u6811\u6784\u5efa<\/strong><\/p>\n<p>\u4f5c\u8005\u5f53\u524d\u5bf9\u7279\u5f81\u9694\u79bb\u7684\u63cf\u8ff0\u5047\u8bbe\u5df2\u5b58\u5728\u4e00\u4e2a\u516b\u53c9\u6811, \u7528\u4e8e\u5c06\u7a7a\u95f4\u5212\u5206\u4e3a\u5355\u5143\u683c. \u7136\u800c, \u6709\u65f6\u9700\u8981\u5bf9\u9690\u51fd\u6570$f(x, y, z)$\u8fdb\u884c\u8fd1\u4f3c, \u4f7f\u5176\u8bef\u5dee\u5728\u7ed9\u5b9a\u7684\u5bb9\u5dee\u8303\u56f4$\\epsilon$\u5185. \u4e3a\u4e86\u5b9e\u73b0\u8fd9\u4e00\u76ee\u6807, \u4f5c\u8005\u63d0\u51fa\u4e86\u4e00\u79cd\u81ea\u9876\u5411\u4e0b\u7684\u516b\u53c9\u6811\u6784\u5efa\u7b97\u6cd5.<br \/>\n$\\\\$ \u8be5\u516b\u53c9\u6811\u6784\u5efa\u7b97\u6cd5\u4ece\u4e00\u4e2a\u521d\u59cb\u5355\u5143\u683c$c$\u5f00\u59cb\u6784\u5efa\u516b\u53c9\u6811. \u4f5c\u8005\u4f7f\u7528\u4e0a\u8ff0\u91c7\u6837\u7b97\u6cd5\u5728\u5747\u5300\u7f51\u683c\u4e0a\u5bf9\u9690\u51fd\u6570$f(x, y, z)$\u8fdb\u884c\u7cbe\u7ec6\u91c7\u6837(\u4e5f\u53ef\u91c7\u7528\u968f\u673a\u91c7\u6837\u7b56\u7565). \u63a5\u4e0b\u6765, \u5229\u7528\u8fd9\u4e9b\u91c7\u6837\u70b9\u6784\u5efaQEF\u5e76\u5bf9\u5176\u8fdb\u884c\u6700\u5c0f\u5316. \u82e5\u516c\u5f0f$\\eqref{QEF}$\u8ba1\u7b97\u7684\u8bef\u5dee\u5927\u4e8e\u7ed9\u5b9a\u5bb9\u5dee$\\epsilon$, \u5219\u5c06\u8be5\u5355\u5143\u683c\u5212\u5206\u4e3a8\u4e2a\u5b50\u5355\u5143\u683c, \u5e76\u9012\u5f52\u6267\u884c\u6b64\u8fc7\u7a0b. \u5f53\u516b\u53c9\u6811\u7684\u6240\u6709\u53f6\u8282\u70b9\u8bef\u5dee\u5747\u5c0f\u4e8e\u5bb9\u5dee$\\epsilon$\u65f6, \u7b97\u6cd5\u7ec8\u6b62, \u6700\u7ec8\u6784\u5efa\u51fa\u6700\u5c0f\u5316\u7684\u516b\u53c9\u6811\u4ee5\u8fd1\u4f3c\u9690\u51fd\u6570$f(x, y, z)$.<\/p>\n<p><strong>2. \u62d3\u6251\u521b\u5efa<\/strong><\/p>\n<p>\u7ed9\u5b9a\u4e00\u4e2a\u516b\u53c9\u6811, \u7279\u5f81\u9694\u79bb\u4f1a\u751f\u6210\u5bf9\u5076\u7f51\u683c\u7684\u9876\u70b9(\u8868\u9762\u5c06\u4ece\u8fd9\u4e9b\u9876\u70b9\u4e2d\u63d0\u53d6). \u62d3\u6251\u521b\u5efa\u5219\u5b9e\u9645\u751f\u6210\u5bf9\u5076\u7f51\u683c\u7684\u62d3\u6251\u7ed3\u6784. \u8be5\u5bf9\u5076\u7f51\u683c\u5728\u62d3\u6251\u4e0a\u4e0e\u516b\u53c9\u6811\u4e92\u4e3a\u5bf9\u5076. \u56e0\u6b64, \u5bf9\u4e8e\u516b\u53c9\u6811\u4e2d\u7684\u6bcf\u4e2a\u9876\u70b9, \u4f1a\u5728\u5bf9\u5076\u7f51\u683c\u4e2d\u751f\u6210\u4e00\u4e2a\u5355\u5143\u683c, \u5176\u9876\u70b9\u662f\u5305\u542b\u8be5\u516b\u53c9\u6811\u9876\u70b9\u7684\u6240\u6709\u7acb\u65b9\u4f53\u5185\u7684\u7279\u5f81\u9876\u70b9. \u56fe3\u5c55\u793a\u4e86\u4ece\u4e00\u4e2a\u793a\u4f8b\u56db\u53c9\u6811\u751f\u6210\u7684\u5bf9\u5076\u7f51\u683c\u62d3\u6251(\u5176\u4e2d\u5bf9\u5076\u7f51\u683c\u7684\u6bcf\u4e2a\u9876\u70b9\u4f4d\u4e8e\u5176\u5bf9\u5e94\u7acb\u65b9\u4f53\u7684\u4e2d\u5fc3). \u5c3d\u7ba1\u4f5c\u8005\u7684\u7b97\u6cd5\u64cd\u4f5c\u4e8e\u516b\u53c9\u6811, \u4f46\u672c\u8282\u8ba8\u8bba\u5c06\u7b80\u5316\u4e3a\u56db\u53c9\u6811\u4ee5\u65b9\u4fbf\u8bf4\u660e; \u7136\u800c, \u6b64\u5904\u63cf\u8ff0\u7684\u7b97\u6cd5\u53ef\u81ea\u7136\u6269\u5c55\u5230\u516b\u53c9\u6811.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/connectivity_dual_grid_primal_quadtree.png\" alt=\"\" width=\"572\" height=\"581\" class=\"aligncenter size-full wp-image-4646\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/connectivity_dual_grid_primal_quadtree.png 572w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/connectivity_dual_grid_primal_quadtree-295x300.png 295w\" sizes=\"(max-width: 572px) 100vw, 572px\" \/><center><em><strong>\u56fe3:<\/strong> \u539f\u59cb\u56db\u53c9\u6811(\u7ec6\u84dd\u7ebf) \u5bf9\u5e94\u7684\u5bf9\u5076\u7f51\u683c(\u7c97\u9ed1\u7ebf) \u7684\u8fde\u901a\u6027.<\/em><\/center><\/p>\n<p>\u7ed9\u5b9a\u4e00\u4e2a\u56db\u53c9\u6811$q$, \u4efb\u52a1\u662f\u4ee5\u9ad8\u6548\u7684\u65b9\u5f0f\u679a\u4e3e\u5bf9\u5076\u7f51\u683c\u7684\u6bcf\u4e2a\u5355\u5143\u683c, \u800c\u65e0\u9700\u5728\u56db\u53c9\u6811\u4e2d\u663e\u5f0f\u67e5\u627e\u90bb\u5c45\u8282\u70b9. \u4f5c\u8005\u7684\u89e3\u51b3\u65b9\u6848\u662f\u4e00\u79cd\u9012\u5f52\u904d\u5386\u7b97\u6cd5, \u901a\u8fc7\u679a\u4e3e\u6240\u6709\u5171\u4eab\u540c\u4e00\u9876\u70b9\u7684\u53f6\u65b9\u5757\u5143\u7ec4\u6765\u5b9e\u73b0\u5bf9\u5076\u7f51\u683c\u5355\u5143\u683c\u7684\u751f\u6210. \u8be5\u65b9\u6cd5\u662f\u5bf9\u6587\u732eT. Ju, F. Losasso, S. Schaefer, and J. Warren. Dual contouring of hermite data. ACM Transactions on Graphics, 21(3):339\u2013346, July 2002. ISSN 0730-0301 (Proceedings of ACM SIGGRAPH 2002).\u4e2d\u56db\u53c9\u6811\u904d\u5386\u7b56\u7565\u7684\u7b80\u5355\u6269\u5c55.<br \/>\n$\\\\$ \u8be5\u904d\u5386\u8fc7\u7a0b\u6d89\u53ca\u4e09\u4e2a\u9012\u5f52\u51fd\u6570: faceProc[$q_1$], edgeProc[$q_1$, $q_2$]\u548cvertProc[$q_1$, $q_2$, $q_3$, $q_4$].<br \/>\n$\\\\$ $\\cdot$ \u7ed9\u5b9a\u56db\u53c9\u6811\u4e2d\u7684\u4e00\u4e2a\u5185\u90e8\u8282\u70b9$q_1$, faceProc[$q_1$]\u51fd\u6570\u4f1a\u9012\u5f52\u8c03\u7528\u5176\u56db\u4e2a\u5b50\u8282\u70b9, \u5e76\u5bf9$q_1$\u7684\u6240\u6709\u56db\u5bf9\u8fb9\u76f8\u90bb\u5b50\u8282\u70b9\u8c03\u7528edgeProc, \u4ee5\u53ca\u5bf9$q_1$\u7684\u56db\u4e2a\u5b50\u8282\u70b9\u8c03\u7528\u4e00\u6b21vertProc.<br \/>\n$\\\\$ $\\cdot$ \u7ed9\u5b9a\u4e00\u5bf9\u8fb9\u76f8\u90bb\u7684\u5185\u90e8\u8282\u70b9$q_1$\u548c$q_2$, edgeProc[$q_1$, $q_2$]\u51fd\u6570\u4f1a\u9012\u5f52\u8c03\u7528\u8de8\u8d8a$q_1$\u548c$q_2$\u5171\u540c\u8fb9\u7684\u4e24\u5bf9\u8fb9\u76f8\u90bb\u5b50\u8282\u70b9, \u5e76\u5bf9$q_1$\u548c$q_2$\u4e2d\u63a5\u89e6\u8be5\u5171\u540c\u8fb9\u4e2d\u70b9\u7684\u56db\u4e2a\u5b50\u8282\u70b9\u8c03\u7528\u4e00\u6b21vertProc.<br \/>\n$\\\\$ $\\cdot$ \u7ed9\u5b9a\u56db\u4e2a\u5171\u4eab\u540c\u4e00\u9876\u70b9\u7684\u5185\u90e8\u8282\u70b9$q_1$, $q_2$, $q_3$\u548c$q_4$, vertProc[$q_1$, $q_2$, $q_3$, $q_4$]\u51fd\u6570\u4f1a\u9012\u5f52\u8c03\u7528\u8fd9\u56db\u4e2a\u5b50\u8282\u70b9(\u5b83\u4eec\u5728\u8be5\u5171\u4eab\u9876\u70b9\u5904\u4ea4\u6c47).<br \/>\n$\\\\$ \u5728\u8fd9\u4e9b\u9012\u5f52\u8c03\u7528\u4e2d, \u82e5\u67d0\u4e2a\u8282\u70b9$q_i$\u662f\u53f6\u8282\u70b9, \u5219\u5176\u5b50\u8282\u70b9\u4e0d\u53ef\u7528\u4e8e\u540e\u7eed\u9012\u5f52\u8c03\u7528. \u6b64\u65f6, \u5bf9\u4e8e\u9012\u5f52\u6240\u9700\u7684\u6bcf\u4e2a$q_i$\u7684\u5b50\u8282\u70b9, \u5747\u4f20\u9012\u4e00\u4e2a$q_i$\u7684\u526f\u672c\u4f5c\u4e3a\u53c2\u6570. \u56fe4\u5c55\u793a\u4e86\u8fd9\u4e09\u4e2a\u51fd\u6570\u7684\u76f8\u4e92\u9012\u5f52\u7ed3\u6784.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/recursive_functions_faceProc_edgeProc_vertProc_used_enumerating_cells_dual_grid.png\" alt=\"\" width=\"1879\" height=\"1145\" class=\"aligncenter size-full wp-image-4649\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/recursive_functions_faceProc_edgeProc_vertProc_used_enumerating_cells_dual_grid.png 1879w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/recursive_functions_faceProc_edgeProc_vertProc_used_enumerating_cells_dual_grid-300x183.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/recursive_functions_faceProc_edgeProc_vertProc_used_enumerating_cells_dual_grid-768x468.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/recursive_functions_faceProc_edgeProc_vertProc_used_enumerating_cells_dual_grid-1536x936.png 1536w\" sizes=\"(max-width: 1879px) 100vw, 1879px\" \/><center><em><strong>\u56fe4:<\/strong> \u7528\u4e8e\u679a\u4e3e\u5bf9\u5076\u7f51\u683c\u5355\u5143\u683c\u7684\u9012\u5f52\u51fd\u6570faceProc(\u9ed1\u8272), edgeProc(\u6df1\u7070\u8272) \u548cvertProc(\u6d45\u7070\u8272).<\/em><\/center><\/p>\n<p>\u5f53\u6240\u6709\u8282\u70b9$q_i$\u5747\u4e3a\u56db\u53c9\u6811\u53f6\u8282\u70b9\u65f6, \u8fd9\u4e9b\u51fd\u6570\u7684\u9012\u5f52\u8c03\u7528\u7ec8\u6b62. \u5728vertProc\u7684\u7ec8\u6b62\u6761\u4ef6\u4e2d, \u53ef\u5f97\u5230\u56db\u4e2a\u5171\u4eab\u540c\u4e00\u89d2\u70b9\u7684\u53f6\u8282\u70b9. \u6b64\u65f6, \u901a\u8fc7\u8fde\u63a5\u7279\u5f81\u9694\u79bb\u9636\u6bb5\u4e2d\u4e3a\u6bcf\u4e2a\u53f6\u8282\u70b9\u751f\u6210\u7684\u9876\u70b9, \u62d3\u6251\u4e0a\u5f62\u6210\u4e00\u4e2a\u6b63\u65b9\u5f62, \u4ece\u800c\u6784\u5efa\u4e00\u4e2a\u5bf9\u5076\u7f51\u683c\u5355\u5143\u683c. \u5728\u81ea\u9002\u5e94\u60c5\u51b5\u4e0b, \u5176\u4e2d\u4e00\u4e2a\u53f6\u8282\u70b9$q_i$\u53ef\u80fd\u88ab\u91cd\u590d\u4f7f\u7528, \u5bfc\u81f4\u51e0\u4f55\u4e0a\u5f62\u6210\u4e09\u89d2\u5f62. \u9700\u6ce8\u610f, \u8be5\u65b9\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u4e0e\u56db\u53c9\u6811\u7684\u89c4\u6a21\u5448\u7ebf\u6027\u5173\u7cfb, \u56e0\u4e3a\u56db\u53c9\u6811\u4e2d\u7684\u6bcf\u4e2a\u65b9\u5757\u5bf9\u5e94\u4e00\u6b21faceProc\u8c03\u7528, \u6bcf\u6761\u8fb9\u5bf9\u5e94\u4e00\u6b21edgeProc\u8c03\u7528, \u6bcf\u4e2a\u9876\u70b9\u5bf9\u5e94\u4e00\u6b21vertProc\u8c03\u7528.<br \/>\n$\\\\$ \u5982\u4e0a\u6240\u8ff0\uff0c\u8c03\u7528faceProc[$q_1$]\u4ec5\u751f\u6210\u4f4d\u4e8e\u56db\u53c9\u6811\u6839\u8282\u70b9\u5185\u90e8\u7684\u5355\u5143\u683c. \u4e3a\u4e86\u5c06\u5bf9\u5076\u7f51\u683c\u6269\u5c55\u5230\u56db\u53c9\u6811\u7684\u8fb9\u754c, \u4f5c\u8005\u5c06\u56db\u53c9\u6811\u89c6\u4e3a\u4e00\u4e2a$3 \\times 3$\u9000\u5316\u77e9\u5f62\u7f51\u683c\u4e2d\u7684\u4e2d\u5fc3\u65b9\u5757. \u5728\u6b64\u7f51\u683c\u4e2d, \u56db\u4e2a\u65b9\u5757\u9000\u5316\u4e3a\u56db\u53c9\u6811\u7684\u8fb9, \u800c\u5269\u4f59\u56db\u4e2a\u65b9\u5757\u9000\u5316\u4e3a\u56db\u53c9\u6811\u7684\u9876\u70b9. \u901a\u8fc7\u5c06\u56db\u53c9\u6811\u7684\u6839\u8282\u70b9\u4e0e\u5176\u6bcf\u4e2a\u9000\u5316\u8fb9\u76f8\u90bb\u7684\u90bb\u5c45\u8282\u70b9\u4f5c\u4e3a\u53c2\u6570\u8c03\u7528edgeProc, \u5373\u53ef\u751f\u6210\u4e0e\u56db\u53c9\u6811\u5bf9\u5e94\u8fb9\u63a5\u89e6\u7684\u5355\u5143\u683c. \u540c\u6837\u5730, \u901a\u8fc7\u5c06\u6839\u8282\u70b9\u7684\u6bcf\u4e2a\u9876\u70b9\u4e0e\u5176\u4e09\u4e2a\u9000\u5316\u9876\u70b9\u90bb\u5c45\u8282\u70b9\u4f5c\u4e3a\u53c2\u6570\u8c03\u7528vertProc, \u5373\u53ef\u751f\u6210\u4e0e\u8be5\u9876\u70b9\u63a5\u89e6\u7684\u5355\u4e00\u5bf9\u5076\u5355\u5143\u683c. \u56fe4\u5c55\u793a\u4e86\u4f7f\u7528\u6b64\u65b9\u6cd5\u751f\u6210\u7684\u5bf9\u5076\u7f51\u683c, \u5176\u4e2d\u5305\u542b\u4f4d\u4e8e\u7f51\u683c\u8fb9\u548c\u89d2\u70b9\u7684\u9876\u70b9.<\/p>\n<p><strong>3. \u5bf9\u5076\u7f51\u683c\u7b49\u503c\u9762\u63d0\u53d6<\/strong><\/p>\n<p>\u5728\u751f\u6210\u5bf9\u5076\u7f51\u683c\u540e, \u4f5c\u8005\u901a\u8fc7\u5c06Marching Cubes\u7b97\u6cd5\u7b80\u5355\u6269\u5c55\u81f3\u8fd9\u4e9b\u5bf9\u5076\u7f51\u683c\u6765\u63d0\u53d6\u8868\u9762. Marching Cubes\u6700\u521d\u4ec5\u8bbe\u8ba1\u7528\u4e8e\u5904\u7406\u8f74\u5bf9\u9f50\u7684\u7acb\u65b9\u4f53, \u800c\u8bba\u6587\u6240\u4f7f\u7528\u7684\u5bf9\u5076\u7f51\u683c\u5355\u5143\u683c\u5728\u7a7a\u95f4\u4e2d\u5177\u6709\u4efb\u610f\u4f4d\u7f6e\u7684\u9876\u70b9. \u4f5c\u8005\u7684\u89e3\u51b3\u65b9\u6848\u662f: \u6ce8\u610f\u5230\u7531\u4e0a\u8ff0\u9012\u5f52\u7b97\u6cd5\u751f\u6210\u7684\u5bf9\u5076\u7f51\u683c\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c\u5728\u62d3\u6251\u7ed3\u6784\u4e0a\u7b49\u4ef7\u4e8e\u4e00\u4e2a\u7acb\u65b9\u4f53(\u5c3d\u7ba1\u7531\u4e8e\u516b\u53c9\u6811\u7684\u591a\u5206\u8fa8\u7387\u7ed3\u6784, \u67d0\u4e9b\u9876\u70b9\u53ef\u80fd\u91cd\u590d). \u5229\u7528\u5bf9\u5076\u7f51\u683c\u9876\u70b9\u5904\u7684\u6807\u91cf\u503c, \u4f5c\u8005\u8ba1\u7b97\u6240\u6709\u5305\u542b\u7b26\u53f7\u53d8\u5316\u7684\u8fb9\u4e0a\u7684\u4ea4\u70b9. \u968f\u540e, \u901a\u8fc7Marching Cubes\u63d0\u4f9b\u7684\u67e5\u627e\u8868\u751f\u6210\u8be5\u5355\u5143\u683c\u5185\u90e8\u8868\u9762\u7684\u62d3\u6251\u7ed3\u6784.<br \/>\n$\\\\$ \u7531\u4e8e\u5bf9\u5076\u7f51\u683c\u7684\u9876\u70b9\u4f4d\u4e8e\u9690\u51fd\u6570$f(x, y, z)$\u7684\u7279\u5f81\u4e0a, \u8be5\u5bf9\u5076\u7f51\u683c\u7684\u7b49\u503c\u9762\u540c\u6837\u8868\u73b0\u51fa\u5c16\u9510\u7279\u5f81, \u7c7b\u4f3c\u4e8e\u6269\u5c55Marching Cubes\u548c\u5bf9\u5076\u8f6e\u5ed3\u5316\u65b9\u6cd5\u751f\u6210\u7684\u7ed3\u679c. \u5bf9\u5076Marching Cubes\u76f8\u8f83\u4e8e\u8fd9\u4e9b\u65b9\u6cd5\u7684\u4e3b\u8981\u4f18\u52bf\u5728\u4e8e: \u751f\u6210\u7b49\u6548\u7b49\u503c\u9762\u6240\u4f9d\u8d56\u7684\u5e95\u5c42\u516b\u53c9\u6811\u66f4\u52a0\u7a00\u758f.<br \/>\n$\\\\$ \u4e3a\u4e86\u7406\u89e3\u8fd9\u4e00\u73b0\u8c61, \u8003\u8651\u4e00\u4e2a\u4e8c\u7ef4\u793a\u4f8b. \u56fe5(\u5de6) \u5c55\u793a\u4e86\u4e24\u6761\u7d27\u5bc6\u6392\u5217\u7684\u5e73\u884c\u7ebf(\u5982\u56fe5\u53f3\u4fa7\u6240\u793a) \u7684\u8ddd\u79bb\u51fd\u6570. \u8fd9\u4e24\u6761\u7ebf\u6027\u51fd\u6570\u901a\u8fc7\u53d6\u6bcf\u6761\u7ebf\u8ddd\u79bb\u51fd\u6570\u7684\u6700\u5927\u503c\u5f97\u5230(\u6700\u5927\u503c\u8fd0\u7b97\u7528\u4e8e\u6784\u9020\u4ea4\u96c6\u533a\u57df: $max(d_1, d_2) \\le 0$\u65f6, \u8868\u793a\u540c\u65f6\u5728\u4e24\u6761\u7ebf\u7684\u5185\u4fa7\u533a\u57df, \u5373\u4e24\u6761\u7ebf\u7684\u4ea4\u96c6). \u56fe\u5de6\u4fa7\u663e\u793a\u4e86$f(x, y)$\u7684\u5bf9\u5076\u7f51\u683c\u7684\u7ed3\u6784, \u7531\u56db\u4e2a\u56db\u8fb9\u5f62\u5355\u5143\u683c\u7ec4\u6210, \u5176\u4e2d\u9876\u70b9\u7684\u9ad8\u5ea6\u8bb0\u4e3a\u6807\u91cf\u503c$w$. \u5bf9\u6bcf\u4e2a\u56db\u8fb9\u5f62\u8fdb\u884c\u7b49\u503c\u7ebf\u63d0\u53d6, \u4f1a\u751f\u6210\u56db\u6761\u7ebf\u6bb5, \u4e14\u7ed3\u679c\u4e0e\u4e24\u6761\u5e73\u884c\u7ebf\u7684\u95f4\u8ddd\u65e0\u5173.<br \/>\n$\\\\$ \u5728EMC(\u6269\u5c55Marching Cubes) \u548c\u5bf9\u5076\u8f6e\u5ed3\u5316\u7b49\u65b9\u6cd5\u4e2d, \u7528\u4e8e\u89e3\u6790\u8be5\u51fd\u6570\u4e24\u6761\u72ec\u7acb\u7b49\u503c\u7ebf\u7684\u539f\u59cb\u7f51\u683c\u5bc6\u5ea6\u4f9d\u8d56\u4e8e\u4e24\u6761\u7b49\u503c\u7ebf\u7684\u5206\u79bb\u8ddd\u79bb(\u4f9d\u8d56\u539f\u59cb\u7f51\u683c\u7684\u5206\u8fa8\u7387\u5206\u79bb\u4e24\u6761\u7d27\u5bc6\u5e73\u884c\u7ebf. \u82e5\u4e24\u6761\u7ebf\u95f4\u8ddd\u5c0f\u4e8e\u7f51\u683c\u5206\u8fa8\u7387, \u539f\u59cb\u7f51\u683c\u4f1a\u5c06\u5b83\u4eec\u5408\u5e76\u4e3a\u4e00\u6761\u8f6e\u5ed3\u7ebf; \u5b83\u4eec\u9700\u8981\u9ad8\u5206\u8fa8\u7387\u7f51\u683c\u5373\u66f4\u591a\u7ec6\u5206\u5c42\u7ea7\u624d\u80fd\u89e3\u6790\u7d27\u5bc6\u7279\u5f81, \u5bfc\u81f4\u8ba1\u7b97\u5f00\u9500\u5267\u589e). \u5728\u56fe5\u793a\u4f8b\u4e2d, \u4e24\u6761\u7ebf\u7684\u4ea4\u96c6\u533a\u57df\u5f62\u6210\u4e00\u4e2a\u5c16\u9510\u7684V\u5f62\u51f9\u89d2, \u4f20\u7edf\u65b9\u6cd5\u96be\u4ee5\u7528\u4f4e\u5206\u8fa8\u7387\u7f51\u683c\u89e3\u6790\u8be5\u51f9\u89d2.<br \/>\n$\\\\$ \u800c\u5728\u5bf9\u5076Marching Cubes\u4e2d, \u7528\u4e8e\u7b49\u503c\u7ebf\u63d0\u53d6\u7684\u7f51\u683c\u4f1a\u81ea\u9002\u5e94\u8ddd\u79bb\u51fd\u6570\u7684\u7279\u5f81(\u5982\u68af\u5ea6\u65b9\u5411), \u800c\u975e\u7b49\u503c\u7ebf\u672c\u8eab\u7684\u7279\u5f81. \u8fd9\u4e00\u533a\u522b\u901a\u5e38\u907f\u514d\u4e86\u56e0\u7b49\u503c\u7ebf\u7d27\u5bc6\u6392\u5217\u800c\u9700\u8981\u9ad8\u5206\u8fa8\u7387\u7ec6\u5206\u7684\u9700\u6c42. \u901a\u8fc7\u7279\u5f81\u9694\u79bb\u9636\u6bb5, \u9876\u70b9\u4f4d\u7f6e\u7531QEF\u4f18\u5316\u786e\u5b9a, \u5f3a\u5236\u9876\u70b9\u5bf9\u9f50\u8ddd\u79bb\u51fd\u6570\u7684\u68af\u5ea6\u65b9\u5411, \u80fd\u7cbe\u786e\u5bf9\u9f50\u4e24\u6761\u7ebf\u7684\u4ea4\u70b9(\u5982\u56fe5\u4e2d\u56db\u8fb9\u5f62\u9876\u70b9\u4f4d\u4e8e\u4e24\u6761\u7ebf\u7684\u4ea4\u7ebf\u4e0a), \u65e0\u9700\u9ad8\u5206\u8fa8\u7387\u7ec6\u5206. \u5728\u56fe5\u793a\u4f8b\u4e2d, \u5bf9\u5076Marching Cubes\u901a\u8fc7\u5bf9\u5076\u7f51\u683c\u7684\u81ea\u9002\u5e94\u6027\u76f4\u63a5\u6355\u6349\u51f9\u89d2\u7279\u5f81. \u5373\u4f7f\u4e24\u6761\u7ebf\u95f4\u8ddd\u6781\u5c0f, \u5bf9\u5076\u7f51\u683c\u4ecd\u80fd\u751f\u6210\u72ec\u7acb\u7684\u7ebf\u6bb5(\u5982\u56fe5\u5de6\u7684\u56db\u6761\u7ebf\u6bb5), \u800c\u4f20\u7edf\u65b9\u6cd5\u9700\u7ec6\u5206\u7f51\u683c\u624d\u80fd\u8fbe\u5230\u76f8\u540c\u6548\u679c.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/dual_grid_max_two_linear_functions_plane_w_0_drawn_transparently.png\" alt=\"\" width=\"1000\" height=\"433\" class=\"aligncenter size-full wp-image-4657\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/dual_grid_max_two_linear_functions_plane_w_0_drawn_transparently.png 1000w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/dual_grid_max_two_linear_functions_plane_w_0_drawn_transparently-300x130.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/dual_grid_max_two_linear_functions_plane_w_0_drawn_transparently-768x333.png 768w\" sizes=\"(max-width: 1000px) 100vw, 1000px\" \/><center><em><strong>\u56fe5:<\/strong> \u4e24\u4e2a\u7ebf\u6027\u51fd\u6570\u6700\u5927\u503c\u7684\u5bf9\u5076\u7f51\u683c(\u5de6\u56fe), \u5176\u4e2d\u5e73\u9762\u4ee5\u900f\u660e\u65b9\u5f0f\u7ed8\u5236; \u8be5\u51fd\u6570\u7684\u96f6\u7b49\u503c\u7ebf(\u52a0\u7c97\u663e\u793a) \u4ee5\u53ca\u5bf9\u5076\u7f51\u683c\u5728\u7b49\u503c\u7ebf\u5e73\u9762\u4e0a\u7684\u6295\u5f71(\u53f3\u56fe).<\/em><\/center><\/p>\n<p>\u4f5c\u4e3a\u4e00\u4e2a\u66f4\u590d\u6742\u7684\u4e09\u7ef4\u793a\u4f8b, \u8003\u8651\u56fe1\u4e2d\u7684\u8584\u58c1\u623f\u95f4. \u8be5\u623f\u95f4\u901a\u8fc7\u4e00\u7cfb\u5217\u57fa\u4e8e\u5e73\u9762\u57fa\u5143\u7684CSG\u64cd\u4f5c\u8bbe\u8ba1\u800c\u6210. \u5de6\u4e0b\u89d2\u7684\u7f51\u683c\u662f\u4f20\u7edfMarching Cubes\u5728\u7ec6\u7f51\u683c\u4e0a\u5bf9\u8ddd\u79bb\u51fd\u6570\u7684\u7b49\u503c\u9762\u63d0\u53d6\u7ed3\u679c, \u7528\u4e8e\u8fd8\u539f\u8584\u58c1\u7ed3\u6784(\u6ce8\u610f\u5c16\u9510\u8fb9\u7f18\u7684\u5706\u5316\u73b0\u8c61). \u53f3\u4e0b\u89d2\u7684\u7f51\u683c\u7531\u5bf9\u5076\u8f6e\u5ed3\u5316\u751f\u6210, \u80fd\u7cbe\u786e\u8fd8\u539f\u623f\u95f4\u5f62\u72b6, \u4f46\u56e0\u5899\u58c1\u8fc7\u8584, \u65e0\u6cd5\u7b80\u5316\u7f51\u683c\u4e2d\u5927\u91cf\u7684\u5e73\u5766\u533a\u57df. \u4f5c\u8005\u901a\u8fc7\u8ba1\u7b97\u516b\u53c9\u6811, \u4ee5\u8fd1\u4f3c\u539f\u59cbCSG\u6811\u7ed3\u6784. \u53f3\u4e0a\u89d2\u7684\u8868\u9762\u7531\u5bf9\u5076Marching Cubes\u901a\u8fc7\u5bf9\u8be5\u516b\u53c9\u6811\u7684\u5bf9\u5076\u7f51\u683c\u8fdb\u884c\u7b49\u503c\u9762\u63d0\u53d6\u751f\u6210. \u8be5\u8868\u9762\u4e0d\u4ec5\u7cbe\u786e\u8fd8\u539f\u4e86\u623f\u95f4\u5f62\u72b6, \u4e14\u4f7f\u7528\u4e86\u663e\u8457\u66f4\u5c11\u7684\u591a\u8fb9\u5f62. \u503c\u5f97\u6ce8\u610f\u7684\u662f, \u5f53\u623f\u95f4\u5899\u58c1\u539a\u5ea6\u51cf\u5c0f\u65f6, \u4f20\u7edfMarching Cubes\u548c\u5bf9\u5076\u8f6e\u5ed3\u5316\u6240\u9700\u7684\u591a\u8fb9\u5f62\u6570\u91cf\u4f1a\u6025\u5267\u589e\u52a0, \u800c\u5bf9\u5076Marching Cubes\u751f\u6210\u7684\u591a\u8fb9\u5f62\u6570\u91cf\u57fa\u672c\u4fdd\u6301\u6052\u5b9a.<br \/>\n$\\\\$ \u4f7f\u7528Marching Cubes\u5bf9\u8fd9\u4e9b\u5bf9\u5076\u7f51\u683c\u8fdb\u884c\u7b49\u503c\u9762\u63d0\u53d6\u7684\u4e00\u4e2a\u7f3a\u70b9\u662f, \u5f53\u5bf9\u5076\u7f51\u683c\u7684\u9876\u70b9\u53ef\u80fd\u56e0QEF\u4f18\u5316\u8bef\u5dee\u6216\u516b\u53c9\u6811\u7ec6\u5206\u4e0d\u5145\u5206\u800c\u63a5\u8fd1\u5e73\u9762$w $$ = 0$\u65f6, \u751f\u6210\u7684\u4e09\u89d2\u5f62\u5bb9\u6613\u56e0\u9876\u70b9\u5206\u5e03\u4e0d\u5747\u800c\u53d8\u5f97\u6781\u6241\u5e73, \u4ece\u800c\u751f\u6210\u5927\u91cf\u7ec6\u957f\u4e09\u89d2\u5f62(Sliver Polygons). \u89e3\u51b3\u8be5\u95ee\u9898\u7684\u65b9\u6cd5\u662f\u5c3d\u53ef\u80fd\u5c06\u5bf9\u5076\u7f51\u683c\u7684\u9876\u70b9\u7cbe\u786e\u8c03\u6574\u5230\u5e73\u9762$w = $$ 0$\u4e0a. \u8fd9\u4e00\u8c03\u6574\u5bf9\u6700\u7ec8\u7b49\u503c\u9762\u7684\u6548\u679c\u662f\u6d88\u9664\u5927\u90e8\u5206\u7ec6\u957f\u4e09\u89d2\u5f62. \u5177\u4f53\u5b9e\u73b0\u4e2d, \u4f5c\u8005\u9996\u5148\u8ba1\u7b97\u53d7\u9650QEF $E(0, x, y, z)$\u7684\u6700\u5c0f\u5316\u89e3$\\hat{p}$. \u82e5\u8be5\u89e3$\\hat{p}$\u5bf9\u5e94\u7684\u6b8b\u5dee(\u5373$E(\\hat{p})$\u7684\u7edd\u5bf9\u503c) \u5c0f\u4e8e\u9608\u503c$\\epsilon$, \u5219\u4f7f\u7528\u8be5\u89e3\u4f5c\u4e3a\u5bf9\u5e94\u7acb\u65b9\u4f53\u7684\u9876\u70b9; \u5426\u5219, \u6309\u4e4b\u524d\u65b9\u6cd5\u6267\u884c\u7279\u5f81\u9694\u79bb. \u56fe6\u5c55\u793a\u4e86\u5bf9\u5076Marching Cubes\u751f\u6210\u7684\u7b49\u503c\u9762, \u5176\u4e2d\u7ec6\u957f\u4e09\u89d2\u5f62\u5df2\u88ab\u6d88\u9664.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/contoured_sphere_without_sliver_elimination_with_sliver_elimination.png\" alt=\"\" width=\"1050\" height=\"430\" class=\"aligncenter size-full wp-image-4659\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/contoured_sphere_without_sliver_elimination_with_sliver_elimination.png 1050w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/contoured_sphere_without_sliver_elimination_with_sliver_elimination-300x123.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/contoured_sphere_without_sliver_elimination_with_sliver_elimination-768x315.png 768w\" sizes=\"(max-width: 1050px) 100vw, 1050px\" \/><center><em><strong>\u56fe6:<\/strong> \u672a\u8fdb\u884c\u7ec6\u957f\u4e09\u89d2\u5f62\u6d88\u9664\u7684\u7403\u4f53\u8f6e\u5ed3(\u5de6) \u4e0e\u8fdb\u884c\u4e86\u6d88\u9664\u7684\u7403\u4f53\u8f6e\u5ed3(\u53f3).<\/em><\/center><\/p>\n<p>\u56fe2\u5c55\u793a\u4e86\u4e00\u4e2a\u901a\u8fc7CSG\u64cd\u4f5c\u5e8f\u5217\u6784\u5efa\u7684\u706b\u7bad\u6a21\u578b\u793a\u4f8b, \u4ee5\u53ca\u4f7f\u7528\u5bf9\u5076Marching Cubes, Marching Cubes\u548c\u5bf9\u5076\u8f6e\u5ed3\u5316\u751f\u6210\u7684\u8fd1\u4f3c\u6a21\u578b. \u8be5\u706b\u7bad\u662f\u4e00\u4e2a\u5178\u578b\u7684\u6d4b\u8bd5\u6848\u4f8b, \u56e0\u4e3a\u5176\u5e95\u90e8\u8584\u9ccd\u5177\u6709\u4e94\u89d2\u5f62\u65cb\u8f6c\u5bf9\u79f0. \u5f53\u51e0\u4f55\u7279\u5f81\u4e0e\u5750\u6807\u8f74\u5bf9\u9f50\u65f6, \u539f\u59cb\u7f51\u683c(\u5982Marching Cubes) \u8868\u73b0\u826f\u597d, \u4f46\u8fd9\u4e9b\u8584\u9ccd\u4e0e\u539f\u59cb\u7f51\u683c\u7684\u53c2\u6570\u7ebf\u65b9\u5411\u4e0d\u4e00\u81f4. \u56e0\u6b64, Marching Cubes\u548c\u5bf9\u5076\u8f6e\u5ed3\u5316\u7b49\u65b9\u6cd5\u96be\u4ee5\u51c6\u786e\u8fd8\u539f\u8fd9\u4e9b\u8584\u58c1\u7279\u5f81. \u76f8\u6bd4\u4e4b\u4e0b, \u5bf9\u5076Marching Cubes\u901a\u8fc7\u6784\u5efa\u4e0e\u8584\u9ccd\u5bf9\u9f50\u7684\u5bf9\u5076\u7f51\u683c, \u5373\u4f7f\u8584\u9ccd\u672a\u4e0e\u516b\u53c9\u6811\u5bf9\u9f50, \u4e5f\u80fd\u7cbe\u786e\u91cd\u5efa\u8fd9\u4e9b\u7ec6\u8282.<br \/>\n$\\\\$ \u56fe7\u5c55\u793a\u4e86\u4f7f\u7528\u9a6c\u548c\u9f99\u6a21\u578b\u8fdb\u884c\u81ea\u9002\u5e94\u8868\u9762\u63d0\u53d6\u7684\u6548\u679c. \u5728\u9a6c\u7684\u6a21\u578b\u4e2d, \u8eab\u4f53\u90e8\u5206\u7684\u591a\u8fb9\u5f62\u6bd4\u9888\u90e8\u548c\u817f\u90e8\u66f4\u7c97\u7cd9; \u5c3d\u7ba1\u817f\u90e8\u76f8\u5bf9\u4e8e\u8eab\u4f53\u5176\u4ed6\u90e8\u5206\u8f83\u7ec6, \u4f46\u4ecd\u80fd\u88ab\u51c6\u786e\u8fd8\u539f. \u9f99\u7684\u6a21\u578b\u540c\u6837\u5c55\u793a\u4e86\u81ea\u9002\u5e94\u591a\u8fb9\u5f62\u5316, \u5176\u9762\u90e8\u548c\u8db3\u90e8\u7684\u591a\u8fb9\u5f62\u6bd4\u8eab\u4f53\u5176\u4ed6\u90e8\u5206\u66f4\u7cbe\u7ec6. \u5c3d\u7ba1\u4f5c\u8005\u4ec5\u5728\u8fd9\u4e9b\u6a21\u578b\u4e0a\u76f4\u63a5\u8fd0\u884cMarching Cubes\u7b97\u6cd5, \u672a\u6dfb\u52a0\u4efb\u4f55\u591a\u5206\u8fa8\u7387\u6269\u5c55, \u4f46\u5bf9\u5076\u7f51\u683c\u7684\u7ed3\u6784\u81ea\u7136\u751f\u6210\u4e86\u8fd9\u4e9b\u591a\u5206\u8fa8\u7387\u7b49\u503c\u7ebf. \u6b64\u5916, \u7531\u4e8e\u4f5c\u8005\u4f7f\u7528Marching Cubes\u5bf9\u8fd9\u4e9b\u6a21\u578b\u8fdb\u884c\u7b49\u503c\u9762\u63d0\u53d6, \u751f\u6210\u7684\u8868\u9762\u5728\u62d3\u6251\u4e0a\u662f\u6d41\u5f62.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/adaptive_surfaces_extracted_using_dual_marching_cubes.png\" alt=\"\" width=\"900\" height=\"715\" class=\"aligncenter size-full wp-image-4663\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/adaptive_surfaces_extracted_using_dual_marching_cubes.png 900w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/adaptive_surfaces_extracted_using_dual_marching_cubes-300x238.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/11\/adaptive_surfaces_extracted_using_dual_marching_cubes-768x610.png 768w\" sizes=\"(max-width: 900px) 100vw, 900px\" \/><center><em><strong>\u56fe7:<\/strong> \u4f7f\u7528\u5bf9\u5076Marching Cubes\u63d0\u53d6\u7684\u81ea\u9002\u5e94\u8868\u9762.<\/em><\/center><\/p>\n<p><strong>4. \u5b9e\u73b0<\/strong><\/p>\n<p>\u5728\u5b9e\u73b0\u8be5\u65b9\u6cd5\u65f6, \u4f5c\u8005\u5c06\u7b97\u6cd5\u53ef\u80fd\u63a5\u53d7\u7684\u5404\u79cd\u8f93\u5165\u62bd\u8c61\u4e3a\u4e00\u4e2a\u51fd\u6570, \u8be5\u51fd\u6570\u53ef\u88ab\u6c42\u503c\u4ee5\u786e\u5b9a\u91c7\u6837\u70b9\u5904\u7684\u6807\u91cf\u503c\u548c\u68af\u5ea6. \u7531\u4e8e\u4f5c\u8005\u4f7f\u7528QEFs\u4f30\u8ba1\u5c16\u9510\u7279\u5f81, \u4f53\u7d20\u5316\u65b9\u6cd5\u4e2d\u4f1a\u9047\u5230\u4e0e\u8868\u9762\u5efa\u6a21\u65b9\u6cd5\u76f8\u540c\u7684\u95ee\u9898. \u4f8b\u5982, \u5728\u6700\u5c0f\u5316\u516c\u5f0f$\\eqref{QEF}$\u65f6, \u7279\u5f81\u9876\u70b9\u53ef\u80fd\u4f4d\u4e8e\u751f\u6210\u5b83\u7684\u7acb\u65b9\u4f53\u4e4b\u5916. \u6b64\u65f6\uff0c\u4f5c\u8005\u5982\u8bba\u6587P. Lindstrom and C. T. Silva. A memory insensitive technique for large model simplification. In Proceedings of the conference on Visualization \u201901, pages 121\u2013126. IEEE Computer Society, 2001.\u6240\u8ff0, \u5728\u7acb\u65b9\u4f53\u8fb9\u754c\u4e0a\u5bf9QEF\u8fdb\u884c\u6700\u5c0f\u5316.<br \/>\n$\\\\$ \u6b64\u5916, \u9700\u8981\u6ce8\u610f\u7684\u662f, \u5e76\u6ca1\u6709\u5f3a\u5236\u8981\u6c42\u5fc5\u987b\u5c06\u9876\u70b9\u653e\u7f6e\u5728\u8ddd\u79bb\u51fd\u6570\u7684\u7279\u5f81\u4e0a. \u867d\u7136\u8fd9\u79cd\u653e\u7f6e\u65b9\u5f0f\u80fd\u591f\u8fd8\u539f\u5c16\u9510\u7279\u5f81, \u4f46\u4e5f\u53ef\u4ee5\u4fee\u6539\u7279\u5f81\u9694\u79bb\u9636\u6bb5, \u76f4\u63a5\u5728\u6bcf\u4e2a\u7acb\u65b9\u4f53\u7684\u4e2d\u5fc3\u91c7\u6837$f(x, y, z)$. \u8fd9\u6837\u751f\u6210\u7684\u7b97\u6cd5\u4e0e\u8bba\u6587R. N. Perry and S. F. Frisken. Kizamu: A system for sculpting digital characters. In Proceedings of SIGGRAPH 2001, Computer Graphics Proceedings, Annual Conference Series, pages 47\u201356. ACM Press \/ ACM SIGGRAPH, August 2001.\u4e2d\u7684\u7b49\u503c\u9762\u63d0\u53d6\u7b97\u6cd5\u7c7b\u4f3c, \u4f46\u907f\u514d\u4e86\u81ea\u9002\u5e94\u914d\u7f6e\u4e2d\u4ea7\u751f\u7684\u7279\u6b8a\u60c5\u51b5, \u4e14\u59cb\u7ec8\u751f\u6210\u62d3\u6251\u6d41\u5f62\u8868\u9762.<br \/>\n$\\\\$ \u5c31\u6027\u80fd\u800c\u8a00, \u5bf9\u5076\u7f51\u683c\u7684\u7b49\u503c\u9762\u63d0\u53d6\u8017\u65f6\u4e0d\u9ad8\u4e8e\u539f\u59cb\u7f51\u683c. \u7531\u4e8e\u5bf9\u5076\u7f51\u683c\u66f4\u8d34\u5408\u9690\u51fd\u6570$f(x, y, z)$\u7684\u51e0\u4f55\u7279\u5f81, \u56e0\u6b64\u53ef\u4f7f\u7528\u66f4\u7a00\u758f\u7684\u7f51\u683c\u63d0\u53d6\u8868\u9762, \u4ece\u800c\u5b9e\u9645\u51cf\u5c11\u7b49\u503c\u9762\u63d0\u53d6\u65f6\u95f4. \u7136\u800c, \u5bf9\u5076\u7f51\u683c\u7684\u751f\u6210\u4ecd\u5b58\u5728\u4e00\u5b9a\u8017\u65f6. \u57283GHz\u9891\u7387\u7684Pentium\u5904\u7406\u5668\u4e0a, \u57fa\u4e8e\u4e0a\u8ff0\u516b\u53c9\u6811\u6784\u5efa\u7b97\u6cd5, \u7279\u5f81\u9694\u79bb\u9636\u6bb5\u5173\u4e8e\u56fe1\u6240\u793a\u7684\u8f93\u5165\u8017\u65f60.1\u79d2, \u5173\u4e8e\u56fe2\u6240\u793a\u7684\u8f93\u5165\u4e2d\u8017\u65f62.5\u79d2, \u800c\u5173\u4e8e\u56fe7\u6240\u793a\u7684\u8f93\u5165\u5219\u8017\u65f6\u6570\u5206\u949f.<br \/>\n$\\\\$ \u89e3\u51b3\u8fd9\u4e00\u95ee\u9898\u7684\u4e00\u79cd\u65b9\u6cd5\u662f\u8ba4\u8bc6\u5230\u5bf9\u5076\u7f51\u683c\u7684\u5927\u90e8\u5206\u5355\u5143\u5e76\u4e0d\u76f4\u63a5\u53c2\u4e0e\u751f\u6210\u63d0\u53d6\u7684\u8868\u9762. \u53ea\u6709\u5305\u542b\u7b26\u53f7\u53d8\u5316\u7684\u5bf9\u5076\u7f51\u683c\u5355\u5143\u624d\u4f1a\u5305\u542b\u8868\u9762\u7247\u6bb5. \u56e0\u6b64, \u53ef\u4ee5\u5f00\u53d1\u4e00\u79cd\u7b97\u6cd5, \u5c06\u7279\u5f81\u9694\u79bb\u548c\u62d3\u6251\u751f\u6210\u9636\u6bb5\u9650\u5236\u5728\u5305\u542b\u8868\u9762\u7247\u6bb5\u7684\u5355\u5143\u5185. \u4f5c\u8005\u9884\u8ba1\u8be5\u4f18\u5316\u65b9\u6cd5\u7684\u901f\u5ea6\u5c06\u4e0e\u5176\u5b83\u6d41\u884c\u65b9\u6cd5\u76f8\u5f53.<\/p>\n<p><strong>5. \u672a\u6765\u5de5\u4f5c<\/strong><\/p>\n<p>\u4f5c\u8005\u8ba4\u4e3a, \u5bf9\u5076\u7f51\u683c\u751f\u6210\u7684\u6982\u5ff5\u4e0d\u4ec5\u9002\u7528\u4e8e\u7b49\u503c\u9762\u63d0\u53d6\u9886\u57df. \u5b9e\u9645\u4e0a, \u8be5\u7f51\u683c\u751f\u6210\u8fc7\u7a0b\u672c\u8d28\u4e0a\u662f\u5728\u7acb\u65b9\u4f53\u57df\u4e0a\u5bf9\u7ed9\u5b9a\u51fd\u6570$f(x, y, z)$\u8fdb\u884c\u5206\u6bb5\u7ebf\u6027\u903c\u8fd1. \u57fa\u4e8e\u8fd9\u4e00\u7279\u6027, \u5bf9\u5076\u7f51\u683c\u751f\u6210\u5e94\u6210\u4e3a\u4e00\u79cd\u4f18\u79c0\u7684\u81ea\u9002\u5e94\u62fc\u63a5\u65b9\u6cd5, \u7528\u4e8e\u5904\u7406\u591a\u4e2a\u76f8\u4ea4\u7684\u53c2\u6570\u66f2\u9762\u2014\u2014 \u56e0\u5176\u80fd\u5c06\u5bf9\u5076\u7f51\u683c\u7684\u8fb9\u81ea\u9002\u5e94\u4e8e\u4ea4\u7ebf. \u5728\u673a\u5668\u4eba\u9886\u57df, \u8ddd\u79bb\u51fd\u6570\u5728\u8def\u5f84\u89c4\u5212\u7b49\u5e94\u7528\u4e2d\u4e5f\u8d77\u7740\u5173\u952e\u4f5c\u7528. \u56e0\u6b64, \u8be5\u5bf9\u5076\u7f51\u683c\u7684\u53e6\u4e00\u6f5c\u5728\u5e94\u7528\u662f: \u8ba1\u7b97\u8ddd\u79bb\u51fd\u6570\u7684\u7cbe\u786e\u81ea\u9002\u5e94\u903c\u8fd1, \u5e76\u5728\u5bf9\u5076\u7f51\u683c\u4e0a\u6267\u884c\u8def\u5f84\u89c4\u5212.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4e00\u5468\u524d\u7ec8\u4e8e\u7ed3\u675f\u4e86\u957f\u8fbe3\u4e2a\u6708\u76842025\u4e0b\u534a\u5e74\u7cfb\u7edf\u67b6\u6784\u8bbe\u8ba1\u5e08\u8f6f\u8003\u5907\u8003\u5386\u7a0b, \u81ea\u5df1\u5bf9\u4e8eUE\u51e0\u4f55\u90e8\u5206\u7684\u5b66\u4e60\u8282\u594f\u4e5f\u65ad\u4e86\u597d &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2025\/11\/21\/dual_marching_cubes_primal_contouring_dual_grids_paper_interpretation\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u300aDual Marching Cubes: Primal Contouring of Dual Grids\u300b\u8bba\u6587\u89e3\u8bfb<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[24,10,5],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/4624"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=4624"}],"version-history":[{"count":75,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/4624\/revisions"}],"predecessor-version":[{"id":5079,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/4624\/revisions\/5079"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=4624"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=4624"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=4624"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}