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Lecture15.html
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</style><title>Lecture15</title>
</head>
<body class='typora-export os-windows typora-export-show-outline typora-export-collapse-outline'><div class='typora-export-content'>
<div class="typora-export-sidebar"><div class="outline-content"><li class="outline-item-wrapper outline-h1 outline-item-open"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#games101-lecture-15---ray-tracing-3-light-transport-and-global-illumination-with-review-on-probability">GAMES101 Lecture 15 - Ray Tracing 3 (Light Transport and Global Illumination, with Review on Probability)</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h2 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#i-radiometry-cont">I. Radiometry Cont.</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h2"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#ii-bidirectional-reflectance-distribution-function-brdf">II. Bidirectional Reflectance Distribution Function (BRDF)</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h3 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#reflection-at-a-point">Reflection at a Point</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h3 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#brdf">BRDF</a></div><ul class="outline-children"></ul></li></ul></li><li class="outline-item-wrapper outline-h2"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#iii-the-rendering-equation">III. The Rendering Equation</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h3 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#the-rendering-equation">The Rendering Equation</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h3"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#transforming-the-rendering-equation">Transforming the Rendering Equation</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h4 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#rendering-equation-as-integral-equation">Rendering Equation as Integral Equation</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h4 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#linear-operator-equation">Linear Operator Equation</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h4 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#simplifying-the-linear-operator-equation">Simplifying the Linear Operator Equation</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h4 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#ray-tracing-and-rasterization">Ray Tracing and Rasterization</a></div><ul class="outline-children"></ul></li></ul></li></ul></li><li class="outline-item-wrapper outline-h2 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#appendix-a-review-on-probability">Appendix A: Review on Probability</a></div><ul class="outline-children"></ul></li></ul></li></div></div><div id='write' class=''><h1 id='games101-lecture-15---ray-tracing-3-light-transport-and-global-illumination-with-review-on-probability'><span>GAMES101 Lecture 15 - Ray Tracing 3 (Light Transport and Global Illumination, with Review on Probability)</span></h1><p><a href='https://sites.cs.ucsb.edu/~lingqi/teaching/resources/GAMES101_Lecture_15.pdf'><span>GAMES101_Lecture_15.pdf</span></a></p><h2 id='i-radiometry-cont'><span>I. Radiometry Cont.</span></h2><p><em><span>Please refer to</span></em><span> </span><code>Lecture14.md</code><span>.</span></p><p> </p><h2 id='ii-bidirectional-reflectance-distribution-function-brdf'><span>II. Bidirectional Reflectance Distribution Function (BRDF)</span></h2><h3 id='reflection-at-a-point'><span>Reflection at a Point</span></h3><p><img src="../images/Lecture14-img-14.png" alt="img-14" style="zoom:50%;" /></p><p><span>Radiance from direction </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.147ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 949 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-45-TEX-I-1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 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9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="OP"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="64" xlink:href="#MJX-35-TEX-N-64"></use></g></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D434" xlink:href="#MJX-35-TEX-I-1D434"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>A</mi></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\dd{A}</script><span> receives. Then the power </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.729ex" height="1.538ex" role="img" focusable="false" viewBox="0 -680 764 680" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-51-TEX-I-1D438" d="M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D438" xlink:href="#MJX-51-TEX-I-1D438"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">E</script><span> will be come the radiance to any other direction </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.371ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1047.9 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-37-TEX-I-1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 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data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45C" xlink:href="#MJX-37-TEX-I-1D45C"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mi>o</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_{o}</script><span>.</span></p><ul><li><p><span>Differential radiance incoming:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n13" cid="n13" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="24.723ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 10927.4 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-1-TEX-N-64" d="M376 495Q376 511 376 535T377 568Q377 613 367 624T316 637H298V660Q298 683 300 683L310 684Q320 685 339 686T376 688Q393 689 413 690T443 693T454 694H457V390Q457 84 458 81Q461 61 472 55T517 46H535V0Q533 0 459 -5T380 -11H373V44L365 37Q307 -11 235 -11Q158 -11 96 50T34 215Q34 315 97 378T244 442Q319 442 376 393V495ZM373 342Q328 405 260 405Q211 405 173 369Q146 341 139 305T131 211Q131 155 138 120T173 59Q203 26 251 26Q322 26 373 103V342Z"></path><path id="MJX-1-TEX-I-1D438" d="M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 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187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z"></path><path id="MJX-1-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-1-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-1-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-1-TEX-I-1D43F" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path><path id="MJX-1-TEX-N-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z"></path><path id="MJX-1-TEX-N-6F" d="M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z"></path><path id="MJX-1-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-1-TEX-N-2061" d=""></path><path id="MJX-1-TEX-I-1D703" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="OP"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="64" xlink:href="#MJX-1-TEX-N-64"></use></g></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D438" xlink:href="#MJX-1-TEX-I-1D438"></use></g><g data-mml-node="mo" transform="translate(1320,0)"><use data-c="28" xlink:href="#MJX-1-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(1709,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-1-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(2658,0)"><use data-c="29" xlink:href="#MJX-1-TEX-N-29"></use></g></g><g data-mml-node="mo" transform="translate(3324.7,0)"><use data-c="3D" xlink:href="#MJX-1-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(4380.5,0)"><use data-c="1D43F" xlink:href="#MJX-1-TEX-I-1D43F"></use></g><g data-mml-node="mo" transform="translate(5061.5,0)"><use data-c="28" xlink:href="#MJX-1-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(5450.5,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-1-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(6399.5,0)"><use data-c="29" xlink:href="#MJX-1-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(6955.1,0)"><use data-c="63" xlink:href="#MJX-1-TEX-N-63"></use><use data-c="6F" xlink:href="#MJX-1-TEX-N-6F" transform="translate(444,0)"></use><use data-c="73" xlink:href="#MJX-1-TEX-N-73" transform="translate(944,0)"></use></g><g data-mml-node="mo" transform="translate(8293.1,0)"><use data-c="2061" xlink:href="#MJX-1-TEX-N-2061"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(8459.8,0)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D703" xlink:href="#MJX-1-TEX-I-1D703"></use></g><g data-mml-node="mi" transform="translate(502,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1-TEX-I-1D456"></use></g></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="OP" transform="translate(9422.4,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="64" xlink:href="#MJX-1-TEX-N-64"></use></g></g><g data-mml-node="msub" transform="translate(556,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-1-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1-TEX-I-1D456"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo>=</mo><mi>L</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><msub><mi>θ</mi><mi>i</mi></msub></mrow><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><msub><mi>ω</mi><mi>i</mi></msub></mrow></math></mjx-assistive-mml></mjx-container></div></div></li><li><p><span>Differential radiance exiting due to </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.894ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3047 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-38-TEX-N-64" d="M376 495Q376 511 376 535T377 568Q377 613 367 624T316 637H298V660Q298 683 300 683L310 684Q320 685 339 686T376 688Q393 689 413 690T443 693T454 694H457V390Q457 84 458 81Q461 61 472 55T517 46H535V0Q533 0 459 -5T380 -11H373V44L365 37Q307 -11 235 -11Q158 -11 96 50T34 215Q34 315 97 378T244 442Q319 442 376 393V495ZM373 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488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-38-TEX-I-1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z"></path><path id="MJX-38-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-38-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="OP"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="64" xlink:href="#MJX-38-TEX-N-64"></use></g></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D438" xlink:href="#MJX-38-TEX-I-1D438"></use></g><g data-mml-node="mo" transform="translate(1320,0)"><use data-c="28" xlink:href="#MJX-38-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(1709,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-38-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-38-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(2658,0)"><use data-c="29" xlink:href="#MJX-38-TEX-N-29"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\dd{E(\omega_i)}</script><span>:</span></p><div contenteditable="false" 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334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-41-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-41-TEX-I-1D45F"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_r</script><span> from each incoming direction </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" 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transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-4-TEX-N-28"></use><use data-c="31" xlink:href="#MJX-4-TEX-N-31" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-4-TEX-N-29" transform="translate(889,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(1)</mtext></mtd><mtd><msub><mi>L</mi><mi>r</mi></msub><mo stretchy="false">(</mo><mtext>p</mtext><mo>,</mo><msub><mi>ω</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msup><mi>H</mi><mn>2</mn></msup></mrow></msub><msub><mi>f</mi><mi>r</mi></msub><mo stretchy="false">(</mo><mtext>p</mtext><mo>,</mo><msub><mi>ω</mi><mi>i</mi></msub><mo accent="false" stretchy="false">→</mo><msub><mi>ω</mi><mi>r</mi></msub><mo stretchy="false">)</mo><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mtext>p</mtext><mo>,</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><msub><mi>θ</mi><mi>i</mi></msub></mrow><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><msub><mi>ω</mi><mi>i</mi></msub></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>In this equation, what we do is essentially </span><strong><span>summing up the contributions</span></strong><span> to this particular outgoing direction </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.317ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1023.9 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path 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287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-41-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-41-TEX-I-1D45F"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_r</script><span> from all other directions, by doing integration on the entire hemisphere. The differential part on the right side, is acquired by multiplying BRDF with the differential of the irradiance.</span></p><p> </p><p><strong><span>Challenge: Recursive Equation</span></strong></p><ul><li><p><span>Incoming radiance depends on reflected radiance, at another point in the scene</span></p></li></ul><p> </p><h2 id='iii-the-rendering-equation'><span>III. The Rendering Equation</span></h2><h3 id='the-rendering-equation'><span>The Rendering Equation</span></h3><p><span>Adding </span><strong><span>an emission term</span></strong><span> on equation </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-42-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><a href="#mjx-eqn%3Arefleq"><g data-mml-node="mrow" class=""><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="500" height="666" y="0"></rect><g data-mml-node="mtext"><use data-c="31" xlink:href="#MJX-42-TEX-N-31"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow href="#mjx-eqn%3Arefleq" class="MathJax_ref"><mtext>1</mtext></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\ref{refleq}</script><span> gives the </span><strong><span>rendering equation</span></strong><span>:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n34" cid="n34" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="["rendeq"]"><div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 67.731ex; position: relative;"><svg 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transform="translate(556,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-57-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-57-TEX-I-1D456"></use></g></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1452.2 1 2404.4"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:rendeq" transform="translate(0,841.2)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-57-TEX-N-28"></use><use data-c="32" xlink:href="#MJX-57-TEX-N-32" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-57-TEX-N-29" transform="translate(889,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(2)</mtext></mtd><mtd><msub><mi>L</mi><mi>o</mi></msub><mo stretchy="false">(</mo><mtext>p</mtext><mo>,</mo><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>L</mi><mi>e</mi></msub><mo stretchy="false">(</mo><mtext>p</mtext><mo>,</mo><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi mathvariant="normal">Ω</mi><mrow data-mjx-texclass="ORD"><mo>+</mo></mrow></msub></mrow></msub><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mtext>p</mtext><mo>,</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mi>f</mi><mi>r</mi></msub><mo stretchy="false">(</mo><mtext>p</mtext><mo>,</mo><msub><mi>ω</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mtext mathvariant="bold">n</mtext><mo>⋅</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><msub><mi>ω</mi><mi>i</mi></msub></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>where </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.138ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 503 636" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-43-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 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xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-44-TEX-N-3A9" d="M55 454Q55 503 75 546T127 617T197 665T272 695T337 704H352Q396 704 404 703Q527 687 596 615T666 454Q666 392 635 330T559 200T499 83V80H543Q589 81 600 83T617 93Q622 102 629 135T636 172L637 177H677V175L660 89Q645 3 644 2V0H552H488Q461 0 456 3T451 20Q451 89 499 235T548 455Q548 512 530 555T483 622T424 656T361 668Q332 668 303 658T243 626T193 560T174 456Q174 380 222 233T270 20Q270 7 263 0H77V2Q76 3 61 89L44 175V177H84L85 172Q85 171 88 155T96 119T104 93Q109 86 120 84T178 80H222V83Q206 132 162 199T87 329T55 454Z"></path><path id="MJX-44-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="3A9" xlink:href="#MJX-44-TEX-N-3A9"></use></g><g data-mml-node="TeXAtom" transform="translate(755,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2B" xlink:href="#MJX-44-TEX-N-2B"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ω</mi><mrow data-mjx-texclass="ORD"><mo>+</mo></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Omega_{+}</script><span> denotes the upper hemisphere.</span></p><ul><li><p><span>All directions are pointing </span><strong><span>outwards</span></strong><span>, e.g. </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.147ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 949 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-45-TEX-I-1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z"></path><path id="MJX-45-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-45-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-45-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_i</script><span>.</span></p></li><li><p><span>Why can we directly do an integration? </span><em><span>From the linearity.</span></em></p></li></ul><p> </p><h3 id='transforming-the-rendering-equation'><span>Transforming the Rendering Equation</span></h3><h4 id='rendering-equation-as-integral-equation'><span>Rendering Equation as Integral Equation</span></h4><p><span>Equation </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-60-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><a href="#mjx-eqn%3Arendeq"><g data-mml-node="mrow" class=""><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="500" height="666" y="0"></rect><g data-mml-node="mtext"><use data-c="32" xlink:href="#MJX-60-TEX-N-32"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow href="#mjx-eqn%3Arendeq" class="MathJax_ref"><mtext>2</mtext></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\ref{rendeq}</script><span> is a </span><strong><span>Fredhold Integral Equation</span></strong><span> of second kind (extensively studied numerically) with </span><strong><span>canonical form</span></strong><span>:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n45" cid="n45" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" 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166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.019382,-0.019382) translate(0, -1816.8)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 1816.8) matrix(1 0 0 -1 0 0) scale(51.6)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="8423.3 -1816.8 1 3133.7"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr" transform="translate(0,455.8)"><g data-mml-node="mtd"><g data-mml-node="mi"><use data-c="1D459" xlink:href="#MJX-6-TEX-I-1D459"></use></g><g data-mml-node="mo" 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transform="translate(6020,0) translate(0 1)"><use data-c="222B" xlink:href="#MJX-6-TEX-LO-222B"></use></g><g data-mml-node="mi" transform="translate(7130.7,0)"><use data-c="1D459" xlink:href="#MJX-6-TEX-I-1D459"></use></g><g data-mml-node="mo" transform="translate(7428.7,0)"><use data-c="28" xlink:href="#MJX-6-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(7817.7,0)"><use data-c="1D463" xlink:href="#MJX-6-TEX-I-1D463"></use></g><g data-mml-node="mo" transform="translate(8302.7,0)"><use data-c="29" xlink:href="#MJX-6-TEX-N-29"></use></g><g data-mml-node="munder" transform="translate(8691.7,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="OP" transform="translate(1889.3,0)"><g data-mml-node="munder"><g data-mml-node="mrow"><g data-mml-node="mi"><use data-c="1D43E" xlink:href="#MJX-6-TEX-I-1D43E"></use></g><g data-mml-node="mo" transform="translate(889,0)"><use data-c="28" xlink:href="#MJX-6-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1278,0)"><use 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xlink:href="#MJX-6-TEX-N-20" transform="translate(3948,0)"></use><use data-c="74" xlink:href="#MJX-6-TEX-N-74" transform="translate(4198,0)"></use><use data-c="68" xlink:href="#MJX-6-TEX-N-68" transform="translate(4587,0)"></use><use data-c="65" xlink:href="#MJX-6-TEX-N-65" transform="translate(5143,0)"></use><use data-c="20" xlink:href="#MJX-6-TEX-N-20" transform="translate(5587,0)"></use><use data-c="45" xlink:href="#MJX-6-TEX-N-45" transform="translate(5837,0)"></use><use data-c="71" xlink:href="#MJX-6-TEX-N-71" transform="translate(6518,0)"></use><use data-c="75" xlink:href="#MJX-6-TEX-N-75" transform="translate(7046,0)"></use><use data-c="61" xlink:href="#MJX-6-TEX-N-61" transform="translate(7602,0)"></use><use data-c="74" xlink:href="#MJX-6-TEX-N-74" transform="translate(8102,0)"></use><use data-c="69" xlink:href="#MJX-6-TEX-N-69" transform="translate(8491,0)"></use><use data-c="6F" xlink:href="#MJX-6-TEX-N-6F" transform="translate(8769,0)"></use><use data-c="6E" xlink:href="#MJX-6-TEX-N-6E" transform="translate(9269,0)"></use></g></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="OP" transform="translate(15805.7,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="64" xlink:href="#MJX-6-TEX-N-64"></use></g></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D463" xlink:href="#MJX-6-TEX-I-1D463"></use></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1816.8 1 3133.7"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:fredhold" transform="translate(0,1205.8)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-6-TEX-N-28"></use><use data-c="33" xlink:href="#MJX-6-TEX-N-33" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-6-TEX-N-29" transform="translate(889,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3)</mtext></mtd><mtd><mi>l</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><mi>e</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>+</mo><mo data-mjx-texclass="OP">∫</mo><mi>l</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><munder><mrow data-mjx-texclass="OP"><munder><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder></mrow><mrow data-mjx-texclass="ORD"><mtext>Kernel of the Equation</mtext></mrow></munder><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>v</mi></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><h4 id='linear-operator-equation'><span>Linear Operator Equation</span></h4><p><span>Equation </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.554ex" role="img" focusable="false" viewBox="0 -665 500 687" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-47-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><a href="#mjx-eqn%3Afredhold"><g data-mml-node="mrow" class=""><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="500" height="687" y="-22"></rect><g data-mml-node="mtext"><use data-c="33" xlink:href="#MJX-47-TEX-N-33"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow href="#mjx-eqn%3Afredhold" class="MathJax_ref"><mtext>3</mtext></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\ref{fredhold}</script><span> can be further transformed by applying the </span><strong><span>Light Transport Operator</span></strong><span>.</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n48" cid="n48" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="12.604ex" height="1.731ex" role="img" focusable="false" viewBox="0 -683 5571 765" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-7-TEX-I-1D43F" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path><path id="MJX-7-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-7-TEX-I-1D438" d="M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z"></path><path id="MJX-7-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-7-TEX-I-1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D43F" xlink:href="#MJX-7-TEX-I-1D43F"></use></g><g data-mml-node="mo" transform="translate(958.8,0)"><use data-c="3D" xlink:href="#MJX-7-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2014.6,0)"><use data-c="1D438" xlink:href="#MJX-7-TEX-I-1D438"></use></g><g data-mml-node="mo" transform="translate(3000.8,0)"><use data-c="2B" xlink:href="#MJX-7-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(4001,0)"><use data-c="1D43E" xlink:href="#MJX-7-TEX-I-1D43E"></use></g><g data-mml-node="mi" transform="translate(4890,0)"><use data-c="1D43F" xlink:href="#MJX-7-TEX-I-1D43F"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>L</mi><mo>=</mo><mi>E</mi><mo>+</mo><mi>K</mi><mi>L</mi></math></mjx-assistive-mml></mjx-container></div></div><p><span>which can be then </span><strong><span>discretized</span></strong><span> to a simple matrix equation, where:</span></p><ul><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.541ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 681 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-48-TEX-I-1D43F" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D43F" xlink:href="#MJX-48-TEX-I-1D43F"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">L</script><span>, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.729ex" height="1.538ex" role="img" focusable="false" viewBox="0 -680 764 680" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-51-TEX-I-1D438" d="M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D438" xlink:href="#MJX-51-TEX-I-1D438"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">E</script><span> are vectors, and </span></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.011ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 889 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-50-TEX-I-1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use 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WTF? </span><strong><span>TODO</span></strong><span>: Needs further explanation.</span></em></p><p> </p><h4 id='simplifying-the-linear-operator-equation'><span>Simplifying the Linear Operator Equation</span></h4><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n58" cid="n58" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="73.339ex" height="17.877ex" role="img" focusable="false" viewBox="0 -4200.9 32415.9 7901.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -8.373ex;"><defs><path id="MJX-8-TEX-I-1D43F" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 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stretchy="false">)</mo><mi>L</mi></mtd><mtd><mi></mi><mo>=</mo><mi>E</mi></mtd></mtr><mtr><mtd><mi>L</mi></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">(</mo><mi>I</mi><mo>−</mo><mi>K</mi><msup><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mi>E</mi></mtd></mtr><mtr><mtd><mi>L</mi></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mi>K</mi><mo>+</mo><msup><mi>K</mi><mn>2</mn></msup><mo>+</mo><msup><mi>K</mi><mn>3</mn></msup><mo>+</mo><mo>⋯</mo><mo stretchy="false">)</mo><mi>E</mi><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mtext>(Applying the binomial theorem)</mtext></mtd></mtr><mtr><mtd><mi>L</mi></mtd><mtd><mi></mi><mo>=</mo><mi>E</mi><mo>+</mo><mi>K</mi><mi>E</mi><mo>+</mo><msup><mi>K</mi><mn>2</mn></msup><mi>E</mi><mo>+</mo><msup><mi>K</mi><mn>3</mn></msup><mi>E</mi><mo>+</mo><mo>⋯</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>Those expanded terms has physical meanings:</span></p><ul><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.729ex" height="1.538ex" role="img" focusable="false" viewBox="0 -680 764 680" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-51-TEX-I-1D438" d="M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D438" xlink:href="#MJX-51-TEX-I-1D438"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">E</script><span> - Emission directly from light sources</span></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.74ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 1653 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-52-TEX-I-1D43E" d="M285 628Q285 635 228 637Q205 637 198 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id="MJX-52-TEX-I-1D438" d="M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D43E" xlink:href="#MJX-52-TEX-I-1D43E"></use></g><g data-mml-node="mi" transform="translate(889,0)"><use data-c="1D438" xlink:href="#MJX-52-TEX-I-1D438"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi><mi>E</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">KE</script><span> - Direct illumination on surfaces</span></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.845ex" height="1.887ex" role="img" focusable="false" viewBox="0 -833.9 2141.6 833.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-53-TEX-I-1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 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565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D43E" xlink:href="#MJX-53-TEX-I-1D43E"></use></g><g data-mml-node="mn" transform="translate(974,363) scale(0.707)"><use data-c="32" xlink:href="#MJX-53-TEX-N-32"></use></g></g><g data-mml-node="mi" transform="translate(1377.6,0)"><use data-c="1D438" xlink:href="#MJX-53-TEX-I-1D438"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>K</mi><mn>2</mn></msup><mi>E</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">K^2E</script><span> - Indirect illumination, </span><strong><span>one</span></strong><span> bounce</span></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.845ex" height="1.885ex" role="img" focusable="false" viewBox="0 -833.2 2141.6 833.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-54-TEX-I-1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 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data-mml-node="mn" transform="translate(974,363) scale(0.707)"><use data-c="33" xlink:href="#MJX-54-TEX-N-33"></use></g></g><g data-mml-node="mi" transform="translate(1377.6,0)"><use data-c="1D438" xlink:href="#MJX-54-TEX-I-1D438"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>K</mi><mn>3</mn></msup><mi>E</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">K^3E</script><span> - </span><strong><span>Two</span></strong><span> bounces</span></p></li><li><p><span>...</span></p></li></ul><p> </p><h4 id='ray-tracing-and-rasterization'><span>Ray Tracing and Rasterization</span></h4><p><strong><span>Shading in rasterization</span></strong><span> is equivalent to solving the rendering equation using </span><strong><span>zero</span></strong><span> bounces.</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" 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data-mml-node="mi" transform="translate(4890,0)"><use data-c="1D438" xlink:href="#MJX-9-TEX-I-1D438"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>L</mi><mo>=</mo><mi>E</mi><mo>+</mo><mi>K</mi><mi>E</mi></math></mjx-assistive-mml></mjx-container></div></div><figure><table><thead><tr><th><span>Type</span></th><th><span>Effect</span></th></tr></thead><tbody><tr><td><span>Direct Illumination</span></td><td><img src="../images/Lecture15-img-1.png" referrerpolicy="no-referrer" alt="img-1"></td></tr><tr><td><span>One-bounce global illumination</span></td><td><img src="../images/Lecture15-img-2.png" referrerpolicy="no-referrer" alt="img-2"></td></tr><tr><td><span>Two-bounce global illumination</span></td><td><img src="../images/Lecture15-img-3.png" referrerpolicy="no-referrer" alt="img-3"></td></tr><tr><td><span>Four-bounce global illumination</span></td><td><img src="../images/Lecture15-img-4.png" referrerpolicy="no-referrer" alt="img-4"></td></tr><tr><td><span>Eight-bounce global illumination</span></td><td><img src="../images/Lecture15-img-5.png" referrerpolicy="no-referrer" alt="img-5"></td></tr><tr><td><span>Sixteen-bounce global illumination</span></td><td><img src="../images/Lecture15-img-6.png" referrerpolicy="no-referrer" alt="img-6"></td></tr></tbody></table></figure><ul><li><p><span>Why the lantern is light/dark in some rendered results?</span></p><ul><li><p><span>Light cannot escape the lantern only after several bounces (so that it reaches an illuminated surface)</span></p></li></ul></li><li><p><span>Is the result guaranteed to converge if an infinity number of passes can be calculated?</span></p></li></ul><p> </p><h2 id='appendix-a-review-on-probability'><span>Appendix A: Review on Probability</span></h2><ul><li><p><strong><span>Random Variable</span></strong></p></li><li><p><strong><span>Probability Density/Distribution Function, PDF</span></strong></p></li><li><p><span>Expected Value</span></p></li><li><p><strong><span>Function of a Random Variable</span></strong></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n116" cid="n116" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="31.965ex" height="5.027ex" role="img" focusable="false" viewBox="0 -1361 14128.4 2222" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -1.948ex;"><defs><path id="MJX-10-TEX-I-1D438" d="M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 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