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\u672c\u6587\u65b9\u6cd5\u7684\u7b2c\u4e00\u9636\u6bb5\u5c06\u4f17\u6240\u5468\u77e5\u7684A^{*}\u7b97\u6cd5\u7684\u4e00\u79cd\u53d8\u4f53\u5e94\u7528\u4e8e\u8f66\u8f86\u76843D\u8fd0\u52a8\u5b66\u72b6\u6001\u7a7a\u95f4\uff0c\u4f46\u662f\u4fee\u6539\u4e86\u72b6\u6001\u66f4\u65b0\u89c4\u5219\uff0c\u8be5\u89c4\u5219\u5728A^{*}\u7684\u79bb\u6563\u641c\u7d22\u8282\u70b9\u4e2d\u6355\u83b7\u8fde\u7eed\u72b6\u6001\u6570\u636e\u3002\u5982\u4f20\u7edf\u7684A^{*}\u4e00\u6837\uff0c\u79bb\u6563\u5316\u641c\u7d22\u7a7a\u95f4\uff0c\u4f46\u4e0e\u4f20\u7edf\u7684A^{*}\u4e0d\u540c\uff0c\u4f20\u7edfA^{*}\u53ea\u5141\u8bb8\u8bbf\u95ee\u6805\u683c\u7684\u4e2d\u5fc3\uff0c\u800c\u672c\u6587\u7684\u6df7\u5408\u72b6\u6001A^{*}\u5c06\u6bcf\u4e2a\u6805\u683c\u4e0e\u8f66\u8f86\u7684\u4e00\u4e2a\u8fde\u7eed3D\u72b6\u6001\u76f8\u5173\u8054\uff0c\u5982\u56fe2\u6240\u793a\u3002<\/p>\n

<\/p>\n

\u5982\u4e0a\u6240\u8ff0\uff0c\u7531\u4e8e\u6df7\u5408\u72b6\u6001A^{*}\u5408\u5e76\u4e86\u5728\u79bb\u6563\u7a7a\u95f4\u4e2d\u5360\u636e\u76f8\u540c\u6805\u683c\u7684\u8fde\u7eed\u5750\u6807\u7cfb\u4e0b\u72b6\u6001\uff0c\u56e0\u6b64\u5b83\u4e0d\u80fd\u591f\u4fdd\u8bc1\u627e\u5230\u6700\u5c0f\u4ee3\u4ef7\u7684\u89e3\u3002\u7136\u800c\uff0c\u7ed3\u679c\u8def\u5f84\u80fd\u591f\u4fdd\u8bc1\u662f\u53ef\u884c\u9a76\u7684\uff08\u800c\u4e0d\u662f\u50cf\u6807\u51c6A^{*}\u60c5\u51b5\u4e0b\u662f\u5206\u6bb5\u7ebf\u6027\u7684\uff09\u3002\u6b64\u5916\uff0c\u5b9e\u9645\u4e0a\u6df7\u5408A^{*}\u7684\u89e3\u901a\u5e38\u4f4d\u4e8e\u5168\u5c40\u6700\u4f18\u7684\u90bb\u57df\u5185\uff0c\u8fd9\u5141\u8bb8\u6211\u4eec\u901a\u8fc7\u672c\u6587\u7b97\u6cd5\u7684\u7b2c\u4e8c\u9636\u6bb5\uff08\u4f7f\u7528\u68af\u5ea6\u4e0b\u964d\u6765\u5c40\u90e8\u5730\u6539\u8fdb\u8def\u5f84\uff0c\u5982\u4e0b\u6240\u8ff0\uff09\u6765\u9891\u7e41\u5730\u83b7\u5f97\u5168\u5c40\u6700\u4f18\u89e3\u3002<\/p>\n

\u6df7\u5408\u72b6\u6001A^{*}\u7684\u4e3b\u8981\u4f18\u52bf\u4f53\u73b0\u5728\u7d27\u51d1\u7a7a\u95f4\u4e2d\u5b83\u7684\u884c\u4e3a\uff0c\u5176\u4e2d\u79bb\u6563\u8bef\u5dee\u53d8\u5f97\u81f3\u5173\u91cd\u8981\u3002\u672c\u6587\u7b97\u6cd5\u89c4\u5212\u6b63\u5411\u548c\u53cd\u5411\u8fd0\u52a8\uff0c\u5176\u5bf9\u53cd\u5411\u884c\u9a76\u548c\u5207\u6362\u8fd0\u52a8\u65b9\u5411\u505a\u51fa\u60e9\u7f5a\u3002<\/p>\n

A.\u542f\u53d1\u51fd\u6570\u672c\u6587\u641c\u7d22\u7b97\u6cd5\u7531\u4e24\u79cd\u542f\u53d1\u51fd\u6570\u5f15\u5bfc\uff0c\u5982\u56fe3\u6240\u793a\u3002\u8fd9\u4e9b\u542f\u53d1\u51fd\u6570\u4e0d\u4f9d\u8d56\u4e8e\u6df7\u5408\u72b6\u6001A^{*}\u7684\u4efb\u4f55\u6027\u8d28\uff0c\u5e76\u4e14\u8fd8\u80fd\u88ab\u5e94\u7528\u4e8e\u5176\u5b83\u641c\u7d22\u65b9\u6cd5\uff08\u4f8b\u5982\u79bb\u6563A^{*}\uff09\u3002<\/p>\n

<\/p>\n

\u7b2c\u4e00\u79cd\u542f\u53d1\u51fd\u6570\uff08\u672c\u6587\u79f0\u4e3a\u201c\u4e0d\u8003\u8651\u969c\u788d\u7269\u7684\u975e\u5b8c\u6574\u6027\u542f\u53d1\u201d\uff09\u5ffd\u7565\u969c\u788d\u7269\uff0c\u4f46\u662f\u8003\u8651\u8f66\u8f86\u7684\u975e\u5b8c\u6574\u7279\u6027\u3002\u4e3a\u4e86\u8ba1\u7b97\u5b83\uff0c\u5047\u5b9a\u76ee\u6807\u72b6\u6001\u4e3a(x_g, y_g, theta_g) = (0, 0, 0)\uff0c\u5e76\u4e14\u5728\u5b8c\u5168\u6ca1\u6709\u969c\u788d\u7269\u7684\u5047\u8bbe\u4e0b\u8ba1\u7b97\u4ece\u76ee\u6807\u7684\u67d0\u4e2a\u79bb\u6563\u90bb\u57df\u4e2d\u7684\u6bcf\u4e2a\u70b9(x, y, theta)\u5230\u76ee\u6807\u7684\u6700\u77ed\u8def\u5f84\uff08\u4e0d\u8003\u8651\u969c\u788d\u7269\u7684\u975e\u5b8c\u6574\u6027\u4ee3\u4ef7\uff09\u3002\u672c\u6587\u4f7f\u7528160×160\u7684\u6805\u683c\u5730\u56fe\uff0c\u5176x-y\u65b9\u5411\u7684\u5206\u8fa8\u7387\u4e3a1\u7c73\uff0c\u89d2\u5ea6\u5206\u8fa8\u7387\u4e3a5\u00b0\u3002\u663e\u7136\uff0c\u8fd9\u662f\u4e00\u79cd\u53ef\u63a5\u53d7\u7684\u542f\u53d1\u51fd\u6570\u3002\u63a5\u7740\uff0c\u4f7f\u7528\u4e0d\u8003\u8651\u969c\u788d\u7269\u7684\u975e\u5b8c\u6574\u6027\u4ee3\u4ef7\uff08\u4e0a\u8ff0\uff09\u548c\u4e24\u7ef4\u7684\u6b27\u5f0f\u8ddd\u79bb\u4e2d\u6700\u5927\u503c\u4f5c\u4e3a\u672c\u6587\u7684\u542f\u53d1\u51fd\u6570\u3002\u8fd9\u79cd\u542f\u53d1\u51fd\u6570\u7684\u6548\u679c\u5728\u4e8e\uff0c\u5b83\u5220\u9664\u4e86\u4ee5\u9519\u8bef\u671d\u5411\u63a5\u8fd1\u76ee\u6807\u7684\u641c\u7d22\u5206\u652f\u3002\u6ce8\u610f\uff0c\u7531\u4e8e\u8fd9\u79cd\u542f\u53d1\u51fd\u6570\u4e0d\u4f9d\u8d56\u4e8e\u8fd0\u884c\u65f6\u7684\u4f20\u611f\u5668\u4fe1\u606f\uff08\u8fd9\u662f\u56e0\u4e3a\u8be5\u542f\u53d1\u51fd\u6570\u4e0d\u8003\u8651\u969c\u788d\u7269\uff09\uff0c\u56e0\u6b64\u5b83\u80fd\u591f\u88ab\u5b8c\u5168\u5730\u79bb\u7ebf\u9884\u5148\u8ba1\u7b97\uff0c\u5e76\u4e14\u63a5\u7740\u88ab\u7b80\u5355\u5730\u5e73\u79fb\u548c\u65cb\u8f6c\u6765\u5339\u914d\u5f53\u524d\u7684\u76ee\u6807\u3002\u5728\u6211\u4eec\u7684\u771f\u5b9e\u9a7e\u9a76\u573a\u666f\u5b9e\u9a8c\u4e2d\uff0c\u8fd9\u79cd\u542f\u53d1\u51fd\u6570\u6bd4\u76f4\u63a5\u4f7f\u7528\u4e24\u7ef4\u6b27\u6c0f\u8ddd\u79bb\u4ee3\u4ef7\u5728\u8282\u70b9\u6269\u5c55\u6570\u91cf\u65b9\u9762\u7ed9\u51fa\u4e86\u63a5\u8fd1\u4e00\u4e2a\u6570\u91cf\u7ea7\u7684\u6539\u8fdb\u3002<\/p>\n

\u7b2c\u4e8c\u79cd\u542f\u53d1\u51fd\u6570\u662f\u7b2c\u4e00\u79cd\u542f\u53d1\u51fd\u6570\u7684\u5bf9\u5076\u7248\u672c\uff0c\u56e0\u4e3a\u5b83\u5ffd\u7565\u4e86\u8f66\u8f86\u7684\u975e\u5b8c\u6574\u7279\u6027\uff0c\u4f46\u662f\u4f7f\u7528\u969c\u788d\u7269\u5730\u56fe\u901a\u8fc7\u5728\u4e24\u7ef4\u7a7a\u95f4\u4e2d\u6267\u884c\u52a8\u6001\u89c4\u5212\u6765\u8ba1\u7b97\u5230\u76ee\u6807\u7684\u6700\u77ed\u8ddd\u79bb\u3002\u8fd9\u79cd\u542f\u53d1\u51fd\u6570\u7684\u4f18\u52bf\u5728\u4e8e\uff0c\u5b83\u53d1\u73b0\u4e24\u7ef4\u7a7a\u95f4\u4e2d\u6240\u6709\u7684U\u578b\u969c\u788d\u7269\u548c\u6b7b\u80e1\u540c\uff0c\u5e76\u4e14\u63a5\u7740\u5f15\u5bfc\u8ba1\u7b97\u4ee3\u4ef7\u66f4\u6602\u8d35\u7684\u4e09\u7ef4\u641c\u7d22\u8fdc\u79bb\u8fd9\u4e9b\u533a\u57df\u3002<\/p>\n

\u8fd9\u4e24\u79cd\u542f\u53d1\u51fd\u6570\u4eceA^{*}\u610f\u4e49\u8003\u8651\u662f\u6570\u5b66\u4e0a\u5408\u7406\u7684\uff0c\u6240\u4ee5\u4f7f\u7528\u8fd9\u4e24\u79cd\u542f\u53d1\u51fd\u6570\u7684\u6700\u5927\u503c\u3002<\/p>\n

B.\u89e3\u6790\u6269\u5c55<\/p>\n

\u4e0a\u8ff0\u524d\u5411\u641c\u7d22\u4f7f\u7528\u4e86\u63a7\u5236\u52a8\u4f5c\uff08\u8f6c\u5411\uff09\u7684\u79bb\u6563\u7a7a\u95f4\u3002\u8fd9\u610f\u5473\u7740\u641c\u7d22\u5c06\u6c38\u8fdc\u65e0\u6cd5\u8fbe\u5230\u7cbe\u786e\u7684\u8fde\u7eed\u5750\u6807\u7cfb\u4e0b\u7684\u76ee\u6807\u72b6\u6001\uff08\u7cbe\u5ea6\u4f9d\u8d56\u4e8eA^{*}\u6805\u683c\u7684\u5206\u8fa8\u7387\uff09\u3002\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u7cbe\u5ea6\u95ee\u9898\u5e76\u4e14\u8fdb\u4e00\u6b65\u63d0\u9ad8\u641c\u7d22\u901f\u5ea6\uff0c\u6211\u4eec\u4f7f\u7528\u57fa\u4e8eReed-Shepp\u6a21\u578b\u7684\u89e3\u6790\u6269\u5c55\u6765\u589e\u5f3a\u641c\u7d22\u3002\u5728\u4e0a\u8ff0\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u901a\u8fc7\u4f7f\u7528\u7279\u5b9a\u7684\u63a7\u5236\u52a8\u4f5c\u5728\u4e00\u5c0f\u6bb5\u65f6\u95f4\u5185\uff08\u5bf9\u5e94\u4e8e\u6805\u683c\u7684\u5206\u8fa8\u7387\uff09\u6a21\u62df\u8f66\u8f86\u7684\u8fd0\u52a8\u5b66\u6a21\u578b\u6765\u6269\u5c55\u6811\u4e2d\u7684\u8282\u70b9\u3002<\/p>\n

\u9664\u4e86\u4ee5\u8fd9\u79cd\u65b9\u5f0f\u751f\u6210\u5b50\u8282\u70b9\u5916\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u8282\u70b9\uff0c\u901a\u8fc7\u8ba1\u7b97\u4ece\u5f53\u524d\u72b6\u6001\u5230\u76ee\u6807\u7684\u6700\u4f18Reed-Shepp\u8def\u5f84\u6765\u751f\u6210\u989d\u5916\u7684\u5b50\u8282\u70b9\uff08\u5047\u8bbe\u73af\u5883\u4e2d\u6ca1\u6709\u969c\u788d\u7269\uff09\u3002\u63a5\u7740\uff0c\u68c0\u67e5Reed-Shepp\u8def\u5f84\u662f\u5426\u548c\u5f53\u524d\u969c\u788d\u7269\u5730\u56fe\u53d1\u751f\u78b0\u649e\uff0c\u5e76\u4e14\u4ec5\u5f53\u8def\u5f84\u6ca1\u6709\u53d1\u751f\u78b0\u649e\u65f6\u624d\u5c06\u5b50\u8282\u70b9\u52a0\u5165\u5230\u6811\u4e2d\u3002\u51fa\u4e8e\u8ba1\u7b97\u6548\u7387\u7684\u539f\u56e0\uff0c\u4e0d\u5e0c\u671b\u5c06Reed-Shepp\u6269\u5c55\u5e94\u7528\u4e8e\u6bcf\u4e2a\u8282\u70b9\uff08\u7279\u522b\u662f\u79bb\u76ee\u6807\u8f83\u8fdc\u7684\u8282\u70b9\uff0c\u56e0\u4e3a\u8fd9\u79cd\u60c5\u51b5\u4e0b\u7684\u5927\u591a\u6570Reed-Shepp\u8def\u5f84\u90fd\u4f1a\u4e0e\u969c\u788d\u7269\u53d1\u751f\u78b0\u649e\uff09\u3002\u5728\u672c\u6587\u7684\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u4e00\u79cd\u7b80\u5355\u7684\u9009\u62e9\u89c4\u5219\uff0c\u5176\u4e2dReed-Shepp\u66f2\u7ebf\u88ab\u5e94\u7528\u4e8e\u6bcfN\u4e2a\u8282\u70b9\u4e2d\u7684\u4e00\u4e2a\u8282\u70b9\uff0c\u5e76\u4e14N\u4f5c\u4e3a\u5230\u76ee\u6807\u4ee3\u4ef7\u7684\u542f\u53d1\u51fd\u6570\u800c\u51cf\u5c0f\uff08\u5f53\u63a5\u8fd1\u76ee\u6807\u65f6\uff0c\u5c06\u5bf9\u8282\u70b9\u8fdb\u884c\u66f4\u4e3a\u9891\u7e41\u5730\u89e3\u6790\u6269\u5c55\uff09\u3002<\/p>\n

Reed-Shepp\u6269\u5c55\u7684\u641c\u7d22\u6811\u5982\u56fe4\u6240\u793a\u3002\u7531\u8282\u70b9\u77ed\u589e\u91cf\u6269\u5c55\u751f\u6210\u7684\u641c\u7d22\u6811\u663e\u793a\u4e3a\u9ec4\u8272\u5230\u7eff\u8272\u7684\u989c\u8272\u8303\u56f4\uff0c\u5e76\u4e14Reed-Shepp\u6269\u5c55\u663e\u793a\u4e3a\u4e00\u6761\u8fde\u63a5\u76ee\u6807\u7684\u5355\u4e00\u7d2b\u8272\u66f2\u7ebf\u3002\u6211\u4eec\u53d1\u73b0\uff0c\u641c\u7d22\u6811\u7684\u8fd9\u79cd\u89e3\u6790\u6269\u5c55\u5728\u7cbe\u5ea6\u548c\u89c4\u5212\u65f6\u95f4\u65b9\u9762\u90fd\u5177\u6709\u663e\u8457\u7684\u4f18\u52bf\u3002<\/p>\n

<\/p>\n

\u4f7f\u7528Voronoi\u573a\u7684\u8def\u5f84\u4ee3\u4ef7\u51fd\u6570<\/p>\n

\u672c\u6587\u4f7f\u7528\u5982\u4e0b\u52bf\u573a\uff08\u6211\u4eec\u79f0\u4e3aVoronoi\u573a\uff09\u6765\u5b9a\u4e49\u8def\u5f84\u957f\u5ea6\u548c\u8f66\u8f86\u63a5\u8fd1\u969c\u788d\u7269\u4e4b\u95f4\u7684\u5e73\u8861\u3002Voronoi\u573a\u88ab\u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n

rho_V = (frac{alpha}{alpha + d_{mathcal O}(x,y)})(frac{d_{mathcal V}}{d_{mathcal O}(x,y) + d_{mathcal V}(x,y)}) frac{(d_{mathcal O} – d^{max}_{mathcal O})^2}{(d^{max}_{mathcal O})^2} tag{1}<\/p>\n

\u5176\u4e2d\uff0cd_{mathcal O}\u548cd_{mathcal V}\u5206\u522b\u4e3a\u5230\u6700\u8fd1\u969c\u788d\u7269\u548c\u5e7f\u4e49Voronoi\u56fe\uff08GVD\uff09\u8fb9\u7684\u8ddd\u79bb\uff0c\u5e76\u4e14alpha > 0\uff0cd_{mathcal O} > 0\u4e3a\u5e38\u6570\uff0c\u7528\u4e8e\u63a7\u5236\u52bf\u573a\u7684\u8870\u51cf\u7387\u548c\u6700\u5927\u6709\u6548\u8303\u56f4\u3002\u516c\u5f0f\uff081\uff09\u4e2d\u7684\u8868\u8fbe\u5f0f\u662f\u5bf9\u4e8ed_{mathcal O} leq d^{max}_{mathcal O}\u7684\uff0c\u5426\u5219\uff0crho_V(x, y) = 0\u3002<\/p>\n

\u8fd9\u4e2a\u52bf\u573a\u5177\u6709\u5982\u4e0b\u6027\u8d28\uff1ai\uff09\u5f53d_{mathcal O} geq d^{max}_{mathcal O}\u65f6\uff0c\u5176\u503c\u4e3a\u96f6\uff1bii\uff09rho_V(x, y) in [0,1]\uff0c\u5e76\u4e14\u5728(x,y)\u7a7a\u95f4\u4e0a\u8fde\u7eed\uff1b\u56e0\u4e3a\u4e0d\u80fd\u540c\u65f6\u6ee1\u8db3d_{mathcal O} = d_{mathcal V} = 0\uff1biii\uff09\u53ea\u6709\u5728\u969c\u788d\u7269\u5185\u65f6\u5b83\u624d\u8fbe\u5230\u6700\u5927\u503c\uff1biv\uff09\u53ea\u6709\u5728GVD\u7684\u8fb9\u4e0a\u65f6\uff0c\u5b83\u624d\u8fbe\u5230\u6700\u5c0f\u503c\u3002<\/p>\n

Voronoi\u573a\u76f8\u5bf9\u4e8e\u4f20\u7edf\u52bf\u573a\u7684\u5173\u952e\u4f18\u52bf\u5728\u4e8e\uff0c\u52bf\u573a\u503c\u4e0e\u5bfc\u822a\u603b\u5171\u53ef\u901a\u884c\u533a\u57df\u6210\u6b63\u6bd4\u3002\u56e0\u6b64\uff0c\u5373\u4f7f\u72ed\u7a84\u7684\u5f00\u653e\u533a\u57df\u4e5f\u4fdd\u6301\u53ef\u901a\u884c\uff0c\u8fd9\u4e0d\u662f\u6807\u51c6\u7684\u52bf\u573a\u603b\u662f\u80fd\u591f\u505a\u5230\u7684\u3002<\/p>\n

\u56fe5\u8bf4\u660e\u4e86\u8fd9\u4e2a\u6027\u8d28\u3002\u56fe5a\u5c55\u793a\u4e86Voronoi\u573a\u7684\u4e24\u7ef4\u6295\u5f71\uff0c\u5e76\u4e14\u56fe5b\u7ed9\u51fa\u4e86\u5bf9\u5e94\u7684\u5e7f\u4e49Voronoi\u56fe\u3002\u6ce8\u610f\uff0c\u5f7c\u6b64\u9760\u8fd1\u7684\u969c\u788d\u7269\u95f4\u7684\u72ed\u7a84\u901a\u9053\u6ca1\u6709\u88ab\u52bf\u573a\u6240\u963b\u585e\uff0c\u5e76\u4e14\u5728\u5b83\u4eec\u4e4b\u95f4\u603b\u662f\u5b58\u5728\u4e00\u6761\u8fde\u7eed\u7684rho_V = 0\u7684\u8def\u5f84\u3002\u5c06\u5b83\u4e0e\u56fe5c\u4e2d\u663e\u793a\u7684\u7b80\u5355\u6734\u7d20\u7684\u52bf\u573arho(x, y) = alpha (alpha + d_{mathcal O}(x, y))^{-1}\u76f8\u6bd4\uff0c\u8be5\u6734\u7d20\u7684\u52bf\u573a\u5728\u969c\u788d\u7269\u4e4b\u95f4\u7684\u72ed\u7a84\u901a\u9053\u4e2d\u5177\u6709\u9ad8\u52bf\u573a\u533a\u57df\u3002<\/p>\n

<\/p>\n

\u56fe6\u5c55\u793a\u4e86\u771f\u5b9e\u505c\u8f66\u573a\u7684Voronoi\u573a\u548c\u4e00\u6761\u53ef\u884c\u9a76\u7684\u8f68\u8ff9\u3002<\/p>\n

<\/p>\n

\u6211\u4eec\u5e94\u8be5\u6ce8\u610f\u5230\uff0cVoronoi\u56fe\u548c\u52bf\u573a\u7684\u4f7f\u7528\u65e9\u5df2\u7ecf\u5728\u673a\u5668\u4eba\u8fd0\u52a8\u89c4\u5212\u4e2d\u63d0\u51fa\u3002\u4f8b\u5982\uff0cVoronoi\u56fe\u80fd\u591f\u88ab\u7528\u4e8e\u83b7\u53d6\u53ef\u901a\u884c\u7a7a\u95f4\u7684\u9aa8\u67b6\uff082000\u5e74Choset\u548cBurdick\uff09\u3002\u7136\u800c\uff0c\u5bf9\u4e8e\u5177\u6709\u975e\u5b8c\u6574\u7279\u6027\u7684\u6c7d\u8f66\u800c\u8a00\uff0c\u6cbf\u7740Voronoi\u56fe\u5bfc\u822a\u662f\u4e0d\u53ef\u80fd\u7684\u3002<\/p>\n

\u5bfc\u822a\u51fd\u6570\uff081987\u5e74Koditschek\u30011992\u5e74Rimon\u548cKoditschek\uff09\u548c\u62c9\u666e\u62c9\u65af\u52bf\u80fd\uff081990\u5e74Connolly\u3001Burns\u548cWeiss\uff09\u4e5f\u7c7b\u4f3c\u4e8e\u6211\u4eec\u7684Voronoi\u573a\uff0c\u56e0\u4e3a\u4ed6\u4eec\u4e3a\u5168\u5c40\u5bfc\u822a\u6784\u5efa\u4e86\u65e0\u5c40\u90e8\u6781\u5c0f\u503c\u7684\u52bf\u80fd\u51fd\u6570\u3002\u6211\u4eec\u6ca1\u6709\u5c06Voronoi\u573a\u7528\u4e8e\u5168\u5c40\u5bfc\u822a\u3002\u7136\u800c\uff0c\u6211\u4eec\u89c2\u6d4b\u5230\uff0c\u5bf9\u4e8e\u5177\u6709\u51f8\u969c\u788d\u7269\u7684\u5de5\u4f5c\u7a7a\u95f4\uff0c\u80fd\u591f\u901a\u8fc7\u5168\u5c40\u5438\u5f15\u52bf\u80fd\u6765\u589e\u5f3aVoronoi\u573a\uff0c\u4ece\u800c\u4ea7\u751f\u4e00\u4e2a\u65e0\u5c40\u90e8\u6781\u5c0f\u503c\u7684\u573a\uff0c\u5e76\u56e0\u6b64\u9002\u5408\u4e8e\u5168\u5c40\u5bfc\u822a\u3002<\/p>\n

\u5c40\u90e8\u4f18\u5316\u548c\u5e73\u6ed1<\/p>\n

\u7531\u6df7\u5408\u72b6\u6001A^{*}\u4ea7\u751f\u7684\u8def\u5f84\u901a\u5e38\u4ecd\u7136\u662f\u6b21\u4f18\u7684\uff0c\u5e76\u4e14\u503c\u5f97\u8fdb\u4e00\u6b65\u4f18\u5316\u3002\u6839\u636e\u7ecf\u9a8c\uff0c\u6211\u4eec\u53d1\u73b0\u8fd9\u79cd\u8def\u5f84\u662f\u53ef\u884c\u9a76\u7684\uff0c\u4f46\u662f\u53ef\u80fd\u5305\u542b\u9700\u8981\u975e\u5fc5\u8981\u8f6c\u5411\u7684\u4e0d\u6b63\u5e38\u7684\u6025\u8f6c\u5f2f\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u901a\u8fc7\u5e94\u7528\u4ee5\u4e0b\u4e24\u9636\u6bb5\u7684\u4f18\u5316\u8fc7\u7a0b\u6765\u540e\u5904\u7406\u6df7\u5408\u72b6\u6001A^{*}\u7684\u89e3\u3002\u5728\u7b2c\u4e00\u9636\u6bb5\uff0c\u6211\u4eec\u5728\u8def\u5f84\u9876\u70b9\u7684\u5750\u6807\u4e0a\u6784\u5efa\u4e00\u4e2a\u975e\u7ebf\u6027\u4f18\u5316\u8fc7\u7a0b\uff0c\u4ece\u800c\u63d0\u9ad8\u89e3\u7684\u957f\u5ea6\u548c\u5e73\u6ed1\u6027\u3002\u7b2c\u4e8c\u9636\u6bb5\u4f7f\u7528\u5177\u6709\u66f4\u9ad8\u5206\u8fa8\u7387\u8def\u5f84\u79bb\u6563\u7684\u5171\u8f6d\u68af\u5ea6\u7684\u53e6\u4e00\u79cd\u8fed\u4ee3\u6765\u8fdb\u884c\u975e\u53c2\u6570\u5316\u63d2\u503c\u3002<\/p>\n

\u7ed9\u5b9a\u4e00\u7cfb\u5217\u9876\u70b9mathbf{x}_i = (x_i, y_i), i in [1,N]\uff0c\u6211\u4eec\u5b9a\u4e49\u4e00\u4e9b\u53d8\u91cf\uff1amathbf{o}_i\u4e3a\u79bb\u9876\u70b9\u6700\u8fd1\u7684\u969c\u788d\u7269\u7684\u4f4d\u7f6e\uff1bDelta mathbf{x}_i = mathbf{x}_i – mathbf{x}_{i-1}\u4e3a\u9876\u70b9\u7684\u4f4d\u79fb\u5411\u91cf\uff1bDelta phi_i = lvert tan^{-1}frac{Delta y_{i+1}}{Delta x_{i+1}} – tan^{-1}frac{Delta y_i}{Delta x_i} rvert\u4e3a\u9876\u70b9\u7684\u5207\u5411\u89d2\u5ea6\u53d8\u5316\u3002<\/p>\n

\u76ee\u6807\u51fd\u6570\u4e3a\uff1abegin{matrix} w_{rho} sum^N_{i = 1} rho_V(x_i, y_i) + w_o sum^N_{i = 1}sigma_o(lvert mathbf{x}_i – mathbf{o}_i rvert – d_{max}) \\ + w_{kappa} sum^{N-1}_{i=1} sigma_{kappa}(frac{Delta phi_i}{lvert Delta mathbf{x}_i rvert} – kappa_{max}) + w_s sum^{N-1}_{i=1}(Delta mathbf{x}_{i+1} – Delta mathbf{x}_i)^2 end{matrix}<\/p>\n

\u5176\u4e2d\uff0crho_V\u4e3aVoronoi\u573a\uff0ckappa_{max}\u4e3a\u8def\u5f84\u7684\u6700\u5927\u5141\u8bb8\u66f2\u7387\uff08\u7531\u8f66\u8f86\u7684\u6700\u5c0f\u8f6c\u5f2f\u534a\u5f84\u5b9a\u4e49\uff09\uff0csigma_o\u548csigma_{kappa}\u4e3a\u60e9\u7f5a\u51fd\u6570\uff08\u6839\u636e\u7ecf\u9a8c\uff0c\u53d1\u73b0\u7b80\u5355\u7684\u5e73\u65b9\u60e9\u7f5a\u9879\u5c31\u80fd\u5f88\u597d\u5de5\u4f5c\uff09\uff0c\u4ee5\u53caw_{rho}\u3001w_o\u3001w_{kappa}\u548cw_s\u4e3a\u6743\u91cd\u3002<\/p>\n

\u4ee3\u4ef7\u51fd\u6570\u7684\u7b2c\u4e00\u9879\u6709\u6548\u5730\u5f15\u5bfc\u673a\u5668\u4eba\u5728\u72ed\u7a84\u901a\u9053\u548c\u5bbd\u655e\u901a\u9053\u8fdc\u79bb\u969c\u788d\u7269\u3002\u7b2c\u4e8c\u9879\u60e9\u7f5a\u4e0e\u969c\u788d\u7269\u53d1\u751f\u7684\u78b0\u649e\u3002\u7b2c\u4e09\u9879\u7ea6\u675f\u4e86\u8f68\u8ff9\u5728\u6bcf\u4e2a\u8282\u70b9\u5904\u77ac\u65f6\u66f2\u7387\u7684\u4e0a\u9650\uff0c\u5e76\u4e14\u65bd\u52a0\u8f66\u8f86\u7684\u975e\u5b8c\u6574\u6027\u7ea6\u675f\u3002\u7b2c\u56db\u9879\u4e3a\u8def\u5f84\u5e73\u6ed1\u6027\u7684\u5ea6\u91cf\u3002<\/p>\n

\u4e0a\u8ff0\u4ee3\u4ef7\u51fd\u6570\u7684\u68af\u5ea6\u4ee5\u5982\u4e0b\u63cf\u8ff0\u7684\u76f4\u63a5\u65b9\u5f0f\u8ba1\u7b97\u3002\u5bf9\u4e8eVoronoi\u573a\u9879\uff0c\u5f53d_{mathcal O} leq d^{max}_{mathcal O}\u65f6\uff0c\u6709\uff1afrac{partial rho_V}{partial mathbf{x}_i} = frac{partial rho_V}{partial d_{mathcal O}}frac{partial d_{mathcal O}}{partial mathbf{x}_i} + frac{partial rho_V}{partial d_{mathcal V}}frac{partial d_{mathcal V}}{partial mathbf{x}_i} frac{partial d_{mathcal O}}{partial mathbf{x}_i} = frac{mathbf{x}_i – mathbf{o}_i}{lvert mathbf{x}_i – mathbf{o}_i rvert} frac{partial d_{mathcal V}}{partial mathbf{x}_i} = frac{mathbf{x}_i – mathbf{v}_i}{lvert mathbf{x}_i – mathbf{v}_i rvert} frac{partial rho_V}{partial d_{mathcal V}} = frac{alpha}{alpha + d_{mathcal O}} frac{(d_{mathcal O} – d^{max}_{mathcal O})^2}{(d^{max}_{mathcal O})^2} frac{d_{mathcal O}}{(d_{mathcal O}+d_{mathcal V})^2} frac{partial rho_V}{partial d_{mathcal O}} = frac{alpha}{alpha + d_{mathcal O}} frac{d_{mathcal V}}{d_{mathcal O} + d_{mathcal V}} frac{(d_{mathcal O} – d^{max}_{mathcal O})}{(d^{max}_{mathcal O})^2} left[ frac{-(d_{mathcal O} – d^{max}_{mathcal O})}{alpha + d_{mathcal O}} – frac{d_{mathcal O} – d^{max}_{mathcal O}}{d_{mathcal O} + d_{mathcal V}} + 2 right]<\/p>\n

\u5176\u4e2d\uff0cmathbf{v}_i\u4e3a\u5e7f\u4e49Voronoi\u56fe\uff08GVD\uff09\u8fb9\u7f18\u4e0a\u7684\u70b9\u5750\u6807\u7684\u4e24\u7ef4\u5411\u91cf\uff0c\u5b83\u79bb\u9876\u70b9i\u6700\u8fd1\u3002\u6211\u4eec\u901a\u8fc7\u7ef4\u62a4\u6240\u6709\u969c\u788d\u7269\u70b9\u548cGVD\u8fb9\u7f18\u70b9\u7684kd\u6811\u5e76\u4e14\u5728\u5171\u8f6d\u68af\u5ea6\u7684\u6bcf\u6b21\u8fed\u4ee3\u4e2d\u66f4\u65b0\u9876\u70b9\u7684\u6700\u8fd1\u90bb\u6765\u8ba1\u7b97\u6700\u8fd1\u7684\u969c\u788d\u7269mathbf{o}_i\u548c\u6700\u8fd1\u7684GVD\u8fb9\u7f18\u70b9mathbf{v}_i\u3002<\/p>\n

\u5bf9\u4e8e\u4e00\u4e2a\u5e73\u65b9\u5f62\u5f0fsigma_o\u7684\u78b0\u649e\u60e9\u7f5a\u9879\uff0c\u5982\u679clvert mathbf{x}_i – mathbf{o}_i rvert leq d_{max}\uff0c\u6709\uff1afrac{partial sigma_o}{partial mathbf{x}_i} = 2(lvert mathbf{x}_i – mathbf{o}_i rvert – d_{max}) frac{mathbf{x}_i – mathbf{o}_i}{lvert mathbf{x}_i – mathbf{o}_i rvert}<\/p>\n

\u5bf9\u4e8e\u9876\u70b9i\u5904\u7684\u6700\u5927\u66f2\u7387\u9879\uff0c\u5fc5\u987b\u6c42\u5b83\u76f8\u5bf9\u4e8e\u5f71\u54cd\u70b9i\u5904\u66f2\u7387\u7684\u4e09\u4e2a\u70b9\uff08\u70b9i-1\u3001\u70b9i\u548c\u70b9i+1\uff09\u7684\u5bfc\u6570\u3002\u51fa\u4e8e\u8ba1\u7b97\u8003\u8651\uff0c\u8282\u70b9i\u5904\u5207\u5411\u89d2\u5ea6\u7684\u53d8\u5316\u6700\u597d\u8868\u793a\u4e3a\uff1aDelta phi_i = cos^{-1} frac{Delta mathbf{x}_i^T Delta mathbf{x}_{i+1}}{lvert Delta mathbf{x}_i rvert lvert Delta mathbf{x}_{i+1} rvert} tag{2}<\/p>\n

\u66f2\u7387kappa_i = Delta phi_i \/ lvert Delta mathbf{x}_i rvert\u76f8\u5bf9\u4e8e\u4e09\u4e2a\u8282\u70b9\u5750\u6807\u7684\u5bfc\u6570\u4e3a\uff1afrac{partial kappa_i}{partial mathbf{x}_i} = -frac{1}{lvert Delta mathbf{x}_i rvert}frac{partial Delta phi_i}{partial cos(Delta phi_i)} frac{partial cos(Delta phi_i)}{partial mathbf{x}_i} – frac{Delta phi_i}{(Delta mathbf{x}_i)^2} frac{partial Delta mathbf{x}_i}{partial mathbf{x}_i} frac{partial kappa_i}{partial mathbf{x}_{i-1}} = -frac{1}{lvert Delta mathbf{x}_i rvert}frac{partial Delta phi_i}{partial cos(Delta phi_i)} frac{partial cos(Delta phi_i)}{partial mathbf{x}_{i-1}} – frac{Delta phi_i}{(Delta mathbf{x}_i)^2} frac{partial Delta mathbf{x}_i}{partial mathbf{x}_{i-1}} frac{partial kappa_i}{partial mathbf{x}_{i+1}} = -frac{1}{lvert Delta mathbf{x}_i rvert}frac{partial Delta phi_i}{partial cos(Delta phi_i)} frac{partial cos(Delta phi_i)}{partial mathbf{x}_{i+1}} \u5176\u4e2d\uff1afrac{partial Delta phi_i}{partial cos(Delta phi_i)} = frac{partial cos^{-1}(cos(Delta phi_i))}{partial cos(Delta phi_i)} = frac{-1}{(1-cos^2(Delta phi_i))^{1\/2}}<\/p>\n

\u5176\u4e2d\uff0ccos(Delta phi_i)\u76f8\u5bf9\u4e8e\u4e09\u4e2a\u9876\u70b9\u5750\u6807\u7684\u5bfc\u6570\u6700\u5bb9\u6613\u7528\u6b63\u4ea4\u8865\u5f62\u5f0f\u8868\u793a\uff1aa bot b = a – frac{a^T b}{lvert b rvert} frac{b}{lvert b rvert} tag{3}<\/p>\n

\u5f15\u5165\u5982\u4e0b\u5f52\u4e00\u5316\u6b63\u4ea4\u8865\uff1amathbf{p}_1 = frac{mathbf{x}_i bot (-mathbf{x}_{i+1})}{lvert mathbf{x}_i rvert lvert mathbf{x}_{i+1} rvert}, mathbf{p}_2 = frac{(-mathbf{x}_{i+1}) bot mathbf{x}_i}{lvert mathbf{x}_i rvert lvert mathbf{x}_{i+1} rvert} tag{4}<\/p>\n

\u63a5\u7740\uff0c\u80fd\u591f\u5c06\u5bfc\u6570\u8868\u793a\u4e3a\uff1abegin{matrix} frac{partial cos(Delta phi_i)}{partial mathbf{x}_i} = -mathbf{p}_1 – mathbf{p}_2 \\ frac{partial cos(Delta phi_i)}{partial mathbf{x}_{i-1}} = mathbf{p}_2, frac{partial cos(Delta phi_i)}{partial mathbf{x}_{i+1}} = mathbf{p}_1 end{matrix} tag{5}<\/p>\n

\u56fe7\u5c55\u793a\u4e86\u7b2c\u4e8c\u6b21\u4f18\u5316\u548c\u5e73\u6ed1\u6b65\u9aa4\u7684\u6548\u679c\uff0c\u56fe\u4e2d\u7ea2\u7ebf\u4e3aA^{*}\u7684\u89e3\uff0c\u84dd\u7ebf\u4e3a\u7531CG\u4f18\u5316\u83b7\u5f97\u7684\u8def\u5f84\u3002<\/p>\n

\u4f7f\u7528\u4e0a\u8ff0CG\u5e73\u6ed1\uff0c\u672c\u6587\u83b7\u5f97\u4e00\u6761\u6bd4A^{*}\u7684\u89e3\u5e73\u6ed1\u5f88\u591a\u7684\u8def\u5f84\uff0c\u4f46\u662f\u5b83\u4ecd\u7136\u662f\u5206\u6bb5\u7ebf\u6027\u7684\uff0c\u5176\u9876\u70b9\u4e4b\u95f4\u5b58\u5728\u660e\u663e\u7684\u8ddd\u79bb\uff08\u5728\u672c\u6587\u5b9e\u73b0\u4e2d\u4e3a0.5m-1m\uff09\u3002\u8fd9\u4f1a\u5bfc\u81f4\u7269\u7406\u8f66\u8f86\u53d1\u751f\u975e\u5e38\u7a81\u7136\u7684\u8f6c\u5411\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4f7f\u7528CG\u89e3\u7684\u9876\u70b9\u4e4b\u95f4\u7684\u63d2\u503c\u6765\u8fdb\u4e00\u6b65\u5e73\u6ed1\u8def\u5f84\u3002\u5f88\u591a\u53c2\u6570\u5316\u63d2\u503c\u6280\u672f\u5bf9\u8f93\u5165\u4e2d\u7684\u566a\u58f0\u975e\u5e38\u654f\u611f\uff0c\u5e76\u4e14\u52a0\u5267\u4e86\u8f93\u51fa\u4e2d\u7684\u4efb\u4f55\u6b64\u7c7b\u566a\u58f0\uff08\u4f8b\u5982\uff0c\u7531\u4e8e\u8f93\u5165\u9876\u70b9\u5f7c\u6b64\u95f4\u9694\u8f83\u8fd1\uff0c\u56e0\u6b64\u4e09\u6b21\u6837\u6761\u66f2\u7ebf\u5c06\u5bfc\u81f4\u8f93\u51fa\u4e2d\u5b58\u5728\u4efb\u610f\u5927\u7684\u632f\u8361\uff09\u3002<\/p>\n

\u56e0\u6b64\uff0c\u6211\u4eec\u4f7f\u7528\u975e\u53c2\u6570\u5316\u7684\u63d2\u503c\u65b9\u5f0f\uff0c\u901a\u8fc7\u52a0\u5165\u65b0\u7684\u9876\u70b9\u3001\u4f7f\u7528CG\u6700\u5c0f\u5316\u8def\u5f84\u7684\u66f2\u7387\u540c\u65f6\u4fdd\u6301\u539f\u59cb\u9876\u70b9\u56fa\u5b9a\u6765\u5bf9\u8def\u5f84\u8d85\u91c7\u6837\u3002\u56fe8a\u4e2d\u8def\u5f84\u7684\u63d2\u503c\u7ed3\u679c\u5728\u56fe8b\u4e2d\u6240\u793a\u3002<\/p>\n

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\u56fe9\u63cf\u8ff0\u4e86\u4e00\u4e9b\u5728DARPA\u57ce\u5e02\u6311\u6218\u8d5b\u4e2d\u7531Junior\u884c\u9a76\u7684\u8f68\u8ff9\u3002\u56fe9a-c\u5c55\u793a\u4e86\u5728\u963b\u585e\u9053\u8def\u6389\u5934\u7684\u60c5\u51b5\uff0c\u56fe9d\u5c55\u793a\u6d89\u53ca\u505c\u8f66\u573a\u5bfc\u822a\u7684\u4efb\u52a1\u3002<\/p>\n

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