{"id":3581,"date":"2025-03-31T18:17:28","date_gmt":"2025-03-31T10:17:28","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=3581"},"modified":"2025-05-08T13:56:26","modified_gmt":"2025-05-08T05:56:26","slug":"new_quadric_metric_describing_meshes_appearance_attributes_paper_points_interpretation","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2025\/03\/31\/new_quadric_metric_describing_meshes_appearance_attributes_paper_points_interpretation\/","title":{"rendered":"\u300aNew Quadric Metric for Simplifying Meshes with Appearance Attributes\u300b\u8bba\u6587\u8981\u70b9\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>\u6700\u8fd1\u5728\u590d\u4e60Mesh\u51cf\u9762\u76f8\u5173\u7684\u77e5\u8bc6\u70b9, \u5f53\u521d\u8bfb\u7f62Hoppe H. New quadric metric for simplifying meshes with appearance attributes[C]\/\/Proceedings Visualization&#8217;99 (Cat. No. 99CB37067). IEEE, 1999: 59-510.\u8fd9\u7bc7\u8bba\u6587\u540e\u5f00\u53d1\u4e86Mesh\u81ea\u52a8\u51cf\u9762\u6a21\u5757, \u4f46\u5176\u5b9e\u5e76\u6ca1\u6709\u5b8c\u5168\u5403\u900f\u8fd9\u7bc7\u8bba\u6587, \u5bf9\u8bb8\u591a\u7406\u8bba\u7ec6\u8282\u4ea6\u662f\u56eb\u56f5\u541e\u67a3. \u65f6\u9694\u8fd1\u4e24\u5e74, \u91cd\u65b0\u9605\u8bfb\u8fd9\u7bc7\u8bba\u6587, &#8220;\u67f3\u6697\u82b1\u660e\u53c8\u4e00\u6751&#8221;, \u5bf9\u4e8e\u4e4b\u524d\u8bb8\u591a\u4ee4\u81ea\u5df1\u8ff7\u60d1\u7684\u7406\u8bba\u7ec6\u8282\u8c41\u7136\u5f00\u6717~<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. Hoppe H. New quadric metric for simplifying meshes with appearance attributes[C]\/\/Proceedings Visualization&#8217;99 (Cat. No. 99CB37067). IEEE, 1999: 59-510.<br \/>\n2. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/686734856\">\u70b9\u4e91\u6cd5\u5411\u91cf\u548c\u5e73\u9762\u65b9\u7a0b<\/a><\/p>\n<p><strong>1. \u4f20\u7edf\u7684\u5173\u4e8e\u51e0\u4f55\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf<\/strong><\/p>\n<p>\u4f20\u7edf\u7684\u5173\u4e8e\u51e0\u4f55\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf\u5728\u539f\u59cbMesh\u7684\u6bcf\u4e2a\u9762$f$\u4e0a\u5b9a\u4e49\u4e86\u4e00\u4e2a\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf$Q^f(\\mathbf{v})$, \u8be5\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf\u503c\u7b49\u4e8e\u70b9$\\mathbf{v} = (\\mathbf{p}) \\in \\mathbb{R}^3$\u5230\u5305\u542b\u9762$f$\u7684\u5e73\u9762\u7684\u8ddd\u79bb\u7684\u5e73\u65b9. \u539f\u59cbMesh\u7684\u6bcf\u4e2a\u9876\u70b9$v$\u4e0a\u5b9a\u4e49\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf\u503c\u7b49\u4e8e\u9876\u70b9$v$\u7684\u76f8\u90bb\u9762\u7684\u6309\u9762\u79ef\u52a0\u6743\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf\u503c\u4e4b\u548c:\\begin{equation}Q^v(\\mathbf{v}) = \\sum_{f \\ni v} area(f) \\cdot Q^f(\\mathbf{v}).\\label{quadratic_error_functions_defined_on_vertices}\\end{equation}\u6bcf\u6b21\u8fb9\u584c\u7f29$(v_1, v_2) \\to v$\u540e, \u65b0\u9876\u70b9$v$\u7684\u4f4d\u7f6e\u5c06\u88ab\u66f4\u65b0\u4e3a$\\mathbf{v}$ s.t. $Q^v(\\mathbf{v}) = $$ Q^{v_1}( $$ \\mathbf{v}) + Q^{v_2}(\\mathbf{v})$\u6781\u5c0f\u5316, \u4e14\u4e0b\u4e00\u6761\u584c\u7f29\u7684\u8fb9\u5c06\u5177\u6709\u6700\u4f4e\u7684$Q^v(\\mathbf{v})$\u6781\u5c0f\u503c.<br \/>\n$\\\\$ \u63a5\u4e0b\u6765\u63a8\u5bfc\u7ed9\u5b9a\u9762$f = (v_1, v_2, v_3)$\u4e0a\u7684$Q^f(\\mathbf{v})$. \u4ee4$\\mathbf{v} = (\\mathbf{p})$, \u5219$\\mathbf{p}$\u5230\u5305\u542b$f$\u7684\u5e73\u9762$P \\subset \\mathbb{R}^3$\u7684\u6709\u5411\u8ddd\u79bb\u4e3a$\\mathbf{n}^T \\mathbf{p} + d$, \u5176\u4e2d,$$\\mathbf{n} = (\\mathbf{p}_2 &#8211; \\mathbf{p}_1) \\times (\\mathbf{p}_3 &#8211; \\mathbf{p}_1) \/ \\left \\| \\mathbf{n} = (\\mathbf{p}_2 &#8211; \\mathbf{p}_1) \\times (\\mathbf{p}_3 &#8211; \\mathbf{p}_1) \\right \\| $$\u4e3a\u9762\u6cd5\u7ebf, \u6807\u91cf$d = -\\mathbf{n}^T \\mathbf{p}_1$. \u6b64\u5916, \u53e6\u5916\u4e00\u79cd\u8ba1\u7b97\u4e0a\u8ff0\u53c2\u6570\u7684\u65b9\u5f0f\u662f\u6c42\u89e3\u4e0b\u8ff0\u7ebf\u6027\u65b9\u7a0b\u7ec4$$\\begin{pmatrix}<br \/>\n\\mathbf{p}_1^T &#038; 1 \\\\<br \/>\n\\mathbf{p}_2^T &#038; 1 \\\\<br \/>\n\\mathbf{p}_3^T &#038; 1 \\\\<br \/>\n\\end{pmatrix} \\begin{pmatrix}<br \/>\n\\mathbf{n} \\\\<br \/>\n\\\\<br \/>\nd<br \/>\n\\end{pmatrix} = \\begin{pmatrix}<br \/>\n0 \\\\<br \/>\n0 \\\\<br \/>\n0<br \/>\n\\end{pmatrix},$$\u9644\u52a0\u7684\u7ea6\u675f\u6761\u4ef6\u4e3a$\\left \\| \\mathbf{n} \\right \\| = 1$.<br \/>\n$\\\\$ \u6545\u70b9$\\mathbf{p}$\u5230\u5e73\u9762$P$\u7684\u8ddd\u79bb\u7684\u5e73\u65b9\u4e3a$$Q^f(\\mathbf{v} = (\\mathbf{p})) = (\\mathbf{n}^T\\mathbf{v} + d)^2 = \\mathbf{v}^T (\\mathbf{n} \\mathbf{n}^T) \\mathbf{v} + 2d\\mathbf{n}^T \\mathbf{v} + d^2,$$\u5176\u53ef\u4ee5\u8868\u793a\u4e3a\u4e00\u4e2a\u4e8c\u6b21\u51fd\u6570$\\mathbf{v}^T \\mathbf{A} \\mathbf{v} + 2\\mathbf{b}^T \\mathbf{v} + c$, \u5176\u4e2d, $\\mathbf{A}$\u4e3a$3 \\times 3$\u5bf9\u79f0\u77e9\u9635, $\\mathbf{b}$\u4e3a3\u7ef4\u5217\u5411\u91cf, $c$\u4e3a\u6807\u91cf. \u4ece\u800c,$$Q^f = (\\mathbf{A}, \\mathbf{b}, c) = (\\mathbf{n} \\mathbf{n}^T, d\\mathbf{n}, d^2)$$\u53ef\u75286 + 3 + 1 = 10\u4e2a\u7cfb\u6570\u8fdb\u884c\u5b58\u50a8. \u8fd9\u79cd\u8868\u793a\u7684\u4f18\u70b9\u662f, \u5f0f$\\ref{quadratic_error_functions_defined_on_vertices}$\u4e2d\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf$Q^v$\u662f\u901a\u8fc7\u4e0a\u8ff0\u7cfb\u6570\u5411\u91cf\u7684\u7b80\u5355\u7ebf\u6027\u7ec4\u5408\u5f97\u5230\u7684.<br \/>\n$\\\\$ \u5728\u4e00\u6b21\u8fb9\u584c\u7f29\u540e, \u53ef\u5f97\u65b0\u7684\u9876\u70b9\u4f4d\u7f6e$\\mathbf{v}_{min}$ s.t. $Q^v(\\mathbf{v})$\u6781\u5c0f\u5316, \u5373\u68af\u5ea6$$\\bigtriangledown Q^v(\\mathbf{v}) = 2 \\mathbf{A} \\mathbf{v} + 2\\mathbf{b} = 0,$$\u8ba1\u7b97\u65b0\u7684\u9876\u70b9\u4f4d\u7f6e$\\mathbf{v}_{min}$\u7684\u65b9\u5f0f\u662f\u6c42\u89e3\u4e0b\u8ff0\u7ebf\u6027\u65b9\u7a0b\u7ec4\\begin{equation}\\mathbf{A} \\mathbf{v}_{min} = -\\mathbf{b}.\\label{found_vertex_position}\\end{equation}<br \/>\n<strong>2. \u4f20\u7edf\u7684\u5173\u4e8e\u51e0\u4f55\u4e0e\u5176\u5b83\u9876\u70b9\u5c5e\u6027\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf<\/strong><\/p>\n<p>Garland\u4e0eHeckbert\u6269\u5c55\u4e86\u4ed6\u4eec\u7684\u6846\u67b6\u7528\u4e8e\u8ba1\u7b97\u5305\u542b\u5176\u5b83\u9876\u70b9\u5c5e\u6027\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf. \u4ed6\u4eec\u7684\u65b9\u6cd5\u662f\u5c06$\\mathbb{R}^3$\u4e2d\u70b9\u5230\u5e73\u9762\u7684\u8ddd\u79bb\u5ea6\u91cf\u63a8\u5e7f\u81f3$\\mathbb{R}^{3 + m}$\u4e2d\u7684\u70b9\u5230\u8d85\u5e73\u9762\u7684\u8ddd\u79bb\u5ea6\u91cf, \u5176\u4e2d, $m$\u4e3a\u5176\u5b83\u9876\u70b9\u5c5e\u6027\u7684\u7ef4\u5ea6\u548c. \u5373, \u5bf9\u4e8e$\\mathbf{v} = \\begin{pmatrix}<br \/>\n\\mathbf{p} \\\\<br \/>\n\\mathbf{s}<br \/>\n\\end{pmatrix} \\in \\mathbb{R}^{3 + m}$, $Q^f(\\mathbf{v})$\u88ab\u5b9a\u4e49\u4e3a$\\mathbb{R}^{3 + m}$\u4e2d\u4ece$\\mathbf{v}$\u5230\u7531\u4e09\u4e2a\u9876\u70b9$(\\mathbf{v}_1, $$ \\mathbf{v}_2, \\mathbf{v}_3)$\u5f20\u6210\u7684\u4eff\u5c04\u5b50\u7a7a\u95f4$P&#8217; $$ \\subset \\mathbb{R}^{3 + m}$\u7684\u8ddd\u79bb.<br \/>\n$\\\\$ \u4ee4$\\mathbf{v}&#8217;$\u8868\u793a$\\mathbf{v}$\u5728\u8be5\u4eff\u5c04\u5b50\u7a7a\u95f4\u4e0a\u7684\u6295\u5f71. \u8bef\u5dee$Q^f(\\mathbf{v}) = \\left \\| \\mathbf{v} &#8211; \\mathbf{v}&#8217; \\right \\|^2$\u53ef\u4ee5\u770b\u4f5c\u662f\u51e0\u4f55\u8ddd\u79bb\u8bef\u5dee$\\left \\| \\mathbf{p} &#8211; \\mathbf{p}&#8217; \\right \\|^2$\u4e0e\u5c5e\u6027\u8ddd\u79bb\u8bef\u5dee$\\left \\| \\mathbf{s} &#8211; \\mathbf{s}&#8217; \\right \\|^2$\u8fd9\u4e24\u9879\u7684\u603b\u548c. \u6ce8\u610f\u5230\u70b9$\\mathbf{p}&#8217;$\u5e76\u975e\u5982\u524d\u6240\u8ff0\u4e3a\u70b9$\\mathbf{p}$\u5728\u5e73\u9762$P \\subset \\mathbb{R}^3$\u4e0a\u7684\u6295\u5f71. Garland\u4e0eHeckbert\u5e76\u975e\u5c06$\\mathbf{v}$\u4e0e$\\mathbb{R}^3$\u4e2d\u6b27\u5f0f\u8ddd\u79bb\u6700\u8fd1\u7684\u70b9\u8fdb\u884c\u6bd4\u8f83, \u800c\u662f\u4e0e\u5c5e\u6027\u503c\u66f4\u63a5\u8fd1\u7684\u70b9\u8fdb\u884c\u5bf9\u6bd4. \u7136\u800c, \u8be5\u5bf9\u6bd4\u7b56\u7565\u53ef\u80fd\u4f4e\u4f30\u4e86$\\mathbf{v}$\u4e0e$\\mathbf{v}&#8217;$\u5728$\\mathbb{R}^3$\u4e2d\u7684\u6b27\u6c0f\u8ddd\u79bb.<br \/>\n$\\\\$ \u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf$Q^f(\\mathbf{v})$\u7531\u4e00\u4e2a$(3 + m) \\times (3 + m)$\u77e9\u9635$\\mathbf{A}$, \u4e00\u4e2a$3 + m$\u7ef4\u5217\u5411\u91cf$\\mathbf{b}$\u4e0e\u4e00\u4e2a\u6807\u91cf$c$\u7ec4\u6210. \u7531\u4e8e\u7531\u4e0a\u8ff0\u516c\u5f0f\u5f97\u5230\u7684\u77e9\u9635$\\mathbf{A}$\u662f\u7a20\u5bc6\u7684\u5b9e\u5bf9\u79f0\u77e9\u9635, \u6545$Q$\u7684\u5b58\u50a8\u9700\u8981\u5171\u8ba1$$(1 + (3 + m))(3 + m) \/ 2 + (3 + m) + 1 = (4 + m)(5 + m) \/ 2$$\u4e2a\u7cfb\u6570, \u5373\u7cfb\u6570\u6570\u91cf\u662f\u4e00\u4e2a\u5173\u4e8e$m$\u7684\u4e8c\u6b21\u51fd\u6570.<br \/>\n$\\\\$ \u4e3a\u4e86\u6743\u8861\u51e0\u4f55\u7cbe\u5ea6\u548c\u5c5e\u6027\u7cbe\u5ea6, \u7528\u6237\u4e3a\u6bcf\u4e2a\u5c5e\u6027$j \\in \\{ 1 \\dots m \\}$\u6307\u5b9a\u76f8\u5bf9\u91cd\u8981\u6027\u6743\u91cd$\\lambda_j$, \u8be5\u76f8\u5bf9\u91cd\u8981\u6027\u6743\u91cd$\\lambda_j$\u9884\u5148\u4e58\u4ee5\u5c5e\u6027\u503c, \u5bf9$\\mathbb{R}^{3 + m}$\u4e2d\u7684\u90e8\u5206\u8f74\u8fdb\u884c\u4e86\u6709\u6548\u7684\u7f29\u653e. \u4e3a\u4e86\u89c4\u907fMesh\u5927\u5c0f\u5bf9\u4e8e\u7b97\u6cd5\u7684\u5f71\u54cd, \u5e94\u5bf9Mesh\u8fdb\u884c\u5f52\u4e00\u5316, \u4f7f\u5176\u5916\u63a5\u7403\u4e3a\u5355\u4f4d\u7403.<\/p>\n<p><strong>3. \u672c\u6587\u63d0\u51fa\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf<\/strong><\/p>\n<p>\u672c\u6587\u7684\u8d21\u732e\u662f\u5f15\u5165\u4e86\u4e00\u79cd\u65b0\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf, \u8be5\u5ea6\u91cf\u57fa\u4e8e3D\u7a7a\u95f4\u4e2d\u7684\u51e0\u4f55\u5bf9\u5e94\u5173\u7cfb\u5b9a\u4e49\u4e86\u51e0\u4f55\u8bef\u5dee\u4e0e\u5c5e\u6027\u8bef\u5dee. \u672c\u6587\u4e0d\u662f\u5c06\u7ed9\u5b9a\u70b9\u6295\u5f71\u5230\u62bd\u8c61\u7684\u9ad8\u7ef4\u7a7a\u95f4$\\mathbb{R}^{3 + m}$\u4e2d\u7684Mesh\u9762\u4e0a, \u800c\u662f\u5728$\\mathbb{R}^3$\u4e2d\u8fdb\u884c\u6295\u5f71, \u5e76\u6839\u636e\u8fd9\u79cd\u5bf9\u5e94\u5173\u7cfb\u8ba1\u7b97\u51e0\u4f55\u8bef\u5dee\u4e0e\u5c5e\u6027\u8bef\u5dee.<br \/>\n$\\\\$ \u9762$f$\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf\u5b9a\u4e49\u4e3a$$Q^f(\\mathbf{v} = \\begin{pmatrix}<br \/>\n\\mathbf{p} \\\\<br \/>\n\\mathbf{s}<br \/>\n\\end{pmatrix}) = Q^f_p(\\mathbf{v}) + \\sum^m_{j = 1} Q^f_{s_j}(\\mathbf{v}),$$\u5176\u4e2d, \u51e0\u4f55\u8bef\u5dee$Q^f_p(\\mathbf{v})$\u4e3a\u4ece$\\mathbf{p}$\u5230\u5728\u5305\u542b$f$\u7684\u5e73\u9762$P \\subset \\mathbb{R}^3$\u4e0a\u7684\u6295\u5f71\u70b9$\\mathbf{p}&#8217;$\u7684\u8ddd\u79bb\u7684\u5e73\u65b9, \u5c5e\u6027\u8bef\u5dee$Q^f_{s_j}(\\mathbf{v})$\u4e3a\u5728\u6295\u5f71\u70b9$\\mathbf{p}&#8217;$\u5904\u6839\u636e\u9762$f$\u7684\u9876\u70b9\u5c5e\u6027\u503c\u63d2\u503c\u7684\u5c5e\u6027\u503c$\\mathbf{s}&#8217;$\u4e0e$\\mathbf{s}$\u7684\u8ddd\u79bb\u7684\u5e73\u65b9. \u63a5\u4e0b\u6765\u63a8\u5bfc\u8fd9\u4e9b\u9879\u7684\u8ba1\u7b97\u8fc7\u7a0b.<br \/>\n$\\\\$ \u672c\u6587\u6240\u4f7f\u7528\u7684\u51e0\u4f55\u8bef\u5dee\u9879\u4ec5\u662f\u901a\u8fc7\u5c06\u4f20\u7edf\u7684\u51e0\u4f55\u8bef\u5dee\u9879\u8fdb\u884c\u7b80\u5355\u6269\u5c55\u5f97\u5230\u7684:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/geometric_error_term_zero_extended_version.jpg\" alt=\"\" width=\"623\" height=\"127\" class=\"aligncenter size-full wp-image-3710\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/geometric_error_term_zero_extended_version.jpg 623w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/geometric_error_term_zero_extended_version-300x61.jpg 300w\" sizes=\"(max-width: 623px) 100vw, 623px\" \/><\/p>\n<p>\u5176\u4e2d, $\\mathbf{A}$\u4e0e$\\mathbf{b}$\u4e2d\u7684\u5206\u9694\u7ebf\u6807\u8bb0\u4e86\u524d3\u884c\u4e0e\u524d3\u5217.<br \/>\n$\\\\$ \u4e3a\u4e86\u8ba1\u7b97\u5c5e\u6027\u8bef\u5dee\u9879$Q^f_{s_j}$, \u672c\u6587\u9996\u5148\u5b9a\u4e49\u4e00\u4e2a\u7ebf\u6027\u6cdb\u51fd$$\\widehat{s}_j(\\mathbf{p}) = \\mathbf{g}^T_j \\mathbf{p} + d_j$$\u7528\u4e8e\u8868\u793a\u4e8e\u70b9$\\mathbf{p} \\in \\mathbb{R}^{3}$\u5904\u7684\u671f\u671b\u5c5e\u6027\u503c. \u68af\u5ea6$\\mathbf{g}_j$\u4e0e\u6807\u91cf$d_j$\u5b9a\u4e49\u5982\u4e0b\u6240\u793a. \u81ea\u7136\u5730, $\\widehat{s}_j(\\mathbf{p})$\u5e94\u8be5\u7531\u9762\u9876\u70b9$f = (\\begin{pmatrix}<br \/>\n\\mathbf{p}_1 \\\\<br \/>\n\\mathbf{s}_1<br \/>\n\\end{pmatrix}, \\begin{pmatrix}<br \/>\n\\mathbf{p}_2 \\\\<br \/>\n\\mathbf{s}_2<br \/>\n\\end{pmatrix}, \\begin{pmatrix}<br \/>\n\\mathbf{p}_3 \\\\<br \/>\n\\mathbf{s}_3<br \/>\n\\end{pmatrix})$\u63d2\u503c\u5f97\u5230, \u4ece\u800c\u5339\u914d\u5e73\u9762$P$\u4e0a\u7684\u7ebf\u6027\u63d2\u503c\u7ed3\u679c. \u6b64\u5916, \u5bf9\u4e8e\u4efb\u610f$\\mathbf{p} \\in \\mathbb{R}^3$, $\\widehat{s}_j(\\mathbf{p})$\u5e94\u4e0e$\\mathbf{p}$\u5728$P$\u4e0a\u7684\u51e0\u4f55\u6295\u5f71\u5904\u7684\u5c5e\u6027\u503c$\\widehat{s}_j(\\mathbf{p}&#8217;)$\u76f8\u540c; \u8fd9\u7b49\u4ef7\u4e8e\u4ee4$\\mathbf{n}^T \\mathbf{g}_j = $$ 0$. \u6545$\\mathbf{g}_j$\u4e3a\u6807\u91cf\u51fd\u6570\u5728\u4e09\u89d2\u9762\u4e0a\u7684\u68af\u5ea6. \u53ef\u901a\u8fc7\u6c42\u89e3\u4e0b\u8ff0$4 \\times 4$\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u8ba1\u7b97\u53c2\u6570$(\\mathbf{g}_j, d_j)$$$\\begin{pmatrix}<br \/>\n\\mathbf{p}^T_1 &#038; 1 \\\\<br \/>\n\\mathbf{p}^T_2 &#038; 1 \\\\<br \/>\n\\mathbf{p}^T_3 &#038; 1 \\\\<br \/>\n\\mathbf{n}^T &#038; 1<br \/>\n\\end{pmatrix} \\begin{pmatrix}<br \/>\n\\mathbf{g}_j \\\\<br \/>\nd_j<br \/>\n\\end{pmatrix} = \\begin{pmatrix}<br \/>\ns_{1, j} \\\\<br \/>\ns_{2, j} \\\\<br \/>\ns_{3, j} \\\\<br \/>\n0<br \/>\n\\end{pmatrix}.$$\u56e0\u4e3a$Q^f_{s_j}(\\mathbf{v}) = (\\widehat{s}_j(\\mathbf{p}) &#8211; s_j)^2 = (\\mathbf{g}^T_j \\mathbf{p} + d_j &#8211; s_j)^2$, \u901a\u8fc7\u4ee3\u6570\u91cd\u6392\u53ef\u5f97$Q^f_{s_j} = $$ (\\mathbf{A}, \\mathbf{b}, c) = $<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/attribute_error_term.jpg\" alt=\"\" width=\"721\" height=\"256\" class=\"aligncenter size-full wp-image-3724\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/attribute_error_term.jpg 721w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/attribute_error_term-300x107.jpg 300w\" sizes=\"(max-width: 721px) 100vw, 721px\" \/><\/p>\n<p>\u5176\u4e2d, \u503c1\u51fa\u73b0\u4e8e$\\mathbf{A}_{3 + j, 3 + j}$\u4e2d, $-d_j$\u51fa\u73b0\u4e8e$\\mathbf{b}_{3 + j}$\u4e2d. \u4e0d\u96be\u9a8c\u8bc1, $$Q^f_{s_j} = \\begin{bmatrix}<br \/>\n\\begin{array}{c | c}<br \/>\n\\mathbf{p}^T &#038; \\cdots s_j \\cdots<br \/>\n\\end{array}<br \/>\n\\end{bmatrix} \\mathbf{A} \\begin{bmatrix}<br \/>\n\\begin{array}{c}<br \/>\n\\mathbf{p} \\\\ \\hline<br \/>\n\\vdots \\\\<br \/>\ns_j \\\\<br \/>\n\\vdots<br \/>\n\\end{array}<br \/>\n\\end{bmatrix} + \\mathbf{b}^T  \\begin{bmatrix}<br \/>\n\\begin{array}{c}<br \/>\n\\mathbf{p} \\\\ \\hline<br \/>\n\\vdots \\\\<br \/>\ns_j \\\\<br \/>\n\\vdots<br \/>\n\\end{array}<br \/>\n\\end{bmatrix} + c.$$\u5c06\u6240\u6709\u5173\u4e8e$s_j$\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf\u7d2f\u52a0\u53ef\u5f97$Q^f = (\\mathbf{A}, \\mathbf{b}, c) = $<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/new_quadric_error_metric.jpg\" alt=\"\" width=\"639\" height=\"208\" class=\"aligncenter size-full wp-image-3728\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/new_quadric_error_metric.jpg 639w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/new_quadric_error_metric-300x98.jpg 300w\" sizes=\"(max-width: 639px) 100vw, 639px\" \/><\/p>\n<p>\u6ce8\u610f\u5230, $\\mathbf{A}$\u7684\u524d3\u884c\u4e0e\u524d3\u5217\u662f\u7a20\u5bc6\u7684, \u5269\u4f59\u7684$m \\times m$\u5b50\u77e9\u9635\u4e3a\u5355\u4f4d\u77e9\u9635$I$. \u56de\u987e\u4e00\u4e0b, \u4e00\u7ec4\u6743\u91cd$\\lambda_j$\u7528\u4e8e\u7f29\u653e\u5c5e\u6027\u8bef\u5dee\u76f8\u5bf9\u4e8e\u51e0\u4f55\u8bef\u5dee\u7684\u5927\u5c0f. \u82e5\u5b9a\u4e49$Q = Q_p + $$ \\sum_j \\lambda_j^2 Q_{s_j}$, \u5219\u5b50\u77e9\u9635\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u5bf9\u5e94\u7684\u6743\u91cd\u4e3a$\\lambda_j^2$. \u672c\u6587\u4f7f\u7528\u66f4\u7b80\u5355\u7684\u65b9\u6cd5, \u5728\u6784\u9020\u548c\u8ba1\u7b97$Q$\u4e4b\u524d, \u901a\u8fc7$\\lambda_j$\u5bf9\u5c5e\u6027\u503c$s_j$\u8fdb\u884c\u9884\u7f29\u653e. \u5728\u4efb\u4f55\u60c5\u51b5\u4e0b, $m \\times m$\u5b50\u77e9\u9635\u603b\u4e3a\u4e00\u4e2a\u5355\u4f4d\u77e9\u9635\u4e0e\u4e00\u4e2a\u6807\u91cf\u56e0\u5b50\u7684\u4e58\u79ef, \u56e0\u6b64\u4ec5\u9700\u5b58\u50a8\u4e00\u4e2a\u7cfb\u6570. \u603b\u4f53\u800c\u8a00, \u5b58\u50a8$Q$\u9700\u8981\u5b58\u50a8$11 + 4m$\u4e2a\u7cfb\u6570, \u8fd9\u5173\u4e8e$m$\u662f\u7ebf\u6027\u7684, \u5982\u4e0b\u8868\u6240\u793a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/number_of_coefficients_necessary_to_represent_Q_for_various_numbers_m_of_scalar_arributes.png\" alt=\"\" width=\"676\" height=\"200\" class=\"aligncenter size-full wp-image-3787\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/number_of_coefficients_necessary_to_represent_Q_for_various_numbers_m_of_scalar_arributes.png 676w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/number_of_coefficients_necessary_to_represent_Q_for_various_numbers_m_of_scalar_arributes-300x89.png 300w\" sizes=\"(max-width: 676px) 100vw, 676px\" \/><\/p>\n<p>$\\\\$ \u67d0\u4e9b\u5c5e\u6027\u503c(\u5982\u989c\u8272\u901a\u9053) \u662f\u6709\u754c\u7684(\u5982$0 \\le r, g, b \\le 1$). \u5f0f$\\ref{found_vertex_position}$\u4e2d\u7684$\\mathbf{v}_{min}$\u53ef\u80fd\u5305\u542b\u8fd9\u4e9b\u7ebf\u6027\u4e0d\u7b49\u5f0f\u7ea6\u675f\u4e4b\u5916\u7684\u5c5e\u6027\u503c\u5206\u91cf. \u5f53\u8fd9\u79cd\u60c5\u51b5\u53d1\u751f\u65f6, \u4e00\u79cd\u56de\u9000\u7b56\u7565\u53ef\u80fd\u662f\u89e3\u51b3\u8ba1\u7b97\u4ee3\u4ef7\u66f4\u6602\u8d35\u7684\u5e26\u7ea6\u675f\u7684\u4e8c\u6b21\u89c4\u5212\u95ee\u9898. \u7136\u800c, \u672c\u6587\u9009\u62e9\u7b80\u5355\u5730\u5c06\u5c5e\u6027\u503c\u622a\u65ad\u5230\u5176\u8fb9\u754c, \u5e76\u91cd\u65b0\u8ba1\u7b97$Q^v(\\mathbf{v})$. \u540c\u7406, \u66f2\u9762\u4e0a\u7684\u6cd5\u7ebf\u5c5e\u6027\u5411\u91cf\u5e94\u4fdd\u6301\u6a21\u957f\u4e3a1\u7684\u7279\u5f81. \u7136\u800c, \u4e8c\u6b21\u7ea6\u675f\u4e8c\u6b21\u89c4\u5212\u95ee\u9898\u662f\u4e00\u4e2a\u66f4\u56f0\u96be\u7684\u95ee\u9898, \u56e0\u6b64\u672c\u6587\u9009\u62e9\u4e0d\u5bf9\u6cd5\u7ebf\u5c5e\u6027\u5206\u91cf\u8fdb\u884c\u622a\u65ad\u7b49\u64cd\u4f5c, \u5e76\u91cd\u65b0\u5bf9\u6cd5\u7ebf\u5c5e\u6027\u5411\u91cf\u8fdb\u884c\u5f52\u4e00\u5316.<\/p>\n<p><strong>4. \u5c5e\u6027\u7684\u4e0d\u8fde\u7eed\u6027<\/strong><\/p>\n<p>Mesh\u7684\u5c5e\u6027\u901a\u5e38\u662f\u4e0d\u8fde\u7eed\u7684. \u4f8b\u5982, \u6298\u75d5\u662fMesh\u4e0a\u6cd5\u7ebf\u4e0d\u8fde\u7eed\u7684\u8fb9\u7f18\u94fe. \u5728\u8f90\u5c04\u5ea6\u91cf\u5b66\u4e2d, \u5982\u679c\u76f8\u90bbPatches\u4e0d\u5e73\u884c, \u5219Patches\u4e0a\u7684\u5f3a\u5ea6\u901a\u5e38\u4e0d\u540c. \u5bf9\u8fd9\u79cd\u4e0d\u8fde\u7eed\u6027\u8fdb\u884c\u5efa\u6a21\u9700\u8981\u4e3a\u6bcf\u4e2a\u9876\u70b9\u5b58\u50a8\u591a\u7ec4\u5c5e\u6027\u503c. \u4e3a\u6b64, \u672c\u6587\u9009\u62e9\u5728\u6240\u4f7f\u7528\u7684\u6570\u636e\u7ed3\u6784\u4e2d\u5f15\u5165\u6954\u7684\u6982\u5ff5, \u5982\u4e0b\u56fe\u6240\u793a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/wedge_based_mesh_representation.jpg\" alt=\"\" width=\"293\" height=\"231\" class=\"aligncenter size-full wp-image-3744\" \/><\/p>\n<p>\u4e00\u4e2a\u9876\u70b9\u88ab\u5212\u5206\u4e3a$k \\ge 1$\u4e2a\u6954, \u6bcf\u4e2a\u6954$w_i$\u90fd\u6709\u81ea\u5df1\u7684\u5c5e\u6027\u5411\u91cf$\\mathbf{s}^i$. \u4e0e\u9876\u70b9\u76f8\u90bb\u7684\u6bcf\u4e2a\u9762\u7684\u89d2\u88ab\u6307\u5b9a\u7ed9\u5176\u4e2d\u4e00\u4e2a\u6954. \u6954$w_i$\u5904\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf$Q^{w_i}(\\mathbf{p}, $$ \\mathbf{s}^i)$\u4e3a\u5176\u76f8\u90bb\u9762\u5bf9\u5e94\u7684$Q^f$\u7684\u9762\u79ef\u52a0\u6743\u548c,\\begin{equation}Q^w(\\mathbf{v}) = \\sum_{f \\ni w} area(f) \\cdot Q^f(\\mathbf{v}),\\label{quadratic_error_functions_defined_on_wedges}\\end{equation}\u5219\u5f0f$\\ref{quadratic_error_functions_defined_on_vertices}$\u88ab\u66ff\u6362\u4e3a\\begin{equation}Q^v(\\mathbf{p}, \\mathbf{s}^1, \\dots, \\mathbf{s}^k) = \\sum^k_{i = 1} Q^{w_i}(\\mathbf{p}, \\mathbf{s}^i).\\label{replaced_quadratic_error_functions_defined_on_vertices}\\end{equation}\u65b0\u9876\u70b9\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf$Q^v$\u7684\u7ef4\u6570\u4e3a$3 + km$. \u6ce8\u610f\u5230, \u8be5\u7ef4\u6570\u53ef\u53d8\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf$Q^v$\u4e0d\u9700\u8981\u663e\u5f0f\u5b58\u50a8\u5728Mesh\u4e0a, \u56e0\u4e3a\u5f53\u8003\u8651\u8fb9\u584c\u7f29\u65f6, \u53ef\u4ee5\u7b80\u5355\u5730\u4ece$Q^{w_i}$\u6784\u9020\u5f97\u5230$Q^v$. \u6700\u5c0f\u5316\u8be5\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf$Q^v$\u53ef\u540c\u65f6\u5f97\u5230\u65b0\u7684\u9876\u70b9\u4f4d\u7f6e\u53ca\u5176\u6240\u6709\u6954\u7684\u5c5e\u6027.<br \/>\n$\\\\$ \u5bf9\u4e8e\u8fb9\u584c\u7f29, \u5fc5\u987b\u91cd\u65b0\u5b9a\u4e49\u6954\u5408\u5e76\u540e\u5176\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf\u7684\u5408\u5e76\u7b56\u7565(\u7c7b\u4f3c\u4e8e$Q^v(\\mathbf{v}) $$ = Q^{v_1}(\\mathbf{v}) + Q^{v_2}(\\mathbf{v})$). \u53c2\u89c1\u4e0b\u56fe, \u82e5$w_a$\u4e0e$w_b$\u5728\u8fb9\u574d\u584c\u524d\u5747\u4e0e\u9762$f_1$\u76f8\u90bb, \u5219\u5728\u8fb9\u574d\u584c\u540e$w_a$\u4e0e$w_b$\u76f8\u90bb, \u5e76\u91cd\u65b0\u8bb0\u4e3a$w_a&#8217;$\u4e0e$w_b&#8217;$, \u540c\u6837\u5730, \u5728\u9762$f_2$\u7684\u53e6\u4e00\u4fa7\u4ea6\u662f\u5982\u6b64.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/tests_for_wedge_unification_after_edge_collapse.jpg\" alt=\"\" width=\"463\" height=\"235\" class=\"aligncenter size-full wp-image-3756\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/tests_for_wedge_unification_after_edge_collapse.jpg 463w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/tests_for_wedge_unification_after_edge_collapse-300x152.jpg 300w\" sizes=\"(max-width: 463px) 100vw, 463px\" \/><\/p>\n<p>\u5bf9\u4e8e\u6bcf\u5bf9\u88ab\u8fb9\u584c\u7f29\u5f71\u54cd\u7684\u6954(0\u5bf9, 1\u5bf9\u62162\u5bf9), \u672c\u6587\u5c06\u5b83\u4eec\u7684\u6954\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf\u8fdb\u884c\u6c42\u548c. \u5f53\u9876\u70b9$v_1$\u4e0e$v_2$\u5747\u4ec5\u6709\u4e00\u4e2a\u6954\u65f6, \u901a\u8fc7\u8fd9\u4e9b\u89c4\u5219\u5f97\u5230\u7684\u584c\u7f29\u6548\u679c\u4e0e\u539f\u59cb\u65b9\u6848\u4e00\u81f4.<br \/>\n$\\\\$ \u6700\u540e\u4e00\u4e2a\u7ec6\u8282\u662f, \u672c\u6587\u901a\u8fc7\u4e3a\u6bcf\u6761\u5c16\u9510\u8fb9(\u5305\u62ec\u8fb9\u754c\u8fb9) \u6dfb\u52a0\u989d\u5916\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf\u6765\u4fdd\u6301\u4e0d\u8fde\u7eed\u66f2\u7ebf\u7684\u51e0\u4f55\u5f62\u72b6. \u5728\u672c\u6587\u7684\u6846\u67b6\u4e2d, \u8be5\u8fb9\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf\u88ab\u6dfb\u52a0\u5230\u4e0e\u8fb9\u76f8\u90bb\u76844\u4e2a\u89d2\u4e0a\u7684$Q^w(\\mathbf{v})$\u4e0a(\u6216\u8005\u5728\u8fb9\u754c\u8fb9\u7684\u60c5\u51b5\u4e0b\u4e3a2\u4e2a\u89d2).<\/p>\n<p><strong>5. \u51cf\u9762\u4f18\u5316<\/strong><\/p>\n<p>\u9664\u4e86\u5b9a\u4e49\u65b0\u7684QEM, \u672c\u6587\u8fd8\u5c1d\u8bd5\u4e86\u4e24\u79cd\u6280\u672f, \u5360\u7528\u5185\u5b58\u66f4\u5c11\u7684\u51cf\u9762\u7b97\u6cd5\u4e0e\u4fdd\u6301\u4f53\u79ef\u7684\u7b97\u6cd5, \u53d1\u73b0\u5b83\u4eec\u8fdb\u4e00\u6b65\u6539\u5584\u4e86\u7ed3\u679c.<\/p>\n<p><strong>5.1 \u5360\u7528\u5185\u5b58\u66f4\u5c11\u7684\u51cf\u9762\u7b97\u6cd5<\/strong><\/p>\n<p>\u672c\u6587\u6ca1\u6709\u5c06$Q^w(\\mathbf{v})$\u5206\u914d\u7ed9\u539f\u59cbMesh\u4e2d\u7684\u6954, \u5e76\u901a\u8fc7$Q^v(\\mathbf{v}) = Q^{v_1}(\\mathbf{v}) + Q^{v_2}( $$ \\mathbf{v})$\u5728\u6bcf\u6b21\u8fb9\u584c\u7f29\u540e\u5bf9$Q^w(\\mathbf{v})$\u8fdb\u884c\u7d2f\u52a0, \u800c\u662f\u5c1d\u8bd5\u4e86\u53e6\u4e00\u79cd\u5360\u7528\u5185\u5b58\u66f4\u5c11\u7684\u51cf\u9762\u7b97\u6cd5, \u7b97\u6cd5\u6267\u884c\u8fc7\u7a0b\u4e2d\u6839\u636e\u5df2\u7b80\u5316\u7684Mesh\u7684\u51e0\u4f55\u5f62\u72b6\u4e0e\u5c5e\u6027\u4e0d\u65ad\u91cd\u65b0\u8ba1\u7b97$Q^w( $$ \\mathbf{v})$. \u56e0\u6b64, \u5728\u6267\u884c\u8fb9\u584c\u7f29$(v_1, $$ v_2) \\to v$\u65f6, \u6211\u4eec\u4f7f\u7528\u5f0f$\\ref{quadratic_error_functions_defined_on_wedges}$\u4e0e\u5f0f$\\ref{replaced_quadratic_error_functions_defined_on_vertices}$\u5728\u4e0b\u56fe\u6240\u793a\u7684\u9762\u96c6\u5408$F^{i + 1}$\u4e0a\u8ba1\u7b97$Q^v( $$ \\mathbf{v})$.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/edge_collapse_transformation.jpg\" alt=\"\" width=\"457\" height=\"212\" class=\"aligncenter size-full wp-image-3764\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/edge_collapse_transformation.jpg 457w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/edge_collapse_transformation-300x139.jpg 300w\" sizes=\"(max-width: 457px) 100vw, 457px\" \/><\/p>\n<p>\u672c\u6587\u8bc1\u5b9e\u4e86\u5360\u7528\u5185\u5b58\u66f4\u5c11\u7684\u51cf\u9762\u7b97\u6cd5\u63d0\u9ad8\u4e86\u7ed3\u679c\u7684\u51c6\u786e\u6027. \u867d\u7136\u8fd9\u8d77\u521d\u53ef\u80fd\u770b\u8d77\u6765\u8fdd\u53cd\u76f4\u89c9, \u4f46\u53ef\u4ee5\u901a\u8fc7\u4e0b\u56fe\u6765\u89e3\u91ca.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/illustration_of_standard_QEM_and_memoryless_QEM_simplification.jpg\" alt=\"\" width=\"643\" height=\"271\" class=\"aligncenter size-full wp-image-3767\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/illustration_of_standard_QEM_and_memoryless_QEM_simplification.jpg 643w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/illustration_of_standard_QEM_and_memoryless_QEM_simplification-300x126.jpg 300w\" sizes=\"(max-width: 643px) 100vw, 643px\" \/><br \/>\n<center><em>\u865a\u7ebf\u692d\u5706\u8868\u793a\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf$Q^v_p$\u7684\u5f62\u72b6; (a) \u5728\u6807\u51c6\u7b97\u6cd5\u4e2d, \u5b83\u4eec\u5728\u9884\u5904\u7406\u8fc7\u7a0b\u4e2d\u8ba1\u7b97\u4e00\u6b21, \u968f\u540e\u5728\u51cf\u9762\u8fc7\u7a0b\u4e2d\u6c42\u548c; (b) \u5728\u5360\u7528\u5185\u5b58\u66f4\u5c11\u7684\u51cf\u9762\u7b97\u6cd5\u4e2d, \u5b83\u4eec\u662f\u5728\u7b97\u6cd5\u6267\u884c\u8fc7\u7a0b\u4e2d\u6839\u636e\u5df2\u7b80\u5316\u7684Mesh\u7684\u51e0\u4f55\u5f62\u72b6\u4e0e\u5c5e\u6027\u4e0d\u65ad\u91cd\u65b0\u8ba1\u7b97\u7684.<\/em><\/center><\/p>\n<p>\u6807\u51c6QEM\u7b97\u6cd5\u5728\u539f\u59cbMesh\u4e0a\u8ba1\u7b97$Q^v$, \u7136\u540e\u6c42\u548c. \u7ed3\u679c, \u975e\u5e73\u884c\u692d\u5706(\u5bf9\u5e94\u4e8e\u7cbe\u7ec6\u7279\u5f81) \u7684\u5408\u5e76\u4ea7\u751f\u7d27\u5bc6\u7684\u5f62\u72b6\u4e3a\u7403\u9762\u7684\u4e8c\u6b21\u8bef\u5dee\u5ea6\u91cf, \u9501\u5b9a\u9876\u70b9\u5e76\u9632\u6b62\u8fdb\u4e00\u6b65\u7b80\u5316, \u5373\u4f7f\u9876\u70b9\u5747\u5904\u4e8e\u540c\u4e00\u5e73\u9762\u4e0a. \u63a5\u4e0b\u6765\u5206\u6b65\u89e3\u91ca\u4e0a\u56fe.<\/p>\n<p>$\\cdot$ <strong>\u692d\u5706\u51e0\u4f55\u7ea6\u675f\u7684\u672c\u8d28<\/strong><br \/>\n$\\\\$ \u5728QEM\u6846\u67b6\u4e2d, \u6bcf\u4e2a\u692d\u5706\u4ee3\u8868\u9876\u70b9\u5728\u4fdd\u6301\u5c40\u90e8\u5f62\u72b6\u7cbe\u5ea6\u4e0b\u7684\u53ef\u79fb\u52a8\u8303\u56f4(\u5373\u4e0d\u663e\u8457\u589e\u52a0\u4e8c\u6b21\u8bef\u5dee\u7684\u533a\u57df). \u692d\u5706\u7684:<br \/>\n$\\\\$ a) \u65b9\u5411: \u7531\u77e9\u9635$Q^f$\u7684\u6700\u5c0f\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u51b3\u5b9a(\u6cd5\u5411\u91cf\u65b9\u5411, \u8be6\u89c1\u53c2\u8003\u6750\u65992).<br \/>\n$\\\\$ b) \u5f62\u72b6: \u7531\u7279\u5f81\u503c\u7684\u6bd4\u503c\u51b3\u5b9a(\u5404\u5411\u5f02\u6027\u7a0b\u5ea6).<br \/>\n$\\\\$ c) \u8303\u56f4: \u7531\u7279\u5f81\u503c\u7684\u5012\u6570\u5e73\u65b9\u6839\u51b3\u5b9a(\u6cbf\u5404\u8f74\u7684\u79fb\u52a8\u81ea\u7531\u5ea6).<\/p>\n<p>$\\cdot$ <strong>\u975e\u5e73\u884c\u692d\u5706\u7684\u7ea6\u675f\u51b2\u7a81<\/strong><br \/>\n$\\\\$ \u5f53\u4e24\u4e2a\u692d\u5706\u975e\u5e73\u884c\u65f6:<br \/>\n$\\\\$ a) \u65b9\u5411\u5dee\u5f02: \u5b83\u4eec\u7684\u6cd5\u5411\u91cf\u65b9\u5411\u4e0d\u4e00\u81f4.<br \/>\n$\\\\$ b) \u7ea6\u675f\u4e92\u8865\u6027: \u6bcf\u4e2a\u692d\u5706\u5728\u4e0d\u540c\u65b9\u5411\u4e0a\u65bd\u52a0\u7ea6\u675f. \u4f8b\u5982:<br \/>\n$\\\\$ $\\quad$ &#8211; \u692d\u5706A\u53ef\u80fd\u5728$x-y$\u5e73\u9762\u65b9\u5411\u7ea6\u675f\u5f3a.<br \/>\n$\\\\$ $\\quad$ &#8211; \u692d\u5706B\u53ef\u80fd\u5728$y-z$\u5e73\u9762\u65b9\u5411\u7ea6\u675f\u5f3a.<br \/>\n$\\\\$ c) \u4ea4\u96c6\u533a\u57df: \u8fd9\u79cd\u65b9\u5411\u5dee\u5f02\u5bfc\u81f4\u5b83\u4eec\u7684\u4ea4\u96c6\u533a\u57df\u5fc5\u987b\u540c\u65f6\u6ee1\u8db3\u4e24\u4e2a\u65b9\u5411\u7684\u7ea6\u675f, \u76f8\u5f53\u4e8e\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u88ab&#8221;\u53cc\u5411\u6324\u538b&#8221;.<\/p>\n<p>$\\cdot$ <strong>\u77e9\u9635\u5408\u5e76\u7684\u4ee3\u6570\u6548\u5e94<\/strong><br \/>\n$\\\\$ \u5408\u5e76\u4e24\u4e2a\u9876\u70b9\u7684$Q^f$\u77e9\u9635\u65f6:<br \/>\n$\\\\$ a) \u77e9\u9635\u76f8\u52a0: \u65b0\u77e9\u9635$Q_{new} = Q_1 + Q_2$.<br \/>\n$\\\\$ b) \u7279\u5f81\u503c\u589e\u5f3a: \u7531\u4e8e$Q_1$\u4e0e$Q_2$\u5728\u4e0d\u540c\u65b9\u5411\u4e0a\u5b58\u5728\u7ea6\u675f, \u76f8\u52a0\u540e:<br \/>\n$\\\\$ $\\quad$ &#8211; \u5728\u5171\u6709\u7ea6\u675f\u65b9\u5411(\u5982\u6cd5\u5411\u91cf\u65b9\u5411) \u7684\u7279\u5f81\u503c\u4f1a\u663e\u8457\u589e\u5927.<br \/>\n$\\\\$ $\\quad$ &#8211; \u5728\u5176\u4ed6\u65b9\u5411\u7684\u7279\u5f81\u503c\u4e5f\u4f1a\u56e0\u77e9\u9635\u975e\u5bf9\u89d2\u5143\u7d20\u8026\u5408\u800c\u589e\u5927.<br \/>\n$\\\\$ c) \u7403\u5f62\u5316\u8d8b\u52bf: \u5f53\u5404\u65b9\u5411\u7279\u5f81\u503c\u8d8b\u4e8e\u63a5\u8fd1\u65f6, \u692d\u5706\u65b9\u7a0b\u9000\u5316\u4e3a\u7403\u9762\u65b9\u7a0b. \u5408\u5e76\u540e\u7684\u77e9\u9635\u7279\u5f81\u503c\u5206\u5e03\u66f4\u5747\u5300, \u5bfc\u81f4\u7ea6\u675f\u533a\u57df\u63a5\u8fd1\u7403\u5f62.<\/p>\n<p>$\\cdot$ <strong>\u81ea\u7531\u5ea6\u9501\u5b9a\u673a\u5236<\/strong><br \/>\n$\\\\$ a) \u79fb\u52a8\u8303\u56f4\u6536\u7f29: \u539f\u59cb\u692d\u5706\u5141\u8bb8\u9876\u70b9\u5728\u7279\u5b9a\u5e73\u9762\u5185\u79fb\u52a8(\u5982\u6cbf\u692d\u5706\u957f\u8f74), \u4f46\u5408\u5e76\u540e\u7684\u7403\u5f62\u7ea6\u675f\u8981\u6c42\u9876\u70b9\u5728\u6240\u6709\u65b9\u5411\u4e0a\u4fdd\u6301\u63a5\u8fd1\u539f\u59cb\u4f4d\u7f6e.<br \/>\n$\\\\$ b) \u8bef\u5dee\u654f\u611f\u533a\u6269\u5927: \u4efb\u4f55\u504f\u79bb\u539f\u59cb\u4f4d\u7f6e\u7684\u8fd0\u52a8\u90fd\u4f1a\u663e\u8457\u5f71\u54cd\u6240\u6709\u65b9\u5411\u7684\u8bef\u5dee\u5ea6\u91cf, \u5f62\u6210&#8221;\u9501\u5b9a\u6548\u5e94&#8221;.<br \/>\n$\\\\$ c) \u7b80\u5316\u505c\u6ede: \u5373\u4f7f\u6574\u4f53\u5f62\u72b6\u5df2\u63a5\u8fd1\u5e73\u9762, \u9876\u70b9\u4ecd\u88ab\u9650\u5236\u5728\u539f\u59cb\u4f4d\u7f6e\u9644\u8fd1, \u65e0\u6cd5\u8fdb\u4e00\u6b65\u5408\u5e76\u6216\u79fb\u52a8.<\/p>\n<p>\u76f8\u6bd4\u4e4b\u4e0b, \u5728\u4f7f\u7528\u5360\u7528\u5185\u5b58\u66f4\u5c11\u7684\u51cf\u9762\u7b97\u6cd5\u7684\u60c5\u51b5\u4e0b, $Q^v$\u662f\u5728\u7b97\u6cd5\u6267\u884c\u8fc7\u7a0b\u4e2d\u6839\u636e\u5df2\u7b80\u5316\u7684Mesh\u7684\u51e0\u4f55\u5f62\u72b6\u4e0e\u5c5e\u6027\u4e0d\u65ad\u91cd\u65b0\u8ba1\u7b97\u7684. \u8fd9\u79cd\u5fd8\u8bb0\u66f4\u7cbe\u7ec6\u7684\u7ec6\u8282\u7684\u80fd\u529b\u5141\u8bb8\u5728\u9700\u8981\u65f6\u8fdb\u4e00\u6b65\u7b80\u5316.<br \/>\n$\\\\$ \u5c3d\u7ba1\u5360\u7528\u5185\u5b58\u66f4\u5c11\u7684\u51cf\u9762\u7b97\u6cd5\u964d\u4f4e\u4e86\u5b58\u50a8QEM\u8fd9\u4e00\u6b65\u9aa4\u7684\u5fc5\u8981\u6027, \u4f46\u4e3a\u4e86\u52a0\u5feb\u7b97\u6cd5\u901f\u5ea6, \u672c\u6587\u53d1\u73b0\u5728Mesh\u7684\u9762\u4e0a\u7f13\u5b58\u503c($area(f) \\cdot Q^f(\\mathbf{v})$) \u662f\u6709\u7528\u7684, \u5e76\u5728\u8fb9\u584c\u7f29\u65f6\u9002\u5f53\u5730\u66f4\u65b0\u5b83\u4eec. \u56e0\u6b64, \u672c\u6587\u63d0\u51fa\u7684\u65b0QEM\u7684\u7d27\u51d1\u5f62\u5f0f\u4ecd\u7136\u5177\u6709\u4f18\u52bf.<\/p>\n<p><strong>5.2 \u4fdd\u6301\u4f53\u79ef<\/strong><\/p>\n<p>\u5b9e\u9a8c\u8868\u660e, \u65b0\u7684QEM\u6709\u65f6\u4f1a\u5728\u5c5e\u6027\u9ad8\u68af\u5ea6\u533a\u57df\u7f29\u5c0fMesh\u7684\u51e0\u4f55\u5f62\u72b6. \u4e5f\u5c31\u662f\u8bf4, \u5728\u5c16\u9510\u7684\u5c5e\u6027\u503c\u8fc7\u6e21\u5904, \u65b0\u9876\u70b9$\\mathbf{v}$\u53ef\u80fd\u88ab\u63a8\u5411\u66f2\u9762\u4e0a\u65e7\u9876\u70b9\u7684\u66f2\u7387\u4e2d\u5fc3.<br \/>\n$\\\\$ \u4e0b\u56fe\u4ee5$\\mathbb{R}^2$\u4e2d\u591a\u8fb9\u5f62\u66f2\u7ebf\u7684\u7b80\u5316\u4e3a\u4f8b\u76f4\u89c2\u5730\u8bf4\u660e\u4e86\u8fd9\u4e2a\u73b0\u8c61. \u5728\u539f\u59cbMesh\u4e0a\u5b9a\u4e49\u7684\u6807\u91cf\u573a\u4ece$\\mathbf{1}$\u53d8\u5316\u81f3$\\mathbf{0}$. \u663e\u7136, \u5728\u8fd9\u4e2a\u90bb\u57df\u4e0a\u7684\u8fb9\u584c\u7f29\u65e0\u6cd5\u4fdd\u7559\u8fd9\u4e2a\u5c5e\u6027\u5b57\u6bb5. \u63a5\u4e0b\u6765\u8003\u8651\u4e8c\u6b21\u5ea6\u91cf\u6240\u6d4b\u91cf\u7684\u8bef\u5dee.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/illustration_of_trade_off_between_geometric_accuracy_and_attribute_accuracy.jpg\" alt=\"\" width=\"629\" height=\"301\" class=\"aligncenter size-full wp-image-3779\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/illustration_of_trade_off_between_geometric_accuracy_and_attribute_accuracy.jpg 629w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2025\/03\/illustration_of_trade_off_between_geometric_accuracy_and_attribute_accuracy-300x144.jpg 300w\" sizes=\"(max-width: 629px) 100vw, 629px\" \/><br \/>\n<center><em>\u5bf9\u4e8e$\\mathbb{R}^2$\u4e2d\u7684\u591a\u8fb9\u5f62\u66f2\u7ebf, \u4e0a\u56fe\u523b\u753b\u4e86\u5173\u4e8e\u51e0\u4f55\u8bef\u5dee($Q_p = 0$) \u4e0e\u5c5e\u6027\u8bef\u5dee($Q_s = 0$) \u4e4b\u95f4\u7684\u533a\u522b. \u5f53\u5c5e\u6027\u68af\u5ea6\u8f83\u9ad8\u65f6, \u53d7\u5c5e\u6027\u7cbe\u5ea6\u7684\u5f71\u54cd\u53ef\u80fd\u4f1a\u5bfc\u81f4\u65b0\u9876\u70b9\u5411\u65e7\u9876\u70b9\u7684\u66f2\u7387\u4e2d\u5fc3\u504f\u79fb.<\/em><\/center><\/p>\n<p>\u5728\u4e0a\u56fe\u53f3\u4e0a\u89d2\u4e2d, \u8fb9\u584c\u7f29\u540e\u51e0\u4f55\u4f53\u4f53\u79ef\u5c06\u5b8c\u5168\u4fdd\u6301($Q_p = 0$), \u4f46\u5c5e\u6027\u8bef\u5dee$Q_s$\u4e0d\u4e3a0. \u539f\u56e0\u662f\u9876\u70b9$\\mathbf{v} =(2, 1, \\mathbf{a})$\u7684\u5c5e\u6027\u503c$\\mathbf{a}$\u4e0d\u80fd\u540c\u65f6\u5728\u6240\u67093\u4e2a\u539f\u59cb\u7ebf\u6bb5\u4e0a\u63d2\u503c\u5c5e\u6027\u68af\u5ea6. \u7279\u522b\u5730, \u5fc5\u987b\u5c06$\\mathbf{a}$\u8bbe\u7f6e\u4e3a$\\mathbf{-1}$\u6765\u5916\u63d2\u6700\u5de6\u4fa7\u7684\u6bb5\u7ebf\u6bb5$[(0, 1, \\mathbf{1}), (1, 1, \\mathbf{0})]$.<br \/>\n$\\\\$ \u53e6\u4e00\u65b9\u9762, \u5728\u4e0a\u56fe\u53f3\u4e0b\u89d2\u4e2d, \u8fb9\u584c\u7f29\u4f1a\u5bfc\u81f4\u51e0\u4f55\u8bef\u5dee$Q_p > 0$, \u4f46&#8221;\u5b9e\u73b0&#8221; \u4e86$Q_s = $$ 0$, \u56e0\u4e3a$\\mathbf{v} =(1, 0, \\mathbf{0})$\u5728\u539f\u59cb3\u6761\u7ebf\u6bb5\u4e0a\u7684\u6bcf\u4e00\u4e2a\u6295\u5f71\u70b9\u5747\u7cbe\u786e\u5730\u63d2\u503c\u4e86\u539f\u59cb\u5c5e\u6027. \u76f4\u89c2\u4e0a, \u9876\u70b9\u671d\u5411\u5176\u66f2\u7387\u4e2d\u5fc3\u7684\u8fd0\u52a8\u5141\u8bb8\u5b83\u6295\u5f71\u5230\u539f\u59cb\u7ebf\u6bb5\u7684\u5185\u90e8, \u4ece\u800c\u907f\u514d\u5c5e\u6027\u5916\u63d2.<br \/>\n$\\\\$ \u4e3a\u4e86\u62b5\u6d88\u51e0\u4f55\u6536\u7f29\u7684\u5f71\u54cd, \u672c\u6587\u5f15\u5165\u4e86\u4fdd\u6301\u4f53\u79ef\u7684\u7ea6\u675f. \u5728\u8fb9\u584c\u7f29\u8fc7\u7a0b\u4e2d\u4fdd\u6301\u4f53\u79ef\u7b49\u4ef7\u4e8e\u4e0b\u8ff0\u5728\u5408\u5e76\u540e\u7684\u9876\u70b9$\\mathbf{v}$\u7684\u4f4d\u7f6e$\\mathbf{p}$\u4e0a\u7684\u7ebf\u6027\u7ea6\u675f$$\\mathbf{g}^T_{VOL} \\mathbf{p} + d_{VOL} = 0.$$\u5176\u4e2d, \u4f53\u79ef\u68af\u5ea6$\\mathbf{g}^T_{VOL}$\u4e3a$F^{i + 1}$\u7684\u9762\u6cd5\u7ebf\u7684\u52a0\u6743\u548c, \u6bcf\u4e00\u6761\u9762\u6cd5\u7ebf\u7684\u6743\u91cd\u4e3a\u5bf9\u5e94\u9762\u7684\u9762\u79ef\u7684\u4e09\u5206\u4e4b\u4e00. \u672c\u6587\u5229\u7528\u4e0a\u5f0f\u6765\u7ea6\u675fMesh\u4f53\u79ef, \u4e3b\u8981\u662f\u56e0\u4e3a\u4e0a\u5f0f\u5b9a\u4e49\u4e86\u4e00\u4e2a\u5e73\u9762\u65b9\u7a0b, \u901a\u8fc7\u8c03\u6574\u5e73\u9762\u7684\u6cd5\u5411\u91cf\u548c\u4e0e\u539f\u70b9\u7684\u8ddd\u79bb, \u53ef\u4ee5\u63a7\u5236Mesh\u7684\u4f53\u79ef\u53d8\u5316. \u5728Mesh\u7b80\u5316\u8fc7\u7a0b\u4e2d, \u5c06\u8fd9\u4e2a\u5e73\u9762\u65b9\u7a0b\u4f5c\u4e3a\u4f18\u5316\u76ee\u6807\u7684\u4e00\u90e8\u5206, \u6709\u52a9\u4e8e\u4fdd\u6301\u7b80\u5316\u540eMesh\u7684\u4f53\u79ef\u4e0e\u539f\u59cbMesh\u5c3d\u53ef\u80fd\u63a5\u8fd1. \u8fd9\u79cd\u65b9\u6cd5\u4e0d\u4ec5\u63d0\u9ad8\u4e86Mesh\u7b80\u5316\u7684\u8d28\u91cf, \u8fd8\u4fdd\u6301\u4e86Mesh\u7684\u89c6\u89c9\u6548\u679c\u4e0e\u51e0\u4f55\u51c6\u786e\u6027. \u4f7f\u7528\u62c9\u683c\u6717\u65e5\u4e58\u6570\u6cd5(\u5176\u5bf9\u5e94\u7684\u62c9\u683c\u6717\u65e5\u4e58\u6570\u4e3a$\\lambda$) \u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u8ba1\u7b97\u53d7\u7ebf\u6027\u7ea6\u675f\u7684$Q^v(\\mathbf{v})$\u7684\u6781\u5c0f\u503c:$$\\begin{pmatrix}<br \/>\n\\mathbf{A} &#038; \\mathbf{g}_{VOL} \\\\<br \/>\n\\mathbf{g}^T_{VOL} &#038; 0<br \/>\n\\end{pmatrix} \\begin{pmatrix}<br \/>\n\\mathbf{v}_{min} \\\\<br \/>\n\\lambda<br \/>\n\\end{pmatrix} = \\begin{pmatrix}<br \/>\n-\\mathbf{b} \\\\<br \/>\n-d_{VOL}<br \/>\n\\end{pmatrix}.$$\u82e5\u9876\u70b9$\\mathbf{v}$\u5728Mesh\u4e0a\u7684\u90bb\u57df\u7684\u9ad8\u65af\u66f2\u7387\u4e3a0, \u5373\u82e5\u5b83\u7684\u5f62\u72b6\u4e3a\u5e73\u9762\u6216\u5706\u67f1\u5f62, \u5219\u8be5\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u77e9\u9635(\u5373\u5f0f$\\ref{found_vertex_position}$\u4e2d\u7684\u7cfb\u6570\u77e9\u9635) \u53ef\u80fd\u662f\u75c5\u6001\u7684.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u5148\u8bc1, \u5bf9\u4e8e\u4efb\u610f$n$\u7ef4\u975e\u96f6\u5411\u91cf$\\mathbf{a}$, $\\mathbf{b}$, $\\mathbf{a} \\mathbf{b}^T$\u4e3a\u4e00\u4e2a$n \\times n$\u7684\u79e91\u77e9\u9635.<br \/>\n$\\\\$ \u4e0d\u59a8\u4ee4$\\mathbf{b} = (b_1, \\cdots, b_n)^T$, \u5219$$\\mathbf{a} \\mathbf{b}^T = (\\mathbf{a} b_1, \\cdots, \\mathbf{a} b_n),$$\u6bcf\u4e00\u5217\u5411\u91cf\u6784\u6210\u7684\u5411\u91cf\u7ec4\u662f\u7ebf\u6027\u76f8\u5173\u7684(\u6bcf\u4e00\u5217\u5411\u91cf\u5747\u4e3a$\\mathbf{a}$\u7684\u500d\u6570), \u6545$\\mathbf{a} \\mathbf{b}^T$\u4e3a\u4e00\u4e2a$n \\times n$\u7684\u79e91\u77e9\u9635.<br \/>\n$\\\\$ \u7531\u4e8e\u9876\u70b9$\\mathbf{v}$\u5728Mesh\u4e0a\u7684\u90bb\u57df\u4e3a\u5e73\u9762\u6216\u5706\u67f1\u5f62, \u5373$\\mathbf{v}$\u7684\u76f8\u90bb\u9762\u7684\u6cd5\u5411$\\mathbf{n}_i, i $$ \\in \\{1, 2 $$ , \\cdots, k\\}$\u51e0\u4e4e\u4e00\u81f4, \u5176\u4e2d, $k$\u4e3a$\\mathbf{v}$\u7684\u76f8\u90bb\u9762\u6570\u91cf, \u4e0d\u59a8\u4ee4\u5176\u76f8\u90bb\u9762\u7684\u6cd5\u5411\u5747\u4e3a$\\mathbf{n}$, \u5219$Q^v(\\mathbf{v})$\u7684\u7cfb\u6570\u77e9\u9635\u7ea6\u7b49\u4e8e$k \\mathbf{A} \\approx k \\mathbf{n} \\mathbf{n}^T$, \u4e3a\u4e00\u4e2a\u79e91\u77e9\u9635. \u4ece\u800c$Q^v(\\mathbf{v})$\u7684\u7cfb\u6570\u77e9\u9635\u7684\u6700\u5c0f\u5947\u5f02\u503c\u7ea6\u4e3a0, \u6761\u4ef6\u6570\u8d8b\u8fd1\u4e8e\u65e0\u7a77\u5927, \u4e3a\u4e00\u4e2a\u75c5\u6001\u77e9\u9635.<\/p>\n<div style=\"text-align: right;\">$\\square$<\/div>\n<p>$\\\\$ \u8bf7\u6ce8\u610f, \u5c5e\u6027\u4e0d\u4f1a\u8d21\u732e\u4efb\u4f55\u96f6\u5947\u5f02\u503c, \u56e0\u4e3a$\\mathbf{A}$\u7684\u5bf9\u89d2\u7ebf\u4e0a\u5927\u5c0f\u4e3a$m \\times m$\u7684$k$\u4e2a\u5b50\u77e9\u9635\u5747\u4e3a\u5355\u4f4d\u77e9\u9635$\\mathbf{I}$. \u5f53\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u77e9\u9635\u4e3a\u75c5\u6001\u77e9\u9635\u65f6, \u672c\u6587\u4ee4$\\mathbf{p} = \\mathbf{p}_1 + $$ \\mathbf{p}_2$, \u800c\u540e\u518d\u6b21\u6c42\u89e3$\\{ s_1, \\dots, s_k \\}$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u8fd1\u5728\u590d\u4e60Mesh\u51cf\u9762\u76f8\u5173\u7684\u77e5\u8bc6\u70b9, \u5f53\u521d\u8bfb\u7f62Hoppe H. New quadric metric for  &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2025\/03\/31\/new_quadric_metric_describing_meshes_appearance_attributes_paper_points_interpretation\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u300aNew Quadric Metric for Simplifying Meshes with Appearance Attributes\u300b\u8bba\u6587\u8981\u70b9\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\/3581"}],"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=3581"}],"version-history":[{"count":138,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3581\/revisions"}],"predecessor-version":[{"id":3777,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3581\/revisions\/3777"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=3581"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=3581"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=3581"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}