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<!DOCTYPE html><html lang="zh-CN"><head hexo-theme="https://github.com/volantis-x/hexo-theme-volantis/#6.0.0-alpha.0"><meta name="generator" content="Hexo 6.3.0"><meta name="Volantis" content="6.0.0-alpha.0"><meta charset="utf-8"><link rel="canonical" href="https://blog.mhuig.top/p/483290b5/"><meta http-equiv="x-dns-prefetch-control" content="on"><link rel="dns-prefetch" href="https://static.mhuig.top"><link rel="preconnect" href="https://static.mhuig.top" crossorigin><meta name="renderer" content="webkit"><meta name="force-rendering" content="webkit"><meta http-equiv="X-UA-Compatible" content="IE=Edge,chrome=1"><meta name="HandheldFriendly" content="True"><meta name="apple-mobile-web-app-capable" content="yes"><meta name="viewport" content="width=device-width,initial-scale=1,maximum-scale=5"><meta content="black-translucent" name="apple-mobile-web-app-status-bar-style"><meta content="telephone=no" name="format-detection"><script>function registerServiceWorker(e){"localhost"!=window.location.hostname&&"serviceWorker"in navigator&&navigator.serviceWorker.register(`${e}`).then((function(e){"localhost"==window.location.hostname&&(e.onupdatefound=function(){var r=e.installing;r.onstatechange=function(){switch(r.state){case"installed":navigator.serviceWorker.controller?console.log("Updated Service Worker."):console.log("Service Worker Sucess!");break;case"redundant":console.log("The installing service worker became redundant.")}}})})).catch((function(e){console.error("Error during service worker registration:",e),"undefined"==typeof swinstallretry&&(swinstallretry=1,registerServiceWorker("/jquery.js?v=0258ff4f79"))}))}registerServiceWorker("/jquery.js?v=0258ff4f79")</script><link rel="apple-touch-icon" sizes="180x180" href="/lib/favicon/apple-touch-icon.png?v=e035908f18"><link rel="icon" type="image/png" sizes="32x32" href="/lib/favicon/favicon-32x32.png?v=b9d9ae54cc"><link rel="icon" type="image/png" sizes="192x192" href="/lib/favicon/android-chrome-192x192.png?v=8bf02188e3"><link rel="icon" type="image/png" sizes="144x144" href="/lib/favicon/android-chrome-144x144.png?v=5c8eadcbb7"><link rel="icon" type="image/png" sizes="16x16" href="/lib/favicon/favicon-16x16.png?v=f896f2f634"><link rel="manifest" href="/lib/favicon/site.webmanifest?v=f138de5899"><link rel="mask-icon" href="/lib/favicon/safari-pinned-tab.svg?v=87d61e51a5" color="#5bbad5"><meta name="apple-mobile-web-app-title" content="MHuiG's Blog"><meta name="application-name" content="MHuiG's Blog"><meta name="msapplication-TileColor" content="#87ceeb"><meta name="msapplication-TileImage" content="/lib/favicon/mstile-144x144.png"><meta name="theme-color" content="#87ceeb" media="(prefers-color-scheme: light)"><meta name="theme-color" content="#21232f" media="(prefers-color-scheme: dark)"><link href="/opensearch.xml?v=2f4fd4e7e0" rel="search" title="MHuiG's Blog" type="application/opensearchdescription+xml"><link rel="preload" href="/css/style.css?v=2d24f6fe3e" as="style"><link rel="preload" href="https://static.mhuig.top/npm/volantis-static@0.0.1660614606622/media/fonts/VarelaRound/VarelaRound-Regular.ttf" as="font" type="font/ttf" crossorigin="anonymous"><link rel="alternate" href="/atom.xml?v=fe8205151a" title="MHuiG" type="application/atom+xml"><link rel="alternate" href="/rss2.xml?v=6b8706d024" title="MHuiG" type="application/rss+xml"><title>特征和分类器 - MHuiG</title><meta name="keywords" content="机器学习,机器视觉,分类器,特征,Blog, IT, BigData, MHuiG, @MHuiG, iMHuiG, 博客, 梦幻岛, Neverland, 技术博客"><meta desc name="description" content="特征提取和分类是典型计算机视觉系统的两个关键阶段。
视觉系统的准确性、稳健性和效率在很大程度上取决于图像特征，和分类器的质量。因此，目标是从输入图像中提取信息丰富的、可靠的特征，以便能够开发出很大程度上独立于领域理论的分类。 - MHuiG - MHuiG"><meta property="og:type" content="article"><meta property="og:title" content="特征和分类器"><meta property="og:url" content="https://blog.mhuig.top/p/483290b5/"><meta property="og:site_name" content="MHuiG"><meta property="og:description" content="特征提取和分类是典型计算机视觉系统的两个关键阶段。 视觉系统的准确性、稳健性和效率在很大程度上取决于图像特征，和分类器的质量。因此，目标是从输入图像中提取信息丰富的、可靠的特征，以便能够开发出很大程度上独立于领域理论的分类。"><meta property="og:locale" content="zh_CN"><meta property="og:image" content="https://blog.mhuig.top/lib/favicon/android-chrome-192x192.png"><meta property="article:published_time" content="2019-09-12T03:19:19.000Z"><meta property="article:modified_time" content="2019-09-12T03:19:19.000Z"><meta property="article:author" content="MHuiG"><meta property="article:tag" content="机器学习"><meta property="article:tag" content="机器视觉"><meta property="article:tag" content="分类器"><meta property="article:tag" content="特征"><meta name="twitter:card" 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class="menu navigation"><div class="list-h"><a target="_blank" rel="external nofollow noopener noreferrer" href="https://github.com/MHuiG" active-action="action-https:githubcomMHuiG"><i class="fa-brands fa-github color-github fa-fw"></i><p>Github</p></a><a href="/pages/friends/" active-action="action-pagesfriends"><i class="fa-duotone fa-link color-friends fa-fw"></i><p>友链</p></a><a href="/pages/about/" active-action="action-pagesabout"><i class="fa-duotone fa-user-tie color-about fa-fw"></i><p>关于</p></a><a href="https://www.travellings.cn/go-by-clouds.html" target="_blank" active-action="action-https:wwwtravellingscngo-by-cloudshtml" rel="external nofollow noopener noreferrer"><i class="fa-duotone fa-subway color-travellings fa-fw"></i><p>Travelling</p></a></div></div></div></div><div id="scroll-down" style="display:none"><i class="fa fa-chevron-down scroll-down-effects"></i></div></div></div><div id="safearea"><div class="body-wrapper"><div id="l_main" class><article itemscope itemtype="http://schema.org/Article" class="article post white-box reveal md shadow floatable blur article-type-post" id="post" itemprop="blogPost"><link itemprop="mainEntityOfPage" href="https://blog.mhuig.top/p/483290b5/"><span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization"><meta itemprop="name" content="MHuiG"></span><span hidden itemprop="post" itemscope itemtype="http://schema.org/Post"><meta itemprop="name" content="MHuiG"><meta itemprop="description" content="MHuiG&#39;s Blog (MHuiG的博客) MHuiG&#39;s Neverland（MHuiG的梦幻岛） —— MHuiG(@MHuiG) 随便写写画画的地方 - 技术博客"></span><span hidden><meta itemprop="image" content="/lib/favicon/android-chrome-192x192.png"></span><div class="article-meta" id="top"><h1 class="title" itemprop="name headline"> 特征和分类器</h1><div class="new-meta-box"><div class="new-meta-item author" itemprop="author" itemscope itemtype="http://schema.org/Person"> <a itemprop="url" class="author" target="_blank" href="https://mhuig.top/" rel="nofollow noopener"><img itemprop="image" src="/lib/avatar/avatar-16.webp?v=2199236952" class="lazyload" data-srcset="/lib/avatar/avatar-16.webp" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw=="><p itemprop="name">MHuiG</p></a></div><div class="new-meta-item wordcount"><a class="notlink"><i class="fa-duotone fa-keyboard fa-fw" aria-hidden="true"></i><p>字数：1.4k 字</p></a></div><div class="new-meta-item readtime"><a class="notlink"><i class="fa-duotone fa-hourglass-half fa-fw" aria-hidden="true"></i><p>时长：4 分钟</p></a></div></div></div><div id="layoutHelper-page-plugins"></div><div id="post-body" itemprop="articleBody"><p>特征提取和分类是典型计算机视觉系统的两个关键阶段。</p><p>视觉系统的准确性、稳健性和效率在很大程度上取决于图像特征，和分类器的质量。因此，目标是从输入图像中提取信息丰富的、可靠的特征，以便能够开发出很大程度上独立于领域理论的分类。</p><span id="more"></span><div class="story post-story"><h2 id="特征"><a href="#特征" class="headerlink" title="特征"></a>特征</h2><p>特征是任何独特的方面或特性，用于解决与特定应用相关的计算任务。</p><p>n 个特征的组合可以表示成 n 维向量，称为特征向量。特征向量的质量取决于其区分不同类别的图像样本的能力。</p><p>来自同一类的图像样本应该有相似的特征值，来自不同类的图像应具有不同的特征值。</p></div><div class="story post-story"><h2 id="分类器"><a href="#分类器" class="headerlink" title="分类器"></a>分类器</h2><p>分类器是现代计算机视觉和模式识别的核心。</p><p>分类器的任务是使用特征向量对图像或感兴趣区域（RoI）划分类别。</p><p>分类任务的困难程度取决于来自相同类别的图像的特征值的可变性，以及相对于来自不同类别图像的特征值的差异性。</p><p>但是，完美的分类性能通常是不可能的。这主要是因为：</p><ul><li>噪声（以阴影、遮挡、透视扭曲等形式）</li><li>异常值</li><li>模糊性</li><li>缺少标签</li><li>仅有小训练样本可用</li><li>训练数据样本中正 / 负覆盖的不平衡</li></ul></div><div class="story post-story"><h2 id="传统特征描述符"><a href="#传统特征描述符" class="headerlink" title="传统特征描述符"></a>传统特征描述符</h2><p>传统（手工设计）特征提取方法可分为两大类：</p><ul><li>全局</li><li>局部</li></ul><p>全局特征提取方法定义了一组有效描述整个图像的全局特征。因此形状细节被忽略。全局特征也不适用于识别部分遮挡的对象。</p><p>局部特征提取方法提取关键点周围的局部区域，由此可以更好的处理遮挡。</p><p>检测关键点，并在他们周围构建描述符的方法：</p><p>局部描述符（如 HOG、SIFT、SURF、FREAK、ORB、BRISK、BRIEF、LIOP）</p></div><div class="story post-story"><h2 id="方向梯度直方图"><a href="#方向梯度直方图" class="headerlink" title="方向梯度直方图"></a>方向梯度直方图</h2><p>HOG 是一个特征描述符，用于自动检测图像中的对象。HOG 描述符对图像中局部部分的梯度方向的分布进行编码。</p><p>HOG 背后的想法是可以通过边缘方向的直方图来描述图像内的对象外观和形状。</p><h3 id="1-梯度计算"><a href="#1-梯度计算" class="headerlink" title="1.梯度计算"></a>1. 梯度计算</h3><p>第一步是计算梯度值。在图像的水平和垂直方向上，执行一维中心点离散微分模板。具体的说，该方法需要用以下滤波器内核处理灰度图像：</p><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:44.929ex"><svg style="vertical-align:-.777ex;min-width:44.929ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.685ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -843.3)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 843.3) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="7851.2 -843.3 1 1186.7"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr" transform="translate(0,-48.4)"><g data-mml-node="mtd"><g data-mml-node="msub"><g 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437Q21 437 21 445 21 450 37 501T71 602L88 651Q93 669 101 677H569 659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437 640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631 469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46 418 46 427 45T436 36Q436 31 433 22 429 4 424 1L422 0Q419 0 415 0 410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83 94Q174 46 189 55 190 56 191 56 196 59 201 76T241 233Q258 301 269 344 339 619 339 625 339 630 310 630H279Q212 630 191 624 146 614 121 583T67 467Q60 445 57 441T43 437H40Z"/></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -843.3 1 1186.7"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:1" transform="translate(0,701.6)"><text data-id-align="true"/><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><path data-c="28" d="M94 250Q94 319 104 381T127 488 164 576 202 643 244 695 277 729 302 750H315 319Q333 750 333 741 333 738 316 720T275 667 226 581 184 443 167 250 184 58 225-81 274-167 316-220 333-241Q333-250 318-250H315 302L274-226Q180-141 137-14T94 250Z"/><path data-c="31" d="M213 578 200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641 273 663Q275 666 285 666 294 666 302 660V361L303 61Q310 54 315 52T339 48 401 46H427V0H416Q395 3 257 3 121 3 100 0H88V46H114Q136 46 152 46T177 47 193 50 201 52 207 57 213 61V578Z" transform="translate(389,0)"/><path data-c="29" d="M60 749 64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12 224-76 186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z" transform="translate(889,0)"/></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>因此给定一个图像<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.14ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 504 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D43C" d="M43 1Q26 1 26 10 26 12 29 24 34 43 39 45 42 46 54 46H60Q120 46 136 53 137 53 138 54 143 56 149 77T198 273Q210 318 216 344 286 624 286 626 284 630 284 631 274 637 213 637H193Q184 643 189 662 193 677 195 680T209 683H213Q285 681 359 681 481 681 487 683H497Q504 676 504 672T501 655 494 639Q491 637 471 637 440 637 407 634 393 631 388 623 381 609 337 432 326 385 315 341 245 65 245 59 245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2 143 2 113 2T65 2 43 1Z"/></g></g></g></svg></mjx-container>，以下卷积操作（表示为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:.079ex" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="0.973ex" role="img" focusable="false" viewBox="0 -465 500 430"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><path data-c="2217" d="M229 286Q216 420 216 436 216 454 240 464 241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315 420 312 357 282T289 250L355 219 425 184Q434 175 434 161 434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63 283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127 106 124 100 124 87 124 76 134T64 161Q64 166 64 169T67 175 72 181 81 188 94 195 113 204 138 215 170 230 210 250L74 315Q65 324 65 338 65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z"/></g></g></g></svg></mjx-container>）得出图像<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.14ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 504 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D43C" d="M43 1Q26 1 26 10 26 12 29 24 34 43 39 45 42 46 54 46H60Q120 46 136 53 137 53 138 54 143 56 149 77T198 273Q210 318 216 344 286 624 286 626 284 630 284 631 274 637 213 637H193Q184 643 189 662 193 677 195 680T209 683H213Q285 681 359 681 481 681 487 683H497Q504 676 504 672T501 655 494 639Q491 637 471 637 440 637 407 634 393 631 388 623 381 609 337 432 326 385 315 341 245 65 245 59 245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2 143 2 113 2T65 2 43 1Z"/></g></g></g></svg></mjx-container>在<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442 467 442 494 420T522 361Q522 332 508 314T481 292 458 288Q439 288 427 299T415 328Q415 374 465 391 454 404 425 404 412 404 406 402 368 386 350 336 290 115 290 78 290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145 504 144 502 134 486 77 440 33T333-11Q263-11 227 52 186-10 133-10H127Q78-10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101 142 81 130 66T107 46 94 41L91 40Q91 39 97 36T113 29 132 26Q168 26 194 71 203 87 217 139T245 247 261 313Q266 340 266 352 266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"/></g></g></g></svg></mjx-container>和<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.464ex" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D466" d="M21 287Q21 301 36 335T84 406 158 442Q199 442 224 419T250 355Q248 336 247 334 247 331 231 288T198 191 182 105Q182 62 196 45T238 27Q261 27 281 38T312 61 339 94Q339 95 344 114T358 173 377 247Q415 397 419 404 432 431 462 431 475 431 483 424T494 412 496 403Q496 390 447 193T391-23Q363-106 294-155T156-205Q111-205 77-183T43-117Q43-95 50-80T69-58 89-48 106-45Q150-45 150-87 150-107 138-122T115-142 102-147L99-148Q101-153 118-160T152-167H160Q177-167 186-165 219-156 247-127T290-65 313-9 321 21L315 17Q309 13 296 6T270-6Q250-11 231-11 185-11 150 11T104 82Q103 89 103 113 103 170 138 262T173 379Q173 380 173 381 173 390 173 393T169 400 158 404H154Q131 404 112 385T82 344 65 302 57 280Q55 278 41 278H27Q21 284 21 287Z"/></g></g></g></svg></mjx-container>方向的导数：</p><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:31.354ex"><svg style="vertical-align:-.616ex;min-width:31.354ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.364ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -772.5)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 772.5) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="4851.3 -772.5 1 1045"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr" transform="translate(0,22.5)"><g data-mml-node="mtd"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D43C" d="M43 1Q26 1 26 10 26 12 29 24 34 43 39 45 42 46 54 46H60Q120 46 136 53 137 53 138 54 143 56 149 77T198 273Q210 318 216 344 286 624 286 626 284 630 284 631 274 637 213 637H193Q184 643 189 662 193 677 195 680T209 683H213Q285 681 359 681 481 681 487 683H497Q504 676 504 672T501 655 494 639Q491 637 471 637 440 637 407 634 393 631 388 623 381 609 337 432 326 385 315 341 245 65 245 59 245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2 143 2 113 2T65 2 43 1Z"/></g><g data-mml-node="mi" transform="translate(473,-150) scale(0.707)"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442 467 442 494 420T522 361Q522 332 508 314T481 292 458 288Q439 288 427 299T415 328Q415 374 465 391 454 404 425 404 412 404 406 402 368 386 350 336 290 115 290 78 290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145 504 144 502 134 486 77 440 33T333-11Q263-11 227 52 186-10 133-10H127Q78-10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101 142 81 130 66T107 46 94 41L91 40Q91 39 97 36T113 29 132 26Q168 26 194 71 203 87 217 139T245 247 261 313Q266 340 266 352 266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"/></g></g><g data-mml-node="mo" transform="translate(1205.2,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347 722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153 722 140 707 133H70Q56 140 56 153Z"/></g><g data-mml-node="mi" transform="translate(2261,0)"><path data-c="1D43C" d="M43 1Q26 1 26 10 26 12 29 24 34 43 39 45 42 46 54 46H60Q120 46 136 53 137 53 138 54 143 56 149 77T198 273Q210 318 216 344 286 624 286 626 284 630 284 631 274 637 213 637H193Q184 643 189 662 193 677 195 680T209 683H213Q285 681 359 681 481 681 487 683H497Q504 676 504 672T501 655 494 639Q491 637 471 637 440 637 407 634 393 631 388 623 381 609 337 432 326 385 315 341 245 65 245 59 245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2 143 2 113 2T65 2 43 1Z"/></g><g data-mml-node="mo" transform="translate(2987.2,0)"><path data-c="2217" d="M229 286Q216 420 216 436 216 454 240 464 241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315 420 312 357 282T289 250L355 219 425 184Q434 175 434 161 434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63 283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127 106 124 100 124 87 124 76 134T64 161Q64 166 64 169T67 175 72 181 81 188 94 195 113 204 138 215 170 230 210 250L74 315Q65 324 65 338 65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z"/></g><g data-mml-node="msub" transform="translate(3709.5,0)"><g data-mml-node="mi"><path data-c="1D453" d="M118-162Q120-162 124-164T135-167 147-168Q160-168 171-155T187-126Q197-99 221 27T267 267 289 382V385H242Q195 385 192 387 188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432 298 434 307 482T319 540Q356 705 465 705 502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603 443 622 454 636T478 657L487 662Q471 668 457 668 445 668 434 658T419 630Q412 601 403 552T387 469 380 433Q380 431 435 431 480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282-47 255-132 212-173 175-205 139-205 107-205 81-186T55-132Q55-95 76-78T118-61Q162-61 162-103 162-122 151-136T127-157L118-162Z"/></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442 467 442 494 420T522 361Q522 332 508 314T481 292 458 288Q439 288 427 299T415 328Q415 374 465 391 454 404 425 404 412 404 406 402 368 386 350 336 290 115 290 78 290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145 504 144 502 134 486 77 440 33T333-11Q263-11 227 52 186-10 133-10H127Q78-10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101 142 81 130 66T107 46 94 41L91 40Q91 39 97 36T113 29 132 26Q168 26 194 71 203 87 217 139T245 247 261 313Q266 340 266 352 266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"/></g></g><g data-mml-node="mo" transform="translate(4686.9,0)"><path data-c="2C" d="M78 35T78 60 94 103 137 121Q165 121 187 96T210 8Q210-27 201-60T180-117 154-158 130-185 117-194Q113-194 104-185T95-172Q95-168 106-156T131-126 157-76 173-3V9L172 8Q170 7 167 6T161 3 152 1 140 0Q113 0 96 17Z"/></g><g data-mml-node="msub" transform="translate(5131.6,0)"><g data-mml-node="mi"><path data-c="1D43C" d="M43 1Q26 1 26 10 26 12 29 24 34 43 39 45 42 46 54 46H60Q120 46 136 53 137 53 138 54 143 56 149 77T198 273Q210 318 216 344 286 624 286 626 284 630 284 631 274 637 213 637H193Q184 643 189 662 193 677 195 680T209 683H213Q285 681 359 681 481 681 487 683H497Q504 676 504 672T501 655 494 639Q491 637 471 637 440 637 407 634 393 631 388 623 381 609 337 432 326 385 315 341 245 65 245 59 245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2 143 2 113 2T65 2 43 1Z"/></g><g data-mml-node="mi" transform="translate(473,-150) scale(0.707)"><path data-c="1D466" d="M21 287Q21 301 36 335T84 406 158 442Q199 442 224 419T250 355Q248 336 247 334 247 331 231 288T198 191 182 105Q182 62 196 45T238 27Q261 27 281 38T312 61 339 94Q339 95 344 114T358 173 377 247Q415 397 419 404 432 431 462 431 475 431 483 424T494 412 496 403Q496 390 447 193T391-23Q363-106 294-155T156-205Q111-205 77-183T43-117Q43-95 50-80T69-58 89-48 106-45Q150-45 150-87 150-107 138-122T115-142 102-147L99-148Q101-153 118-160T152-167H160Q177-167 186-165 219-156 247-127T290-65 313-9 321 21L315 17Q309 13 296 6T270-6Q250-11 231-11 185-11 150 11T104 82Q103 89 103 113 103 170 138 262T173 379Q173 380 173 381 173 390 173 393T169 400 158 404H154Q131 404 112 385T82 344 65 302 57 280Q55 278 41 278H27Q21 284 21 287Z"/></g></g><g data-mml-node="mo" transform="translate(6278.9,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347 722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153 722 140 707 133H70Q56 140 56 153Z"/></g><g data-mml-node="mi" transform="translate(7334.6,0)"><path data-c="1D43C" d="M43 1Q26 1 26 10 26 12 29 24 34 43 39 45 42 46 54 46H60Q120 46 136 53 137 53 138 54 143 56 149 77T198 273Q210 318 216 344 286 624 286 626 284 630 284 631 274 637 213 637H193Q184 643 189 662 193 677 195 680T209 683H213Q285 681 359 681 481 681 487 683H497Q504 676 504 672T501 655 494 639Q491 637 471 637 440 637 407 634 393 631 388 623 381 609 337 432 326 385 315 341 245 65 245 59 245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2 143 2 113 2T65 2 43 1Z"/></g><g data-mml-node="mo" transform="translate(8060.9,0)"><path data-c="2217" d="M229 286Q216 420 216 436 216 454 240 464 241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315 420 312 357 282T289 250L355 219 425 184Q434 175 434 161 434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63 283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127 106 124 100 124 87 124 76 134T64 161Q64 166 64 169T67 175 72 181 81 188 94 195 113 204 138 215 170 230 210 250L74 315Q65 324 65 338 65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z"/></g><g data-mml-node="msub" transform="translate(8783.1,0)"><g data-mml-node="mi"><path data-c="1D453" d="M118-162Q120-162 124-164T135-167 147-168Q160-168 171-155T187-126Q197-99 221 27T267 267 289 382V385H242Q195 385 192 387 188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432 298 434 307 482T319 540Q356 705 465 705 502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603 443 622 454 636T478 657L487 662Q471 668 457 668 445 668 434 658T419 630Q412 601 403 552T387 469 380 433Q380 431 435 431 480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282-47 255-132 212-173 175-205 139-205 107-205 81-186T55-132Q55-95 76-78T118-61Q162-61 162-103 162-122 151-136T127-157L118-162Z"/></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><path data-c="1D466" d="M21 287Q21 301 36 335T84 406 158 442Q199 442 224 419T250 355Q248 336 247 334 247 331 231 288T198 191 182 105Q182 62 196 45T238 27Q261 27 281 38T312 61 339 94Q339 95 344 114T358 173 377 247Q415 397 419 404 432 431 462 431 475 431 483 424T494 412 496 403Q496 390 447 193T391-23Q363-106 294-155T156-205Q111-205 77-183T43-117Q43-95 50-80T69-58 89-48 106-45Q150-45 150-87 150-107 138-122T115-142 102-147L99-148Q101-153 118-160T152-167H160Q177-167 186-165 219-156 247-127T290-65 313-9 321 21L315 17Q309 13 296 6T270-6Q250-11 231-11 185-11 150 11T104 82Q103 89 103 113 103 170 138 262T173 379Q173 380 173 381 173 390 173 393T169 400 158 404H154Q131 404 112 385T82 344 65 302 57 280Q55 278 41 278H27Q21 284 21 287Z"/></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -772.5 1 1045"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:2" transform="translate(0,772.5)"><text data-id-align="true"/><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><path data-c="28" d="M94 250Q94 319 104 381T127 488 164 576 202 643 244 695 277 729 302 750H315 319Q333 750 333 741 333 738 316 720T275 667 226 581 184 443 167 250 184 58 225-81 274-167 316-220 333-241Q333-250 318-250H315 302L274-226Q180-141 137-14T94 250Z"/><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315 301 241Q265 210 201 149L142 93 218 92Q375 92 385 97 392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19 31Q50 38 56 46T86 81Q115 113 136 137 145 147 170 174T204 211 233 244 261 278 284 308 305 340 320 369 333 401 340 431 343 464Q343 527 309 573T212 619Q179 619 154 602T119 569 109 550Q109 549 114 549 132 549 151 535T170 489Q170 464 154 447T109 429Z" transform="translate(389,0)"/><path data-c="29" d="M60 749 64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12 224-76 186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z" transform="translate(889,0)"/></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>因此，梯度的方向<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.023ex" xmlns="http://www.w3.org/2000/svg" width="1.061ex" height="1.618ex" role="img" focusable="false" viewBox="0 -705 469 715"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D703" d="M35 200Q35 302 74 415T180 610 319 704Q320 704 327 704T339 705Q393 701 423 656 462 596 462 495 462 380 417 261T302 66 168-10H161Q125-10 99 10T60 63 41 130 35 200ZM383 566Q383 668 330 668 294 668 260 623T204 521 170 421 157 371Q206 370 254 370L351 371Q352 372 359 404T375 484 383 566ZM113 132Q113 26 166 26 181 26 198 36T239 74 287 161 335 307L340 324H145Q145 321 136 286T120 208 113 132Z"/></g></g></g></svg></mjx-container>和梯度的大小<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.564ex" xmlns="http://www.w3.org/2000/svg" width="2.337ex" height="2.26ex" role="img" focusable="false" viewBox="0 -749.5 1033 999"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><path data-c="7C" d="M139-249H137Q125-249 119-235V251L120 737Q130 750 139 750 152 750 159 735V-235Q151-249 141-249H139Z"/></g><g data-mml-node="mi" transform="translate(278,0)"><path data-c="1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412 431 419 447 422Q461 422 470 413T480 394Q480 379 423 152T363-80Q345-134 286-169T151-205Q10-205 10-137 10-111 28-91T74-71Q89-71 102-80T116-111Q116-121 114-130T107-144 99-154 92-162L90-164H91Q101-167 151-167 189-167 211-155 234-144 254-122T282-75Q288-56 298-13 311 35 311 43ZM384 328 380 339Q377 350 375 354T369 368 359 382 346 393 328 402 306 405Q262 405 221 352 191 313 171 233T151 117Q151 38 213 38 269 38 323 108L331 118 384 328Z"/></g><g data-mml-node="mo" transform="translate(755,0) translate(0 -0.5)"><path data-c="7C" d="M139-249H137Q125-249 119-235V251L120 737Q130 750 139 750 152 750 159 735V-235Q151-249 141-249H139Z"/></g></g></g></svg></mjx-container>计算如下：</p><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:39.174ex"><svg style="vertical-align:-2.016ex;min-width:39.174ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="5.162ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1390.9)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 1390.9) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="6579.4 -1390.9 1 2281.7"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr" transform="translate(0,-47.1)"><g data-mml-node="mtd"><g data-mml-node="mi"><path data-c="1D703" d="M35 200Q35 302 74 415T180 610 319 704Q320 704 327 704T339 705Q393 701 423 656 462 596 462 495 462 380 417 261T302 66 168-10H161Q125-10 99 10T60 63 41 130 35 200ZM383 566Q383 668 330 668 294 668 260 623T204 521 170 421 157 371Q206 370 254 370L351 371Q352 372 359 404T375 484 383 566ZM113 132Q113 26 166 26 181 26 198 36T239 74 287 161 335 307L340 324H145Q145 321 136 286T120 208 113 132Z"/></g><g data-mml-node="mo" transform="translate(746.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347 722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153 722 140 707 133H70Q56 140 56 153Z"/></g><g data-mml-node="mi" transform="translate(1802.6,0)"><path data-c="61" d="M137 305T115 305 78 320 63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59 481 29 455 12T400-6 353 12 329 54V58L327 55Q325 52 322 49T314 40 302 29 287 17 269 6 247-2 221-8 190-11Q130-11 82 20T34 107Q34 128 41 147T68 188 116 225 194 253 304 268H318V290Q318 324 312 340 290 411 215 411 197 411 181 410T156 406 148 403Q170 388 170 359 170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175 318 233 317 233 309 233 296 232T251 223 193 203 147 166 126 106Z"/><path data-c="72" d="M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122 98 161 98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441 171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 373Q364 352 351 337T313 322Q288 322 276 338T263 372Q263 381 265 388T270 400 273 405Q271 407 250 401 234 393 226 386 179 341 179 207V154Q179 141 179 127T179 101 180 81 180 66V61Q181 59 183 57T188 54 193 51 200 49 207 48 216 47 225 47 235 46 245 46H276V0H267Q249 3 140 3 37 3 28 0H20V46H36Z" transform="translate(500,0)"/><path data-c="63" d="M370 305T349 305 313 320 297 358Q297 381 312 396 317 401 317 402T307 404Q281 408 258 408 209 408 178 376 131 329 131 219 131 137 162 90 203 29 272 29 313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120 415 116 411 104T395 71 366 33 318 2 249-11Q163-11 99 53T34 214Q34 318 99 383T250 448 370 421 404 357Q404 334 387 320Z" transform="translate(892,0)"/><path data-c="74" d="M27 422Q80 426 109 478T141 600V615H181V431H316V385H181V241Q182 116 182 100T189 68Q203 29 238 29 282 29 292 100 293 108 293 146V181H333V146 134Q333 57 291 17 264-10 221-10 187-10 162 2T124 33 105 68 98 100Q97 107 97 248V385H18V422H27Z" transform="translate(1336,0)"/><path data-c="61" d="M137 305T115 305 78 320 63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59 481 29 455 12T400-6 353 12 329 54V58L327 55Q325 52 322 49T314 40 302 29 287 17 269 6 247-2 221-8 190-11Q130-11 82 20T34 107Q34 128 41 147T68 188 116 225 194 253 304 268H318V290Q318 324 312 340 290 411 215 411 197 411 181 410T156 406 148 403Q170 388 170 359 170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175 318 233 317 233 309 233 296 232T251 223 193 203 147 166 126 106Z" transform="translate(1725,0)"/><path data-c="6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122 103 161 103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441 178 442H181V402Q181 364 182 364T187 369 199 384 218 402 247 421 285 437Q305 442 336 442 450 438 463 329 464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62 380 64 380 200 379 335 378 343 372 371 358 385T334 402 308 404Q263 404 229 370 202 343 195 315T187 232V168 108Q187 78 188 68T191 55 200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z" transform="translate(2225,0)"/></g><g data-mml-node="mo" transform="translate(4583.6,0)"><path data-c="2061" d=""/></g><g data-mml-node="mfrac" transform="translate(4750.2,0)"><g data-mml-node="msub" transform="translate(249,755)"><g data-mml-node="mi"><path data-c="1D43C" d="M43 1Q26 1 26 10 26 12 29 24 34 43 39 45 42 46 54 46H60Q120 46 136 53 137 53 138 54 143 56 149 77T198 273Q210 318 216 344 286 624 286 626 284 630 284 631 274 637 213 637H193Q184 643 189 662 193 677 195 680T209 683H213Q285 681 359 681 481 681 487 683H497Q504 676 504 672T501 655 494 639Q491 637 471 637 440 637 407 634 393 631 388 623 381 609 337 432 326 385 315 341 245 65 245 59 245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2 143 2 113 2T65 2 43 1Z"/></g><g data-mml-node="mi" transform="translate(473,-150) scale(0.707)"><path data-c="1D466" d="M21 287Q21 301 36 335T84 406 158 442Q199 442 224 419T250 355Q248 336 247 334 247 331 231 288T198 191 182 105Q182 62 196 45T238 27Q261 27 281 38T312 61 339 94Q339 95 344 114T358 173 377 247Q415 397 419 404 432 431 462 431 475 431 483 424T494 412 496 403Q496 390 447 193T391-23Q363-106 294-155T156-205Q111-205 77-183T43-117Q43-95 50-80T69-58 89-48 106-45Q150-45 150-87 150-107 138-122T115-142 102-147L99-148Q101-153 118-160T152-167H160Q177-167 186-165 219-156 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430 442 467 442 494 420T522 361Q522 332 508 314T481 292 458 288Q439 288 427 299T415 328Q415 374 465 391 454 404 425 404 412 404 406 402 368 386 350 336 290 115 290 78 290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145 504 144 502 134 486 77 440 33T333-11Q263-11 227 52 186-10 133-10H127Q78-10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101 142 81 130 66T107 46 94 41L91 40Q91 39 97 36T113 29 132 26Q168 26 194 71 203 87 217 139T245 247 261 313Q266 340 266 352 266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"/></g></g><rect width="1127.5" height="60" x="120" y="220"/></g><g data-mml-node="mo" transform="translate(6117.7,0)"><path data-c="2C" d="M78 35T78 60 94 103 137 121Q165 121 187 96T210 8Q210-27 201-60T180-117 154-158 130-185 117-194Q113-194 104-185T95-172Q95-168 106-156T131-126 157-76 173-3V9L172 8Q170 7 167 6T161 3 152 1 140 0Q113 0 96 17Z"/></g><g data-mml-node="mo" transform="translate(6562.4,0) translate(0 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41L91 40Q91 39 97 36T113 29 132 26Q168 26 194 71 203 87 217 139T245 247 261 313Q266 340 266 352 266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"/></g></g><g data-mml-node="mo" transform="translate(1216,0)"><path data-c="2B" d="M56 237T56 250 70 270H369V420L370 570Q380 583 389 583 402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401-82 391-82H389 387Q375-82 369-68V230H70Q56 237 56 250Z"/></g><g data-mml-node="msubsup" transform="translate(2216.2,0)"><g data-mml-node="mi"><path data-c="1D43C" d="M43 1Q26 1 26 10 26 12 29 24 34 43 39 45 42 46 54 46H60Q120 46 136 53 137 53 138 54 143 56 149 77T198 273Q210 318 216 344 286 624 286 626 284 630 284 631 274 637 213 637H193Q184 643 189 662 193 677 195 680T209 683H213Q285 681 359 681 481 681 487 683H497Q504 676 504 672T501 655 494 639Q491 637 471 637 440 637 407 634 393 631 388 623 381 609 337 432 326 385 315 341 245 65 245 59 245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2 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524Q69 579 117 622T233 665Q268 665 277 664 351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336 375 315Q457 262 457 175 457 96 395 37T238-22Q158-22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94 148 82 136 74 126 70 118 67L114 66Q165 21 238 21 293 21 321 74 338 107 338 175V195Q338 290 274 322 259 328 213 329L171 330 168 332Q166 335 166 348 166 366 174 366 202 366 232 371 266 376 294 413T322 525V533Q322 590 287 612 265 626 240 626 208 626 181 615T143 592 132 580H135Q138 579 143 578T153 573 165 566 175 555 183 540 186 520Q186 498 172 481T127 463Z" transform="translate(389,0)"/><path data-c="29" d="M60 749 64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12 224-76 186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z" transform="translate(889,0)"/></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><ul><li>CNN 也在层中使用卷积运算，然而主要区别在于不使用手工设计的滤波器，CNN 使用可训练的滤波器，使其具有高度的自适应性。</li><li>CNN 也在层中使用卷积运算，然而主要区别在于不使用手工设计的滤波器，CNN 使用可训练的滤波器，使其具有高度的自适应性。</li></ul><p><img src="https://static.mhuig.top/npm/mhgoos@0.0.1652000153902/20191220133534.webp" class="lazyload" data-srcset="https://static.mhuig.top/npm/mhgoos@0.0.1652000153902/20191220133534.webp" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" alt="HOG描述符"></p><h3 id="2-单元方向直方图"><a href="#2-单元方向直方图" class="headerlink" title="2.单元方向直方图"></a>2. 单元方向直方图</h3><p>第二步是计算单元直方图。首先将图像分成小的（通常是 8X8 像素）单元。每个单元都有固定数量的梯度方向区间，他们均匀分布在<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.186ex" xmlns="http://www.w3.org/2000/svg" width="9.078ex" height="2.275ex" role="img" focusable="false" viewBox="0 -923.4 4012.6 1005.4"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="30" d="M96 585Q152 666 249 666 297 666 345 640T423 548Q460 465 460 320 460 165 417 83 397 41 362 16T301-15 250-22Q224-22 198-16T137 16 82 83Q39 165 39 320 39 494 96 585ZM321 597Q291 629 250 629 208 629 178 597 153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16 290 16 318 46 347 76 354 130T362 333Q362 478 354 524T321 597Z"/></g><g data-mml-node="mo" transform="translate(722.2,0)"><path data-c="2212" d="M84 237T84 250 98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"/></g><g data-mml-node="msup" transform="translate(1722.4,0)"><g data-mml-node="mn"><path data-c="31" d="M213 578 200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641 273 663Q275 666 285 666 294 666 302 660V361L303 61Q310 54 315 52T339 48 401 46H427V0H416Q395 3 257 3 121 3 100 0H88V46H114Q136 46 152 46T177 47 193 50 201 52 207 57 213 61V578Z"/><path data-c="38" d="M70 417T70 494 124 618 248 666Q319 666 374 624T429 515Q429 485 418 459T392 417 361 389 335 371 324 363L338 354Q352 344 366 334T382 323Q457 264 457 174 457 95 399 37T249-22Q159-22 101 29T43 155Q43 263 172 335L154 348Q133 361 127 368 70 417 70 494ZM286 386 292 390Q298 394 301 396T311 403 323 413 334 425 345 438 355 454 364 471 369 491 371 513Q371 556 342 586T275 624Q268 625 242 625 201 625 165 599T128 534Q128 511 141 492T167 463 217 431Q224 426 228 424L286 386ZM250 21Q308 21 350 55T392 137Q392 154 387 169T375 194 353 216 330 234 301 253 274 270Q260 279 244 289T218 306L210 311Q204 311 181 294T133 239 107 157Q107 98 150 60T250 21Z" transform="translate(500,0)"/><path data-c="30" d="M96 585Q152 666 249 666 297 666 345 640T423 548Q460 465 460 320 460 165 417 83 397 41 362 16T301-15 250-22Q224-22 198-16T137 16 82 83Q39 165 39 320 39 494 96 585ZM321 597Q291 629 250 629 208 629 178 597 153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16 290 16 318 46 347 76 354 130T362 333Q362 478 354 524T321 597Z" transform="translate(1000,0)"/></g><g data-mml-node="TeXAtom" transform="translate(1533,393.1) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">。</text></g></g></g></g></g></svg></mjx-container>或<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.186ex" xmlns="http://www.w3.org/2000/svg" width="9.078ex" height="2.275ex" role="img" focusable="false" viewBox="0 -923.4 4012.6 1005.4"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="30" d="M96 585Q152 666 249 666 297 666 345 640T423 548Q460 465 460 320 460 165 417 83 397 41 362 16T301-15 250-22Q224-22 198-16T137 16 82 83Q39 165 39 320 39 494 96 585ZM321 597Q291 629 250 629 208 629 178 597 153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16 290 16 318 46 347 76 354 130T362 333Q362 478 354 524T321 597Z"/></g><g data-mml-node="mo" transform="translate(722.2,0)"><path data-c="2212" d="M84 237T84 250 98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"/></g><g data-mml-node="msup" transform="translate(1722.4,0)"><g data-mml-node="mn"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664 351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336 375 315Q457 262 457 175 457 96 395 37T238-22Q158-22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94 148 82 136 74 126 70 118 67L114 66Q165 21 238 21 293 21 321 74 338 107 338 175V195Q338 290 274 322 259 328 213 329L171 330 168 332Q166 335 166 348 166 366 174 366 202 366 232 371 266 376 294 413T322 525V533Q322 590 287 612 265 626 240 626 208 626 181 615T143 592 132 580H135Q138 579 143 578T153 573 165 566 175 555 183 540 186 520Q186 498 172 481T127 463Z"/><path data-c="36" d="M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601 351 623 311 626 240 626 194 566 147 500 147 364L148 360Q153 366 156 373 197 433 263 433H267Q313 433 348 414 372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149 440 90 387 34T253-22Q225-22 199-14T143 16 92 75 56 172 42 313ZM257 397Q227 397 205 380T171 335 154 278 148 216Q148 133 160 97T198 39Q222 21 251 21 302 21 329 59 342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z" transform="translate(500,0)"/><path data-c="30" d="M96 585Q152 666 249 666 297 666 345 640T423 548Q460 465 460 320 460 165 417 83 397 41 362 16T301-15 250-22Q224-22 198-16T137 16 82 83Q39 165 39 320 39 494 96 585ZM321 597Q291 629 250 629 208 629 178 597 153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16 290 16 318 46 347 76 354 130T362 333Q362 478 354 524T321 597Z" transform="translate(1000,0)"/></g><g data-mml-node="TeXAtom" transform="translate(1533,393.1) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">。</text></g></g></g></g></g></svg></mjx-container>之间，具体取决于梯度是有符号的还是无符号的。</p><p>单元内的每一个像素基于该像素处梯度的模对每一个梯度方向区间偷加权票。对于投票权重，可以是梯度大小，梯度大小的平方根或梯度大小的平方。</p><h3 id="3-描述符块"><a href="#3-描述符块" class="headerlink" title="3.描述符块"></a>3. 描述符块</h3><p>为了处理光照和对比度的变化，通过将单元组合在一起形成更大的空间上相连的块，局部的归一化梯度强度。然后，HOG 描述符是来自所有块区域内的、归一化的单元直方图部件的向量。</p><h3 id="4-块的归一化"><a href="#4-块的归一化" class="headerlink" title="4.块的归一化"></a>4. 块的归一化</h3><p>最后一步是块描述符的归一化。设 v 是包含给定块中所有直方图的非归一化向量，‖为它的 (k) 阶范数（(k=1,2) ），(\epsilon) 是一个小常量。归一化因子可以是如下之一：</p><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:33.918ex"><svg style="vertical-align:-2.759ex;min-width:33.918ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="6.649ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1719.5)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 1719.5) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="5417.8 -1719.5 1 2939"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr" transform="translate(0,600.5)"><g data-mml-node="mtd"><g data-mml-node="mi"><path data-c="1D43F" d="M228 637Q194 637 192 641 191 643 191 649 191 673 202 682 204 683 217 683 271 680 344 680 485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628 387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54 478 72 522 106 564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203 583 83 557 9Q555 4 553 3T537 0 494-1Q483-1 418-1T294 0H116Q32 0 32 10 32 17 34 24 39 43 44 45 48 46 59 46H65Q92 46 125 49 139 52 144 61 147 65 216 339T285 628Q285 635 228 637Z"/></g><g data-mml-node="mn" transform="translate(681,0)"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315 301 241Q265 210 201 149L142 93 218 92Q375 92 385 97 392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19 31Q50 38 56 46T86 81Q115 113 136 137 145 147 170 174T204 211 233 244 261 278 284 308 305 340 320 369 333 401 340 431 343 464Q343 527 309 573T212 619Q179 619 154 602T119 569 109 550Q109 549 114 549 132 549 151 535T170 489Q170 464 154 447T109 429Z"/></g><g data-mml-node="mi" transform="translate(1181,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">范</text></g><g data-mml-node="mi" transform="translate(2181,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">数</text></g><g data-mml-node="mo" transform="translate(3458.8,0)"><path data-c="3A" d="M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310 96 327 78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0 96 17 78 60Z"/></g><g data-mml-node="mi" transform="translate(4014.6,0)"><path data-c="1D463" d="M173 380Q173 405 154 405 130 405 104 376T61 287Q60 286 59 284T58 281 56 279 53 278 49 278 41 278H27Q21 284 21 287 21 294 29 316T53 368 97 419 160 441Q202 441 225 417T249 361Q249 344 246 335 246 329 231 291T200 202 182 113Q182 86 187 69 200 26 250 26 287 26 319 60T369 139 398 222 409 277Q409 300 401 317T383 343 365 361 357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159 347 40 241-11Q177-11 139 22 102 54 102 117 102 148 110 181T151 298Q173 362 173 380Z"/></g><g data-mml-node="mo" transform="translate(4777.3,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347 722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153 722 140 707 133H70Q56 140 56 153Z"/></g><g data-mml-node="mfrac" transform="translate(5833.1,0)"><g data-mml-node="mi" transform="translate(2258.8,676)"><path data-c="1D463" d="M173 380Q173 405 154 405 130 405 104 376T61 287Q60 286 59 284T58 281 56 279 53 278 49 278 41 278H27Q21 284 21 287 21 294 29 316T53 368 97 419 160 441Q202 441 225 417T249 361Q249 344 246 335 246 329 231 291T200 202 182 113Q182 86 187 69 200 26 250 26 287 26 319 60T369 139 398 222 409 277Q409 300 401 317T383 343 365 361 357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159 347 40 241-11Q177-11 139 22 102 54 102 117 102 148 110 181T151 298Q173 362 173 380Z"/></g><g data-mml-node="msqrt" transform="translate(220,-1251.1)"><g transform="translate(1020,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><path data-c="7C" d="M139-249H137Q125-249 119-235V251L120 737Q130 750 139 750 152 750 159 735V-235Q151-249 141-249H139Z"/></g><g data-mml-node="mi" transform="translate(278,0)"><path data-c="1D463" d="M173 380Q173 405 154 405 130 405 104 376T61 287Q60 286 59 284T58 281 56 279 53 278 49 278 41 278H27Q21 284 21 287 21 294 29 316T53 368 97 419 160 441Q202 441 225 417T249 361Q249 344 246 335 246 329 231 291T200 202 182 113Q182 86 187 69 200 26 250 26 287 26 319 60T369 139 398 222 409 277Q409 300 401 317T383 343 365 361 357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159 347 40 241-11Q177-11 139 22 102 54 102 117 102 148 110 181T151 298Q173 362 173 380Z"/></g><g data-mml-node="msubsup" transform="translate(763,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><path data-c="7C" d="M139-249H137Q125-249 119-235V251L120 737Q130 750 139 750 152 750 159 735V-235Q151-249 141-249H139Z"/></g><g data-mml-node="mn" transform="translate(311,353.6) scale(0.707)"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315 301 241Q265 210 201 149L142 93 218 92Q375 92 385 97 392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19 31Q50 38 56 46T86 81Q115 113 136 137 145 147 170 174T204 211 233 244 261 278 284 308 305 340 320 369 333 401 340 431 343 464Q343 527 309 573T212 619Q179 619 154 602T119 569 109 550Q109 549 114 549 132 549 151 535T170 489Q170 464 154 447T109 429Z"/></g><g data-mml-node="mn" transform="translate(311,-297.3) scale(0.707)"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315 301 241Q265 210 201 149L142 93 218 92Q375 92 385 97 392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19 31Q50 38 56 46T86 81Q115 113 136 137 145 147 170 174T204 211 233 244 261 278 284 308 305 340 320 369 333 401 340 431 343 464Q343 527 309 573T212 619Q179 619 154 602T119 569 109 550Q109 549 114 549 132 549 151 535T170 489Q170 464 154 447T109 429Z"/></g></g><g data-mml-node="mo" transform="translate(1699.8,0)"><path data-c="2B" d="M56 237T56 250 70 270H369V420L370 570Q380 583 389 583 402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401-82 391-82H389 387Q375-82 369-68V230H70Q56 237 56 250Z"/></g><g data-mml-node="msup" transform="translate(2700,0)"><g data-mml-node="mi"><path data-c="1D716" d="M227-11Q149-11 95 41T40 174Q40 262 87 322 121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411 382 385 341 385H325 312Q191 385 154 277L150 265H327Q340 256 340 246 340 228 320 219H138V217Q128 187 128 143 128 77 160 52T231 26Q258 26 284 36T326 57 343 68Q350 68 354 58T358 39Q358 36 357 35 354 31 337 21T289 0 227-11Z"/></g><g data-mml-node="mn" transform="translate(439,289) scale(0.707)"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315 301 241Q265 210 201 149L142 93 218 92Q375 92 385 97 392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19 31Q50 38 56 46T86 81Q115 113 136 137 145 147 170 174T204 211 233 244 261 278 284 308 305 340 320 369 333 401 340 431 343 464Q343 527 309 573T212 619Q179 619 154 602T119 569 109 550Q109 549 114 549 132 549 151 535T170 489Q170 464 154 447T109 429Z"/></g></g></g><g data-mml-node="mo" transform="translate(0,81.1)"><path data-c="221A" d="M1001 1150Q1017 1150 1020 1132 1020 1127 741 244L460-643Q453-650 436-650H424Q423-647 423-645T421-640 419-631 415-617 408-594 399-560 385-512 367-448 343-364 312-259L203 119 138 41 111 67 212 188 264 248 472-474 983 1140Q988 1150 1001 1150Z"/></g><rect width="3542.6" height="60" x="1020" y="1171.1"/></g><rect width="4762.6" height="60" x="120" y="220"/></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1719.5 1 2939"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:4" transform="translate(0,1350.5)"><text data-id-align="true"/><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><path data-c="28" d="M94 250Q94 319 104 381T127 488 164 576 202 643 244 695 277 729 302 750H315 319Q333 750 333 741 333 738 316 720T275 667 226 581 184 443 167 250 184 58 225-81 274-167 316-220 333-241Q333-250 318-250H315 302L274-226Q180-141 137-14T94 250Z"/><path data-c="34" d="M462 0Q444 3 333 3 217 3 199 0H190V46H221Q241 46 248 46T265 48 279 53 286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675 336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212 183 211H293Z" transform="translate(389,0)"/><path data-c="29" d="M60 749 64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12 224-76 186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z" transform="translate(889,0)"/></g></g></g></g></svg></g></g></g></g></svg></mjx-container><br>或者</p><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:30.622ex"><svg style="vertical-align:-1.785ex;min-width:30.622ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="4.701ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1289)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 1289) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="4689.6 -1289 1 2078"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr" transform="translate(0,170)"><g data-mml-node="mtd"><g data-mml-node="mi"><path data-c="1D43F" d="M228 637Q194 637 192 641 191 643 191 649 191 673 202 682 204 683 217 683 271 680 344 680 485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628 387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54 478 72 522 106 564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203 583 83 557 9Q555 4 553 3T537 0 494-1Q483-1 418-1T294 0H116Q32 0 32 10 32 17 34 24 39 43 44 45 48 46 59 46H65Q92 46 125 49 139 52 144 61 147 65 216 339T285 628Q285 635 228 637Z"/></g><g data-mml-node="mn" transform="translate(681,0)"><path data-c="31" d="M213 578 200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641 273 663Q275 666 285 666 294 666 302 660V361L303 61Q310 54 315 52T339 48 401 46H427V0H416Q395 3 257 3 121 3 100 0H88V46H114Q136 46 152 46T177 47 193 50 201 52 207 57 213 61V578Z"/></g><g data-mml-node="mi" transform="translate(1181,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">范</text></g><g data-mml-node="mi" transform="translate(2181,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">数</text></g><g data-mml-node="mo" transform="translate(3458.8,0)"><path data-c="3A" d="M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310 96 327 78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0 96 17 78 60Z"/></g><g data-mml-node="mi" transform="translate(4014.6,0)"><path data-c="1D463" d="M173 380Q173 405 154 405 130 405 104 376T61 287Q60 286 59 284T58 281 56 279 53 278 49 278 41 278H27Q21 284 21 287 21 294 29 316T53 368 97 419 160 441Q202 441 225 417T249 361Q249 344 246 335 246 329 231 291T200 202 182 113Q182 86 187 69 200 26 250 26 287 26 319 60T369 139 398 222 409 277Q409 300 401 317T383 343 365 361 357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159 347 40 241-11Q177-11 139 22 102 54 102 117 102 148 110 181T151 298Q173 362 173 380Z"/></g><g data-mml-node="mo" transform="translate(4777.3,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347 722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153 722 140 707 133H70Q56 140 56 153Z"/></g><g data-mml-node="mfrac" transform="translate(5833.1,0)"><g data-mml-node="mi" transform="translate(1530.5,676)"><path data-c="1D463" d="M173 380Q173 405 154 405 130 405 104 376T61 287Q60 286 59 284T58 281 56 279 53 278 49 278 41 278H27Q21 284 21 287 21 294 29 316T53 368 97 419 160 441Q202 441 225 417T249 361Q249 344 246 335 246 329 231 291T200 202 182 113Q182 86 187 69 200 26 250 26 287 26 319 60T369 139 398 222 409 277Q409 300 401 317T383 343 365 361 357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159 347 40 241-11Q177-11 139 22 102 54 102 117 102 148 110 181T151 298Q173 362 173 380Z"/></g><g data-mml-node="mrow" transform="translate(220,-709.5)"><g data-mml-node="mo" transform="translate(0 -0.5)"><path data-c="7C" d="M139-249H137Q125-249 119-235V251L120 737Q130 750 139 750 152 750 159 735V-235Q151-249 141-249H139Z"/></g><g data-mml-node="mi" transform="translate(278,0)"><path data-c="1D463" d="M173 380Q173 405 154 405 130 405 104 376T61 287Q60 286 59 284T58 281 56 279 53 278 49 278 41 278H27Q21 284 21 287 21 294 29 316T53 368 97 419 160 441Q202 441 225 417T249 361Q249 344 246 335 246 329 231 291T200 202 182 113Q182 86 187 69 200 26 250 26 287 26 319 60T369 139 398 222 409 277Q409 300 401 317T383 343 365 361 357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159 347 40 241-11Q177-11 139 22 102 54 102 117 102 148 110 181T151 298Q173 362 173 380Z"/></g><g data-mml-node="msub" transform="translate(763,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><path data-c="7C" d="M139-249H137Q125-249 119-235V251L120 737Q130 750 139 750 152 750 159 735V-235Q151-249 141-249H139Z"/></g><g data-mml-node="mn" transform="translate(311,-150) scale(0.707)"><path data-c="31" d="M213 578 200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641 273 663Q275 666 285 666 294 666 302 660V361L303 61Q310 54 315 52T339 48 401 46H427V0H416Q395 3 257 3 121 3 100 0H88V46H114Q136 46 152 46T177 47 193 50 201 52 207 57 213 61V578Z"/></g></g><g data-mml-node="mo" transform="translate(1699.8,0)"><path data-c="2B" d="M56 237T56 250 70 270H369V420L370 570Q380 583 389 583 402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401-82 391-82H389 387Q375-82 369-68V230H70Q56 237 56 250Z"/></g><g data-mml-node="mi" transform="translate(2700,0)"><path data-c="1D716" d="M227-11Q149-11 95 41T40 174Q40 262 87 322 121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411 382 385 341 385H325 312Q191 385 154 277L150 265H327Q340 256 340 246 340 228 320 219H138V217Q128 187 128 143 128 77 160 52T231 26Q258 26 284 36T326 57 343 68Q350 68 354 58T358 39Q358 36 357 35 354 31 337 21T289 0 227-11Z"/></g></g><rect width="3306" height="60" x="120" y="220"/></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1289 1 2078"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:5" transform="translate(0,920)"><text data-id-align="true"/><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><path data-c="28" d="M94 250Q94 319 104 381T127 488 164 576 202 643 244 695 277 729 302 750H315 319Q333 750 333 741 333 738 316 720T275 667 226 581 184 443 167 250 184 58 225-81 274-167 316-220 333-241Q333-250 318-250H315 302L274-226Q180-141 137-14T94 250Z"/><path data-c="35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22 294 22 326 82 345 115 345 210 345 313 318 349 292 382 260 382H254Q176 382 136 314 132 307 129 306T114 304Q97 304 95 310 93 314 93 485V614Q93 664 98 664 100 666 102 666 103 666 123 658T178 642 253 634Q324 634 389 662 397 666 402 666 410 666 410 648V635Q328 538 205 538 174 538 149 544L139 546V374Q158 388 169 396T205 412 256 420Q337 420 393 355T449 201Q449 109 385 44T229-22Q148-22 99 32T50 154Q50 178 61 192T84 210 107 214Q132 214 148 197T164 157Z" transform="translate(389,0)"/><path data-c="29" d="M60 749 64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12 224-76 186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z" transform="translate(889,0)"/></g></g></g></g></svg></g></g></g></g></svg></mjx-container><br>或者</p><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:39.717ex"><svg style="vertical-align:-2.217ex;min-width:39.717ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="5.566ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1480)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 1480) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="6699.6 -1480 1 2460"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr" transform="translate(0,24.7)"><g data-mml-node="mtd"><g data-mml-node="mi"><path data-c="1D43F" d="M228 637Q194 637 192 641 191 643 191 649 191 673 202 682 204 683 217 683 271 680 344 680 485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628 387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54 478 72 522 106 564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203 583 83 557 9Q555 4 553 3T537 0 494-1Q483-1 418-1T294 0H116Q32 0 32 10 32 17 34 24 39 43 44 45 48 46 59 46H65Q92 46 125 49 139 52 144 61 147 65 216 339T285 628Q285 635 228 637Z"/></g><g data-mml-node="mn" transform="translate(681,0)"><path data-c="31" d="M213 578 200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641 273 663Q275 666 285 666 294 666 302 660V361L303 61Q310 54 315 52T339 48 401 46H427V0H416Q395 3 257 3 121 3 100 0H88V46H114Q136 46 152 46T177 47 193 50 201 52 207 57 213 61V578Z"/></g><g data-mml-node="mi" transform="translate(1181,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">范</text></g><g data-mml-node="mi" transform="translate(2181,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">数</text></g><g data-mml-node="mi" transform="translate(3181,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">平</text></g><g data-mml-node="mi" transform="translate(4181,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">方</text></g><g data-mml-node="mi" transform="translate(5181,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">根</text></g><g data-mml-node="mo" transform="translate(6458.8,0)"><path data-c="3A" d="M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310 96 327 78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0 96 17 78 60Z"/></g><g data-mml-node="mi" transform="translate(7014.6,0)"><path data-c="1D463" d="M173 380Q173 405 154 405 130 405 104 376T61 287Q60 286 59 284T58 281 56 279 53 278 49 278 41 278H27Q21 284 21 287 21 294 29 316T53 368 97 419 160 441Q202 441 225 417T249 361Q249 344 246 335 246 329 231 291T200 202 182 113Q182 86 187 69 200 26 250 26 287 26 319 60T369 139 398 222 409 277Q409 300 401 317T383 343 365 361 357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159 347 40 241-11Q177-11 139 22 102 54 102 117 102 148 110 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transform="translate(220,-709.5)"><g data-mml-node="mo" transform="translate(0 -0.5)"><path data-c="7C" d="M139-249H137Q125-249 119-235V251L120 737Q130 750 139 750 152 750 159 735V-235Q151-249 141-249H139Z"/></g><g data-mml-node="mi" transform="translate(278,0)"><path data-c="1D463" d="M173 380Q173 405 154 405 130 405 104 376T61 287Q60 286 59 284T58 281 56 279 53 278 49 278 41 278H27Q21 284 21 287 21 294 29 316T53 368 97 419 160 441Q202 441 225 417T249 361Q249 344 246 335 246 329 231 291T200 202 182 113Q182 86 187 69 200 26 250 26 287 26 319 60T369 139 398 222 409 277Q409 300 401 317T383 343 365 361 357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159 347 40 241-11Q177-11 139 22 102 54 102 117 102 148 110 181T151 298Q173 362 173 380Z"/></g><g data-mml-node="msub" transform="translate(763,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><path data-c="7C" d="M139-249H137Q125-249 119-235V251L120 737Q130 750 139 750 152 750 159 735V-235Q151-249 141-249H139Z"/></g><g data-mml-node="mn" transform="translate(311,-150) scale(0.707)"><path data-c="31" d="M213 578 200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641 273 663Q275 666 285 666 294 666 302 660V361L303 61Q310 54 315 52T339 48 401 46H427V0H416Q395 3 257 3 121 3 100 0H88V46H114Q136 46 152 46T177 47 193 50 201 52 207 57 213 61V578Z"/></g></g><g data-mml-node="mo" transform="translate(1699.8,0)"><path data-c="2B" d="M56 237T56 250 70 270H369V420L370 570Q380 583 389 583 402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401-82 391-82H389 387Q375-82 369-68V230H70Q56 237 56 250Z"/></g><g data-mml-node="mi" transform="translate(2700,0)"><path data-c="1D716" d="M227-11Q149-11 95 41T40 174Q40 262 87 322 121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411 382 385 341 385H325 312Q191 385 154 277L150 265H327Q340 256 340 246 340 228 320 219H138V217Q128 187 128 143 128 77 160 52T231 26Q258 26 284 36T326 57 343 68Q350 68 354 58T358 39Q358 36 357 35 354 31 337 21T289 0 227-11Z"/></g></g><rect width="3306" height="60" x="120" y="220"/></g></g><g data-mml-node="mo" transform="translate(0,-54.7)"><path data-c="221A" d="M424-948Q422-947 313-434T202 80L170 31Q165 24 157 10 137-21 137-21 131-16 124-8L111 5 264 248 473-720Q473-717 727 359T983 1440Q989 1450 1001 1450 1007 1450 1013 1445T1020 1433Q1020 1425 742 244T460-941Q458-950 439-950H436Q424-950 424-948Z"/></g><rect width="3546" height="60" x="1020" y="1335.3"/></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1480 1 2460"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:6" transform="translate(0,774.7)"><text data-id-align="true"/><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><path data-c="28" d="M94 250Q94 319 104 381T127 488 164 576 202 643 244 695 277 729 302 750H315 319Q333 750 333 741 333 738 316 720T275 667 226 581 184 443 167 250 184 58 225-81 274-167 316-220 333-241Q333-250 318-250H315 302L274-226Q180-141 137-14T94 250Z"/><path data-c="36" d="M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601 351 623 311 626 240 626 194 566 147 500 147 364L148 360Q153 366 156 373 197 433 263 433H267Q313 433 348 414 372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149 440 90 387 34T253-22Q225-22 199-14T143 16 92 75 56 172 42 313ZM257 397Q227 397 205 380T171 335 154 278 148 216Q148 133 160 97T198 39Q222 21 251 21 302 21 329 59 342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z" transform="translate(389,0)"/><path data-c="29" d="M60 749 64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12 224-76 186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z" transform="translate(889,0)"/></g></g></g></g></svg></g></g></g></g></svg></mjx-container><br>还有另一个归一化因子 L2-Hys, 它通过削减 v 的 L2 范数得到（将 v 的最大值限制为 0.2），然后重新归一化。</p><p>最终的图像 / RoI 描述符是通过连接所有归一化的块描述符而形成的。</p><p>L2 范数、L2-Hys 和 L1 范数平方根（L1-sqrt）归一化方法提供了类似的性能，而 L1 范数提供了可靠性稍差的性能。</p></div><div class="story post-story"><h2 id="尺度不变特征变换"><a href="#尺度不变特征变换" class="headerlink" title="尺度不变特征变换"></a>尺度不变特征变换</h2><p>SIFT [Lowe,2004] 提供了一组对象的特征，这些特征对于对象缩放和旋转是健壮的。</p><p>SIFT 算法由四个主要步骤组成。</p><h3 id="1-尺度空间的极值侦测"><a href="#1-尺度空间的极值侦测" class="headerlink" title="1.尺度空间的极值侦测"></a>1. 尺度空间的极值侦测</h3><p>第一步旨在确定对缩放和方向不变的潜在关键点。</p><p>SIFT 使用高斯差分（DoG）来检测尺度空间中关键点中的位置。</p><p>高斯差分是将两个不同尺度的图像（其中一个尺度为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.292ex" height="1ex" role="img" focusable="false" viewBox="0 -431 571 442"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D70E" d="M184-11Q116-11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427 562 425 565 422 567 420 569 416 570 412 571 407 572 401Q572 357 507 357 500 357 490 357T476 358H416L421 348Q439 310 439 263 439 153 359 71T184-11ZM361 278Q361 358 276 358 152 358 115 184 114 180 114 178 106 141 106 117 106 67 131 47T188 26Q242 26 287 73 316 103 334 153T356 233 361 278Z"/></g></g></g></svg></mjx-container> , 另一个是其 k 倍，即<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="5.236ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 2314.4 705"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686 294 679 244 477 194 279 194 272 213 282 223 291 247 309 292 354T362 415Q402 442 438 442 468 442 485 423T503 369Q503 344 496 327T477 302 456 291 438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316 228 255Q230 254 243 252T267 246 293 238 320 224 342 206 359 180 365 147Q365 130 360 106T354 66Q354 26 381 26 429 26 459 145 461 153 479 153H483Q499 153 499 144 499 139 496 130 455-11 378-11 333-11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206 200 217 182 220H180Q168 178 159 139T145 81 136 44 129 20 122 7 111-2Q98-11 83-11 66-11 57-1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640 121 647Z"/></g><g data-mml-node="mo" transform="translate(743.2,0)"><path data-c="D7" d="M630 29Q630 9 609 9 604 9 587 25T493 118L389 222 284 117Q178 13 175 11 171 9 168 9 160 9 154 15T147 29Q147 36 161 51T255 146L359 250 255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489 178 487 284 383L389 278 493 382Q570 459 587 475T609 491Q630 491 630 471 630 464 620 453T522 355L418 250 522 145Q606 61 618 48T630 29Z"/></g><g data-mml-node="mi" transform="translate(1743.4,0)"><path data-c="1D70E" d="M184-11Q116-11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427 562 425 565 422 567 420 569 416 570 412 571 407 572 401Q572 357 507 357 500 357 490 357T476 358H416L421 348Q439 310 439 263 439 153 359 71T184-11ZM361 278Q361 358 276 358 152 358 115 184 114 180 114 178 106 141 106 117 106 67 131 47T188 26Q242 26 287 73 316 103 334 153T356 233 361 278Z"/></g></g></g></svg></mjx-container>）的高斯模糊进行差分得到的。</p><h3 id="2-关键点精确定位"><a href="#2-关键点精确定位" class="headerlink" title="2.关键点精确定位"></a>2. 关键点精确定位</h3><h3 id="3-方向定位"><a href="#3-方向定位" class="headerlink" title="3.方向定位"></a>3. 方向定位</h3><h3 id="4-关键点描述符"><a href="#4-关键点描述符" class="headerlink" title="4.关键点描述符"></a>4. 关键点描述符</h3></div></div><div class="footer"><div class="copyright license"><div class="license-title">特征和分类器</div><div class="license-link"><a href="https://blog.mhuig.top/p/483290b5/">https://blog.mhuig.top/p/483290b5/</a></div><div class="license-meta"><div class="license-meta-item"><div class="license-meta-title">本文作者</div><div class="license-meta-text">MHuiG</div></div><div class="license-meta-item"><div class="license-meta-title">发布于</div><div class="license-meta-text">2019年9月12日</div></div><div class="license-meta-item"><div class="license-meta-title">许可协议</div><div class="license-meta-text"><a href="https://creativecommons.org/licenses/by-nc-sa/4.0/deed.zh#" target="_blank" rel="external nofollow noopener noreferrer">CC BY-NC-SA 4.0</a></div></div></div><div class="license-statement">署名-非商业性使用-相同方式共享 4.0 国际。</div></div></div><div class="article-meta" id="bottom"><div class="new-meta-box"><div class="new-meta-item category"><i class="fa-duotone fa-folder-open fa-fw" aria-hidden="true"></i> <a class="category-link" href="/categories/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/">机器学习</a><span class="sep"></span><a class="category-link" href="/categories/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/%E6%9C%BA%E5%99%A8%E8%A7%86%E8%A7%89/">机器视觉</a> <span hidden itemprop="about" itemscope itemtype="http://schema.org/Thing"><a href="/categories/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/" itemprop="url"><span itemprop="name">机器学习</span></a></span> <span hidden itemprop="about" itemscope itemtype="http://schema.org/Thing"><a href="/categories/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/%E6%9C%BA%E5%99%A8%E8%A7%86%E8%A7%89/" itemprop="url"><span itemprop="name">机器视觉</span></a></span></div><div class="new-meta-item meta-tags"><a class="tag" href="/tags/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/" rel="nofollow"><i class="fa-duotone fa-hashtag fa-fw" aria-hidden="true"></i><p>机器学习</p></a></div><div class="new-meta-item meta-tags"><a class="tag" href="/tags/%E6%9C%BA%E5%99%A8%E8%A7%86%E8%A7%89/" rel="nofollow"><i class="fa-duotone fa-hashtag fa-fw" aria-hidden="true"></i><p>机器视觉</p></a></div><div class="new-meta-item meta-tags"><a class="tag" href="/tags/%E5%88%86%E7%B1%BB%E5%99%A8/" rel="nofollow"><i class="fa-duotone fa-hashtag fa-fw" aria-hidden="true"></i><p>分类器</p></a></div><div class="new-meta-item meta-tags"><a class="tag" href="/tags/%E7%89%B9%E5%BE%81/" rel="nofollow"><i class="fa-duotone fa-hashtag fa-fw" aria-hidden="true"></i><p>特征</p></a></div> <span hidden itemprop="keywords">机器学习 机器视觉 分类器 特征</span><div class="new-meta-item date" itemprop="dateCreated datePublished" datetime="2019-09-12T11:19:19+08:00"><a class="notlink"><i class="fa-duotone fa-calendar-alt fa-fw" aria-hidden="true"></i><p>发布于：2019年9月12日</p></a></div><div class="new-meta-item date" itemprop="dateModified" datetime="2019-09-12T11:19:19+08:00"><a class="notlink"><i class="fa-duotone fa-edit fa-fw" aria-hidden="true"></i><p>更新于：2019年9月12日</p></a></div></div></div><div class="prev-next"><a class="prev" href="/p/notes-zookeeper/"><p class="title"><i class="fa-solid fa-chevron-left" aria-hidden="true"></i>Zookeeper 系列专题</p><p class="content">.fa-secondary{opacity:.4} Zookeeper Zookeeper 开始阅读</p></a><a class="next" href="/p/9bc3b11e/"><p class="title">图像导数<i class="fa-solid fa-chevron-right" aria-hidden="true"></i></p><p class="content">在图像中，边缘可以看做是位于一阶导数较大的像素处，因此，我们可以求图像的一阶导数来确定图像的边缘，像 sobel 算子等一系列算子都是基于这个思想的。 如下图 a 表示函数在边沿的时候关系，求...</p></a></div><div class="recommended-article"><div class="recommended-article-header"><i class="fa-solid fa-bookmark fa-fw" aria-hidden="true"></i> <span>推荐阅读</span></div><div class="recommended-article-group"> <a class="recommended-article-item" href="/p/9bc3b11e/" title="图像导数" rel="bookmark"><img src="https://static.mhuig.top/npm/imbox@0.0.15/c/48.webp" class="lazyload" data-srcset="https://static.mhuig.top/npm/imbox@0.0.15/c/48.webp" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" alt="图像导数"> <span class="title">图像导数</span></a> <a class="recommended-article-item" href="/p/xa6297acc/" title="理解卷积" rel="bookmark"><img src="https://static.mhuig.top/npm/imbox@0.0.15/c/112.webp" class="lazyload" data-srcset="https://static.mhuig.top/npm/imbox@0.0.15/c/112.webp" 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id="layoutHelper-comments"></div></article></div><aside id="l_side" itemscope itemtype="http://schema.org/WPSideBar"><div class="widget-sticky pjax"><section class="widget toc-wrapper desktop mobile" id="toc-div"><header><i class="fa-duotone fa-list fa-fw" aria-hidden="true"></i> <span class="name">本文目录</span></header><div class="content"><ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%89%B9%E5%BE%81"><span class="toc-text">特征</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%88%86%E7%B1%BB%E5%99%A8"><span class="toc-text">分类器</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E4%BC%A0%E7%BB%9F%E7%89%B9%E5%BE%81%E6%8F%8F%E8%BF%B0%E7%AC%A6"><span class="toc-text">传统特征描述符</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%96%B9%E5%90%91%E6%A2%AF%E5%BA%A6%E7%9B%B4%E6%96%B9%E5%9B%BE"><span class="toc-text">方向梯度直方图</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#1-%E6%A2%AF%E5%BA%A6%E8%AE%A1%E7%AE%97"><span class="toc-text">1. 梯度计算</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#2-%E5%8D%95%E5%85%83%E6%96%B9%E5%90%91%E7%9B%B4%E6%96%B9%E5%9B%BE"><span class="toc-text">2. 单元方向直方图</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#3-%E6%8F%8F%E8%BF%B0%E7%AC%A6%E5%9D%97"><span class="toc-text">3. 描述符块</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#4-%E5%9D%97%E7%9A%84%E5%BD%92%E4%B8%80%E5%8C%96"><span class="toc-text">4. 块的归一化</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%B0%BA%E5%BA%A6%E4%B8%8D%E5%8F%98%E7%89%B9%E5%BE%81%E5%8F%98%E6%8D%A2"><span class="toc-text">尺度不变特征变换</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#1-%E5%B0%BA%E5%BA%A6%E7%A9%BA%E9%97%B4%E7%9A%84%E6%9E%81%E5%80%BC%E4%BE%A6%E6%B5%8B"><span class="toc-text">1. 尺度空间的极值侦测</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#2-%E5%85%B3%E9%94%AE%E7%82%B9%E7%B2%BE%E7%A1%AE%E5%AE%9A%E4%BD%8D"><span class="toc-text">2. 关键点精确定位</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#3-%E6%96%B9%E5%90%91%E5%AE%9A%E4%BD%8D"><span class="toc-text">3. 方向定位</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#4-%E5%85%B3%E9%94%AE%E7%82%B9%E6%8F%8F%E8%BF%B0%E7%AC%A6"><span class="toc-text">4. 关键点描述符</span></a></li></ol></li></ol></div></section></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div 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