-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathGAMES10114.html
508 lines (399 loc) · 82.5 KB
/
GAMES10114.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
<!DOCTYPE html>
<html lang="zh-CN">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width">
<meta name="theme-color" content="#222"><meta name="generator" content="Hexo 7.2.0">
<link rel="apple-touch-icon" sizes="180x180" href="/images/apple-touch-icon-next.png">
<link rel="icon" type="image/png" sizes="32x32" href="/images/favicon-32x32-next.png">
<link rel="icon" type="image/png" sizes="16x16" href="/images/favicon-16x16-next.png">
<link rel="mask-icon" href="/images/logo.svg" color="#222">
<link rel="stylesheet" href="/css/main.css">
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.7.2/css/all.min.css" integrity="sha256-dABdfBfUoC8vJUBOwGVdm8L9qlMWaHTIfXt+7GnZCIo=" crossorigin="anonymous">
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/animate.css/3.1.1/animate.min.css" integrity="sha256-PR7ttpcvz8qrF57fur/yAx1qXMFJeJFiA6pSzWi0OIE=" crossorigin="anonymous">
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/pace/1.2.4/themes/white/pace-theme-minimal.css">
<script src="https://cdnjs.cloudflare.com/ajax/libs/pace/1.2.4/pace.min.js" integrity="sha256-gqd7YTjg/BtfqWSwsJOvndl0Bxc8gFImLEkXQT8+qj0=" crossorigin="anonymous"></script>
<script class="next-config" data-name="main" type="application/json">{"hostname":"example.com","root":"/","images":"/images","scheme":"Pisces","darkmode":false,"version":"8.22.0","exturl":false,"sidebar":{"position":"left","width_expanded":320,"width_dual_column":240,"display":"post","padding":18,"offset":12},"hljswrap":true,"copycode":{"enable":true,"style":"default"},"fold":{"enable":true,"height":500},"bookmark":{"enable":false,"color":"#222","save":"auto"},"mediumzoom":false,"lazyload":false,"pangu":false,"comments":{"style":"tabs","active":"disqus","storage":true,"lazyload":false,"nav":{"disqus":{"text":"Load Disqus","order":-1}},"activeClass":"disqus"},"stickytabs":false,"motion":{"enable":true,"async":false,"duration":200,"transition":{"menu_item":"fadeInDown","post_block":"fadeIn","post_header":"fadeInDown","post_body":"fadeInDown","coll_header":"fadeInLeft","sidebar":"fadeInUp"}},"prism":false,"i18n":{"placeholder":"搜索...","empty":"没有找到任何搜索结果：${query}","hits_time":"找到 ${hits} 个搜索结果（用时 ${time} 毫秒）","hits":"找到 ${hits} 个搜索结果"},"path":"/search.xml","localsearch":{"enable":true,"top_n_per_article":1,"unescape":false,"preload":false,"trigger":"auto"}}</script><script src="/js/config.js"></script>
<meta name="description" content="前言 GAMES101-16:蒙特卡罗积分、路径追踪">
<meta property="og:type" content="article">
<meta property="og:title" content="GAMES101-16：辐射度量学与路径追踪">
<meta property="og:url" content="http://example.com/GAMES10114.html">
<meta property="og:site_name" content="LeeKa 的酒馆">
<meta property="og:description" content="前言 GAMES101-16:蒙特卡罗积分、路径追踪">
<meta property="og:locale" content="zh_CN">
<meta property="og:image" content="http://example.com/assets/101-glossyReflection.png">
<meta property="og:image" content="http://example.com/assets/101-diffusePathTrace.png">
<meta property="og:image" content="http://example.com/assets/101-pathtracing.png">
<meta property="og:image" content="http://example.com/assets/101-pathtracingLowSPP.png">
<meta property="og:image" content="http://example.com/assets/101-samplingInLight.png">
<meta property="og:image" content="http://example.com/assets/101-samplingInLightAlgorithm.png">
<meta property="og:image" content="http://example.com/assets/101-pathtracingIsCorrect.png">
<meta property="article:published_time" content="2023-08-06T10:11:52.000Z">
<meta property="article:modified_time" content="2024-04-20T18:58:06.439Z">
<meta property="article:author" content="LeeKa">
<meta property="article:tag" content="笔记">
<meta property="article:tag" content="计算机图形学">
<meta property="article:tag" content="GAMES">
<meta property="article:tag" content="GAMES101">
<meta property="article:tag" content="光线追踪">
<meta property="article:tag" content="辐射度量学">
<meta property="article:tag" content="路径追踪">
<meta property="article:tag" content="蒙特卡洛积分">
<meta name="twitter:card" content="summary">
<meta name="twitter:image" content="http://example.com/assets/101-glossyReflection.png">
<link rel="canonical" href="http://example.com/GAMES10114.html">
<script class="next-config" data-name="page" type="application/json">{"sidebar":"","isHome":false,"isPost":true,"lang":"zh-CN","comments":true,"permalink":"http://example.com/GAMES10114.html","path":"/GAMES10114.html","title":"GAMES101-16：辐射度量学与路径追踪"}</script>
<script class="next-config" data-name="calendar" type="application/json">""</script>
<title>GAMES101-16：辐射度量学与路径追踪 | LeeKa 的酒馆</title>
<noscript>
<link rel="stylesheet" href="/css/noscript.css">
</noscript>
</head>
<body itemscope itemtype="http://schema.org/WebPage" class="use-motion">
<div class="headband"></div>
<main class="main">
<div class="column">
<header class="header" itemscope itemtype="http://schema.org/WPHeader"><div class="site-brand-container">
<div class="site-nav-toggle">
<div class="toggle" aria-label="切换导航栏" role="button">
<span class="toggle-line"></span>
<span class="toggle-line"></span>
<span class="toggle-line"></span>
</div>
</div>
<div class="site-meta">
<a href="/" class="brand" rel="start">
<i class="logo-line"></i>
<p class="site-title">LeeKa 的酒馆</p>
<i class="logo-line"></i>
</a>
<p class="site-subtitle" itemprop="description">欢迎，旅人！坐下来享受一下暖烘烘的炉火吧。</p>
</div>
<div class="site-nav-right">
<div class="toggle popup-trigger" aria-label="搜索" role="button">
<i class="fa fa-search fa-fw fa-lg"></i>
</div>
</div>
</div>
<nav class="site-nav">
<ul class="main-menu menu"><li class="menu-item menu-item-home"><a href="/" rel="section"><i class="fa fa-home fa-fw"></i>首页</a></li><li class="menu-item menu-item-about"><a href="/about/" rel="section"><i class="fa fa-user fa-fw"></i>关于</a></li><li class="menu-item menu-item-tags"><a href="/tags/" rel="section"><i class="fa fa-tags fa-fw"></i>标签</a></li><li class="menu-item menu-item-categories"><a href="/categories/" rel="section"><i class="fa fa-th fa-fw"></i>分类</a></li><li class="menu-item menu-item-archives"><a href="/archives/" rel="section"><i class="fa fa-archive fa-fw"></i>归档</a></li><li class="menu-item menu-item-友链"><a href="/links/" rel="section"><i class="fa-solid fa-link fa-fw"></i>友链</a></li>
<li class="menu-item menu-item-search">
<a role="button" class="popup-trigger"><i class="fa fa-search fa-fw"></i>搜索
</a>
</li>
</ul>
</nav>
<div class="search-pop-overlay">
<div class="popup search-popup">
<div class="search-header">
<span class="search-icon">
<i class="fa fa-search"></i>
</span>
<div class="search-input-container">
<input autocomplete="off" autocapitalize="off" maxlength="80"
placeholder="搜索..." spellcheck="false"
type="search" class="search-input">
</div>
<span class="popup-btn-close" role="button">
<i class="fa fa-times-circle"></i>
</span>
</div>
<div class="search-result-container">
<div class="search-result-icon">
<i class="fa fa-spinner fa-pulse fa-5x"></i>
</div>
</div>
</div>
</div>
</header>
<aside class="sidebar">
<div class="sidebar-inner sidebar-nav-active sidebar-toc-active">
<ul class="sidebar-nav">
<li class="sidebar-nav-toc">
文章目录
</li>
<li class="sidebar-nav-overview">
站点概览
</li>
</ul>
<div class="sidebar-panel-container">
<!--noindex-->
<div class="post-toc-wrap sidebar-panel">
<div class="post-toc animated"><ol class="nav"><li class="nav-item nav-level-2"><a class="nav-link" href="#%E5%89%8D%E8%A8%80"><span class="nav-number">1.</span> <span class="nav-text"> 前言</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#%E8%92%99%E7%89%B9%E5%8D%A1%E7%BD%97%E7%A7%AF%E5%88%86"><span class="nav-number">2.</span> <span class="nav-text"> 蒙特卡罗积分</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#%E8%B7%AF%E5%BE%84%E8%BF%BD%E8%B8%AA-path-tracing"><span class="nav-number">3.</span> <span class="nav-text"> 路径追踪 Path Tracing</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#%E9%80%90%E6%AD%A5%E6%8E%A8%E5%87%BA%E8%B7%AF%E5%BE%84%E8%BF%BD%E8%B8%AA%E7%AE%97%E6%B3%95"><span class="nav-number">3.1.</span> <span class="nav-text"> 逐步推出路径追踪算法</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E5%9C%A8%E5%85%89%E6%BA%90%E9%87%87%E6%A0%B7%E7%9A%84%E9%AB%98%E6%95%88%E8%B7%AF%E5%BE%84%E8%BF%BD%E8%B8%AA"><span class="nav-number">3.2.</span> <span class="nav-text"> 在光源采样的高效路径追踪</span></a></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#%E4%B8%80%E4%BA%9B%E7%9C%BA%E6%9C%9Bsidenotes"><span class="nav-number">4.</span> <span class="nav-text"> 一些眺望：SideNotes</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#%E8%B7%B3%E8%BD%AC"><span class="nav-number">5.</span> <span class="nav-text"> 跳转</span></a></li></ol></div>
</div>
<!--/noindex-->
<div class="site-overview-wrap sidebar-panel">
<div class="site-author animated" itemprop="author" itemscope itemtype="http://schema.org/Person">
<img class="site-author-image" itemprop="image" alt="LeeKa"
src="https://s2.loli.net/2022/03/24/zcq6l9KENbRJtDi.jpg">
<p class="site-author-name" itemprop="name">LeeKa</p>
<div class="site-description" itemprop="description">代码、音乐和游戏，一起来聊聊吧</div>
</div>
<div class="site-state-wrap animated">
<nav class="site-state">
<div class="site-state-item site-state-posts">
<a href="/archives/">
<span class="site-state-item-count">63</span>
<span class="site-state-item-name">日志</span>
</a>
</div>
<div class="site-state-item site-state-categories">
<a href="/categories/">
<span class="site-state-item-count">15</span>
<span class="site-state-item-name">分类</span></a>
</div>
<div class="site-state-item site-state-tags">
<a href="/tags/">
<span class="site-state-item-count">160</span>
<span class="site-state-item-name">标签</span></a>
</div>
</nav>
</div>
<div class="links-of-author animated">
<span class="links-of-author-item">
<a href="https://github.com/KXAND" title="GitHub → https://github.com/KXAND" rel="noopener me" target="_blank">GitHub</a>
</span>
<span class="links-of-author-item">
<a href="mailto:leeka.Pub@outlook.com" title="E-Mail → mailto:leeka.Pub@outlook.com" rel="noopener me" target="_blank">E-Mail</a>
</span>
<span class="links-of-author-item">
<a href="https://twitter.com/QuiXand" title="X → https://twitter.com/QuiXand" rel="noopener me" target="_blank">X</a>
</span>
<span class="links-of-author-item">
<a href="https://pinhua.leeka.pub/" title="宁远平话 → https://pinhua.leeka.pub" rel="noopener me" target="_blank">宁远平话</a>
</span>
</div>
<div class="cc-license animated" itemprop="license">
<a href="https://creativecommons.org/licenses/by-nc-sa/4.0/deed.zh-hans" class="cc-opacity" rel="noopener" target="_blank"><img src="https://cdnjs.cloudflare.com/ajax/libs/creativecommons-vocabulary/2020.11.3/assets/license_badges/big/by_nc_sa.svg" alt="Creative Commons"></a>
</div>
</div>
</div>
</div>
</aside>
</div>
<div class="main-inner post posts-expand">
<div class="post-block">
<article itemscope itemtype="http://schema.org/Article" class="post-content" lang="zh-CN">
<link itemprop="mainEntityOfPage" href="http://example.com/GAMES10114.html">
<span hidden itemprop="author" itemscope itemtype="http://schema.org/Person">
<meta itemprop="image" content="https://s2.loli.net/2022/03/24/zcq6l9KENbRJtDi.jpg">
<meta itemprop="name" content="LeeKa">
</span>
<span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization">
<meta itemprop="name" content="LeeKa 的酒馆">
<meta itemprop="description" content="代码、音乐和游戏，一起来聊聊吧">
</span>
<span hidden itemprop="post" itemscope itemtype="http://schema.org/CreativeWork">
<meta itemprop="name" content="GAMES101-16：辐射度量学与路径追踪 | LeeKa 的酒馆">
<meta itemprop="description" content="">
</span>
<header class="post-header">
<h1 class="post-title" itemprop="name headline">
GAMES101-16：辐射度量学与路径追踪<a href="https://github.com/KXAND/BlogSource/edit/source/_posts/GAMES101/GAMES101-14.md" class="post-edit-link" title="编辑" rel="noopener" target="_blank"><i class="fa fa-pen-nib"></i></a>
</h1>
<div class="post-meta-container">
<div class="post-meta">
<span class="post-meta-item">
<span class="post-meta-item-icon">
<i class="far fa-calendar"></i>
</span>
<span class="post-meta-item-text">发表于</span>
<time title="创建时间：2023-08-06 18:11:52" itemprop="dateCreated datePublished" datetime="2023-08-06T18:11:52+08:00">2023-08-06</time>
</span>
<span class="post-meta-item">
<span class="post-meta-item-icon">
<i class="far fa-calendar-check"></i>
</span>
<span class="post-meta-item-text">更新于</span>
<time title="修改时间：2024-04-21 02:58:06" itemprop="dateModified" datetime="2024-04-21T02:58:06+08:00">2024-04-21</time>
</span>
<span class="post-meta-item">
<span class="post-meta-item-icon">
<i class="far fa-folder"></i>
</span>
<span class="post-meta-item-text">分类于</span>
<span itemprop="about" itemscope itemtype="http://schema.org/Thing">
<a href="/categories/GAMES101/" itemprop="url" rel="index"><span itemprop="name">GAMES101</span></a>
</span>
</span>
<span class="post-meta-item">
<span class="post-meta-item-icon">
<i class="far fa-comment"></i>
</span>
<span class="post-meta-item-text">Disqus：</span>
<a title="disqus" href="/GAMES10114.html#disqus_thread" itemprop="discussionUrl">
<span class="post-comments-count disqus-comment-count" data-disqus-identifier="/GAMES10114.html" itemprop="commentCount"></span>
</a>
</span>
<span class="post-meta-item" title="本文字数">
<span class="post-meta-item-icon">
<i class="far fa-file-word"></i>
</span>
<span class="post-meta-item-text">本文字数：</span>
<span>1.8k</span>
</span>
</div>
</div>
</header>
<div class="post-body" itemprop="articleBody"><h2 id="前言"><a class="markdownIt-Anchor" href="#前言"></a> 前言</h2>
<p><a href="https://www.bilibili.com/video/BV1X7411F744/?p=16">GAMES101-16</a>:蒙特卡罗积分、路径追踪 <span id="more"></span></p>
<h2 id="蒙特卡罗积分"><a class="markdownIt-Anchor" href="#蒙特卡罗积分"></a> 蒙特卡罗积分</h2>
<p>数学中将随机采样来求解问题的算法统称统称为蒙特卡罗方法，如果应用在积分上就是蒙特卡罗积分。在高数中，我们采取的是求不定积分的原函数在 a,b 的差的方法来积分，但是现在我们不关心原函数又或者说求原函数过于复杂，这种时候蒙特卡罗方法就很有用了。</p>
<p>蒙特卡罗积分的思想如下：对于每次采样 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> ，求得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。以该值估算积分：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mi>b</mi><mi>a</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\int^a_b f(x)=f(x_i)(a-b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.215112em;vertical-align:-0.35582em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8592920000000001em;"><span style="top:-2.34418em;margin-left:-0.19445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span><span style="top:-3.2579000000000002em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35582em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span>，随机取 N 个点进行 N 次估算取平均值，则有：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mo>∫</mo><mi>b</mi><mi>a</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>∑</mo><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\int^a_b f(x)=\frac{1}{N}\sum \frac{f(x_i)}{p(x_i)}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.326242em;vertical-align:-0.9119499999999999em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.414292em;"><span style="top:-1.7880500000000001em;margin-left:-0.44445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span><span style="top:-3.8129000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9119499999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 是取到该点的概率密度。这就是蒙特卡罗积分的公式，这里我们只需要关心采样方法的概率密度函数就可以积分了。</p>
<blockquote>
<p>一个点的概率密度就是在该点附近的面积占整个概率函数的面积。如果均匀取点，就有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">p(x_i) = \frac{1}{a-b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.2484389999999999em;vertical-align:-0.403331em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.403331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>，那么式子就变成熟悉的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>∑</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{N}\sum f(x)p(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 了。</p>
</blockquote>
<h2 id="路径追踪-path-tracing"><a class="markdownIt-Anchor" href="#路径追踪-path-tracing"></a> 路径追踪 Path Tracing</h2>
<p>Whitted-Style 光线追踪对光线如下处理：</p>
<ul>
<li>如果是光滑面的镜面反射/折射，就进行光线追踪；</li>
<li>如果是漫反射，就停止追踪；</li>
</ul>
<p><strong>而这两种处理都是存在问题的</strong>。</p>
<ul>
<li>对于镜面反射，并不是只有能映出影像的镜面，也有抛光（Glossy）面只有高光没有清晰倒影的情况。这也就是说，这种模糊的镜面反射的反射区间比镜面反射略大。<br />
<img src="./../../assets/101-glossyReflection.png" alt="抛光面反射" /></li>
<li>对于漫反射面，其事实上也能产生反射效果。下图左侧展示了不考虑漫反射出的光的后果（天花板与两个立方体侧面），右侧则展现了考略的后果。<br />
<img src="./../../assets/101-diffusePathTrace.png" alt="漫反射路径追踪" />
<blockquote>
<p>上一节的辐射度量学已经指出漫反射将光向四面八方折射的情况</p>
</blockquote>
</li>
</ul>
<p>这也正是通过辐射度量学来构建光线追踪的意义所在。</p>
<h3 id="逐步推出路径追踪算法"><a class="markdownIt-Anchor" href="#逐步推出路径追踪算法"></a> 逐步推出路径追踪算法</h3>
<p>接下来考虑一个简单场景中的情况：场景中有一个面光源、几个遮挡物体与待渲染点。在这个不自发光的点有</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>L</mi><mi>r</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><msub><mi>ω</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo><msup><mi>H</mi><mn>2</mn></msup></msub><msub><mi>f</mi><mi>r</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><msub><mi>ω</mi><mi>i</mi></msub><mo>→</mo><msub><mi>ω</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mi>L</mi><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo></mo><msub><mi>θ</mi><mi>i</mi></msub><mi mathvariant="normal">d</mi><msub><mi>ω</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">L_r(p,\omega_r)=\int_{H^2}f_r(p,\omega_i\rarr \omega_r)L(p,\omega_i)\cos\theta_i\mathrm{d}\omega_i
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.27195em;vertical-align:-0.9119499999999999em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.38952999999999993em;"><span style="top:-1.7880500000000001em;margin-left:-0.44445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463142857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9119499999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">d</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<p>应用蒙特卡罗方法进行积分，均匀在半球面上进行采样：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>L</mi><mi>r</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><msub><mi>ω</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo><msup><mi mathvariant="normal">Ω</mi><mo>+</mo></msup></msub><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mi>ω</mi><mi>i</mi></msub><mo>≈</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><mspace linebreak="newline"></mspace><mi>p</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>π</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">L_r(p,\omega_r)=\int_{\Omega^+}f(x)\mathrm{d}\omega_i
\approx \frac{1}{N}\sum^N_{i=1}\frac{f(x)}{p(\omega_i)}
\\
p(\omega_i)=\frac{1}{2\pi}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.27195em;vertical-align:-0.9119499999999999em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.42008499999999993em;"><span style="top:-1.7880500000000001em;margin-left:-0.44445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">Ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7026642857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9119499999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathrm">d</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.106005em;vertical-align:-1.277669em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>也就是说，在该点向外随机选择 N 个方向进行采样，如果该方向遇见一个光源，则着色结果加上该方向经过蒙特卡洛积分后的结果，就得到该点的<em>直接光渲染</em>结果。</p>
<p>如果发出射线的方向（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ω</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\omega_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>）遇见的是物体，那么我们就将该物体往渲染点方向（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msub><mi>ω</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">-\omega_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>）反射的光视为光源的光进行着色，这样得到的就是<em>全局光照</em>。</p>
<p>这种算法的名字叫<strong>分布式光线追踪</strong>，这种算法并没有错，但是显然，只需要数次折射，追踪的光线的数量就会指数级增长（折射 k 次，则有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>N</mi><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">N^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.849108em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span>）条光路。这使得这种算法并不实用。</p>
<p>人们对这个问题的解决办法非常简单粗暴：如果每个着色点只有一个采样方向（N = 1），那么折射多少次都不会发生指数爆炸。这样的一个经过多次折射形成的线路就是一个<em>路径</em>。至于噪声问题，则用在一个像素上发出多条路径并取平均的方式进行解决。这就是<strong>路径追踪</strong>。</p>
<p>路径递归的终止条件被称为俄罗斯轮盘赌方法（Russian Roulette，RR）。即类似于俄罗斯轮盘赌，我们取定一个概率 P，每次只有 P 的概率返回值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><msub><mi>L</mi><mi>o</mi></msub><mi>P</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{L_o}{P}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.233431em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8884309999999999em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4101em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">o</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> ，而 （1-P） 的概率返回 0。这样，返回值的期望不变，但是光线不能无限次的折射下去。</p>
<p>此时，我们的算法如下：</p>
<p><img src="./../../assets/101-pathtracing.png" alt="路径追踪" /></p>
<p>这是一个<strong>正确的路径追踪算法</strong>了，唯一的问题在于不够高效，低采样率的情况下效果并不好。</p>
<p><img src="./../../assets/101-pathtracingLowSPP.png" alt="路径追踪低采样率效果" /></p>
<p>造成这种问题的原因在于我们采取了所有方向随机采样的采样方式，低采样率时到光源的方向的可能性低。</p>
<h3 id="在光源采样的高效路径追踪"><a class="markdownIt-Anchor" href="#在光源采样的高效路径追踪"></a> 在光源采样的高效路径追踪</h3>
<p>既然我们想采样到打到光源的路径，那么直接在光源上采样不就好了吗？</p>
<p>此时，由于我们是在光源上采样，那么也就需要在光源上进行积分，因此需要将渲染方程改写到光源上。设想场景如下：</p>
<p><img src="./../../assets/101-samplingInLight.png" alt="光源采样" /></p>
<p>在光源上进行平均采样，则概率密度是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mi>A</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>。将立体角换到光源的立体角上有</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">d</mi><mi>ω</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi>A</mi><mi>cos</mi><mo></mo><msup><mi>θ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><mrow><mi mathvariant="normal">∥</mi><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>−</mo><mi>x</mi><msup><mi mathvariant="normal">∥</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\mathrm{d}\omega = \frac{\mathrm{d}A\cos\theta'}{\|x'-x\|^2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathrm">d</span></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.364892em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∥</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6778919999999999em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">x</span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">d</span></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>于是（在不被遮挡的时候）</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>L</mi><mi>r</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><msub><mi>ω</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo><msup><mi>H</mi><mn>2</mn></msup></msub><msub><mi>f</mi><mi>r</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><msub><mi>ω</mi><mi>i</mi></msub><mo>→</mo><msub><mi>ω</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mi>L</mi><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi>A</mi><mi>cos</mi><mo></mo><msup><mi>θ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><mrow><mi mathvariant="normal">∥</mi><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>−</mo><mi>x</mi><msup><mi mathvariant="normal">∥</mi><mn>2</mn></msup></mrow></mfrac><mi mathvariant="normal">d</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">L_r(p,\omega_r)=\int_{H^2}f_r(p,\omega_i\rarr \omega_r)L(p,\omega_i)\frac{\mathrm{d}A\cos\theta'}{\|x'-x\|^2}\mathrm{d}A
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.27195em;vertical-align:-0.9119499999999999em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.38952999999999993em;"><span style="top:-1.7880500000000001em;margin-left:-0.44445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463142857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9119499999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.364892em;vertical-align:-0.936em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∥</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6778919999999999em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">x</span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">d</span></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathrm">d</span></span><span class="mord mathnormal">A</span></span></span></span></span></p>
<p>不过这只是对光源情况的特殊考虑，对于间接光我们还是采用之前的算法并依然采用俄罗斯轮盘赌的算法，所以这时的算法形如 ：</p>
<p><img src="./../../assets/101-samplingInLightAlgorithm.png" alt="光源采样算法" /></p>
<blockquote>
<p>路径追踪是很难处理点光源的情况的，我们可以把它做成很小的面光源</p>
</blockquote>
<p>路径追踪是几乎 100% 正确的一种算法，其效果和照片几乎一致：</p>
<p><img src="../../assets/101-pathtracingIsCorrect.png" alt="路径追踪的效果" /></p>
<h2 id="一些眺望sidenotes"><a class="markdownIt-Anchor" href="#一些眺望sidenotes"></a> 一些眺望：SideNotes</h2>
<ul>
<li>现代光线追踪的定义：不止于 Whitted-Style 光线追踪<br />
以前，光线追踪仅仅指 Whitted-Style，现在，这个词还可以指各类光线传播的算法，例如（单向、双向）路径追踪、VCM/UPBP、光子映射（Photon Mapping）……</li>
<li>如何均匀地半球面上采样：本课不涉及</li>
<li>如何在蒙特卡洛积分中选择最佳的概率密度函数：拓展至“重要性采样理论”</li>
<li>RR 中随机数的生成方式很重要：Low Discrepancy Sequences （<a href="https://en.wikipedia.org/wiki/Low-discrepancy_sequence">低差异序列/拟随机序列</a>， 一种均匀分布随机数的替代序列）</li>
<li>结合着色点半球面与光源上的采样方法：可行，MIS（Multiple Importance Sampling） 理论</li>
<li>考虑对一个像素的多个路径进行加权处理</li>
<li>路径追踪得到的最后 <strong>Radiance 不是着色结果，也和着色结果没有线性对应关系</strong>。
<ul>
<li>色彩空间、伽马校正……</li>
</ul>
</li>
</ul>
<h2 id="跳转"><a class="markdownIt-Anchor" href="#跳转"></a> 跳转</h2>
<p>Home:<a href="GAMES10101.html">GAMES101-1：课程总览与笔记导航</a></p>
<p>Prev:<a href="GAMES10113.html">GAMES101-14&15：辐射度量学</a></p>
<p>Next：<a href="GAMES10115.html">GAMES101-17：基于物理的材质（BRDF）</a></p>
</div>
<footer class="post-footer">
<div class="post-tags">
<a href="/tags/%E7%AC%94%E8%AE%B0/" rel="tag"><i class="fa fa-tag"></i> 笔记</a>
<a href="/tags/%E8%AE%A1%E7%AE%97%E6%9C%BA%E5%9B%BE%E5%BD%A2%E5%AD%A6/" rel="tag"><i class="fa fa-tag"></i> 计算机图形学</a>
<a href="/tags/GAMES/" rel="tag"><i class="fa fa-tag"></i> GAMES</a>
<a href="/tags/GAMES101/" rel="tag"><i class="fa fa-tag"></i> GAMES101</a>
<a href="/tags/%E5%85%89%E7%BA%BF%E8%BF%BD%E8%B8%AA/" rel="tag"><i class="fa fa-tag"></i> 光线追踪</a>
<a href="/tags/%E8%BE%90%E5%B0%84%E5%BA%A6%E9%87%8F%E5%AD%A6/" rel="tag"><i class="fa fa-tag"></i> 辐射度量学</a>
<a href="/tags/%E8%B7%AF%E5%BE%84%E8%BF%BD%E8%B8%AA/" rel="tag"><i class="fa fa-tag"></i> 路径追踪</a>
<a href="/tags/%E8%92%99%E7%89%B9%E5%8D%A1%E6%B4%9B%E7%A7%AF%E5%88%86/" rel="tag"><i class="fa fa-tag"></i> 蒙特卡洛积分</a>
</div>
<div class="post-nav">
<div class="post-nav-item">
<a href="/GAMES10113.html" rel="prev" title="GAMES101-14&15：辐射度量学">
<i class="fa fa-angle-left"></i> GAMES101-14&15：辐射度量学
</a>
</div>
<div class="post-nav-item">
<a href="/GAMES10115.html" rel="next" title="GAMES101-17：基于物理的材质（BRDF）">
GAMES101-17：基于物理的材质（BRDF） <i class="fa fa-angle-right"></i>
</a>
</div>
</div>
</footer>
</article>
</div>
<div class="comments" id="disqus_thread">
<noscript>Please enable JavaScript to view the comments powered by Disqus.</noscript>
</div>
</div>
</main>
<footer class="footer">
<div class="footer-inner">
<div class="copyright">
© 2020 –
<span itemprop="copyrightYear">2025</span>
<span class="with-love">
<i class="fa fa-heart"></i>
</span>
<span class="author" itemprop="copyrightHolder">LeeKa</span>
</div>
<div class="wordcount">
<span class="post-meta-item">
<span class="post-meta-item-icon">
<i class="fa fa-chart-line"></i>
</span>
<span>站点总字数：</span>
<span title="站点总字数">126k</span>
</span>
</div>
<div class="powered-by">由 <a href="https://hexo.io/" rel="noopener" target="_blank">Hexo</a> & <a href="https://theme-next.js.org/pisces/" rel="noopener" target="_blank">NexT.Pisces</a> 强力驱动
</div>
</div>
</footer>
<div class="toggle sidebar-toggle" role="button">
<span class="toggle-line"></span>
<span class="toggle-line"></span>
<span class="toggle-line"></span>
</div>
<div class="sidebar-dimmer"></div>
<div class="back-to-top" role="button" aria-label="返回顶部">
<i class="fa fa-arrow-up fa-lg"></i>
<span>0%</span>
</div>
<noscript>
<div class="noscript-warning">Theme NexT works best with JavaScript enabled</div>
</noscript>
<script src="https://cdnjs.cloudflare.com/ajax/libs/animejs/3.2.1/anime.min.js" integrity="sha256-XL2inqUJaslATFnHdJOi9GfQ60on8Wx1C2H8DYiN1xY=" crossorigin="anonymous"></script>
<script src="/js/comments.js"></script><script src="/js/utils.js"></script><script src="/js/motion.js"></script><script src="/js/sidebar.js"></script><script src="/js/next-boot.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/hexo-generator-searchdb/1.4.1/search.js" integrity="sha256-1kfA5uHPf65M5cphT2dvymhkuyHPQp5A53EGZOnOLmc=" crossorigin="anonymous"></script>
<script src="/js/third-party/search/local-search.js"></script>
<script src="/js/third-party/pace.js"></script>
<script class="next-config" data-name="disqus" type="application/json">{"enable":true,"shortname":"leekapub","count":true,"i18n":{"disqus":"disqus"}}</script>
<script src="/js/third-party/comments/disqus.js"></script>
</body>
</html>