{"id":271,"date":"2024-04-08T19:56:00","date_gmt":"2024-04-08T19:56:00","guid":{"rendered":"https:\/\/boochlin.com\/?p=271"},"modified":"2026-09-16T09:54:00","modified_gmt":"2026-09-16T09:54:00","slug":"slam-note-ndt_tku","status":"publish","type":"post","link":"https:\/\/boochlin.com\/?p=271","title":{"rendered":"SLAM Note \u2013 NDT_TKU"},"content":{"rendered":"

** NDT-TKU<\/a> ** from Booch Lin<\/a><\/strong><\/p>\n

NDT \u2013 TKU Introduction<\/h3>\n

\u777d\u9055\u8a31\u4e45\u6c92\u6709\u5beb blog \uff0c\u4e3b\u8981\u9084\u662f\u592a\u5fd9\u3002\u9019\u5468\u7d42\u65bc\u7a7a\u51fa\u6642\u9593\u53ef\u4ee5\u4f86\u88dc\u88dc\u4e4b\u524d\u7684\u7b46\u8a18\uff0c\u7531\u65bc\u6700\u8fd1\u624d\u5c0d\u5167\u90e8\u5206\u4eab\u6b64\u4e3b\u984c\u3002\u6240\u4ee5\u9019\u500b\u4e3b\u984c\u5370\u8c61\u6700\u6df1\u523b\u3002
\n\u96d6\u7136\u5f88\u60f3\u518d\u6b21\u5c31\u9032\u5165\u6b63\u984c\uff0c\u4f46\u662f\u7531\u65bc\u9019 blog \u7684\u98a8\u683c\u5c31\u662f\u5ee2\u8a71\u591a\uff0c\u6240\u4ee5\u6211\u7e7c\u7e8c\u5beb\u5ee2\u8a71\uff0c\u8ac7\u5230\u7684\u9019\u500b\u4e3b\u984c\uff0c\u7528\u5230\u5927\u91cf\u7684\u6578\u5b78\uff0c\u800c\u5c0f\u5f1f\u6211\u5728\u5927\u5b78\u57fa\u672c\u4e0a\u300e\u6a5f\u7387\u300f\u300e\u7dda\u6027\u4ee3\u6578\u300f\u300e\u5fae\u7a4d\u5206\u300f\u5927\u6982\u90fd\u6709\u91cd\u4fee\u904e\uff0c\u6240\u4ee5\u5982\u679c\u5ba2\u500c\u5c0d\u65bc\u6578\u5b78\u7684\u90e8\u4efd\u6709\u554f\u984c\uff0c\u8acb\u81ea\u884c\u4e0a\u7db2\u67e5\uff0c\u56e0\u70ba\u6211\u7b54\u7684\u4e0d\u6703\u6bd4google \u5927\u795e\u597d\u3002<\/p>\n

\u9019\u6b21\u4e3b\u984c\u7684\u6700\u7d42\u76ee\u6a19 SLAM \uff08<\/strong>\u00a0Simultaneous localization and mapping\uff09 \u540c\u6b65\u5b9a\u4f4d\u8207\u5730\u5716\u69cb\u5efa\uff0c\u662f\u4e00\u500b\u95dc\u65bc\u6a5f\u5668\u4eba\u6216\u662f\u81ea\u52d5\u99d5\u99db\u5fc5\u5b9a\u7814\u7a76\u7684\u4e3b\u984c\uff0c\u800c\u5f04\u61c2\u524d\u4e5f\u5fc5\u9808\u5148\u5177\u6709\u5927\u91cf\u7684\u76f8\u95dc\u77e5\u8b58\uff0c\u524d\u7f6e\u90e8\u4efd\u6211\u6c92\u6709\u8fa6\u6cd5\u7d30\u7d30\u8b1b\u89e3\uff0c\u4e3b\u8981\u4e5f\u662f\u6240\u5b78\u592a\u5c11\uff0c\u56e0\u6b64\u9019\u6b21\u5c31\u53ea\u5c08\u6ce8\u65bc SLAM \u7684\u5176\u4e2d\u4e00\u500b\u65b9\u6cd5 NDT , \u800c TKU\u00a0\u5247\u662f\u518d NDT \u9032\u4e00\u6b65\u7684\u63a8\u5c55\u3002<\/strong><\/p>\n

\u5728\u6b64\u4e4b\u524d\uff0c\u6709\u8208\u8da3\u7684\u4eba\u53ef\u4ee5\u5148\u770b\u9019\u7bc7\u8ad6\u6587<\/p>\n

****Magnusson, M. (2009). The Three-Dimensional Normal-Distributions Transform \u2014 an Ef\ufb01cient Representation for Registration, Surface Analysis, and Loop Detection<\/a><\/p>\n

\u9019\u5927\u6982\u662f\u76ee\u524d\u5c0d\u65bc NDT \u6700\u5165\u9580\u7684\u6587\u4ef6\uff0c\u5f04\u61c2\u4ed6\u4e0d\u6703\u5403\u8667\uff0c\u751a\u81f3\u4e0d\u7528\u770b\u6211\u7684\u6587\u7ae0\u5566\u3002<\/p>\n

SLAM \u7684\u6240\u6709\u6700\u7d42\u76ee\u7684\u5c31\u662f\u8981\u627e\u51fa\u81ea\u8eab\u5230\u5e95\u4f4d\u65bc\u4e16\u754c\u4e2d\u7684\u4f55\u8655\uff0c\u9019\u53ef\u80fd\u5927\u5bb6\u6703\u76f4\u89ba\u8a8d\u70ba\u4ea4\u7d66 GPS \u5c31\u597d\u5566\uff0c\u4f46\u662f\u5be6\u969b\u4e0a\u4ed6\u7684\u4e0d\u78ba\u5b9a\u56e0\u7d20\u5be6\u5728\u592a\u591a\uff0c\u5404\u4f4d\u81ea\u8eab\u7528\u624b\u6a5f\u5c0e\u822a\u90fd\u6703\u5e38\u5e38\u7f75\u8aaa\u5b9a\u4e0d\u6e96\uff0c\u90a3\u9ebc\u8981\u6c42\u66f4\u9ad8\u5b89\u5168\u6027\u7684\u81ea\u52d5\u99d5\u99db\u66f4\u4e0d\u53ef\u80fd\u5b8c\u5168\u4f9d\u8a17\u5728\u9019\u4e0a\u9762\uff0c\u7576\u7136\u76ee\u524d\u5169\u6d3e\u722d\u8ad6\u4e5f\u662f\u6c92\u6709\u505c\u904e\uff0c\u4e0d\u904e\u722d\u8ad6\u7684\u9ede\u4e3b\u8981\u5c31\u662f\u5728\u65bc\u53ea\u8981\u767c\u5c55 GPS \u6280\u8853\u5230\u975e\u5e38\u9ad8\u7cbe\u5ea6\uff0c\u5c31\u53ef\u4ee5\u4e0d\u7528\u7814\u7a76\u5176\u4ed6\u76f8\u95dc\u5b9a\u4f4d\u6280\u8853\u3002\u4f46\u662f\u7528\u819d\u84cb\u60f3\u4e5f\u77e5\u9053\u5c31\u9084\u4e0d\u5920\u7cbe\u5bc6\uff0c\u6240\u4ee5\u5b9a\u4f4d\u6f14\u7b97\u6cd5\u81f3\u4eca\u80fd\u7136\u662f\u4e00\u500b\u4e3b\u8981\u7814\u7a76\u4e3b\u984c\u3002<\/p>\n

\u5b9a\u4f4d\u6f14\u7b97\u6cd5\u4e00\u500b\u91cd\u9ede\u5c31\u662f Sensing\uff0c\u6a5f\u5668\u4eba\u5fc5\u9700\u8981\u80fd\u611f\u77e5\u5468\u570d\u74b0\u5883\uff0c\u5f97\u5230\u76f8\u95dc\u60c5\u5831\uff0c\u6700\u5e38\u898b\u7684\u5c31\u662f\u76f8\u6a5f\uff0c\u9019\u78ba\u5be6\u662f\u500b\u4e3b\u529b\uff0c\u800c\u4e14\u76f8\u8f03\u5176\u4ed6\u50b3\u611f\u5668\u78ba\u5be6\u4fbf\u5b9c\uff0c\u4e0d\u904e\u9019\u6b21\u8a0e\u8ad6\u7684\u76ee\u6a19\u4e26\u975e\u76f8\u6a5f\uff0c\u800c\u662f LIDAR (\u5149\u9054)\uff0c\u901a\u5e38\u6383\u63cf\u5230\u7684\u756b\u9762\u9577\u9019\u6a23<\/p>\n

\u5f71\u7247\u662f VELODYNE LIDAR \u7684\u793a\u610f\u5716\uff0c\u9806\u4ee3\u4e00\u984c\u9019\u6771\u897f\u8d85\u8cb4\uff0c\u6578\u767e\u842c\u6709\u4e4b\u3002\u76ee\u524d\u4e0d\u5c11\u7814\u7a76\u90fd\u662f\u91dd\u5c0d\u9019\u7a2e\u9ede\u96f2\u8cc7\u6599\u4f5c SLAM \u7814\u7a76\u3002<\/p>\n

\u5176\u5be6\u9019\u985e\u8cc7\u6599\u7814\u7a76\u4e3b\u8981\u5c31\u570d\u7e5e\u5728\u300e\u5229\u7528\u9019\u4e00\u6b21\u6383\u7784\u5230\u7684\u9ede\u96f2\u8ddf\u4e0b\u4e00\u6b21\u6383\u7784\u5230\u9ede\u96f2\uff0c\u5224\u65b7\u51fa\u6a5f\u5668\u4eba\u7684\u4f4d\u5b50\u8207\u59ff\u614b\u300f\u3002\u8aaa\u8d77\u4f86\u7c21\u55ae\u662f\u500b\u5339\u914d\u554f\u984c\uff0c\u4f46\u662f\u505a\u8d77\u4f86\u8d85\u7d1a\u9ebb\u7169\uff0c\u5149\u662f\u5206\u985e\u5c31\u53ef\u4ee5\u641e\u6b7b\u81ea\u5df1\u4e86\u3002(\u5beb\u5230\u9019\u908a\u6709\u9ede\u7d2f\u4e86)\u3002<\/p>\n

\u770b\u5230\u9019\u88e1\uff0c\u6709\u4eba\u53ef\u80fd\u76f4\u89ba\u9019\u6703\u591a\u96e3\uff0c\u628a\u9019\u6b21 Scan \u6bcf\u500b\u9ede\u62ff\u4f86\u8ddf\u4e0b\u4e00\u500b Scan \u6bd4\u5c0d\u4e0d\u5c31\u597d\u4e86\uff0c\u90a3\u6211\u5c31\u6703\u60f3\u554f\u600e\u9ebc\u6bd4\uff0c\u832b\u832b\u4eba\u6d77\u4e2d\uff0c\u7e3d\u8981\u6709\u500b\u4f9d\u64da\uff0c\u5565-\u4f60\u8aaa\u6311\u8fd1\u7684\u6bd4\u963f\uff0c\u9019\u500b\u6211\u807d\u904e\uff0c\u5c08\u696d\u9ede\u8aaa\u6cd5\u300eICP \uff08 Iterative Closest Point \uff09\u300f\uff0c\u9019\u6211\u5c31\u63d0\u5230\u9019\u88e1\uff0c\u4e0a\u9762\u90a3\u7bc7\u6587\u7ae0\u6709\u4ecb\u7d39\u3002\u4f86\u5f35\u5716\u544a\u8a34\u4f60\u9019\u6c34\u53ef\u6df1\u4e86<\/p>\n

http:\/\/www.cnblogs.com\/gaoxiang12\/p\/3695962.html<\/a>\u00a0\u4e4b\u547c\u4e0a\u7684\u5927\u795e<\/p>\n

http:\/\/ivory-cavern.blogspot.tw\/2009\/11\/icp-iterative-closest-point-c.html<\/a>\u00a0ICP \u5be6\u505a<\/p>\n

graph LR\n    subgraph ICP [\u50b3\u7d71 ICP: \u9ede\u5c0d\u9ede\u6700\u8fd1\u9130\u6bd4\u5c0d]\n    P1[\u6383\u63cf\u9ede\u96f2 P] -->|\u641c\u5c0b\u6700\u8fd1\u9ede (\u8017\u6642 O(N log N))| Q1[\u53c3\u8003\u9ede\u96f2 Q]\n    end\n    subgraph NDT [NDT: \u9ede\u5c0d\u9ad8\u65af\u5206\u4f48\u6bd4\u5c0d]\n    P2[\u6383\u63cf\u9ede\u96f2 P] -->|\u6295\u5f71\u81f3\u9ad4\u7d20\u6a5f\u7387\u5206\u4f48 (\u76f4\u63a5\u8a55\u4f30\u9023\u7e8c\u7a7a\u9593)| V[3D \u9ad4\u7d20\u5e38\u614b\u5206\u4f48 N(\u03bc, \u03a3)]\n    end<\/pre>\n
\u597d\u5566\uff0cICP \u626f\u9060\u4e86\uff0c\u53ea\u662f\u8981\u544a\u8a34\u5927\u5bb6\u4e26\u975e\u53ea\u6709 NDT \u800c\u5df2\uff0c\u56de\u5230\u6b63\u984c\u00a0NDT \u2013 TKU\uff0c\u9019\u90e8\u4efd\u6211\u5c31\u7167\u8005\u6295\u5f71\u7247\u4f86\u8ac7\uff0c\u4e5f\u8acb\u5404\u4f4d\u540c\u6642\u770b\u6295\u5f71\u7247\u6703\u6bd4\u8f03\u597d\u61c2<\/p>\n
PAGE 2:<\/h4>\n
What is ndt_tku<\/h4>\n
NDT \u5728 SLAM \u4e0a\u7684\u57fa\u672c\u73a9\u6cd5\u7b97\u662f\u884c\u4e4b\u591a\u5e74\uff0c\u5728 ROS \u7cfb\u7d71\u4e0a\u4e5f\u5df2\u7d93\u6709 PCL \u6574\u6210\u76f8\u7576\u65b9\u4fbf\u7684 CLASS\uff0cHow to use Normal Distributions<\/a>\u9019\u7bc7\u6587\u7ae0\u4e5f\u5df2\u7d93\u8a73\u7d30\u6559\u5b78\u904e\uff0c\u800c NDT-TKU\uff0c\u5247\u662f\u6839\u64da \u7af9\u5167\u5148\u751f(TAKEUCHI) \u6559\u6388\u63d0\u51fa\u7684\u8ad6\u6587\u00a0A 3-D Scan Matching using Improved 3-D Normal Distributions Transform for Mobile Robotic Mapping<\/a>(\u8ad6\u6587\u7db2\u8def\u4e0d\u516c\u958b) NDT \u518d\u9032\u5316\u7248\u672c\uff0c\u76f8\u95dc\u5be6\u505a\u5247\u662f\u5728\u4ed6\u9580\u7684 Open source project\u300eAutoware<\/a>\u300f\u4e2d\u7684 ROS node<\/a><\/p>\n
PAGE3:<\/h4>\n
Why is ndt_tku<\/h4>\n
\u7531\u65bc\u6211\u662f\u7814\u7a76\u4ed6\u9580\u7684\u6574\u500b Autoware \u5c08\u6848\uff0c\u88e1\u9762\u5176\u5be6\u5df2\u7d93\u6709\u5be6\u505a PCL NDT \u7684 Node \uff0c\u5728\u975e\u5e38\u597d\u5947\u7684\u60c5\u6cc1\u4e0b\u6df7\u9032\u4ed6\u9580\u7684 slack \uff0c\u63d0\u554f\u70ba\u4f55\u8981\u81ea\u5df1\u4f5c\u4e00\u5957 NDT Tku \u7248\u672c\uff0c\u4e3b\u8981\u539f\u56e0\u5c31\u662f<\/p>\n
    \n
  1. \n
    \u662f\u6307\u5c0e\u6559\u6388\u63d0\u51fa\u7684<\/p>\n<\/li>\n
  2. \n
    PCL \u7684 cuda \u5316\u6975\u5ea6\u9ebb\u7169\uff0c\u5de5\u7a0b\u5e2b\u8868\u793a\u4e0d\u5982\u81ea\u5e79\u4e00\u5957\uff0c\u7136\u5f8c cuda \u5316<\/p>\n<\/li>\n<\/ol>\n
    1 \u5f88\u597d\u7406\u89e3\uff0c2 \u5247\u662f\u70ba\u4e86\u5229\u7528\u986f\u793a\u5361\u7684 GPU\u52a0\u901f\u5316\u3002\u5c0d\u65bc\u81ea\u52d5\u99d5\u99db\u4f86\u8aaa\uff0c\u80fd\u5feb\u4e00\u9ede\u7b97\u51fa\uff0c\u5c31\u53ef\u4ee5\u8b93\u5f8c\u7e8c\u7684\u5224\u65b7\u66f4\u5bb9\u6613\u57f7\u884c\uff0c\u63d0\u9ad8\u5b89\u5168\u6027<\/p>\n
    PAGE4:<\/h4>\n
    Outline<\/h4>\n
    \u7406\u89e3\u9019\u6771\u897f\u662f\u8981\u7167\u9806\u5e8f\u4f86\uff0c\u5148\u77e5\u9053 Normal-Distributions \uff0c\u624d\u80fd\u77e5\u9053\u00a0Normal-Distributions Transform \u5230\u5e95\u5728\u73a9\u5565\u3002\u6700\u5f8c\u624d\u80fd\u5c0e\u5230 TKU \u505a\u4e86\u5565\u6539\u9032\u3002<\/p>\n
    PAGE 5:<\/h4>\n
    Normal Distribution<\/h4>\n
    \u6b63\u614b\u5206\u4f48 \u2013 \u662f\u4e00\u500b\u5728\u6578\u5b78<\/a>\u3001\u7269\u7406<\/a>\u53ca\u5de5\u7a0b<\/a>\u7b49\u9818\u57df<\/a>\u90fd\u975e\u5e38\u91cd\u8981\u7684\u6a5f\u7387\u5206\u4f48\uff0c\u7531\u65bc\u9019\u500b\u5206\u5e03\u51fd\u6578<\/a>\u5177\u6709\u5f88\u591a\u975e\u5e38\u6f02\u4eae\u7684\u6027\u8cea.\u4f7f\u5f97\u5176\u5728\u8af8\u591a\u6d89\u53ca\u7d71\u8a08\u79d1\u5b78\u96e2\u6563\u79d1\u5b78\u7b49\u9818\u57df\u7684\u8a31\u591a\u65b9\u9762\u90fd\u6709\u8457\u91cd\u5927\u7684\u5f71\u97ff\u529b<\/p>\n
    https:\/\/en.wikipedia.org\/wiki\/Normal_distribution<\/a><\/p>\n
    $$
    \np(\\mathbf{x}) = \\frac{1}{(2\\pi)^{D\/2}\\sqrt{|\\mathbf{\\Sigma}|}} \\exp\\left(-\\frac{(\\mathbf{x}-\\mathbf{\\mu})^T \\mathbf{\\Sigma}^{-1}(\\mathbf{x}-\\mathbf{\\mu})}{2}\\right)
    \n$$
    \n(\u591a\u5143\u5e38\u614b\u6a5f\u7387\u5bc6\u5ea6\u5206\u4f48\u51fd\u6578\uff0c\u5176\u4e2d $\\mathbf{\\mu}$ \u70ba\u9ad4\u7d20\u5167\u9ede\u4e4b\u5747\u503c\u5411\u91cf\uff0c$\\mathbf{\\Sigma}$ \u70ba\u5171\u8b8a\u7570\u6578\u77e9\u9663)<\/em><\/p>\n
    \u9023\u7e8c\u7684\u6771\u897f\uff0c\u6578\u5b78\u53ea\u8981\u80fd\u8868\u793a\u51fa\u4f86\u5c31\u80fd\u7b97\uff0c\u5148\u7406\u89e3\u9019\u6a23\u7684\u5206\u4f48\u5c0d\u65bc\u9ede\u96f2\u7684\u904b\u7b97\u662f\u6709\u610f\u7fa9\u7684<\/p>\n
    PAGE 6:<\/h4>\n
    Normal Distribution Transform<\/strong><\/h4>\n
    \u9019\u88e1\u8ac7\u8ac7\u600e\u9ebc\u4f5c\u4e00\u822c\u7248\u7684 NDT\uff0c\u5c07\u6240\u6709\u7684\u9ede\u96f2\u5148\u5207\u6210\u4e00\u683c\u4e00\u683c ND Voxel<\/p>\n
    The normal-distributions transform can be\u00a0 described as a method for compactly representing a surface.<\/p>\n
    \u6b63\u614b\u5206\u4f48\u7d66\u51fa\u4e86\u9ede\u96f2\u7684\u5206\u6bb5\u5e73\u6ed1\u8868\u793a\uff0c\u5177\u6709\u9023\u7e8c\u7684\u5c0e\u6578\u3002 \u6bcf\u500bPDF\uff08\u6a5f\u7387\u5bc6\u5ea6\u51fd\u6578\uff09<\/a>\u53ef\u4ee5\u770b\u4f5c\u662f\u5c40\u90e8\u8868\u9762\u7684\u8fd1\u4f3c\u503c\uff0c\u63cf\u8ff0\u4e86\u8868\u9762\u7684\u4f4d\u7f6e\u4ee5\u53ca\u5176\u53d6\u5411\u548c\u5e73\u6ed1\u5ea6\u3002<\/p>\n
    A 2D laser scan from a mine tunnel (shown as points) and the PDFs describing the surface shape. Each cell is a square with 2 m side length in this case. Brighter areas represent a higher probability. PDFs have been computed only for cells with more than five points.<\/p>\n
    graph TD\n    Cloud[\u8f38\u5165\u5149\u9054 LiDAR \u9ede\u96f2] --> Voxelize[3D \u9ad4\u7d20\u7db2\u683c\u5283\u5206 (Voxel Grid)]\n    Voxelize --> Cell1[Voxel Cell 1: \u8a08\u7b97 \u03bc1, \u03a31]\n    Voxelize --> Cell2[Voxel Cell 2: \u8a08\u7b97 \u03bc2, \u03a32]\n    Voxelize --> CellN[Voxel Cell N: \u8a08\u7b97 \u03bcn, \u03a3n]<\/pre>\n
    PAGE7 :<\/h4>\n
    NDT in tunnel \u2013 3D<\/h4>\n
    \u793a\u610f\u5716\uff0c\u81ea\u5df1\u770b\u6295\u5f71\u7247\u5566<\/p>\n
    PAGE 8:<\/h4>\n
    \u5982\u4f55\u8868\u793a\u9ede\u96f2 Cell \u7684\u6a5f\u7387\u5206\u5e03<\/h4>\n
    D-dimensional normal random process, the likelihood of having measured ~x is\u00a0 where ~yk=1,\u2026, m are the positions of the reference scan points contained in the cell.<\/p>\n
    \u6b63\u614b\u5206\u4f48\u7d66\u51fa\u4e86\u9ede\u96f2\u7684\u5206\u6bb5\u5e73\u6ed1\u8868\u793a\uff0c\u5177\u6709\u9023\u7e8c\u7684\u5c0e\u6578\u3002 \u6bcf\u500bPDF\u53ef\u4ee5\u770b\u4f5c\u662f\u5c40\u90e8\u8868\u9762\u7684\u8fd1\u4f3c\u503c\uff0c\u63cf\u8ff0\u4e86\u8868\u9762\u7684\u4f4d\u7f6e\u4ee5\u53ca\u5176\u53d6\u5411\u548c\u5e73\u6ed1\u5ea6\u3002<\/p>\n
    \u5354\u65b9\u5dee\u77e9\u9663\u7684\u7279\u5fb5\u5411\u91cf\u548c\u7279\u5fb5\u503c\u53ef\u4ee5\u8868\u9054\u8868\u9762\u4fe1\u606f<\/p>\n
    Each PDF can be seen as an approximation of the local surface, describing the position of the surface as well as its orientation and smoothness.<\/p>\n
    PAGE 9, 10:<\/h4>\n
    Scan registration<\/h4>\n
    \u70ba\u4e86\u627e\u51fa\u5169\u6b21 SCAN \u7684 spatial transformation function T(~p, ~x) that moves a point ~x in space by the pose ~p. \u53ef\u4ee5\u5229\u7528\u525b\u525b\u5f97\u5230\u7684 CELL PDF \uff0c\u53bb\u627e\u51fa\u6700\u63a5\u8fd1\uff08max\uff09\u7684\u5206\u4f48\u51fd\u6578<\/p>\n
    NDT score \u53ef\u4ee5\u7528\u65bc\u725b\u9813\u6cd5\uff0c\u8868\u793a\u662f\u5426\u9054\u5230\u6700\u597d\u7684\u5206\u6578\u3002Gaussian approximation \u5247\u53ef\u4ee5\u6e1b\u5c11\u904b\u7b97<\/p>\n
    (\u770b\u6700\u4e0a\u9762\u63a8\u85a6\u7684\u8ad6\u6587\u6216\u662f\u6295\u5f71\u7247\uff0c\u6c92\u8fa6\u6cd5\u591a\u52a0\u8a3b\u89e3)<\/p>\n
    PAGE 11:<\/h4>\n
    Newton\u2019s algorithm for<\/h4>\n