{"id":4931,"date":"2026-01-30T15:34:09","date_gmt":"2026-01-30T07:34:09","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=4931"},"modified":"2026-01-30T17:57:14","modified_gmt":"2026-01-30T09:57:14","slug":"interpretation_paper_dual_contouring_hermite_data","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2026\/01\/30\/interpretation_paper_dual_contouring_hermite_data\/","title":{"rendered":"\u300aDual Contouring of Hermite Data\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>\u76ee\u524d\u5df2\u7ecf\u5728UE\u63d2\u4ef6\u4e2d\u5b9e\u73b0\u4e86GPU\u7248\u672c\u7684Marching Cube\u7b97\u6cd5, \u63a5\u4e0b\u6765\u7ee7\u7eed\u653b\u514bDual Contouring\u7b97\u6cd5, \u4ee5\u672c\u6587\u89e3\u8bfb\u53c8\u4e00\u7bc7\u5728\u7f51\u683c\u751f\u6210\u9886\u57df\u5341\u5206\u7ecf\u5178\u7684\u8bba\u6587\u300aDual Contouring of Hermite Data\u300b. \u76f8\u6bd4\u8f83Dual Contouring\u7b97\u6cd5, Marching Cube\u7b97\u6cd5\u7684\u4e00\u4e2a\u6bd4\u8f83\u5927\u7684\u7f3a\u70b9\u662f\u96be\u4ee5\u8868\u8fbe\u5c16\u9510\u7684\u7279\u5f81.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. Ju T, Losasso F, Schaefer S, et al. Dual contouring of hermite data[C]\/\/Proceedings of the 29th annual conference on Computer graphics and interactive techniques. 2002: 339-346.<br \/>\n2. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/665597711\">Dual Contour\u901a\u4fd7\u89e3\u91ca<\/a><br \/>\n3. <a href=\"https:\/\/www.boristhebrave.com\/2018\/04\/15\/dual-contouring-tutorial\/\">Dual Contouring Tutorial<\/a><br \/>\n4. <a href=\"https:\/\/swiftcoder.wordpress.com\/planets\/isosurface-extraction\/\">Isosurface Extraction<\/a><br \/>\n5. <a href=\"https:\/\/ccjou.wordpress.com\/2010\/02\/18\/givens-%E6%97%8B%E8%BD%89%E6%96%BC-qr-%E5%88%86%E8%A7%A3%E7%9A%84%E6%87%89%E7%94%A8\/\">Givens \u65cb\u8f49\u65bc QR \u5206\u89e3\u7684\u61c9\u7528<\/a><br \/>\n6. <a href=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/slides_ch03_ls01.pdf\">slides_ch03_ls01<\/a><br \/>\n7. <a href=\"https:\/\/zh.wikipedia.org\/wiki\/%E6%95%B0%E5%80%BC%E7%A8%B3%E5%AE%9A%E6%80%A7#%E5%89%8D%E5%90%91%E3%80%81%E5%90%8E%E5%90%91%E4%B8%8E%E6%B7%B7%E5%90%88%E7%A8%B3%E5%AE%9A%E6%80%A7\">\u6570\u503c\u7a33\u5b9a\u6027<\/a><br \/>\n8. <a href=\"https:\/\/www.zhihu.com\/tardis\/zm\/art\/136627868?source_id=1003\">\u5982\u4f55\u7406\u89e3\u65bd\u5bc6\u7279\uff08Schmidt\uff09\u6b63\u4ea4\u5316<\/a><br \/>\n9. <a href=\"https:\/\/arxiv.org\/pdf\/math\/0609346\">SPACES OF POLYTOPES AND COBORDISM OF QUASITORIC MANIFOLDS<\/a><br \/>\n10. <a href=\"https:\/\/sympa.inria.fr\/sympa\/arc\/cgal-discuss\/2014-06\/msg00052.html\">cgal-discuss &#8211; Re: [cgal-discuss] Difference between Manifold and Quasi-manifold<\/a><\/p>\n<p><strong>1. \u5747\u5300\u7f51\u683c\u4e0a\u7684Dual Contouring<\/strong><\/p>\n<p>\u5c3d\u7ba1\u4f5c\u8005\u7684\u6700\u7ec8\u76ee\u6807\u662f\u5f00\u53d1\u4e00\u79cd\u9002\u7528\u4e8e\u516b\u53c9\u6811\u7684\u7b80\u5355\u7b49\u503c\u9762\u65b9\u6cd5, \u4f46\u4f5c\u8005\u9996\u5148\u63a2\u8ba8\u4e86\u9002\u7528\u4e8e\u5e26\u7b26\u53f7\u7684\u5747\u5300\u7f51\u683c(\u7c7b\u4f3c\u4e8e\u7b26\u53f7\u8ddd\u79bb\u573a) \u7684\u591a\u79cd\u7b49\u503c\u9762\u6280\u672f. \u56fe1\u5de6\u4e0a\u89d2\u5c55\u793a\u4e86\u5e26\u7b26\u53f7\u7684\u5747\u5300\u7f51\u683c\u7684\u5178\u578b\u793a\u4f8b. \u7f51\u683c\u4e2d\u53d1\u751f\u7b26\u53f7\u53d8\u5316\u7684\u8fb9\u4f1a\u88abHermite\u6570\u636e(\u4ea6\u79f0\u4e3a\u6cd5\u5411\u91cf\u6216\u68af\u5ea6) \u6807\u8bb0, \u8fd9\u4e9b\u6570\u636e\u5305\u542b\u7cbe\u786e\u7684\u4ea4\u70b9\u5750\u6807\u53ca\u5bf9\u5e94\u7684\u6cd5\u5411\u91cf. \u8fd9\u4e9bHermite\u6570\u636e\u53ef\u901a\u8fc7\u8f6e\u5ed3\u7684\u9690\u5f0f\u5b9a\u4e49\u76f4\u63a5\u8ba1\u7b97, \u6216\u901a\u8fc7\u626b\u63cf\u8f6c\u6362\u5c01\u95ed\u591a\u8fb9\u5f62\u7f51\u683c\u83b7\u5f97.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/typical_example_of_signed_uniform_grid.png\" alt=\"\" width=\"618\" height=\"684\" class=\"aligncenter size-full wp-image-4940\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/typical_example_of_signed_uniform_grid.png 618w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/typical_example_of_signed_uniform_grid-271x300.png 271w\" sizes=\"(max-width: 618px) 100vw, 618px\" \/><center><em>\u56fe1: \u5de6\u4e0a\u89d2\u4e3a\u8fb9\u88abHermite\u6570\u636e\u6807\u8bb0\u7684\u5e26\u7b26\u53f7\u7684\u7f51\u683c, \u53f3\u4e0a\u89d2\u4e3a\u5176Marching Cubes\u7b49\u503c\u9762, \u5de6\u4e0b\u89d2\u4e3a\u5176\u6269\u5c55\u7684Marching Cubes\u7b49\u503c\u9762, \u53f3\u4e0b\u89d2\u4e3a\u5176\u5bf9\u5076\u7b49\u503c\u9762.<\/em><\/center><\/p>\n<p><strong>1.1 \u65e9\u671f\u7b49\u503c\u9762\u65b9\u6cd5<\/strong><\/p>\n<p>\u57fa\u4e8e\u7acb\u65b9\u4f53\u7684\u65b9\u6cd5(\u5982Marching Cubes(MC) \u7b97\u6cd5\u53ca\u5176\u53d8\u4f53) \u4f1a\u4e3a\u7f51\u683c\u4e2d\u6bcf\u4e2a\u4e0e\u7b49\u503c\u9762\u76f8\u4ea4\u7684\u7acb\u65b9\u4f53\u751f\u6210\u4e00\u4e2a\u6216\u591a\u4e2a\u591a\u8fb9\u5f62. \u901a\u5e38, \u8fd9\u4e9b\u65b9\u6cd5\u4f1a\u4e3a\u7b49\u503c\u9762\u4e0e\u67d0\u4e2a\u7279\u5b9a\u7acb\u65b9\u4f53\u76f8\u4ea4\u7684\u90e8\u5206\u751f\u6210\u4e00\u4e2a\u5bf9\u5e94\u591a\u8fb9\u5f62, \u5176\u9876\u70b9\u4f4d\u4e8e\u7b49\u503c\u9762\u4e0e\u7acb\u65b9\u4f53\u8fb9\u7684\u4ea4\u70b9\u5904. \u56fe1\u53f3\u4e0a\u89d2\u5c55\u793a\u4e86\u7531\u5de6\u4fa7\u5e26\u7b26\u53f7\u7684\u7f51\u683c\u751f\u6210\u7684MC\u7b49\u503c\u9762\u7684\u4e8c\u7ef4\u793a\u4f8b. \u56fe2\u5de6\u4fa7\u663e\u793a\u4e86\u4e00\u4e2a\u4e09\u7ef4\u793a\u4f8b: \u5c06\u51fd\u6570$f(x, y, $$ z) = 1 &#8211; x^2 &#8211; y^2 &#8211; z^2$\u7684\u96f6\u7b49\u503c\u9762\u751f\u6210\u4e3a\u4e00\u4e2a\u7403\u4f53. \u8be5\u7b49\u503c\u9762\u7531\u4e00\u7cfb\u5217\u591a\u8fb9\u5f62\u7ec4\u6210, \u8fd9\u4e9b\u591a\u8fb9\u5f62\u8fd1\u4f3c\u8868\u793a\u7b49\u503c\u9762\u5728\u7f51\u683c\u4e2d\u6bcf\u4e2a\u7acb\u65b9\u4f53\u5185\u7684\u5c40\u90e8\u5f62\u6001.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/3d_example_of_sphere_generated_as_zero_contour_of_function-.png\" alt=\"\" width=\"529\" height=\"199\" class=\"aligncenter size-full wp-image-4943\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/3d_example_of_sphere_generated_as_zero_contour_of_function-.png 529w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/3d_example_of_sphere_generated_as_zero_contour_of_function--300x113.png 300w\" sizes=\"(max-width: 529px) 100vw, 529px\" \/><center><em>\u56fe2: \u4f7f\u7528Marching Cubes\u65b9\u6cd5(\u5de6\u56fe) \u548cSurfaceNets\u65b9\u6cd5(\u53f3\u56fe) \u751f\u6210\u7684\u7403\u4f53.<\/em><\/center><\/p>\n<p>\u5bf9\u5076\u65b9\u6cd5(\u5982Gibson 1998\u5e74\u63d0\u51fa\u7684SurfaceNets\u7b97\u6cd5) \u4f1a\u4e3a\u6bcf\u4e2a\u4e0e\u7b49\u503c\u9762\u76f8\u4ea4\u7684\u7acb\u65b9\u4f53\u751f\u6210\u4e00\u4e2a\u4f4d\u4e8e\u7b49\u503c\u9762\u6216\u5176\u9644\u8fd1\u7684\u9876\u70b9. \u5bf9\u4e8e\u7f51\u683c\u4e2d\u53d1\u751f\u7b26\u53f7\u53d8\u5316\u7684\u6bcf\u6761\u8fb9, \u4e0e\u5305\u542b\u8be5\u8fb9\u7684\u56db\u4e2a\u7acb\u65b9\u4f53\u76f8\u5173\u8054\u7684\u9876\u70b9\u4f1a\u88ab\u8fde\u63a5\u8d77\u6765, \u5f62\u6210\u4e00\u4e2a\u56db\u8fb9\u5f62. \u6700\u7ec8\u751f\u6210\u4e00\u4e2a\u8fde\u7eed\u7684\u591a\u8fb9\u5f62\u8868\u9762\u4ee5\u903c\u8fd1\u7b49\u503c\u9762. \u56fe2\u53f3\u4fa7\u5c55\u793a\u4e86\u4f7f\u7528SurfaceNets\u65b9\u6cd5\u751f\u6210\u7684\u540c\u4e00\u7403\u4f53\u793a\u4f8b. \u9700\u8981\u6ce8\u610f\u7684\u662f, SurfaceNets\u65b9\u6cd5\u751f\u6210\u7684\u591a\u8fb9\u5f62\u7f51\u683c\u5728\u6807\u51c6\u62d3\u6251\u610f\u4e49\u4e0a\u4e0eMC\u65b9\u6cd5\u751f\u6210\u7684\u7f51\u683c\u5bf9\u5076: SurfaceNets\u7f51\u683c\u7684\u9876\u70b9\u5bf9\u5e94MC\u7f51\u683c\u7684\u9762, \u53cd\u4e4b\u4ea6\u7136. \u5bf9\u5076\u65b9\u6cd5\u901a\u5e38\u80fd\u751f\u6210\u5177\u6709\u66f4\u4f18\u7eb5\u6a2a\u6bd4\u7684\u591a\u8fb9\u5f62\u7f51\u683c, \u56e0\u4e3a\u7f51\u683c\u9876\u70b9\u53ef\u4ee5\u81ea\u7531\u5730\u5728\u7acb\u65b9\u4f53\u5185\u79fb\u52a8, \u800c\u4e0d\u50cf\u57fa\u4e8e\u7acb\u65b9\u4f53\u7684\u65b9\u6cd5\u90a3\u6837\u53d7\u9650\u4e8e\u7f51\u683c\u8fb9.<br \/>\n$\\\\$ \u6269\u5c55\u7684Marching Cubes(Extended Marching Cubes, EMC) \u65b9\u6cd5\u662f\u4e00\u79cd\u7ed3\u5408\u57fa\u4e8e\u7acb\u65b9\u4f53\u65b9\u6cd5\u548c\u5bf9\u5076\u65b9\u6cd5\u7684\u6df7\u5408\u7b97\u6cd5. EMC\u901a\u8fc7\u5206\u6790\u7acb\u65b9\u4f53\u8fb9\u4e0a\u7684\u4ea4\u70b9\u6cd5\u5411\u91cf\u6765\u68c0\u6d4b\u7acb\u65b9\u4f53\u5185\u662f\u5426\u5b58\u5728\u5c16\u9510\u7279\u5f81. \u82e5\u67d0\u4e2a\u7acb\u65b9\u4f53\u7684\u6cd5\u5411\u91cf\u4f4d\u4e8e\u7528\u6237\u6307\u5b9a\u7684\u9525\u5f62\u533a\u57df\u5185, \u5219\u5224\u5b9a\u8be5\u7acb\u65b9\u4f53\u65e0\u5c16\u9510\u7279\u5f81. \u6b64\u65f6, EMC\u4f7f\u7528\u6807\u51c6Marching Cubes(MC) \u65b9\u6cd5\u751f\u6210\u4e00\u4e2a\u6216\u591a\u4e2a\u591a\u8fb9\u5f62. \u5bf9\u4e8e\u5305\u542b\u5c16\u9510\u7279\u5f81\u7684\u7acb\u65b9\u4f53, \u8be5\u65b9\u6cd5\u4f1a\u751f\u6210\u4e00\u4e2a\u9876\u70b9, \u5176\u4f4d\u7f6e\u4f4d\u4e8e\u4e00\u4e2a\u4e8c\u6b21\u51fd\u6570\u7684\u6781\u5c0f\u5316\u70b9\u5904, \u8be5\u4e8c\u6b21\u51fd\u6570\u5b9a\u4e49\u5982\u4e0b\u6240\u793a:$\\begin{align}E[x] = \\sum_{i} (n_i \\cdot (x &#8211; p_i))^2, \\label{QEF} \\end{align}$\u5176\u4e2d, $(p_i, n_i)$\u5bf9\u5e94\u7b49\u503c\u9762\u4e0e\u7acb\u65b9\u4f53\u7684\u4ea4\u70b9(\u53ca\u5176\u5355\u4f4d\u6cd5\u5411\u91cf). \u4e00\u65e6\u8be5\u9876\u70b9\u5b9a\u4f4d\u5b8c\u6210, \u8be5\u65b9\u6cd5\u4f1a\u901a\u8fc7\u751f\u6210\u4e09\u89d2\u6247(\u6307\u4ee5\u5355\u4e00\u9876\u70b9\u4e3a\u4e2d\u5fc3\u8f90\u5c04\u8fde\u63a5\u591a\u8fb9\u5f62\u9876\u70b9\u7684\u4e09\u89d2\u5316\u65b9\u5f0f) \u8fde\u63a5\u5230\u7acb\u65b9\u4f53\u8fb9\u754c\u4e0a\u7684\u8fb9. \u6700\u7ec8, \u82e5\u4e24\u4e2a\u76f8\u90bb\u7acb\u65b9\u4f53\u5747\u5305\u542b\u5c16\u9510\u7279\u5f81\u9876\u70b9, \u5219\u4e09\u89d2\u6247\u5728\u5b83\u4eec\u7684\u516c\u5171\u9762\u4e0a\u751f\u6210\u7684\u4e09\u89d2\u5f62\u5bf9\u4f1a\u7ffb\u8f6c\u5176\u516c\u5171\u8fb9, \u5f62\u6210\u5c16\u9510\u7279\u5f81\u8fb9. \u56fe1\u5de6\u4e0b\u89d2\u5c55\u793a\u4e86\u7531EMC\u751f\u6210\u7684\u7b49\u503c\u9762\u4e8c\u7ef4\u793a\u4f8b.<\/p>\n<p><strong>1.2 \u57fa\u4e8eHermite\u6570\u636e\u7684Dual Contouring\u7b97\u6cd5<\/strong><\/p>\n<p>EMC\u65b9\u6cd5\u7684\u4e3b\u8981\u4f18\u52bf\u5728\u4e8e\u5176\u5229\u7528Hermite\u6570\u636e\u548c\u4e8c\u6b21\u8bef\u5dee\u51fd\u6570(QEF) \u6765\u5b9a\u4f4d\u5305\u542b\u51e0\u4f55\u7279\u5f81\u7684\u7acb\u65b9\u4f53\u9876\u70b9. \u8fd9\u79cdHermite\u65b9\u6cd5\u80fd\u591f\u751f\u6210\u540c\u65f6\u5305\u542b\u5c16\u9510\u9876\u70b9\u548c\u5c16\u9510\u8fb9\u7684\u7b49\u503c\u9762. \u7136\u800c, \u8fd9\u4e00\u65b9\u6cd5\u7684\u7f3a\u70b9\u662f\u9700\u8981\u663e\u5f0f\u68c0\u6d4b\u6b64\u7c7b\u7279\u5f81, \u5e76\u5728\u8fd9\u4e9b\u60c5\u51b5\u4e0b\u6267\u884c\u7279\u5b9a\u7684\u540e\u5904\u7406\u6b65\u9aa4. \u4f5c\u4e3aEMC\u65b9\u6cd5\u7684\u66ff\u4ee3\u65b9\u6848, \u4f5c\u8005\u63d0\u51fa\u4ee5\u4e0b\u9488\u5bf9Hermite\u6570\u636e\u7684\u5bf9\u5076\u7b49\u503c\u9762\u65b9\u6cd5:<br \/>\n$\\\\$ 1) \u5bf9\u6bcf\u4e2a\u53d1\u751f\u7b26\u53f7\u53d8\u5316\u7684\u7acb\u65b9\u4f53, \u751f\u6210\u4e00\u4e2a\u4f4d\u4e8e\u65b9\u7a0b$\\eqref{QEF}$\u4e8c\u6b21\u51fd\u6570\u6781\u5c0f\u5316\u70b9\u5904\u7684\u9876\u70b9;<br \/>\n$\\\\$ 2) \u5bf9\u6bcf\u6761\u53d1\u751f\u7b26\u53f7\u53d8\u5316\u7684\u8fb9, \u751f\u6210\u4e00\u4e2a\u56db\u8fb9\u5f62, \u8fde\u63a5\u5305\u542b\u8be5\u8fb9\u7684\u56db\u4e2a\u7acb\u65b9\u4f53\u7684\u6781\u5c0f\u5316\u9876\u70b9.<br \/>\n$\\\\$ \u8be5\u65b9\u6cd5\u662fEMC\u65b9\u6cd5\u4e0eSurfaceNets\u65b9\u6cd5\u7684\u4e00\u79cd\u6709\u8da3\u6df7\u5408\u7b56\u7565. \u5b83\u91c7\u7528EMC\u65b9\u6cd5\u7684\u7279\u5f81\u9876\u70b9\u89c4\u5219\u6765\u5b9a\u4f4d\u7b49\u503c\u9762\u7684\u6240\u6709\u9876\u70b9, \u540c\u65f6\u5229\u7528SurfaceNets\u65b9\u6cd5\u786e\u5b9a\u8fd9\u4e9b\u9876\u70b9\u7684\u8fde\u901a\u6027(\u9700\u6ce8\u610f, SurfaceNets\u65b9\u6cd5\u4f7f\u7528\u4e86\u5b8c\u5168\u4e0d\u540c\u7684\u89c4\u5219\u6765\u5b9a\u4f4d\u7b49\u503c\u9762\u4e0a\u7684\u9876\u70b9). \u901a\u8fc7\u4f7f\u7528QEF\u5b9a\u4f4d\u7b49\u503c\u9762\u7684\u6240\u6709\u9876\u70b9, \u8be5\u65b9\u6cd5\u907f\u514d\u4e86\u663e\u5f0f\u68c0\u6d4b\u7279\u5f81\u7684\u9700\u6c42. \u7b49\u503c\u9762\u4e0a\u7684\u9876\u70b9\u4ec5\u9700\u6839\u636eHermite\u6570\u636e\u5bf9\u5e94\u7684\u6cd5\u5411\u91cf\u4e00\u81f4\u6027\u8fdb\u884c\u5b9a\u4f4d. \u56fe1\u53f3\u4e0b\u89d2\u5c55\u793a\u4e86\u7531\u5de6\u4e0a\u89d2Hermite\u6570\u636e\u751f\u6210\u7684\u5bf9\u5076\u7b49\u503c\u9762\u7684\u4e8c\u7ef4\u793a\u4f8b.<br \/>\n$\\\\$ \u56fe3\u5c55\u793a\u4e86\u901a\u8fc7\u5728$64^3$\u7f51\u683c\u4e0aDual Contouring\u7b97\u6cd5\u5904\u7406Hermite\u6570\u636e\u5efa\u6a21\u7684\u673a\u68b0\u96f6\u4ef63D\u793a\u4f8b. \u5de6\u56fe\u663e\u793a\u4e86\u96f6\u4ef6\u7684\u5e73\u6ed1\u7740\u8272\u7248\u672c, \u53f3\u56fe\u5c55\u793a\u4e86Dual Contouring\u7b97\u6cd5\u751f\u6210\u7684\u591a\u8fb9\u5f62\u7f51\u683c. \u8be5\u6a21\u578b\u7684\u4ea4\u70b9\u548c\u6cd5\u5411\u91cf\u7531\u5c01\u95ed\u7ec6\u5206\u66f2\u9762\u751f\u6210, \u6807\u5b9a\u6a21\u578b\u5185\u90e8 \/ \u5916\u90e8\u7684\u7b26\u53f7\u8ddd\u79bb\u573a\u901a\u8fc7\u6807\u51c6\u626b\u63cf\u8f6c\u6362\u7b97\u6cd5\u8ba1\u7b97(\u5982[Foley et al. 1995] \u6240\u8ff0).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/mechanical_part_generated_by_dual_contouring_Hermite.png\" alt=\"\" width=\"1344\" height=\"973\" class=\"aligncenter size-full wp-image-4955\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/mechanical_part_generated_by_dual_contouring_Hermite.png 1344w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/mechanical_part_generated_by_dual_contouring_Hermite-300x217.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/mechanical_part_generated_by_dual_contouring_Hermite-768x556.png 768w\" sizes=\"(max-width: 1344px) 100vw, 1344px\" \/><center><em>\u56fe3: \u5728$\\it{64^3}$\u7f51\u683c\u4e0a\u901a\u8fc7Dual Contouring\u7b97\u6cd5\u5904\u7406\u5e26Hermite\u6570\u636e\u7684\u673a\u68b0\u96f6\u4ef6.<\/em><\/center><\/p>\n<p><strong>1.3 QEF\u7684\u8868\u793a\u4e0e\u6700\u5c0f\u5316<\/strong><\/p>\n<p>\u5728\u6b64, \u9700\u8981\u5bf9QEF\u7684\u8868\u793a\u4e0e\u6700\u5c0f\u5316\u65b9\u6cd5\u4f5c\u51e0\u70b9\u8bf4\u660e. \u65b9\u7a0b$\\eqref{QEF}$\u4e2d\u7684\u51fd\u6570$E[ $$ x]$\u7531\u4e00\u7ec4\u4ea4\u70b9$p_i$\u548c\u6cd5\u5411\u91cf$n_i$\u6784\u5efa\u800c\u6210. \u8be5\u51fd\u6570$E[x]$\u53ef\u8868\u793a\u4e3a\u5185\u79ef\u5f62\u5f0f$( $$ Ax &#8211; b)^T (Ax &#8211; $$ b)$, \u5176\u4e2d\u77e9\u9635$A$\u7684\u884c\u5411\u91cf\u4e3a\u6cd5\u5411\u91cf$n_i$, \u5411\u91cf$b$\u7684\u5143\u7d20\u4e3a$n_i \\cdot $$ p_i$(\u5373\u6cd5\u5411\u91cf\u4e0e\u4ea4\u70b9\u7684\u5185\u79ef). \u901a\u5e38, \u4e8c\u6b21\u51fd\u6570$E[x]$\u4f1a\u88ab\u5c55\u5f00\u4e3a\u4ee5\u4e0b\u6807\u51c6\u5f62\u5f0f:$\\begin{align}E[x] = x^T A^T Ax &#8211; 2x^T A^T b + b^T b, \\end{align}$\u5176\u4e2d, \u77e9\u9635$A^T A$\u662f\u4e00\u4e2a\u5bf9\u79f0\u7684$3 \\times 3$\u77e9\u9635, $A^T b$\u662f\u4e00\u4e2a\u957f\u5ea6\u4e3a3\u7684\u5217\u5411\u91cf, $b^T b$\u662f\u4e00\u4e2a\u6807\u91cf. \u8fd9\u79cd\u5c55\u5f00\u7684\u4f18\u52bf\u5728\u4e8e, \u4ec5\u9700\u5b58\u50a8$A^T A$, $A^T b$\u548c$b^T b$(\u517110\u4e2a\u6d6e\u70b9\u6570), \u800c\u65e0\u9700\u5b58\u50a8\u5b8c\u6574\u7684\u77e9\u9635$A$\u548c\u5411\u91cf$b$. \u6b64\u5916, \u901a\u8fc7\u6c42\u89e3\u6b63\u89c4\u65b9\u7a0b$A^T A\\hat{x} = A^T b$, \u5373\u53ef\u8ba1\u7b97\u51fa$E[x]$\u7684\u6781\u5c0f\u503c\u70b9$\\hat{x}$.<br \/>\n$\\\\$ \u8fd9\u79cd\u8868\u793a\u7684\u4e00\u4e2a\u7f3a\u70b9\u5728\u4e8e\u5176\u6570\u503c\u4e0d\u7a33\u5b9a\u6027. \u4f8b\u5982, \u5f53\u4ece\u5e73\u5766\u533a\u57df\u91c7\u6837\u4ea4\u70b9\u548c\u6cd5\u5411\u91cf\u6784\u5efa$E[x]$\u65f6, \u5728\u6d6e\u70b9\u6570\u8fd0\u7b97\u4e2d\u8ba1\u7b97$E[x]$\u7684\u503c\u53ef\u80fd\u4f1a\u4ea7\u751f\u663e\u8457\u8bef\u5dee. \u5176\u6570\u503c\u4e0d\u7a33\u5b9a\u6027\u7684\u6765\u6e90\u4e3a:<br \/>\n$\\\\$ $\\cdot$ \u6cd5\u5411\u91cf\u5171\u7ebf\u6027: \u5e73\u5766\u533a\u57df\u7684\u6cd5\u5411\u91cf\u51e0\u4e4e\u5e73\u884c, \u5bfc\u81f4\u77e9\u9635$A^T A$\u63a5\u8fd1\u5947\u5f02(\u884c\u5217\u5f0f\u8d8b\u8fd1\u4e8e0), \u6c42\u89e3\u6b63\u89c4\u65b9\u7a0b\u65f6\u8bef\u5dee\u653e\u5927.<br \/>\n$\\\\$ $\\cdot$ \u5927\u6570\u503c(\u5982$b^T b \\approx 10^6$) \u4e0e\u5c0f\u6570\u503c(\u5982$E[x] \\approx 0$) \u7684\u6df7\u5408\u8fd0\u7b97\u4e2d, \u6709\u6548\u6570\u5b57\u88ab\u622a\u65ad, \u5bfc\u81f4\u8ba1\u7b97\u7ed3\u679c\u4e0d\u53ef\u9760.<br \/>\n$\\\\$ \u4ee5$256^3$\u7f51\u683c(\u5982\u56fe4\u6240\u793a) \u4e3a\u4f8b, $b^T b$\u7684\u91cf\u7ea7\u53ef\u80fd\u8fbe\u5230$10^6$\u7ea7\u522b. \u7531\u4e8e\u6d6e\u70b9\u6570\u4ec5\u80fd\u4fdd\u8bc1\u516d\u4f4d\u5341\u8fdb\u5236\u6709\u6548\u6570\u5b57\u7684\u7cbe\u5ea6, \u82e5\u5728\u539f\u59cb\u5e73\u5766\u533a\u57df(\u7406\u8bba\u4e0a$E[x]$\u5e94\u4e3a\u96f6) \u7684\u70b9\u4e0a\u8ba1\u7b97$E[x]$, \u5176\u7ed3\u679c\u8bef\u5dee\u53ef\u80fd\u8fbe\u52301($\\approx 10^6 \\times 10^{-6}$) \u7684\u91cf\u7ea7.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/temple_undergoing_destructive_modifications.png\" alt=\"\" width=\"3360\" height=\"1297\" class=\"aligncenter size-full wp-image-4960\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/temple_undergoing_destructive_modifications.png 3360w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/temple_undergoing_destructive_modifications-300x116.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/temple_undergoing_destructive_modifications-768x296.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/temple_undergoing_destructive_modifications-1536x593.png 1536w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/temple_undergoing_destructive_modifications-2048x791.png 2048w\" sizes=\"(max-width: 3360px) 100vw, 3360px\" \/><center><em>\u56fe4: \u4e00\u5ea7\u7ecf\u5386\u7834\u574f\u6027\u4fee\u6539\u7684\u5bfa\u5e99\u6a21\u578b. \u4e24\u4e2a\u6a21\u578b\u5747\u901a\u8fc7\u5bf9\u5305\u542bHermite\u6570\u636e\u7684\u5e26\u7b26\u53f7\u7684\u516b\u53c9\u6811\u8fdb\u884cDual Contouring\u7b97\u6cd5\u751f\u6210. \u53f3\u4fa7\u7684\u4fee\u6539\u6a21\u578b\u7531\u5de6\u4fa7\u6a21\u578b\u5b9e\u65f6\u8ba1\u7b97\u5f97\u51fa.<\/em><\/center><\/p>\n<p>\u89e3\u51b3\u8fd9\u4e00\u95ee\u9898\u7684\u4e00\u79cd\u53ef\u884c\u65b9\u6848\u662f, \u5728\u8868\u793a$A^T A$, $A^T b$\u548c$b^T b$\u65f6\u6539\u7528\u53cc\u7cbe\u5ea6\u6d6e\u70b9\u6570\u4ee3\u66ff\u5355\u7cbe\u5ea6\u6d6e\u70b9\u6570. \u901a\u8fc7\u4f7f\u7528\u53cc\u7cbe\u5ea6\u6570, \u5e73\u5766\u533a\u57df\u793a\u4f8b\u4e2d\u7684$E[ $$ x]$\u8bef\u5dee\u5c06\u964d\u4f4e\u81f3$10^{-6}$($\\approx 10^6 \\times 10^{-12}$) \u91cf\u7ea7. \u7136\u800c, \u91c7\u7528\u53cc\u7cbe\u5ea6\u7684\u4e3b\u8981\u7f3a\u70b9\u662f\u5b58\u50a8\u4e00\u4e2aQEF\u6240\u9700\u7684\u7a7a\u95f4\u5c06\u7ffb\u500d. \u5728\u672c\u6587\u7684\u5e94\u7528\u4e2d, \u8fd9\u4e00\u65b9\u6848\u5e76\u4e0d\u53ef\u884c, \u56e0\u4e3a\u7a0b\u5e8f\u6027\u80fd\u53d7\u5b58\u50a8\u7a7a\u95f4\u9650\u5236(\u800c\u975e\u65f6\u95f4\u9650\u5236).(\u8be6\u89c1\u6700\u540e\u4e00\u8282\u7684\u8be6\u7ec6\u8bf4\u660e.)<br \/>\n$\\\\$ \u4e00\u79cd\u66ff\u4ee3\u7684QEF\u8868\u793a\u65b9\u6cd5\u53ef\u5728\u4ec5\u4f7f\u7528\u5355\u7cbe\u5ea6\u6d6e\u70b9\u6570\u7684\u540c\u65f6\u8fbe\u5230\u53cc\u7cbe\u5ea6\u7684\u51c6\u786e\u6027, \u5176\u57fa\u4e8eQR\u5206\u89e3[Golub\u4e0eVan Loan 1989]. \u82e5\u5c06\u77e9\u9635$A$\u9644\u52a0\u5217\u5411\u91cf$b$\u540e\u5f62\u6210\u7684\u77e9\u9635\u8bb0\u4e3a$(A \\ | \\ b)$, \u8be5\u5206\u89e3\u7684\u6838\u5fc3\u601d\u60f3\u662f\u8ba1\u7b97\u4e00\u4e2a\u6b63\u4ea4\u77e9\u9635$Q$, \u4f7f\u5f97$Q$\u4e0e$(A \\ | \\ b)$\u7684\u4e58\u79ef\u4e3a\u5982\u4e0b\u5f62\u5f0f\u7684\u4e0a\u4e09\u89d2\u77e9\u9635:$\\begin{align} \\begin{pmatrix}<br \/>\nx &#038; x &#038; x &#038; x \\\\<br \/>\n0 &#038; x &#038; x &#038; x \\\\<br \/>\n0 &#038; 0 &#038; x &#038; x \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; x \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 0 \\\\<br \/>\n\\cdots &#038; \\cdots &#038; \\cdots &#038; \\cdots<br \/>\n\\end{pmatrix} = \\begin{pmatrix}<br \/>\n\\hat{A} &#038; \\hat{b} \\\\<br \/>\n0 &#038; r \\\\<br \/>\n0 &#038; 0 \\\\<br \/>\n\\cdots &#038; \\cdots<br \/>\n\\end{pmatrix}. \\label{upper_triangular_matrix_form_of_augmented_matrix_of_matrix_A} \\end{align}$\u5176\u4e2d, $\\hat{A}$\u662f\u4e00\u4e2a\u4e0a\u4e09\u89d2$3 \\times 3$\u77e9\u9635, $\\hat{b}$\u662f\u4e00\u4e2a\u957f\u5ea6\u4e3a3\u7684\u5217\u5411\u91cf, $r$\u662f\u4e00\u4e2a\u6807\u91cf. \u8be5\u77e9\u9635$Q$\u53ef\u4ee5\u8868\u793a\u4e3a\u4e00\u7cfb\u5217Givens\u65cb\u8f6c\u7684\u4e58\u79ef, \u5176\u4e2d\u6bcf\u4e2a\u65cb\u8f6c\u5c06$(A \\ $$ | \\ b)$\u7684\u4e0b\u4e09\u89d2\u90e8\u5206\u4e2d\u7684\u5355\u4e2a\u5143\u7d20\u5f52\u96f6.<br \/>\n$\\\\$ \u7531\u4e8e\u4efb\u4f55\u6b63\u4ea4\u77e9\u9635$Q$\u5747\u6ee1\u8db3\u5173\u7cfb\u5f0f$Q^T Q = I$, \u51fd\u6570$E[x]$\u53ef\u88ab\u91cd\u5199\u4e3a:$$\\begin{aligned} (Ax &#8211; b)^T (Ax &#8211; b) &#038;= (Ax &#8211; b)^T Q^T Q(Ax &#8211; b) \\\\ &#038;= (QAx &#8211; Qb)^T (QAx &#8211; Qb) \\\\ &#038;= (\\hat{A}x &#8211; \\hat{b})^T (\\hat{A}x &#8211; \\hat{b}) + r^2. \\end{aligned}$$\u4ee5\u8fd9\u79cd\u5f62\u5f0f\u8ba1\u7b97$E[x]$\u65f6, \u9996\u5148\u8ba1\u7b97\u5411\u91cf$\\hat{A}x &#8211; \\hat{b}$\u4e0e\u81ea\u8eab\u7684\u70b9\u79ef, \u7136\u540e\u518d\u52a0$r^2$. \u5982\u6b64\u4e00\u6765, \u95ee\u9898\u8f6c\u5316\u4e3a\u5173\u4e8e\u4e0a\u4e09\u89d2\u7cfb\u7edf$Rx = c$\u7684\u6c42\u89e3, \u800c\u4e0d\u6d89\u53ca\u75c5\u6001\u77e9\u9635$A^T A$; \u6b64\u5916, \u4e00\u822c\u6765\u8bf4, \u4e0a\u4e09\u89d2\u77e9\u9635$R$\u7684\u6761\u4ef6\u6570\u4ea6\u4f4e\u4e8e\u77e9\u9635$A$, \u964d\u4f4e\u4e86\u6570\u503c\u8bef\u5dee\u7684\u5f71\u54cd. \u56de\u5230\u4e4b\u524d\u7684\u5e73\u5766\u533a\u57df\u793a\u4f8b, \u4f5c\u8005\u6ce8\u610f\u5230$b$\u7684\u5143\u7d20\u91cf\u7ea7\u5728$10^3$\u5de6\u53f3, \u56e0\u6b64\u5f53$x$\u9009\u81ea\u5e73\u5766\u533a\u57df\u65f6, $\\hat{A}x &#8211; \\hat{b}$\u7684\u5143\u7d20\u91cf\u7ea7\u4e3a$10^{-3}$. \u8fd9\u662f\u56e0\u4e3a\u7acb\u65b9\u4f53\u7684\u5c3a\u5bf8\u9650\u5236\u4e86\u6b8b\u5dee\u7684\u7406\u8bba\u6700\u5927\u503c:<br \/>\n$\\\\$ $\\cdot$ \u5047\u8bbe\u7acb\u65b9\u4f53\u8fb9\u957f\u4e3a$a$(\u4f8b\u5982$a = 1$, \u5f52\u4e00\u5316\u5355\u4f4d), \u5219$|x &#8211; p_i| \\le s \\sqrt{3} \\approx $$ 1.732$(\u5355\u4f4d\u7acb\u65b9\u4f53\u5bf9\u89d2\u7ebf\u957f\u5ea6).<br \/>\n$\\\\$ $\\cdot$ \u7531\u4e8e$n_i$\u662f\u5355\u4f4d\u6cd5\u5411\u91cf, \u6b8b\u5dee\u7684\u7edd\u5bf9\u503c\u6ee1\u8db3:$$ |\\hat{A}x &#8211; \\hat{b}| = |Q| \\sum_i |n_i \\cdot (x &#8211; p_i)| \\le \\sum_i |x &#8211; p_i| \\le \\sum_i (s \\sqrt{3}) \\approx \\sum_i 1.732. $$\u7531\u6b64, \u8ba1\u7b97\u5f97\u5230\u7684$E[x]$\u503c\u5c06\u5904\u4e8e$10^{-6}$\u91cf\u7ea7.<br \/>\n$\\\\$ \u5982\u679c$\\hat{A}$\u662f\u975e\u5947\u5f02\u77e9\u9635, \u5219\u53ef\u4ee5\u901a\u8fc7\u56de\u4ee3\u6cd5\u6c42\u89e3$\\hat{A}x = \\hat{b}$\u6765\u8ba1\u7b97\u6781\u5c0f\u503c\u70b9$\\hat{x}$. \u7136\u800c, \u5728Dual Contouring\u7b97\u6cd5\u4e2d, $\\hat{A}$\u901a\u5e38\u7531\u8fd1\u4f3c\u5171\u9762\u7684\u566a\u58f0\u6cd5\u5411\u91cf\u8ba1\u7b97\u5f97\u5230. \u6b64\u65f6, \u77e9\u9635$\\hat{A}$\u63a5\u8fd1\u5947\u5f02. \u7ed3\u679c, \u6781\u5c0f\u503c\u70b9$\\hat{x}$\u53ef\u80fd\u4f4d\u4e8e\u5b9a\u4e49\u7acb\u65b9\u4f53\u4e4b\u5916. \u4e3a\u89e3\u51b3\u8fd9\u4e00\u95ee\u9898, \u4f5c\u8005\u5bf9$\\hat{A}$\u8fdb\u884cSVD, \u5e76\u901a\u8fc7\u622a\u65ad\u5176\u8f83\u5c0f\u7684\u5947\u5f02\u503c\u5f62\u6210\u4f2a\u9006\u77e9\u9635(\u65b9\u6cd5\u53c2\u8003Kobbelt et al. 2001; Lindstrom 2000]). \u57fa\u4e8e\u5b9e\u9a8c\u9a8c\u8bc1, \u4f5c\u8005\u901a\u5e38\u622a\u65ad\u6240\u6709\u5947\u5f02\u503c\u91cf\u7ea7\u5c0f\u4e8e0.1\u7684\u90e8\u5206. \u5229\u7528\u6240\u5f97\u7684\u4f2a\u9006\u77e9\u9635, \u4f5c\u8005\u8fd1\u4f3c\u6c42\u89e3$\\hat{A}x = \\hat{b}$, \u540c\u65f6\u6700\u5c0f\u5316$\\hat{x}$\u4e0e\u6240\u6709\u4ea4\u70b9$p_i$\u6784\u6210\u7684\u591a\u8fb9\u5f62\u7684\u8d28\u5fc3\u4e4b\u95f4\u7684\u8ddd\u79bb.<\/p>\n<p><strong>1.4 \u7eb9\u7406\u8f6e\u5ed3\u7ebf\u5efa\u6a21<\/strong><\/p>\n<p>\u5728\u56fe2\u4e2d, \u4e24\u79cd\u7b49\u503c\u9762\u7b97\u6cd5\u751f\u6210\u7684\u8868\u9762\u5747\u5206\u5272\u4e86\u4ece\u8d1f\u7a7a\u95f4(\u7a7a\u7a7a\u95f4) \u5230\u6b63\u7a7a\u95f4(\u5b9e\u4f53\u7a7a\u95f4) \u7684\u8fc7\u6e21. \u5728\u771f\u5b9e\u73af\u5883\u4e2d, \u5b9e\u4f53\u5e76\u975e\u7531\u5355\u4e00\u5747\u5300\u6750\u8d28\u6784\u6210. \u5b9e\u9645\u4e0a, \u5b9e\u4f53\u7531\u591a\u79cd\u6750\u8d28\u5171\u540c\u7ec4\u6210, \u6bcf\u79cd\u6750\u8d28\u4f1a\u5728\u8868\u9762\u4e0a\u5f62\u6210\u72ec\u7279\u7684\u7eb9\u7406. \u56fe5\u5c55\u793a\u4e86\u4e00\u4e2a\u7531\u4e24\u79cd\u6750\u8d28\u7ec4\u6210\u7684\u7acb\u65b9\u4f53\u793a\u4f8b: \u4e00\u79cd\u662f\u91d1\u8272\u6750\u8d28, \u901a\u8fc7\u5c06\u6c49\u5b57\u6cbf\u7acb\u65b9\u4f53\u65b9\u5411\u62c9\u4f38\u8d2f\u7a7f\u800c\u6210; \u53e6\u4e00\u79cd\u662f\u7ea2\u8272\u6750\u8d28, \u6784\u6210\u4e86\u7acb\u65b9\u4f53\u7684\u5176\u4f59\u90e8\u5206. \u9700\u6ce8\u610f, \u91d1\u8272\u6750\u8d28\u662f\u771f\u6b63\u7684\u5b9e\u4f53(\u800c\u975e\u8868\u9762\u7eb9\u7406), \u56e0\u4e3a\u6c49\u5b57\u8d2f\u7a7f\u4e86\u6574\u4e2a\u7acb\u65b9\u4f53(\u5982\u53f3\u56fe\u4e2d\u7403\u5f62\u5207\u5272\u540e\u7684\u7acb\u65b9\u4f53\u6240\u793a).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/solid_cube_undergoing_sequence_of_CSG_operations.png\" alt=\"\" width=\"1800\" height=\"1728\" class=\"aligncenter size-full wp-image-4982\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/solid_cube_undergoing_sequence_of_CSG_operations.png 1800w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/solid_cube_undergoing_sequence_of_CSG_operations-300x288.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/solid_cube_undergoing_sequence_of_CSG_operations-768x737.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/solid_cube_undergoing_sequence_of_CSG_operations-1536x1475.png 1536w\" sizes=\"(max-width: 1800px) 100vw, 1800px\" \/><center><em>\u56fe5: \u901a\u8fc7\u4e00\u7cfb\u5217CSG\u64cd\u4f5c\u5904\u7406\u7684\u5b9e\u4f53\u7acb\u65b9\u4f53.<\/em><\/center><\/p>\n<p>\u901a\u8fc7\u5c06\u5b9e\u4f53\u5206\u5272\u4e3a\u4e0d\u540c\u6750\u8d28, \u8fd9\u79cd\u5212\u5206\u53ef\u4ee5\u9690\u5f0f\u5730\u5efa\u6a21: \u7528\u6750\u8d28\u7d22\u5f15\u66ff\u4ee3\u539f\u672c\u8868\u793a\u7a7a\u7a7a\u95f4(\u8d1f\u503c) \u548c\u5b9e\u4f53\u7a7a\u95f4(\u6b63\u503c) \u7684\u7b26\u53f7-\u548c+. \u5728\u6b64\u8868\u793a\u65b9\u6cd5\u4e2d, \u6bcf\u4e2a\u7f51\u683c\u70b9\u5747\u6709\u4e00\u4e2a\u5bf9\u5e94\u7279\u5b9a\u6750\u8d28\u7684\u7d22\u5f15. (\u53c2\u89c1[Bloomenthal and Ferguson 1995; Bonnell et al. 2000] \u4e86\u89e3\u7c7b\u4f3c\u65b9\u6cd5\u7684\u793a\u4f8b.) \u56fe6\u5c55\u793a\u4e86\u4e00\u4e2a\u5305\u542b\u4e09\u79cd\u4e0d\u540c\u7d22\u5f15\u7684\u4e8c\u7ef4\u7f51\u683c; \u7070\u8272\u548c\u9ed1\u8272\u7f51\u683c\u70b9\u8868\u793a\u4e0d\u540c\u7684\u5b9e\u4f53\u6750\u8d28, \u800c\u767d\u8272\u7f51\u683c\u70b9\u8868\u793a\u7a7a\u7a7a\u95f4. \u4e0e\u4e4b\u524d\u76f8\u540c, \u5b58\u5728\u7d22\u5f15\u53d8\u5316\u7684\u8fb9\u7f18\u7531\u7cbe\u786e\u7684\u4ea4\u70b9\u548c\u6cd5\u5411\u91cf\u6807\u8bb0. \u5728\u7b49\u503c\u9762\u751f\u6210\u8fc7\u7a0b\u4e2d, \u8fd9\u4e9bHermite\u6570\u636e\u901a\u8fc7\u65b9\u7a0b$\\eqref{QEF}$\u5b9a\u4e49\u4e0e\u7f51\u683c\u4e2d\u6bcf\u4e2a\u7acb\u65b9\u4f53\u76f8\u5173\u8054\u7684QEF. \u968f\u540e, \u5bf9\u4e8e\u6bcf\u6761\u5b58\u5728\u7d22\u5f15\u53d8\u5316\u7684\u8fb9\u7f18, Dual Contouring\u7b97\u6cd5\u751f\u6210\u4e00\u4e2a\u56db\u8fb9\u5f62, \u8fde\u63a5\u5305\u542b\u8be5\u8fb9\u7f18\u7684\u56db\u4e2a\u7acb\u65b9\u4f53\u7684QEF\u6781\u5c0f\u503c\u70b9. \u56fe6\u5de6\u4fa7\u5c55\u793a\u4e86\u5206\u9694\u4e09\u79cd\u6750\u8d28\u7684\u5bf9\u5076\u7b49\u503c\u9762.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/2d_grid_with_three_distinct_indices.png\" alt=\"\" width=\"601\" height=\"194\" class=\"aligncenter size-full wp-image-4986\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/2d_grid_with_three_distinct_indices.png 601w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/2d_grid_with_three_distinct_indices-300x97.png 300w\" sizes=\"(max-width: 601px) 100vw, 601px\" \/><center><em>\u56fe6: \u4e09\u6750\u8d28\u7d22\u5f15\u7f51\u683c\u7684\u5bf9\u5076\u7b49\u503c\u9762(\u5de6\u56fe), \u5c06\u4e24\u79cd\u5b9e\u4f53(\u6df1\u8272) \u6750\u8d28\u7d22\u5f15\u5408\u5e76\u4e3a\u5355\u4e00\u7d22\u5f15\u540e\u7684\u7ed3\u679c(\u53f3\u56fe).<\/em><\/center><\/p>\n<p>\u82e5\u89c2\u5bdf\u8005\u4ec5\u80fd\u4ece\u7a7a\u7a7a\u95f4\u89c6\u89d2\u89c2\u5bdf, \u53ef\u4ee5\u9488\u5bf9\u7531\u591a\u79cd\u6750\u8d28\u7ec4\u6210\u7684\u5b9e\u4f53\u4f18\u5316\u6b64\u7b49\u503c\u9762\u65b9\u6cd5. \u7279\u522b\u662f, \u7531\u56fa \/ \u56fa\u8fb9\u7f18\u751f\u6210\u7684\u56db\u8fb9\u5f62\u9762\u7247\u5728\u7a7a\u7a7a\u95f4\u4e2d\u4e0d\u53ef\u89c1(\u5f53\u89c2\u5bdf\u8005\u5904\u4e8e\u7a7a\u7a7a\u95f4\u65f6, \u5b9e\u4f53\u5185\u90e8\u6750\u6599\u4e4b\u95f4\u7684\u8fb9\u754c(\u5982\u7ea2 \/ \u91d1\u5206\u754c) \u65e0\u6cd5\u88ab\u76f4\u63a5\u89c2\u6d4b). \u5269\u4f59\u7684\u56db\u8fb9\u5f62\u9762\u7247\u5bf9\u5e94\u56fa \/ \u7a7a\u8fb9\u7f18, \u5e76\u53ef\u901a\u8fc7\u8fb9\u7f18\u5b9e\u4f53\u7aef\u70b9\u7684\u6750\u8d28\u5c5e\u6027\u8fdb\u884c\u7eb9\u7406\u6620\u5c04. \u5728\u56fe5\u4e2d, \u5e95\u5c42\u7f51\u683c\u5305\u542b\u4e09\u79cd\u6750\u8d28: \u7a7a\u7a7a\u95f4, \u91d1\u8272\u6750\u8d28\u548c\u7ea2\u8272\u6750\u8d28. \u751f\u6210\u7684\u5bf9\u5076\u7b49\u503c\u9762\u7531\u7ea2\u8272 \/ \u7a7a\u8fb9\u7f18\u751f\u6210\u7684\u7ea2\u8272\u56db\u8fb9\u5f62\u548c\u91d1\u8272 \/ \u7a7a\u8fb9\u7f18\u751f\u6210\u7684\u91d1\u8272\u56db\u8fb9\u5f62\u7ec4\u6210. \u7ea2\u8272\u4e0e\u91d1\u8272\u6750\u8d28\u4e4b\u95f4\u7684\u8fb9\u7f18(\u7ea2 \/ \u91d1\u8fb9\u7f18) \u4e0d\u751f\u6210\u56db\u8fb9\u5f62\u9762\u7247(\u901a\u8fc7\u5ffd\u7565\u5185\u90e8\u8fb9\u754c, \u51cf\u5c11\u7f51\u683c\u9762\u6570).<br \/>\n$\\\\$ \u5f53\u4e09\u79cd\u6216\u66f4\u591a\u6750\u8d28\u5728\u5355\u4e2a\u7acb\u65b9\u4f53\u5185\u4ea4\u6c47\u65f6, Dual Contouring\u7b97\u6cd5\u4f1a\u5c06\u6781\u5c0f\u503c\u9876\u70b9\u653e\u7f6e\u5728\u4ea4\u6c47\u70b9\u6216\u5176\u9644\u8fd1. \u8fd9\u79cd\u5b9a\u4f4d\u65b9\u5f0f\u80fd\u591f\u9ad8\u5ea6\u8fd8\u539f\u8868\u9762\u51f8\u8d77\u6587\u5b57\u548c\u5b57\u7b26\u7684\u8f6e\u5ed3(\u56fe7\u53f3\u4fa7\u5c40\u90e8\u653e\u5927\u56fe\u5c55\u793a\u4e86\u8fd9\u4e00\u6548\u679c\u7684\u793a\u4f8b). \u57fa\u4e8e\u7acb\u65b9\u4f53\u7684\u8f6e\u5ed3\u65b9\u6cd5\u5c06\u9876\u70b9\u7ea6\u675f\u57283D\u7f51\u683c\u7684\u8fb9\u4e0a, \u56e0\u6b64\u96be\u4ee5\u5b9e\u73b0\u6b64\u7c7b\u6548\u679c.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/Chinese_cube_after_unsafe_simplification_and_safe_simplification.png\" alt=\"\" width=\"2067\" height=\"1024\" class=\"aligncenter size-full wp-image-4991\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/Chinese_cube_after_unsafe_simplification_and_safe_simplification.png 2067w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/Chinese_cube_after_unsafe_simplification_and_safe_simplification-300x149.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/Chinese_cube_after_unsafe_simplification_and_safe_simplification-768x380.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/Chinese_cube_after_unsafe_simplification_and_safe_simplification-1536x761.png 1536w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/Chinese_cube_after_unsafe_simplification_and_safe_simplification-2048x1015.png 2048w\" sizes=\"(max-width: 2067px) 100vw, 2067px\" \/><center><em>\u56fe7: \u7ecf\u8fc7\u4e0d\u7a33\u5b9a\u4fdd\u6301\u62d3\u6251\u7684\u51cf\u9762\u540e\u7684\u6c49\u5b57\u7acb\u65b9\u4f53(\u5de6) \u4e0e\u7ecf\u8fc7\u7a33\u5b9a\u4fdd\u6301\u62d3\u6251\u7684\u51cf\u9762\u7684\u6c49\u5b57\u7acb\u65b9\u4f53(\u53f3). \u9700\u6ce8\u610f, \u5728\u4e0d\u7a33\u5b9a\u4fdd\u6301\u62d3\u6251\u7684\u51cf\u9762\u8fc7\u7a0b\u4e2d, \u5b57\u7b26\u7684\u4e0d\u540c\u90e8\u5206\u53d1\u751f\u4e86\u5408\u5e76.<\/em><\/center><\/p>\n<p>\u5728\u591a\u6750\u8d28\u573a\u666f\u4e0b, \u53ef\u4ee5\u5bf9\u53cc\u6750\u8d28\u60c5\u51b5\u4e2d\u4f7f\u7528\u7684CSG\u64cd\u4f5c\u8fdb\u884c\u591a\u79cd\u6709\u8da3\u7684\u6269\u5c55. \u53d6\u4ee3\u4f20\u7edf\u7684CSG\u64cd\u4f5c, \u4f5c\u8005\u91c7\u7528\u5355\u4e00\u64cd\u4f5cAdd, \u7528\u65b0\u6750\u8d28\u8986\u76d6\u73b0\u6709\u6a21\u578b\u7684\u4e00\u90e8\u5206. \u4f8b\u5982, \u56fe5\u53f3\u4e0a\u89d2\u5c55\u793a\u7684\u7403\u5f62\u5207\u5272\u7b49\u51cf\u6cd5\u64cd\u4f5c, \u53ef\u901a\u8fc7\u5411\u6a21\u578b\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u7a7a\u7a7a\u95f4\u7403\u4f53\u6765\u5b9e\u73b0. Add\u7684\u4e00\u79cd\u6709\u7528\u53d8\u4f53\u662fReplace\u64cd\u4f5c, \u5b83\u4ec5\u8986\u76d6\u6a21\u578b\u7684\u5b9e\u4f53\u90e8\u5206. \u8be5\u64cd\u4f5c\u8fd8\u53ef\u7528\u4e8e\u6a21\u62df\u5bf9\u8f6e\u5ed3\u90e8\u5206\u533a\u57df\u7684\u7eb9\u7406\u5316. \u4f8b\u5982, \u56fe5\u4e0b\u65b9\u7684\u56fe\u50cf\u5c55\u793a\u4e86\u5c06\u5b9e\u4f53\u90e8\u5206\u66ff\u6362\u4e3a\u84dd\u8272\u6750\u8d28\u540e, \u518d\u901a\u8fc7\u7403\u4f53\u8fdb\u884c\u5207\u5272\u7684\u6548\u679c.<\/p>\n<p><strong>2. \u81ea\u9002\u5e94Dual Contouring\u7b97\u6cd5<\/strong><\/p>\n<p>\u4e4b\u524d\u7684Dual Contouring\u7b97\u6cd5\u5b58\u5728\u4e00\u4e2a\u663e\u800c\u6613\u89c1\u7684\u7f3a\u9677: \u5176\u8bbe\u8ba1\u57fa\u4e8e\u5747\u5300\u7f51\u683c. \u5728\u5b9e\u9645\u5e94\u7528\u4e2d, \u5747\u5300\u7f51\u683c\u7684\u5927\u90e8\u5206\u5b58\u50a8\u7a7a\u95f4\u7528\u4e8e\u4fdd\u5b58\u540c\u8d28\u7acb\u65b9\u4f53(\u5373\u6240\u6709\u9876\u70b9\u7b26\u53f7\u76f8\u540c\u7684\u7acb\u65b9\u4f53), \u4ec5\u6709\u5c11\u91cf\u5f02\u8d28\u7acb\u65b9\u4f53(\u5373\u5305\u542b\u7b49\u503c\u9762\u4ea4\u70b9\u7684\u7acb\u65b9\u4f53) \u53c2\u4e0e\u8868\u9762\u751f\u6210. \u4e3a\u907f\u514d\u8fd9\u79cd\u7a7a\u95f4\u6d6a\u8d39, \u4e00\u79cd\u89e3\u51b3\u65b9\u6848\u662f\u7528\u516b\u53c9\u6811\u66ff\u4ee3\u5747\u5300\u7f51\u683c. \u672c\u8282\u5c06\u63cf\u8ff0\u4e00\u79cd\u57fa\u4e8e\u81ea\u9002\u5e94\u7b80\u5316\u516b\u53c9\u6811\u7684Dual Contouring\u7b97\u6cd5, \u5176\u4e2d\u516b\u53c9\u6811\u7684\u53f6\u8282\u70b9\u5305\u542bQEF. \u8be5\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f[Lindstrom 2000] \u63d0\u51fa\u7684\u5747\u5300\u7b80\u5316\u65b9\u6cd5\u7684\u81ea\u9002\u5e94\u53d8\u4f53. \u4f5c\u8005\u7684\u65b9\u6cd5\u5305\u542b\u4ee5\u4e0b\u4e09\u4e2a\u6b65\u9aa4:<br \/>\n$\\\\$ 1) \u751f\u6210\u5e26\u7b26\u53f7\u7684\u516b\u53c9\u6811, \u5176\u540c\u8d28\u53f6\u8282\u70b9\u5df2\u6700\u5927\u7a0b\u5ea6\u6298\u53e0(\u5408\u5e76).<br \/>\n$\\\\$ 2) \u6784\u9020QEF\u5e76\u7b80\u5316\u516b\u53c9\u6811: \u4e3a\u6bcf\u4e2a\u5f02\u8d28\u53f6\u8282\u70b9\u6784\u5efaQEF, \u5e76\u57fa\u4e8e\u8fd9\u4e9bQEF\u7b80\u5316\u516b\u53c9\u6811.<br \/>\n$\\\\$ 3) \u9012\u5f52\u751f\u6210\u591a\u8fb9\u5f62: \u5bf9\u7b80\u5316\u540e\u7684\u516b\u53c9\u6811\u9012\u5f52\u751f\u6210\u8868\u9762\u591a\u8fb9\u5f62.<br \/>\n$\\\\$ \u9700\u8981\u6ce8\u610f\u7684\u662f, \u4f5c\u8005\u7684\u65b9\u6cd5\u4e0eLindstrom\u7684\u65b9\u6cd5\u4e5f\u6709\u533a\u522b: \u4f5c\u8005\u4ece\u5e26\u7b26\u53f7\u7684\u516b\u53c9\u6811\u4e2d\u76f4\u63a5\u751f\u6210\u591a\u8fb9\u5f62, \u800c\u975e\u5bf9\u73b0\u6709\u7f51\u683c\u4e2d\u7684\u591a\u8fb9\u5f62\u8fdb\u884c\u6298\u53e0\u64cd\u4f5c.<br \/>\n$\\\\$ \u5728\u7b2c\u4e00\u6b65\u4e2d\u751f\u6210\u5e26\u7b26\u53f7\u7684\u516b\u53c9\u6811\u76f8\u5bf9\u7b80\u5355. \u5bf9\u4e8e\u9690\u5f0f\u6a21\u578b\u6216\u9762\u7247\u6a21\u578b, \u53ef\u4ee5\u901a\u8fc7\u5bf9\u6a21\u578b\u8fdb\u884c\u7a7a\u95f4\u5212\u5206, \u4ee5\u81ea\u4e0a\u800c\u4e0b\u7684\u65b9\u5f0f\u9012\u5f52\u6784\u5efa\u8be5\u516b\u53c9\u6811. \u5bf9\u4e8e\u5747\u5300\u7f51\u683c\u4e0a\u7684\u5e26\u7b26\u53f7\u7684\u6570\u636e, \u5219\u53ef\u4ee5\u901a\u8fc7\u81ea\u4e0b\u800c\u4e0a\u7684\u65b9\u5f0f\u9012\u5f52\u5408\u5e76\u5747\u5300\u533a\u57df\u6765\u751f\u6210\u516b\u53c9\u6811. \u63a5\u4e0b\u6765\u7684\u4e24\u4e2a\u5c0f\u8282\u5c06\u66f4\u8be6\u7ec6\u5730\u63a2\u8ba8\u8be5\u6d41\u7a0b\u7684\u7b2c\u4e8c\u6b65\u548c\u7b2c\u4e09\u6b65.(\u9700\u6ce8\u610f, \u6b64\u5904\u63cf\u8ff0\u7684\u81ea\u9002\u5e94\u65b9\u6cd5\u65e0\u9700\u4fee\u6539\u5373\u53ef\u9002\u7528\u4e8e\u591a\u6750\u8d28\u573a\u666f.)<\/p>\n<p><strong>2.1 \u57fa\u4e8eQEF\u7684\u516b\u53c9\u6811\u7b80\u5316<\/strong><\/p>\n<p>\u4f5c\u8005\u5bf9\u81ea\u9002\u5e94\u65b9\u6cd5\u7b2c\u4e8c\u6b65\u7684\u5904\u7406\u662f: \u4f7f\u7528\u65b9\u7a0b$\\eqref{QEF}$\u4e3a\u6bcf\u4e2a\u5f02\u8d28\u53f6\u8282\u70b9\u6784\u5efa\u5bf9\u5e94\u7684QEF. \u9700\u6ce8\u610f, \u8be5QEF\u7684\u6781\u5c0f\u503c\u70b9\u5bf9\u5e94\u7684\u6b8b\u5dee\u53ef\u4f30\u8ba1\u9876\u70b9\u5bf9\u539f\u59cb\u51e0\u4f55\u5f62\u72b6\u7684\u903c\u8fd1\u7cbe\u5ea6[Garland and Heckbert 1998]. \u5bf9\u4e8e\u516b\u53c9\u6811\u7684\u7b80\u5316, \u4f5c\u8005\u7684\u65b9\u6cd5\u662f: \u901a\u8fc7\u7d2f\u52a0\u5f53\u524d\u8282\u70b9\u5b50\u6811\u4e2d\u6240\u6709\u53f6\u8282\u70b9\u7684QEF, \u4e3a\u516b\u53c9\u6811\u5185\u90e8\u8282\u70b9\u751f\u6210\u65b0\u7684QEF. \u82e5\u67d0\u5185\u90e8\u8282\u70b9\u7684QEF\u6b8b\u5dee\u5c0f\u4e8e\u7ed9\u5b9a\u5bb9\u5dee, \u5219\u5c06\u5176\u6298\u53e0\u4e3a\u53f6\u8282\u70b9.<br \/>\n$\\\\$ \u4f5c\u8005\u5bf9\u8fd9\u4e00\u65b9\u6cd5\u6240\u505a\u7684\u552f\u4e00\u4fee\u6539\u662f\u4fee\u6539QEF\u7684\u5185\u90e8\u8868\u793a\u5f62\u5f0f, \u4ee5\u5229\u7528\u524d\u6587\u8ba8\u8bba\u7684QR\u5206\u89e3\u6280\u672f. \u4e3a\u6b64, \u4f5c\u8005\u901a\u8fc7\u65b9\u7a0b$\\eqref{upper_triangular_matrix_form_of_augmented_matrix_of_matrix_A}$\u4e2d\u7684\u4e0a\u4e09\u89d2\u77e9\u9635\u5143\u7d20, \u752810\u4e2a\u6d6e\u70b9\u6570\u8868\u793a\u4e00\u4e2aQEF. \u4e24\u4e2aQEF\u7684\u5408\u5e76\u64cd\u4f5c\u5bf9\u5e94\u4e8e\u5c06\u5176\u5404\u81ea\u7684\u4e24\u4e2a\u4e0a\u4e09\u89d2\u77e9\u9635\u8fdb\u884c\u5408\u5e76, \u5f62\u6210\u4e00\u4e2a$8 \\times 4$\u7684\u8054\u5408\u77e9\u9635, \u5176\u5f62\u5f0f\u5982\u4e0b:$$\\begin{pmatrix}<br \/>\nx &#038; x &#038; x &#038; x \\\\<br \/>\n0 &#038; x &#038; x &#038; x \\\\<br \/>\n0 &#038; 0 &#038; x &#038; x \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; x \\\\<br \/>\nx &#038; x &#038; x &#038; x \\\\<br \/>\n0 &#038; x &#038; x &#038; x \\\\<br \/>\n0 &#038; 0 &#038; x &#038; x \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; x<br \/>\n\\end{pmatrix}.$$\u968f\u540e, \u901a\u8fc7\u4e00\u7cfb\u5217Givens\u65cb\u8f6c\u5c06\u77e9\u9635\u91cd\u65b0\u8f6c\u6362\u4e3a\u65b9\u7a0b$\\eqref{upper_triangular_matrix_form_of_augmented_matrix_of_matrix_A}$\u7684\u4e0a\u4e09\u89d2\u5f62\u5f0f. \u7531\u4e8eGivens\u65cb\u8f6c\u7684\u6b63\u4ea4\u6027, \u5408\u5e76\u7cfb\u7edf\u7684QEF\u7b49\u4e8e\u672a\u5408\u5e76\u7cfb\u7edfQEF\u7684\u548c. \u9700\u6ce8\u610f, \u5c06\u5408\u5e76\u540e\u7684\u77e9\u9635\u91cd\u65b0\u8f6c\u6362\u4e3a\u4e0a\u4e09\u89d2\u5f62\u5f0f\u7684\u901f\u5ea6\u6bd4\u6807\u51c6\u8868\u793a\u6cd5\u4e2d\u76f4\u63a5\u76f8\u52a010\u4e2a\u6d6e\u70b9\u6570\u66f4\u6162(\u7ea6\u9700150\u6b21\u7b97\u672f\u8fd0\u7b97). \u7136\u800c, \u8fd9\u79cd\u8868\u793a\u6cd5\u7684\u6570\u503c\u7a33\u5b9a\u6027\u63d0\u5347\u5e26\u6765\u4e86\u66f4\u4f18\u7684\u51cf\u9762\u6548\u679c.<br \/>\n$\\\\$ \u56fe8\u76f4\u89c2\u5c55\u793a\u4e86QR\u8868\u793a\u6cd5\u7a33\u5b9a\u6027\u7684\u4f18\u52bf. \u8be5\u56fe\u4e2d\u7684\u7f51\u683c\u5c55\u793a\u4e86\u5bf9\u56fe4\u4e2d\u5bfa\u5e99\u6a21\u578b\u5728\u8bef\u5dee\u5bb9\u9650\u4e3a0.014\u65f6\u7684\u4e24\u79cd\u51cf\u9762\u7ed3\u679c: \u5de6\u4fa7\u7f51\u683c\u4f7f\u7528QEF\u7684\u6807\u51c6\u8868\u793a\u6cd5\u8ba1\u7b97, \u53f3\u4fa7\u7f51\u683c\u5219\u4f7f\u7528QEF\u7684QR\u8868\u793a\u6cd5\u8ba1\u7b97.(\u4e3a\u76f4\u89c2\u7406\u89e3\u89c4\u6a21, \u5bfa\u5e99\u6a21\u578b\u5b9a\u4e49\u5728\u4e00\u4e2a$256^3$\u5355\u5143\u7684\u7f51\u683c\u4e0a.) \u7531\u4e8e\u6807\u51c6\u8868\u793a\u6cd5\u7684\u4e0d\u7a33\u5b9a\u6027\u5f15\u5165\u7684\u6570\u503c\u8bef\u5dee(\u56e0\u6b8b\u5dee\u4f30\u8ba1\u5931\u771f, \u5982\u8bef\u5224\u5e94\u5408\u5e76\u533a\u57df\u4e3a\u9700\u4fdd\u7559, \u9700\u989d\u5916\u591a\u8fb9\u5f62\u4fdd\u6301\u51e0\u4f55\u5931\u771f), \u5de6\u4fa7\u7f51\u683c\u5305\u542b78K\u4e2a\u591a\u8fb9\u5f62, \u800c\u53f3\u4fa7\u7f51\u683c\u4ec5\u970036K\u4e2a\u591a\u8fb9\u5f62.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/closeups_of_two_polygonal_approximations_to_temple.png\" alt=\"\" width=\"2100\" height=\"630\" class=\"aligncenter size-full wp-image-5001\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/closeups_of_two_polygonal_approximations_to_temple.png 2100w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/closeups_of_two_polygonal_approximations_to_temple-300x90.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/closeups_of_two_polygonal_approximations_to_temple-768x230.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/closeups_of_two_polygonal_approximations_to_temple-1536x461.png 1536w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/closeups_of_two_polygonal_approximations_to_temple-2048x614.png 2048w\" sizes=\"(max-width: 2100px) 100vw, 2100px\" \/><center><em>\u56fe8: \u4f7f\u7528\u6807\u51c6(\u5de6\u4fa7) \u548cQR(\u53f3\u4fa7) \u8868\u793a\u6cd5\u751f\u6210\u7684\u5bfa\u5e99\u6a21\u578b\u591a\u8fb9\u5f62\u8fd1\u4f3c\u7ed3\u679c\u7684\u5c40\u90e8\u653e\u5927\u56fe.<\/em><\/center><\/p>\n<p><strong>2.2 \u5173\u4e8e\u7b80\u5316\u516b\u53c9\u6811\u7684\u591a\u8fb9\u5f62\u751f\u6210<\/strong><\/p>\n<p>\u57fa\u4e8e\u8fd9\u4e00\u7b80\u5316\u540e\u7684\u516b\u53c9\u6811, \u4f5c\u8005\u7684\u4e0b\u4e00\u4e2a\u4efb\u52a1\u662f\u9002\u5f53\u8c03\u6574Dual Contouring\u7b97\u6cd5\u7684\u591a\u8fb9\u5f62\u751f\u6210\u9636\u6bb5. \u5bf9\u4e8e\u57fa\u4e8e\u7acb\u65b9\u4f53\u7684\u65b9\u6cd5, \u4ece\u516b\u53c9\u6811\u751f\u6210\u7b49\u503c\u9762\u7684\u95ee\u9898\u5df2\u88ab\u5e7f\u6cdb\u7814\u7a76[Bloomenthal 1988; Wilhelms\u548cGelder 1992; Livnat\u7b491996; Shekhar\u7b491996; Westermann\u7b491999; Frisken\u7b492000; Cignoni\u7b492000]. \u901a\u5e38, \u8fd9\u4e9b\u65b9\u6cd5\u4f1a\u9650\u5236\u516b\u53c9\u6811\u7684\u76f8\u90bb\u53f6\u8282\u70b9\u5c42\u7ea7\u5dee\u5f02\u6700\u591a\u4e3a1(\u5373&#8221;\u53d7\u9650&#8221; \u516b\u53c9\u6811), \u5e76\u901a\u5e38\u6267\u884c\u67d0\u79cd\u7c7b\u578b\u7684\u88c2\u7f1d\u4fee\u590d\u4ee5\u786e\u4fdd\u7b49\u503c\u9762\u95ed\u5408. [Perry\u548cFrisken 2001] \u63cf\u8ff0\u4e86\u4e00\u79cd\u57fa\u4e8e\u679a\u4e3e\u516b\u53c9\u6811\u53f6\u8282\u70b9\u76f8\u5173\u8fb9\u7684\u5e26\u7b26\u53f7\u7684\u516b\u53c9\u6811SurfaceNets\u7b97\u6cd5\u53d8\u4f53. \u5177\u4f53\u800c\u8a00:<br \/>\n$\\\\$ $\\cdot$ \u5bf9\u4e8e\u6bcf\u6761\u8868\u73b0\u51fa\u7b26\u53f7\u53d8\u5316\u7684\u8fb9, \u751f\u6210\u6240\u6709\u8fde\u63a5\u8be5\u8fb9\u6240\u5305\u542b\u7684\u4efb\u610f\u4e09\u4e2a\u4e0d\u540c\u7acb\u65b9\u4f53\u5bf9\u5e94\u9876\u70b9\u7684\u4e09\u89d2\u5f62.<br \/>\n$\\\\$ \u5728\u6b64\u8fc7\u7a0b\u4e2d, \u53ef\u80fd\u751f\u6210\u5197\u4f59\u4e09\u89d2\u5f62, \u5176\u6765\u6e90\u4e3b\u8981\u4e3a:<br \/>\n$\\\\$ $\\cdot$ \u5728\u516b\u53c9\u6811\u7b80\u5316\u4e2d, \u53f6\u7acb\u65b9\u4f53\u53ef\u80fd\u5305\u542b\u591a\u4e2a\u76f8\u90bb\u6750\u8d28\u8fb9\u754c(\u5982\u6c49\u5b57\u7b14\u753b\u4ea4\u53c9\u5904).<br \/>\n$\\\\$ $\\cdot$ \u76f4\u63a5\u751f\u6210\u6240\u6709\u53ef\u80fd\u4e09\u89d2\u5f62\u4f1a\u5bfc\u81f4\u91cd\u53e0\u6216\u5197\u4f59(\u5982\u56fe5\u4e2d\u7403\u5f62\u5207\u5272\u7684\u7a7a\u7a7a\u95f4\u533a\u57df\u8bef\u751f\u6210\u4e09\u89d2\u5f62).<br \/>\n$\\\\$ \u4e3a\u907f\u514d\u751f\u6210\u5197\u4f59\u4e09\u89d2\u5f62, \u8be5\u65b9\u6cd5\u6839\u636e\u4e09\u89d2\u5f62\u5bf9\u5e94\u8fb9\u5728\u53f6\u7acb\u65b9\u4f53\u5185\u7684\u76f8\u5bf9\u4f4d\u7f6e\u5254\u9664\u90e8\u5206\u751f\u6210\u7684\u4e09\u89d2\u5f62. \u5728\u6700\u7ec8\u7f51\u683c\u4e2d, \u6bcf\u6761\u8fb9\u5bf9\u5e94\u4e8e\u8f6e\u5ed3\u4e0a\u7684\u4e00\u4e2a\u6216\u4e24\u4e2a\u4e09\u89d2\u5f62(\u5373\u4e00\u4e2a\u56db\u8fb9\u5f62\u9762\u7247). \u8be5\u65b9\u6cd5\u7684\u4e3b\u8981\u7f3a\u70b9(\u4f5c\u8005\u5df2\u627f\u8ba4) \u662f\u5076\u5c14\u4f1a\u4ea7\u751f\u5e26\u6709\u88c2\u7f1d\u7684\u8f6e\u5ed3. \u88c2\u7f1d\u7684\u6210\u56e0\u4e3b\u8981\u4e3a:<br \/>\n$\\\\$ $\\cdot$ \u53d7\u9650\u516b\u53c9\u6811\u7684\u5c42\u7ea7\u5dee\u9650\u5236(\u76f8\u90bb\u53f6\u8282\u70b9\u6700\u591a\u5dee1\u7ea7) \u53ef\u80fd\u65e0\u6cd5\u5b8c\u5168\u8986\u76d6\u6240\u6709\u62d3\u6251\u60c5\u51b5.<br \/>\n$\\\\$ $\\cdot$ \u5254\u9664\u5197\u4f59\u4e09\u89d2\u5f62\u540e, \u90e8\u5206\u8fb9\u53ef\u80fd\u4ec5\u5bf9\u5e94\u5355\u4e2a\u4e09\u89d2\u5f62, \u5bfc\u81f4\u7f51\u683c\u4e0d\u8fde\u7eed(\u5982\u56fe8\u5de6\u4fa7\u6807\u51c6\u8868\u793a\u6cd5\u768478K\u9762\u56e0\u6570\u503c\u8bef\u5dee\u4ea7\u751f\u88c2\u7f1d).<br \/>\n$\\\\$ \u4f5c\u8005\u901a\u8fc7\u8bc6\u522b\u8fd9\u4e9b\u5bfc\u81f4\u95ee\u9898\u7684\u914d\u7f6e, \u5e76\u5728\u516b\u53c9\u6811\u4e0a\u6267\u884c\u989d\u5916\u7ec6\u5206\u6765\u89c4\u907f\u8fd9\u4e9b\u95ee\u9898. \u5176\u7b97\u6cd5\u6d41\u7a0b\u5927\u81f4\u5982\u4e0b\u6240\u793a:<br \/>\n$\\\\$ 1) \u6b65\u9aa41: \u8fb9\u76f8\u5bf9\u4f4d\u7f6e\u68c0\u6d4b.<br \/>\n$\\\\$ $\\quad$ $\\cdot$ \u5bf9\u6bcf\u6761\u8fb9, \u8ba1\u7b97\u5176\u5728\u53f6\u7acb\u65b9\u4f53\u5185\u7684\u4f4d\u7f6e(\u5982\u9760\u8fd1\u7acb\u65b9\u4f53\u4e2d\u5fc3\u6216\u9876\u70b9).<br \/>\n$\\\\$ $\\quad$ $\\cdot$ \u82e5\u591a\u4e2a\u4e09\u89d2\u5f62\u5171\u4eab\u540c\u4e00\u7ec4\u8fb9\u4e14\u4f4d\u7f6e\u91cd\u53e0, \u5219\u5254\u9664\u5197\u4f59\u4e09\u89d2\u5f62(\u5982\u4fdd\u7559\u6700\u9760\u8fd1\u6750\u8d28\u4ea4\u754c\u7684\u4e09\u89d2\u5f62).<br \/>\n$\\\\$ 2) \u6b65\u9aa42: \u56db\u8fb9\u5f62\u9762\u7247\u751f\u6210. \u5bf9\u6bcf\u6761\u8fb9, \u751f\u62101\u62162\u4e2a\u4e09\u89d2\u5f62.<br \/>\n$\\\\$ $\\quad$ $\\cdot$ \u82e5\u8fb9\u4ec5\u88ab\u4e00\u4e2a\u7acb\u65b9\u4f53\u5305\u542b, \u5219\u751f\u6210\u4e00\u4e2a\u4e09\u89d2\u5f62(\u5982\u5e73\u5766\u8868\u9762).<br \/>\n$\\\\$ $\\quad$ $\\cdot$ \u82e5\u8fb9\u88ab\u4e24\u4e2a\u7acb\u65b9\u4f53\u5305\u542b(\u5982\u6750\u8d28\u4ea4\u754c), \u5219\u751f\u6210\u4e24\u4e2a\u4e09\u89d2\u5f62\u7ec4\u6210\u56db\u8fb9\u5f62\u9762\u7247.<br \/>\n$\\\\$ 3) \u6b65\u9aa43: \u88c2\u7f1d\u4fee\u590d.<br \/>\n$\\\\$ $\\quad$ $\\cdot$ \u8bc6\u522b\u5bfc\u81f4\u88c2\u7f1d\u7684\u914d\u7f6e(\u5982\u8fb9\u4ec5\u5bf9\u5e94\u5355\u4e2a\u4e09\u89d2\u5f62\u4e14\u76f8\u90bb\u533a\u57df\u672a\u7ec6\u5206).<br \/>\n$\\\\$ $\\quad$ $\\cdot$ \u5bf9\u76f8\u5173\u53f6\u7acb\u65b9\u4f53\u6267\u884c\u989d\u5916\u7ec6\u5206(\u589e\u52a0\u516b\u53c9\u6811\u5c42\u7ea7), \u5f3a\u5236\u751f\u6210\u5339\u914d\u7684\u4e09\u89d2\u5f62\u5bf9.<br \/>\n$\\\\$ \u4f5c\u8005\u63d0\u51fa\u4e86\u4e00\u79cd\u66f4\u7b80\u5355\u7684Dual Contouring\u7b97\u6cd5\u89c4\u5219, \u7528\u4e8e\u5e26\u7b26\u53f7\u7684\u516b\u53c9\u6811, \u4ece\u800c\u907f\u514d\u989d\u5916\u7ec6\u5206\u7684\u9700\u6c42. \u8be5\u89c4\u5219\u57fa\u4e8e\u4ee5\u4e0b\u89c2\u5bdf: \u53ea\u6709\u90a3\u4e9b\u4e0d\u5305\u542b\u76f8\u90bb\u53f6\u7acb\u65b9\u4f53\u8fb9\u7684\u8fb9\u624d\u4f1a\u751f\u6210\u591a\u8fb9\u5f62. \u4f5c\u8005\u5c06\u8fd9\u7c7b\u8fb9\u79f0\u4e3a\u516b\u53c9\u6811\u7684\u6700\u5c0f\u8fb9. \u56e0\u6b64, \u672c\u6587\u7684\u591a\u8fb9\u5f62\u751f\u6210\u89c4\u5219\u4e3a:<br \/>\n$\\\\$ $\\cdot$ \u5bf9\u4e8e\u6bcf\u6761\u8868\u73b0\u51fa\u7b26\u53f7\u53d8\u5316\u7684\u6700\u5c0f\u8fb9, \u751f\u6210\u8fde\u63a5\u5305\u542b\u8be5\u8fb9\u7684\u7acb\u65b9\u4f53\u7684\u6781\u5c0f\u503c\u9876\u70b9\u7684\u591a\u8fb9\u5f62.<br \/>\n$\\\\$ \u8fd9\u4e00\u89c4\u5219\u7684\u7279\u6027\u5728\u4e8e, \u5b83\u80fd\u591f\u4e3a\u4efb\u4f55\u7b80\u5316\u540e\u7684\u516b\u53c9\u6811\u751f\u6210\u95ed\u5408\u7684\u591a\u8fb9\u5f62\u7f51\u683c. \u5177\u4f53\u800c\u8a00, \u7f51\u683c\u4e2d\u7684\u6bcf\u6761\u8fb9\u5747\u88ab\u5076\u6570\u4e2a\u591a\u8fb9\u5f62\u5305\u542b. \u4e3a\u8bc1\u660e\u8fd9\u4e00\u7279\u6027, \u4f5c\u8005\u89c2\u5bdf\u5230\u5bf9\u5076\u7b49\u503c\u9762\u4e0a\u7684\u8fb9\u7531\u9762\u76f8\u90bb\u7684\u53f6\u7acb\u65b9\u4f53\u5bf9\u751f\u6210. \u8986\u76d6\u5176\u5171\u540c\u6b63\u65b9\u5f62\u9762\u8fb9\u754c\u7684\u6700\u5c0f\u8fb9\u59cb\u7ec8\u8868\u73b0\u51fa\u5076\u6570\u6b21\u7b26\u53f7\u53d8\u5316, \u56e0\u4e3a\u6b63\u65b9\u5f62\u7684\u8fb9\u754c\u662f\u4e00\u6761\u95ed\u5408\u66f2\u7ebf. \u89c4\u5219\u603b\u4f1a\u751f\u6210\u5076\u6570\u4e2a\u591a\u8fb9\u5f62\u5305\u542b\u8be5\u8fb9. \u4f8b\u5982, \u5728\u5747\u5300\u60c5\u51b5\u4e0b, \u5171\u540c\u6b63\u65b9\u5f62\u9762\u7531\u56db\u6761\u8fde\u7eed\u7684\u8fb9\u7ec4\u6210. \u8fd9\u56db\u6761\u8fb9\u94fe\u53ef\u80fd\u8868\u73b0\u51fa\u4e24\u6b21\u6216\u56db\u6b21\u7b26\u53f7\u53d8\u5316, \u4ece\u800c\u751f\u6210\u4e24\u4e2a\u6216\u56db\u4e2a\u5305\u542b\u5171\u540c\u8fb9\u7684\u591a\u8fb9\u5f62.(\u4e5f\u53ef\u6784\u9020\u5e26\u7b26\u53f7\u7684\u516b\u53c9\u6811\u751f\u6210\u5305\u542b6\u4e2a\u6216\u66f4\u591a\u591a\u8fb9\u5f62\u5171\u4eab\u540c\u4e00\u6761\u8fb9\u7684\u5bf9\u5076\u7b49\u503c\u9762.) \u56fe9\u5c55\u793a\u4e86\u673a\u68b0\u96f6\u4ef6\u7684\u4e09\u79cd\u51cf\u9762\u8fd1\u4f3c\u7ed3\u679c. \u9700\u6ce8\u610f, \u6700\u53f3\u4fa7\u7f51\u683c\u7ecf\u5386\u4e86\u62d3\u6251\u53d8\u5316.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/three_simplified_versions_of_mechanical_part.png\" alt=\"\" width=\"1152\" height=\"973\" class=\"aligncenter size-full wp-image-5011\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/three_simplified_versions_of_mechanical_part.png 1152w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/three_simplified_versions_of_mechanical_part-300x253.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/three_simplified_versions_of_mechanical_part-768x649.png 768w\" sizes=\"(max-width: 1152px) 100vw, 1152px\" \/><center><em>\u56fe9: \u673a\u68b0\u96f6\u4ef6\u7684\u4e09\u4e2a\u51cf\u9762\u7248\u672c.<\/em><\/center><\/p>\n<p>\u5728\u516b\u53c9\u6811\u7684\u8fc7\u6e21\u533a\u57df(\u5373\u4e00\u4e2a\u7c97\u7cd9\u7f51\u683c\u7acb\u65b9\u4f53\u4e0e\u56db\u4e2a\u7cbe\u7ec6\u7f51\u683c\u7acb\u65b9\u4f53\u9762\u76f8\u90bb\u7684\u533a\u57df), \u8be5\u89c4\u5219\u4f1a\u751f\u6210\u4e09\u89d2\u5f62\u800c\u975e\u56db\u8fb9\u5f62. \u4f4d\u4e8e\u516c\u5171\u7c97\u7cd9\u7f51\u683c\u9762\u4e2d\u592e\u7684\u6700\u5c0f\u8fb9\u4ec5\u88ab\u4e09\u4e2a\u7acb\u65b9\u4f53\u5305\u542b, \u8fd9\u4e9b\u8fb9\u4f1a\u751f\u6210\u4e09\u89d2\u5f62, \u4ece\u800c\u5728\u7c97\u7cd9\u7f51\u683c\u56db\u8fb9\u5f62\u548c\u7cbe\u7ec6\u7f51\u683c\u56db\u8fb9\u5f62\u4e4b\u95f4\u5f62\u6210\u8fc7\u6e21. \u56fe8\u5c55\u793a\u4e86\u901a\u8fc7\u7b49\u503c\u9762\u751f\u6210\u7b80\u5316\u516b\u53c9\u6811\u65f6\u4ea7\u751f\u7684\u8bb8\u591a\u6b64\u7c7b\u8fc7\u6e21\u4e09\u89d2\u5f62\u7684\u793a\u4f8b.<br \/>\n$\\\\$ \u9700\u6ce8\u610f, Perry \/ Frisken\u89c4\u5219\u901a\u8fc7\u679a\u4e3e\u516b\u53c9\u6811\u4e2d\u7684\u8fb9\u5e76\u5b9a\u4f4d\u5305\u542b\u8be5\u8fb9\u7684\u7acb\u65b9\u4f53\u6765\u5b9e\u73b0. \u8fd9\u4e00\u90bb\u63a5\u67e5\u627e\u8fc7\u7a0b\u9700\u8981\u904d\u5386\u516b\u53c9\u6811\u7684\u4e0a\u4e0b\u5c42\u7ea7\u6216\u663e\u5f0f\u7ef4\u62a4\u76f8\u90bb\u7acb\u65b9\u4f53\u4e4b\u95f4\u7684\u94fe\u63a5. \u76f8\u8f83\u4e8e\u679a\u4e3e\u8fb9\u5e76\u4e3b\u52a8\u67e5\u627e\u90bb\u63a5\u5173\u7cfb, \u4f5c\u8005\u63d0\u51fa\u4e86\u4e00\u79cd\u9012\u5f52\u679a\u4e3e\u65b9\u6cd5, \u7528\u4e8e\u5b9a\u4f4d\u5305\u542b\u5171\u540c\u6700\u5c0f\u8fb9\u7684\u6240\u6709\u7acb\u65b9\u4f53\u96c6\u5408. \u4e3a\u7b80\u5316\u8bf4\u660e, \u4f5c\u8005\u4ee5\u56db\u53c9\u6811\u4e3a\u4f8b\u89e3\u91ca\u8be5\u679a\u4e3e\u65b9\u6cd5, \u540c\u65f6\u8bf4\u660e\u5176\u539f\u7406\u53ef\u81ea\u7136\u6269\u5c55\u81f3\u516b\u53c9\u6811.<br \/>\n$\\\\$ \u8be5\u679a\u4e3e\u8fc7\u7a0b\u7684\u5173\u952e\u5728\u4e8e\u4e24\u4e2a\u9012\u5f52\u51fd\u6570faceProc[$q$]\u548cedgeProc[$q_1$,$q_2$].<br \/>\n$\\\\$ $\\cdot$ \u7ed9\u5b9a\u56db\u53c9\u6811\u4e2d\u7684\u4e00\u4e2a\u5185\u90e8\u8282\u70b9$q$, faceProc[$q$]\u4f1a\u9012\u5f52\u8c03\u7528\u81ea\u8eab\u4e0e$q$\u7684\u56db\u4e2a\u5b50\u8282\u70b9, \u5e76\u5bf9$q$\u7684\u6240\u6709\u56db\u7ec4\u8fb9\u76f8\u90bb\u5b50\u8282\u70b9\u5bf9\u8c03\u7528edgeProc.<br \/>\n$\\\\$ $\\cdot$ \u7ed9\u5b9a\u4e00\u5bf9\u8fb9\u76f8\u90bb\u7684\u5185\u90e8\u8282\u70b9$q_1$\u548c$q_2$, edgeProc[$q_1$,$q_2$]\u4f1a\u9012\u5f52\u8c03\u7528\u81ea\u8eab\u4e0e\u8de8\u8d8a$q_1$\u548c$q_2$\u516c\u5171\u8fb9\u7684\u4e24\u7ec4\u8fb9\u76f8\u90bb\u5b50\u8282\u70b9\u5bf9.<br \/>\n$\\\\$ $\\cdot$ \u56fe10\u5c55\u793a\u4e86\u8fd9\u4e24\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\/2026\/01\/mutually_recursive_structure_of_faceProc_and_edgeProc.png\" alt=\"\" width=\"573\" height=\"240\" class=\"aligncenter size-full wp-image-5016\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/mutually_recursive_structure_of_faceProc_and_edgeProc.png 573w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/mutually_recursive_structure_of_faceProc_and_edgeProc-300x126.png 300w\" sizes=\"(max-width: 573px) 100vw, 573px\" \/><center><em>\u56fe10: \u7528\u4e8e\u679a\u4e3e\u5305\u542b\u516c\u5171\u8fb9\u7684\u53f6\u65b9\u5757\u5bf9\u7684\u9012\u5f52\u51fd\u6570faceProc(\u9ed1\u8272) \u548cedgeProc(\u7070\u8272).<\/em><\/center><\/p>\n<p>\u5f53$q_1$\u548c$q_2$\u5747\u4e3a\u56db\u53c9\u6811\u7684\u53f6\u8282\u70b9\u65f6, \u5bf9edgeProc[$q_1$,$q_2$]\u7684\u9012\u5f52\u8c03\u7528\u7ec8\u6b62. \u6b64\u65f6, edgeProc[$q_1$,$q_2$]\u5df2\u638c\u63e1\u751f\u6210$q_1$\u548c$q_2$\u5171\u4eab\u7684\u6700\u5c0f\u8fb9\u76f8\u5173\u7ebf\u6bb5\u6240\u9700\u7684\u5168\u90e8\u4fe1\u606f. \u9700\u6ce8\u610f, \u8be5\u65b9\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u4e0e\u56db\u53c9\u6811\u7684\u5927\u5c0f\u5448\u7ebf\u6027\u5173\u7cfb, \u56e0\u4e3a\u6bcf\u4e2a\u56db\u53c9\u6811\u65b9\u5757\u5bf9\u5e94\u4e00\u6b21faceProc\u8c03\u7528, \u6bcf\u6761\u8fb9\u5bf9\u5e94\u4e00\u6b21edgeProc\u8c03\u7528.<br \/>\n$\\\\$ \u7b49\u503c\u9762\u751f\u6210\u516b\u53c9\u6811\u9700\u8981\u4e09\u4e2a\u51fd\u6570: cellProc[$q$](\u5355\u5143\u5904\u7406\u51fd\u6570), faceProc[$q_1$, $q_2$](\u9762\u5904\u7406\u51fd\u6570) \u548cedgeProc[$q_1$,$q_2$, $q_3$, $q_4$](\u8fb9\u5904\u7406\u51fd\u6570).<br \/>\n$\\\\$ $\\cdot$ cellProc\u51fd\u6570\u4f1a\u751f\u62108\u6b21cellProc\u8c03\u7528(\u9012\u5f52\u5904\u74068\u4e2a\u5355\u5143), 12\u6b21faceProc\u8c03\u7528(\u5904\u740612\u5bf9\u8fb9\u76f8\u90bb\u5b50\u5355\u5143), 6\u6b21edgeProc\u8c03\u7528(\u5904\u74066\u5bf9\u5171\u4eab\u516c\u5171\u8fb9\u7684\u5b50\u5355\u5143\u5bf9).<br \/>\n$\\\\$ $\\cdot$ faceProc\u4f1a\u751f\u62104\u6b21faceProc\u8c03\u7528(\u9012\u5f52\u5904\u74064\u5bf9\u8fb9\u76f8\u90bb\u5b50\u5355\u5143), 4\u6b21edgeProc\u8c03\u7528(\u5904\u74064\u5bf9\u5171\u4eab\u516c\u5171\u8fb9\u7684\u5b50\u5355\u5143\u7ec4).<br \/>\n$\\\\$ $\\cdot$ edgeProc\u4f1a\u751f\u62102\u6b21edgeProc\u8c03\u7528(\u9012\u5f52\u5904\u7406\u8de8\u8d8a\u516c\u5171\u8fb9\u7684\u4e24\u7ec4\u5b50\u5355\u5143).<br \/>\n$\\\\$ $\\cdot$ \u5f53\u6240\u6709$q_i$\u5747\u4e3a\u516b\u53c9\u6811\u53f6\u8282\u70b9\u65f6, \u5bf9edgeProc[$q_1$,$q_2$, $q_3$, $q_4$]\u7684\u9012\u5f52\u8c03\u7528\u7ec8\u6b62\u4e8e\u516b\u53c9\u6811\u7684\u6700\u5c0f\u8fb9\u5904.<\/p>\n<p><strong>3. \u4fdd\u6301\u62d3\u6251\u7684\u51cf\u9762<\/strong><\/p>\n<p>\u51cf\u9762\u65b9\u6cd5\u5982[Rossignac\u548cBorrell 1993; Lindstrom 2000] \u7684\u7279\u6027\u662f, \u591a\u8fb9\u5f62\u7f51\u683c\u7684\u62d3\u6251\u8fde\u901a\u6027\u53ef\u80fd\u5728\u51cf\u9762\u8fc7\u7a0b\u4e2d\u53d1\u751f\u53d8\u5316. \u66f4\u590d\u6742\u7684\u65b9\u6cd5\u5982[Stander\u548cHart 1997; Gerstner\u548cPajarola 2000; Wood\u7b492000; Guskov\u548cWood 2001] \u88ab\u5f00\u53d1\u7528\u4e8e\u5728\u51cf\u9762\u8fc7\u7a0b\u4e2d\u4fdd\u6301\u7f51\u683c\u7684\u8fde\u901a\u6027. \u9057\u61be\u7684\u662f, \u5728\u672c\u6587\u7684\u573a\u666f\u4e2d, \u4e0d\u4ec5\u8f6e\u5ed3\u7684\u62d3\u6251\u8fde\u901a\u6027, \u5176\u7eb9\u7406\u533a\u57df\u7684\u8fde\u901a\u6027\u4e5f\u53ef\u80fd\u53d8\u5316. \u56fe7\u7684\u5de6\u4fa7\u5c55\u793a\u4e86\u6c49\u5b57\u7b14\u753b\u5728\u51cf\u9762\u4e2d\u5408\u5e76\u7684\u793a\u4f8b, \u5bfc\u81f4\u96be\u4ee5\u8fa8\u8ba4. \u5c3d\u7ba1\u8fd9\u4e9b\u62d3\u6251\u53d8\u5316\u5e76\u975e\u603b\u662f\u4e0d\u53ef\u53d6, \u4f5c\u8005\u5e0c\u671b\u5728\u51cf\u9762\u8fc7\u7a0b\u4e2d\u80fd\u591f\u9009\u62e9\u6027\u5730\u4fdd\u7559\u8f6e\u5ed3\u53ca\u5176\u7eb9\u7406\u533a\u57df\u7684\u62d3\u6251\u8fde\u901a\u6027. \u5177\u4f53\u800c\u8a00, \u7ed9\u5b9a\u516b\u53c9\u6811\u4e2d\u4e00\u4e2a\u5185\u90e8\u8282\u70b9(\u5176\u516b\u4e2a\u5b50\u8282\u70b9\u5747\u4e3a\u53f6\u8282\u70b9), \u9700\u8981\u4e00\u79cd\u57fa\u4e8e\u8fd9\u4e9b\u53f6\u8282\u70b9\u89d2\u70b9\u5904\u7684\u7b26\u53f7(\u6216\u7d22\u5f15) \u7684\u6d4b\u8bd5\u65b9\u6cd5, \u4ee5\u786e\u4fdd\u5728\u8282\u70b9\u6298\u53e0\u65f6\u5bf9\u5076\u7b49\u503c\u9762\u53ca\u5176\u7eb9\u7406\u533a\u57df\u7684\u62d3\u6251\u8fde\u901a\u6027\u5f97\u4ee5\u4fdd\u7559.<\/p>\n<p><strong>3.1 \u53cc\u7b26\u53f7\u60c5\u51b5<\/strong><\/p>\n<p>\u8003\u8651\u4e00\u4e2a\u7531\u516b\u4e2a\u53f6\u7acb\u65b9\u4f53\u7ec4\u6210\u7684\u7c97\u7cd9\u7acb\u65b9\u4f53. \u8fd9\u516b\u4e2a\u53f6\u7acb\u65b9\u4f53\u7684\u89d2\u70b9\u7b26\u53f7\u5b9a\u4e49\u4e86\u4e00\u4e2a$3 \\times 3 \\times 3$\u7684\u7f51\u683c, \u800c\u8be5\u7f51\u683c\u7684\u89d2\u70b9\u53c8\u6784\u6210\u4e86\u4e00\u4e2a$2 \\times 2 \\times 2$\u7684\u7c97\u7cd9\u7f51\u683c, \u5982\u4e0b\u56fe\u6240\u793a. \u4f5c\u8005\u7684\u76ee\u6807\u662f\u5f00\u53d1\u4e00\u79cd\u6d4b\u8bd5\u65b9\u6cd5, \u4ee5\u5224\u65ad\u7531\u8be5\u7cbe\u7ec6\u7f51\u683c\u751f\u6210\u7684\u5bf9\u5076\u7b49\u503c\u9762\u662f\u5426\u4e0e\u7531\u7c97\u7cd9\u7f51\u683c\u751f\u6210\u7684\u5bf9\u5076\u7b49\u503c\u9762\u62d3\u6251\u7b49\u4ef7.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/scene_octree.png\" alt=\"\" width=\"958\" height=\"550\" class=\"aligncenter size-full wp-image-5071\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/scene_octree.png 958w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/scene_octree-300x172.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/scene_octree-768x441.png 768w\" sizes=\"(max-width: 958px) 100vw, 958px\" \/><center><em>\u56fe\u6765\u6e90\u4e8eWikipedia.<\/em><\/center><\/p>\n<p>\u5728\u63d0\u51fa\u6d4b\u8bd5\u65b9\u6cd5\u4e4b\u524d, \u9996\u5148\u56de\u987e: \u82e5\u4e00\u4e2a$d$\u7ef4\u7b49\u503c\u9762\u5728\u5c40\u90e8\u4e0e$d$\u7ef4\u5706\u76d8\u62d3\u6251\u7b49\u4ef7, \u5219\u5176\u4e3a\u6d41\u5f62. \u7531\u4e8e\u7acb\u65b9\u4f53\u670912\u6761\u8fb9, Dual Contouring\u7b97\u6cd5\u6700\u591a\u53ef\u5728\u7acb\u65b9\u4f53\u4e2d\u5fc3\u9876\u70b9\u5904\u751f\u621012\u4e2a\u591a\u8fb9\u5f62. \u5bf9\u4e8e\u7acb\u65b9\u4f53\u4e0a\u5927\u591a\u6570\u5e38\u89c1\u7684\u7b26\u53f7\u914d\u7f6e, \u8fd9\u4e9b\u591a\u8fb9\u5f62\u5728\u8be5\u9876\u70b9\u5904\u5b9a\u4e49\u4e86\u4e00\u4e2a\u6d41\u5f62. \u7136\u800c, \u5b58\u5728\u67d0\u4e9b\u7b26\u53f7\u914d\u7f6e\u4f1a\u4f7f\u5bf9\u5076\u7b49\u503c\u9762\u6210\u4e3a\u975e\u6d41\u5f62.(\u8fd9\u4e9b\u914d\u7f6e\u5bf9\u5e94\u4e8e\u4f20\u7edf\u57fa\u4e8e\u7acb\u65b9\u4f53\u7684\u65b9\u6cd5\u4e2d\u7684&#8221;\u6b67\u4e49&#8221; \u7b26\u53f7\u914d\u7f6e.) \u57fa\u4e8e\u6b64\u5b9a\u4e49, \u5b89\u5168\u6027\u6d4b\u8bd5\u5305\u542b\u4ee5\u4e0b\u4e09\u4e2a\u68c0\u67e5\u6b65\u9aa4:<br \/>\n$\\\\$ 1) \u68c0\u67e5\u7c97\u7cd9\u7acb\u65b9\u4f53\u7684\u5bf9\u5076\u7b49\u503c\u9762\u662f\u5426\u4e3a\u6d41\u5f62. \u82e5\u5426, \u5219\u7ec8\u6b62\u51cf\u9762.<br \/>\n$\\\\$ 2) \u68c0\u67e5\u6bcf\u4e2a\u7cbe\u7ec6\u7acb\u65b9\u4f53\u7684\u5bf9\u5076\u7b49\u503c\u9762\u662f\u5426\u4e3a\u6d41\u5f62. \u82e5\u5426, \u5219\u7ec8\u6b62\u51cf\u9762.<br \/>\n$\\\\$ 3) \u68c0\u67e5\u7cbe\u7ec6\u7b49\u503c\u9762\u5728\u7c97\u7cd9\u7acb\u65b9\u4f53\u7684\u6bcf\u4e2a\u5b50\u9762\u4e0a\u662f\u5426\u4e0e\u7c97\u7cd9\u7b49\u503c\u9762\u62d3\u6251\u7b49\u4ef7. \u82e5\u5426, \u5219\u7ec8\u6b62\u51cf\u9762; \u5426\u5219\u5141\u8bb8\u5b89\u5168\u6298\u53e0.<br \/>\n$\\\\$ \u524d\u4e24\u4e2a\u68c0\u67e5\u6b65\u9aa4\u5c06\u51cf\u9762\u8fc7\u7a0b\u9650\u5236\u5728\u751f\u6210\u6d41\u5f62\u5bf9\u5076\u7b49\u503c\u9762\u7684\u8303\u56f4\u5185.(\u9700\u8981\u6ce8\u610f\u7684\u662f, \u5982\u679c\u7cbe\u7ec6\u53f6\u7acb\u65b9\u4f53\u672c\u8eab\u662f\u524d\u4e00\u6b21\u6298\u53e0\u7684\u7ed3\u679c, \u5219\u53ef\u4ee5\u7701\u7565\u7b2c\u4e8c\u4e2a\u68c0\u67e5.) \u5728\u5b9e\u9645\u5e94\u7528\u4e2d, \u8fd9\u79cd\u9650\u5236\u662f\u53ef\u63a5\u53d7\u7684, \u56e0\u4e3a\u5927\u591a\u6570\u9ad8\u5206\u8fa8\u7387\u7b49\u503c\u9762\u672c\u8d28\u4e0a\u662f\u6d41\u5f62\u7684, \u800c\u975e\u6d41\u5f62\u7ed3\u6784\u901a\u5e38\u7531\u4e0d\u5b89\u5168\u7684\u51cf\u9762\u64cd\u4f5c\u5bfc\u81f4. \u5bf9\u4e8e\u524d\u4e24\u4e2a\u68c0\u67e5\u6b65\u9aa4, [Gerstner\u548cPajarola 2000] \u63cf\u8ff0\u4e86\u4e00\u79cd\u7b80\u5355\u6d4b\u8bd5\u65b9\u6cd5, \u7528\u4e8e\u5224\u65ad\u5355\u4e2a\u7acb\u65b9\u4f53\u7684\u7b49\u503c\u9762\u662f\u5426\u4e3a\u6d41\u5f62. \u5176\u6838\u5fc3\u601d\u60f3\u662f: \u5c06\u7acb\u65b9\u4f53\u89d2\u70b9\u7b26\u53f7\u76f8\u540c\u7684\u8fb9\u53cd\u590d\u6298\u53e0\u4e3a\u5355\u4e2a\u9876\u70b9. \u6b64\u65f6, \u8be5\u7acb\u65b9\u4f53\u7684\u7b49\u503c\u9762\u662f\u6d41\u5f62\u5f53\u4e14\u4ec5\u5f53\u6298\u53e0\u540e\u7684\u7ed3\u679c\u4e3a\u5355\u6761\u8fb9. \u8be5\u6d4b\u8bd5\u7ed3\u679c\u53ef\u5bf9\u6240\u6709\u53ef\u80fd\u7684\u5355\u4e2a\u7acb\u65b9\u4f53\u7b26\u53f7\u914d\u7f6e\u9884\u5148\u8ba1\u7b97, \u5e76\u5b58\u50a8\u5728\u4e00\u4e2a\u5927\u5c0f\u4e3a$2^8$\u7684\u67e5\u627e\u8868\u4e2d.<br \/>\n$\\\\$ \u7b2c\u4e09\u79cd\u68c0\u67e5\u901a\u8fc7\u4ee5\u4e0b\u6b65\u9aa4\u9a8c\u8bc1\u7c97\u7cd9\u7f51\u683c\u4e0e\u7cbe\u7ec6\u7f51\u683c\u7b49\u503c\u9762\u7684\u62d3\u6251\u7b49\u4ef7\u6027:<br \/>\n$\\\\$ 1) \u9996\u5148\u68c0\u67e5\u7c97\u7cd9\u7acb\u65b9\u4f53\u8fb9\u4e0a\u7684\u62d3\u6251\u7b49\u4ef7\u6027.<br \/>\n$\\\\$ 2) \u63a5\u7740\u68c0\u67e5\u7c97\u7cd9\u7acb\u65b9\u4f53\u9762\u4e0a\u7684\u62d3\u6251\u7b49\u4ef7\u6027.<br \/>\n$\\\\$ 3) \u6700\u540e\u68c0\u67e5\u7c97\u7cd9\u7acb\u65b9\u4f53\u5185\u90e8\u7684\u62d3\u6251\u7b49\u4ef7\u6027.<br \/>\n$\\\\$ \u8fd9\u4e9b\u68c0\u67e5\u53ef\u901a\u8fc7\u4e00\u7cfb\u5217\u5bf9$3 \\times 3 \\times 3$\u5e26\u7b26\u53f7\u7684\u7f51\u683c\u7684\u7b26\u53f7\u6bd4\u8f83\u5b9e\u73b0:<br \/>\n$\\\\$ $\\cdot$ \u7c97\u7cd9\u8fb9\u4e2d\u70b9\u7684\u7b26\u53f7\u5fc5\u987b\u4e0e\u8be5\u8fb9\u4e24\u4e2a\u7aef\u70b9\u4e2d\u7684\u81f3\u5c11\u4e00\u4e2a\u7b26\u53f7\u4e00\u81f4.<br \/>\n$\\\\$ $\\cdot$ \u7c97\u7cd9\u9762\u4e2d\u5fc3\u7684\u7b26\u53f7\u5fc5\u987b\u4e0e\u8be5\u9762\u56db\u4e2a\u89d2\u70b9\u4e2d\u7684\u81f3\u5c11\u4e00\u4e2a\u7b26\u53f7\u4e00\u81f4.<br \/>\n$\\\\$ $\\cdot$ \u7c97\u7cd9\u7acb\u65b9\u4f53\u4e2d\u5fc3\u7684\u7b26\u53f7\u5fc5\u987b\u4e0e\u8be5\u7acb\u65b9\u4f53\u516b\u4e2a\u89d2\u70b9\u4e2d\u7684\u81f3\u5c11\u4e00\u4e2a\u7b26\u53f7\u4e00\u81f4.<br \/>\n$\\\\$ \u56fe11\u5c55\u793a\u4e86\u4e09\u4e2a\u5e26\u7b26\u53f7\u7684\u56db\u53c9\u6811\u7684\u51cf\u9762\u5019\u9009\u6848\u4f8b. \u5de6\u4fa7\u56db\u53c9\u6811\u7684\u5bf9\u5076\u7b49\u503c\u9762\u5305\u542b\u4e24\u4e2a\u4e0d\u540c\u7684\u8fde\u901a\u7ec4\u4ef6. \u5728\u6b64\u60c5\u51b5\u4e0b, \u7b2c\u4e00\u4e2a\u68c0\u67e5\u5224\u5b9a\u8be5\u6298\u53e0\u4e3a\u4e0d\u5b89\u5168\u64cd\u4f5c, \u56e0\u4e3a\u6298\u53e0\u540e\u7684\u56db\u53c9\u6811\u7b49\u503c\u9762\u4e3a\u975e\u6d41\u5f62\u7ed3\u6784. \u4e2d\u95f4\u56db\u53c9\u6811\u7684\u5bf9\u5076\u7b49\u503c\u9762\u540c\u6837\u5305\u542b\u4e24\u4e2a\u8fde\u901a\u7ec4\u4ef6, \u4f46\u7b2c\u4e09\u4e2a\u68c0\u67e5\u5224\u5b9a\u5176\u4e0d\u53ef\u5b89\u5168\u6298\u53e0, \u56e0\u4e3a\u8be5\u56db\u53c9\u6811\u7684\u5de6\u4fa7\u8fb9\u65e0\u6cd5\u901a\u8fc7\u62d3\u6251\u7b49\u4ef7\u6027\u9a8c\u8bc1. \u53f3\u4fa7\u56db\u53c9\u6811\u7684\u7b26\u53f7\u914d\u7f6e\u6ee1\u8db3\u6240\u6709\u4e09\u9879\u68c0\u67e5\u6761\u4ef6, \u56e0\u6b64\u8be5\u56db\u53c9\u6811\u53ef\u88ab\u5b89\u5168\u6298\u53e0.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/three_signed_quadtrees_and_their_dual_contours.png\" alt=\"\" width=\"607\" height=\"187\" class=\"aligncenter size-full wp-image-5023\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/three_signed_quadtrees_and_their_dual_contours.png 607w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/three_signed_quadtrees_and_their_dual_contours-300x92.png 300w\" sizes=\"(max-width: 607px) 100vw, 607px\" \/><center><em>\u56fe11: \u4e09\u4e2a\u5e26\u7b26\u53f7\u7684\u56db\u53c9\u6811\u53ca\u5176\u5bf9\u5076\u7b49\u503c\u9762.<\/em><\/center><\/p>\n<p>\u8fd9\u4e9b\u7b26\u53f7\u68c0\u67e5\u7684\u6b63\u786e\u6027\u8bc1\u660e\u57fa\u4e8e\u6309\u7ef4\u5ea6\u9012\u589e\u987a\u5e8f\u5efa\u7acb\u7c97\u7cd9\u7acb\u65b9\u4f53\u5b50\u9762\u7684\u62d3\u6251\u7b49\u4ef7\u6027. \u56fe9\u4e2d\u7684\u53f3\u4fa7\u7f51\u683c\u5c55\u793a\u4e86\u673a\u68b0\u96f6\u4ef6\u51cf\u9762\u7248\u672c\u7684\u4e00\u4e2a\u793a\u4f8b, \u8be5\u51cf\u9762\u7ecf\u5386\u4e86\u5bfc\u81f4\u7f51\u683c\u65ad\u5f00\u7684\u62d3\u6251\u53d8\u5316. \u56fe9\u4e2d\u7684\u4e2d\u95f4\u7f51\u683c\u5c55\u793a\u4e86\u5728\u62d3\u6251\u68c0\u67e5\u4fdd\u62a4\u4e0b\u5b8c\u6210\u5b89\u5168\u51cf\u9762\u540e\u7684\u7ed3\u679c, \u8be5\u68c0\u67e5\u963b\u6b62\u4e86\u8fdb\u4e00\u6b65\u7684\u4e0d\u5b89\u5168\u51cf\u9762.<\/p>\n<p><strong>3.2 \u591a\u6750\u8d28\u60c5\u51b5<\/strong><\/p>\n<p>\u524d\u6587\u8ba8\u8bba\u7684\u7b49\u503c\u9762\u751f\u6210\u4e0e\u51cf\u9762\u65b9\u6cd5\u7684\u4e00\u4e2a\u663e\u8457\u4f18\u52bf\u5728\u4e8e, \u8fd9\u4e9b\u65b9\u6cd5\u80fd\u591f\u8f7b\u677e\u5904\u7406\u591a\u6750\u8d28\u60c5\u51b5(\u65e0\u9700\u989d\u5916\u590d\u6742\u5ea6). \u5e78\u8fd0\u7684\u662f, \u524d\u4e00\u5c0f\u8282\u63cf\u8ff0\u7684\u5b89\u5168\u6027\u6d4b\u8bd5\u4e5f\u53ef\u901a\u8fc7\u4e00\u4e2a\u5fae\u5c0f\u8c03\u6574\u6269\u5c55\u81f3\u591a\u6750\u8d28\u533a\u57df\u7684\u7b49\u503c\u9762.<br \/>\n$\\\\$ \u8868\u9762\u4e0a, \u6311\u6218\u5728\u4e8e\u591a\u6750\u8d28\u533a\u57df\u7684\u7b49\u503c\u9762\u5728\u53cc\u6750\u8d28\u610f\u4e49\u4e0b\u672c\u8d28\u4e0a\u662f\u975e\u6d41\u5f62\u7684. \u4f8b\u5982, \u56fe12\u5c55\u793a\u4e86\u4e09\u4e2a\u5206\u79bb\u4e09\u79cd\u6750\u6599\u7684\u5bf9\u5076\u7b49\u503c\u9762\u793a\u4f8b. \u5176\u4e2d\u4e24\u4e2a\u7b49\u503c\u9762\u5728\u67d0\u4e2a\u9876\u70b9\u5904\u5b58\u5728\u4e09\u79cd\u6750\u8d28\u7684\u4ea4\u754c. \u9700\u8981\u6ce8\u610f\u7684\u662f, \u82e5\u5355\u72ec\u8003\u5bdf\u6bcf\u79cd\u6750\u8d28\u533a\u57df\u7684\u8fb9\u754c, \u5219\u4ecd\u53ef\u5224\u5b9a\u8be5\u90e8\u5206\u5bf9\u5076\u7b49\u503c\u9762\u662f\u5426\u4e3a\u6d41\u5f62. \u5177\u4f53\u800c\u8a00, \u82e5\u6bcf\u79cd\u6750\u8d28\u533a\u57df\u7684\u8fb9\u754c\u5747\u4e3a\u6d41\u5f62, \u5219\u8be5\u591a\u6750\u8d28\u5bf9\u5076\u7b49\u503c\u9762\u4e3aQuasi\u6d41\u5f62(\u6839\u636eQuasi\u6d41\u5f62\u7684\u5b9a\u4e49, \u4e00\u5bf9$d$\u7ef4Cells\u53ea\u80fd\u901a\u8fc7\u5171\u4eab\u7684$d &#8211; 1$\u7ef4Cell\u76f8\u8fde, \u8be6\u89c1\u53c2\u8003\u6750\u65999, 10). \u5728\u53cc\u6750\u8d28\u60c5\u51b5\u4e0b, Quasi\u6d41\u5f62\u7b49\u4e8e\u6d41\u5f62.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/three_multi_material_quadtrees_and_their_dual_contours.png\" alt=\"\" width=\"606\" height=\"190\" class=\"aligncenter size-full wp-image-5026\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/three_multi_material_quadtrees_and_their_dual_contours.png 606w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2026\/01\/three_multi_material_quadtrees_and_their_dual_contours-300x94.png 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><center><em>\u56fe12: \u4e09\u4e2a\u5e26\u7b26\u53f7\u7684\u56db\u53c9\u6811\u53ca\u5176\u5bf9\u5076\u7b49\u503c\u9762.<\/em><\/center><\/p>\n<p>\u591a\u6750\u8d28\u5b89\u5168\u6027\u6d4b\u8bd5\u5224\u65ad\u7ecf\u7b80\u5316\u540e\u7684\u5bf9\u5076\u7b49\u503c\u9762(\u901a\u5e38\u4e3aQuasi\u6d41\u5f62) \u662f\u5426\u662f\u4fdd\u6301\u62d3\u6251\u7684. \u4e0e\u4e4b\u524d\u7c7b\u4f3c, \u8fd9\u4e00\u9650\u5236\u5e76\u975e\u7279\u522b\u6210\u95ee\u9898, \u56e0\u4e3a\u591a\u6750\u8d28\u7b49\u503c\u9762\u7684\u5927\u90e8\u5206\u533a\u57df\u672c\u8d28\u4e0a\u662fQuasi\u6d41\u5f62. \u8fd9\u4e00\u65b0\u6d4b\u8bd5\u540c\u6837\u5206\u4e3a\u4e09\u4e2a\u9636\u6bb5, \u4e0e\u53cc\u6750\u8d28\u6d4b\u8bd5\u5b8c\u5168\u76f8\u540c, \u4f46\u9700\u66ff\u6362\u7b2c\u4e00\u9636\u6bb5\u548c\u7b2c\u4e8c\u9636\u6bb5\u7684\u68c0\u67e5\u89c4\u5219: \u5c06\u5224\u65ad\u7acb\u65b9\u4f53\u5185\u90e8\u7b49\u503c\u9762\u662f\u5426\u4e3a\u6d41\u5f62\u7684\u6d4b\u8bd5\u66ff\u6362\u4e3a\u5224\u65ad\u5176\u662f\u5426\u4e3aQuasi\u6d41\u5f62\u7684\u7b49\u6548\u6d4b\u8bd5, \u800c\u7b2c\u4e09\u9636\u6bb5\u7684\u7d22\u5f15\u6d4b\u8bd5\u4fdd\u6301\u4e0d\u53d8.<br \/>\n$\\\\$ \u4e0e\u6d41\u5f62\u60c5\u51b5\u7c7b\u4f3c, \u591a\u6750\u8d28\u7acb\u65b9\u4f53\u7684Quasi\u6d41\u5f62\u6d4b\u8bd5\u6d89\u53ca\u5c06\u7acb\u65b9\u4f53\u4e2d\u7aef\u70b9\u5177\u6709\u76f8\u540c\u6750\u8d28\u7d22\u5f15\u7684\u8fb9\u8fdb\u884c\u6298\u53e0. \u6b64\u65f6, \u8be5\u7acb\u65b9\u4f53\u7684\u5bf9\u5076\u7b49\u503c\u9762\u662fQuasi\u6d41\u5f62\u5f53\u4e14\u4ec5\u5f53\u6298\u53e0\u540e\u7684\u8fb9\u56fe\u5f62\u6210\u4e00\u4e2a\u5355\u5f62(\u5373\u70b9, \u7ebf\u6bb5, \u4e09\u89d2\u5f62\u6216\u56db\u9762\u4f53). \u4e0e\u53cc\u6750\u8d28\u60c5\u51b5\u7c7b\u4f3c, \u8be5\u6d4b\u8bd5\u7684\u503c\u53ef\u4ee5\u9884\u5148\u8ba1\u7b97\u5e76\u5b58\u50a8\u5728\u4e00\u4e2a\u5927\u5c0f\u4e3a$4^8$\u7684\u67e5\u627e\u8868\u4e2d.(\u82e5\u7acb\u65b9\u4f53\u5305\u542b5\u79cd\u6216\u66f4\u591a\u4e0d\u540c\u7684\u6750\u8d28\u7d22\u5f15, \u5176\u8fb9\u56fe\u65e0\u6cd5\u6298\u53e0\u4e3a\u5355\u5f62.) \u8be5\u6d4b\u8bd5\u7684\u6b63\u786e\u6027\u53ef\u901a\u8fc7\u4ee5\u4e0b\u65b9\u6cd5\u9a8c\u8bc1: \u9009\u62e9\u7acb\u65b9\u4f53\u4e0a\u7684\u4e00\u4e2a\u6750\u8d28\u7d22\u5f15, \u5e76\u5c06\u6240\u6709\u5176\u5b83\u7d22\u5f15\u89c6\u4e3a\u7b49\u6548\u6750\u8d28. \u6b64\u65f6\u6298\u53e0\u540e\u7684\u8fb9\u56fe\u5f62\u6210\u7ebf\u6bb5, \u8868\u660e\u5bf9\u5e94\u6240\u9009\u7d22\u5f15\u7684\u7b49\u503c\u9762\u90e8\u5206\u4e3a\u6d41\u5f62.<br \/>\n$\\\\$ \u56fe12\u5c55\u793a\u4e86\u4e24\u4e2a\u591a\u6750\u8d28\u56db\u53c9\u6811\u7684\u7b80\u5316\u5019\u9009\u6848\u4f8b. \u5de6\u4fa7\u56db\u53c9\u6811\u56e0\u7b2c\u4e09\u9636\u6bb5\u68c0\u67e5\u5728\u5e95\u90e8\u8fb9\u5931\u8d25(\u5982\u4e2d\u70b9\u7b26\u53f7\u4e0e\u7aef\u70b9\u4e0d\u4e00\u81f4) \u800c\u88ab\u62d2\u7edd\u7b80\u5316. \u4e2d\u95f4\u56db\u53c9\u6811\u901a\u8fc7\u4e86\u6240\u6709\u4e09\u4e2a\u68c0\u67e5, \u5e76\u6298\u53e0\u6210\u53f3\u4fa7\u7684\u56db\u53c9\u6811. \u9700\u8981\u6ce8\u610f\u7684\u662f, \u7b80\u5316\u540e\u7684\u56db\u53c9\u6811\u7b49\u503c\u9762\u4e3aQuasi\u6d41\u5f62, \u56e0\u4e3a\u8be5\u6b63\u65b9\u5f62\u7684\u8fb9\u56fe\u6298\u53e0\u540e\u5f62\u6210\u4e86\u4e00\u4e2a\u4e09\u89d2\u5f62.<br \/>\n$\\\\$ \u56fe7\u5c55\u793a\u4e86\u56fe5\u4e2d\u6c49\u5b57\u7acb\u65b9\u4f53\u7684\u4e24\u4e2a\u51cf\u9762\u7248\u672c. \u5de6\u4fa7\u7248\u672c\u5728\u51cf\u9762\u8fc7\u7a0b\u4e2d\u672a\u91c7\u7528\u4efb\u4f55\u4fdd\u6301\u62d3\u6251\u7684\u63aa\u65bd, \u5bfc\u81f4\u539f\u672c\u5206\u79bb\u7684\u6c49\u5b57\u7b14\u753b\u90e8\u5206\u878d\u5408\u5728\u4e00\u8d77. \u53f3\u4fa7\u7248\u672c\u901a\u8fc7\u591a\u6750\u8d28\u6d4b\u8bd5\u5b9e\u73b0\u4e86\u4fdd\u6301\u62d3\u6251\u7684\u51cf\u9762, \u4fdd\u7559\u4e86\u5b57\u7b26\u5404\u533a\u57df\u7684\u72ec\u7acb\u6027(\u51cf\u9762\u540e\u4ecd\u4fdd\u6301\u5206\u79bb).<\/p>\n<blockquote>\n<p>\u9644\u6ce8:<br \/>\n$\\\\$ \u540d\u79f0\u4e2d\u7684&#8221;\u5bf9\u5076&#8221; \u6e90\u4e8e\u7f51\u683c\u4e2d\u7684\u5355\u5143\u5728\u751f\u6210\u7684\u7f51\u683c\u4e2d\u8f6c\u6362\u4e3a\u9876\u70b9, \u8fd9\u4e00\u7279\u6027\u4e0e\u5bf9\u5076\u56fe\u76f8\u5173.<\/p>\n<\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>\u76ee\u524d\u5df2\u7ecf\u5728UE\u63d2\u4ef6\u4e2d\u5b9e\u73b0\u4e86GPU\u7248\u672c\u7684Marching Cube\u7b97\u6cd5, \u63a5\u4e0b\u6765\u7ee7\u7eed\u653b\u514bDual Contour &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2026\/01\/30\/interpretation_paper_dual_contouring_hermite_data\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u300aDual Contouring of Hermite Data\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":[24,5],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/4931"}],"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=4931"}],"version-history":[{"count":134,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/4931\/revisions"}],"predecessor-version":[{"id":5082,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/4931\/revisions\/5082"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=4931"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=4931"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=4931"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}