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Lecture03.html
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Lecture03.html
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</style><title>Lecture03</title>
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<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"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#games202-lecture-03---real-time-shadows-1">GAMES202 Lecture 03 - Real-Time Shadows 1</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h2"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#i-shadow-mapping">I. Shadow Mapping</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="#recap">Recap</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="#cautions">Cautions</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="#common-issues-and-solutions">Common Issues and Solutions</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h4"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#self-occlusion">Self Occlusion</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h5 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#adding-a-variable-bias-to-reduce-self-occlusion">Adding a (variable) bias to reduce self occlusion</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h5 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#second-depth-shadow-mapping">Second-Depth Shadow Mapping</a></div><ul class="outline-children"></ul></li></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="#aliasing">Aliasing</a></div><ul class="outline-children"></ul></li></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="#ii-the-math-behind-shadow-mapping">II. The Math behind Shadow Mapping</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="#approximating-the-rendering-equation">Approximating the Rendering Equation</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-percentage-closer-filtering-pcf-and-percentage-closer-soft-shadows-pcss">III. Percentage Closer Filtering (PCF) and Percentage Closer Soft Shadows (PCSS)</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h3"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#percentage-closer-filtering">Percentage Closer Filtering</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="#the-algorithm">The Algorithm</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="#problems-1">Problems</a></div><ul class="outline-children"></ul></li></ul></li><li class="outline-item-wrapper outline-h3"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#percentage-closer-soft-shadows">Percentage Closer Soft Shadows</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="#pcss---the-complete-algorithm">PCSS - The Complete Algorithm</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="#problems-2">Problems</a></div><ul class="outline-children"></ul></li></ul></li></ul></li></ul></li></div></div><div id='write' class=''><h1 id='games202-lecture-03---real-time-shadows-1'><span>GAMES202 Lecture 03 - Real-Time Shadows 1</span></h1><p><a href='https://sites.cs.ucsb.edu/~lingqi/teaching/resources/GAMES202_Lecture_03.pdf'><span>GAMES202_Lecture_03 (ucsb.edu)</span></a></p><p><img src="../images/Lecture03-img-16.png" alt="image-20230728211702865" style="zoom:33%;" /></p><h2 id='i-shadow-mapping'><span>I. Shadow Mapping</span></h2><p><span>Reference: Real-Time Rendering, 4th Edition</span></p><p><span>The idea of shadow mapping is to render the scene, using the </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.052ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 465 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-384-TEX-I-1D467" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></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="1D467" xlink:href="#MJX-384-TEX-I-1D467"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">z</script><span>-buffer, from the position of the light source that is to cast shadows. Whatever the light </span><em><span>sees</span></em><span> is illuminated, and the rest is in shadow.</span></p><p><span>To use the shadow mapping, the scene is rendered a second time (Pass 2), but this time with respect to the viewer. As each drawing primitive is rendered, its location at each pixel is compared to the shadow mapping. If a rendered point is farther away from the light source than the corresponding value in the shadow mapping, that point is in shadow. Otherwise, it is not.</span></p><h3 id='recap'><span>Recap</span></h3><p><span>Shadow Mapping is:</span></p><ul><li><p><span>A </span><strong><span>2-Pass</span></strong><span> Algorithm</span></p><ul><li><p><span>The light pass generates the shadow mapping, or SM</span></p></li><li><p><span>The camera pass uses the SM</span></p><ul><li><p><span>Check if each fragment can be illuminated by the light source (using the shadow map)</span></p></li></ul></li></ul></li><li><p><span>An </span><strong><span>Image-Space</span></strong><span> Algorithm</span></p><ul><li><p><span>Pro: No required knowledge of scene's geometry</span></p></li><li><p><span>Con: Causing self-occlusion and aliasing issues</span></p></li></ul></li><li><p><span>A well-known shadow rendering technique</span></p><ul><li><p><span>Basic shadowing technique even for early offline renderings, e.g., Toy Story</span></p></li></ul></li></ul><p> </p><figure><table><thead><tr><th><span>Pass 1</span></th><th><span>Pass 2A</span></th><th><span>Pass 2B</span></th></tr></thead><tbody><tr><td><img src="../images/Lecture03-img-1.png" alt="image-20230728161919985" /></td><td><img src="../images/Lecture03-img-2.png" referrerpolicy="no-referrer" alt="image-20230728161944563"></td><td><img src="../images/Lecture03-img-3.png" referrerpolicy="no-referrer" alt="image-20230728161959964"></td></tr></tbody></table></figure><ul><li><p><span>Pass 1</span></p><ul><li><p><span>Output a "depth texture" from the light source</span></p><ul><li><p><span>For each fragment on that depth texture, record the minimum depth seen from the light source</span></p></li></ul></li></ul></li><li><p><span>Pass 2</span></p><ul><li><p><span>Pass </span><strong><span>2A</span></strong><span>: Render a standard image from the eye</span></p></li><li><p><span>Pass </span><strong><span>2B</span></strong><span>: Project visible points in eye view back to light source. Test the depth.</span></p><ul><li><p><span>If the depth does not match the previous one, then the point being rendered is </span><em><span>blocked</span></em><span> from the light source.</span></p></li></ul></li></ul></li></ul><figure><table><thead><tr><th><span>Depth View</span></th><th><span>Eye's View (Depth)</span></th></tr></thead><tbody><tr><td><img src="../images/Lecture03-img-5.png" alt="image-20230728163359092" style="zoom: 50%;" /></td><td><img src="../images/Lecture03-img-6.png" alt="image-20230728163419528" style="zoom: 50%;" /></td></tr></tbody></table></figure><ul><li><p><span>Depth View - Viewed from light source</span></p><ul><li><p><span>In this figure: the darker the fragment is, the closer the corresponding point on that object is to the light source</span></p></li></ul></li><li><p><span>Eye's View - Viewed from camera, projecting the depth map onto the eye's view.</span></p></li></ul><p> </p><p><img src="../images/Lecture03-img-4.png" alt="image-20230728162145321" style="zoom:50%;" /></p><ul><li><p><em><span>Notice how specular highlights never appear in shadows</span></em></p></li><li><p><em><span>Notice how curved surfaces cast shadows on each other</span></em></p></li></ul><h3 id='cautions'><span>Cautions</span></h3><p><span>The definitions of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.052ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 465 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-384-TEX-I-1D467" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></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="1D467" xlink:href="#MJX-384-TEX-I-1D467"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">z</script><span> value used in shadow mapping and in depth-test must be consistent.</span></p><ul><li><p><span>If you are to use the actual </span><em><span>distance</span></em><span>, then use that distance at both 2 passes.</span></p></li></ul><p> </p><h3 id='common-issues-and-solutions'><span>Common Issues and Solutions</span></h3><h4 id='self-occlusion'><span>Self Occlusion</span></h4><p><img src="../images/Lecture03-img-7.png" alt="image-20230728164647761" style="zoom:50%;" /></p><ul><li><p><strong><span>Resolution and floating-point precision count</span></strong><span>. Notice the red slashes on the right figure.</span></p><ul><li><p><span>Each pixel covers an area of the surface, in which </span><strong><span>the depth is constant</span></strong><span> (because it is within a single fragment).</span></p><ul><li><p><span>The actual depth differs from that recorded in the shadow mapping</span></p></li></ul></li><li><p><span>The effect is most severe at </span><strong><span>grazing angle</span></strong><span>. </span></p></li><li><p><span>The effect is mitigated when the plane used for shadow mapping is parallel to the surface being illuminated.</span></p></li></ul></li><li><p><span>The bias is too low, so self-shadowing occurs. Notice the weird pattern on the ground - they are produced because </span><em><span>triangles (that are used to model the ground) shadow themselves</span></em><span>.</span></p></li></ul><p> </p><h5 id='adding-a-variable-bias-to-reduce-self-occlusion'><span>Adding a (variable) bias to reduce self occlusion</span></h5><p><span>Now the shadow occurs </span><strong><span>only if</span></strong><span> the difference exceeds certain threshold:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n642" cid="n642" 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="16.028ex" height="1.932ex" role="img" focusable="false" viewBox="0 -716 7084.6 854" 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If perpendicular to the surface, then little bias. Else, high bias.</span></p></li></ul><p><img src="../images/Lecture03-img-8.png" alt="image-20230728164740245" style="zoom:50%;" /></p><ul><li><p><strong><span>Bias is too high</span></strong><span>: Introducing detached shadow.</span></p><ul><li><p><span>Small object causes the difference between depths to be lied within the bias interval.</span></p><ul><li><p><span>Notice the detached shadow on Laura's shoes.</span></p></li></ul></li></ul></li></ul><p> </p><h5 id='second-depth-shadow-mapping'><span>Second-Depth Shadow Mapping</span></h5><p><span>Store the second-minimum depth when generating the shadow map.</span></p><ul><li><p><span>Using the midpoint between first and second depths in the SM.</span></p></li></ul><p><img src="../images/Lecture03-img-9.png" alt="image-20230728165033670" style="zoom:50%;" /></p><p><em><span>Assume the light source is directly above.</span></em></p><ul><li><p><span>Requires objects to be </span><strong><span>watertight</span></strong><span>:</span></p><ul><li><p><span>Cannot describe planes without volume. (Can be solved with some tricks)</span></p></li></ul></li><li><p><span>The </span><strong><span>overhead</span></strong><span> may not worth it:</span></p><ul><li><p><span>RTR does not trust in </span><strong><span>complexity</span></strong><span>: The actual time may be doubled or tripled or even more.</span></p></li></ul></li></ul><p> </p><h4 id='aliasing'><span>Aliasing</span></h4><p><img src="../images/Lecture03-img-10.png" alt="image-20230728165118670" style="zoom: 67%;" /></p><ul><li><p><strong><span>The resolution still counts</span></strong><span>: The area covered by a single fragment inside the shadow maps has a </span><strong><span>constant depth</span></strong><span>. Solution: change the resolution of shadow maps:</span></p><ul><li><p><span>Shadow Maps with Dynamic Resolution</span></p></li><li><p><span>Cascaded Shadow Maps</span></p></li><li><p><span>Convolutional Shadow Maps</span></p></li></ul></li></ul><p> </p><h2 id='ii-the-math-behind-shadow-mapping'><span>II. 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transform="translate(19003.9,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="64" xlink:href="#MJX-500-TEX-N-64"></use></g></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D465" xlink:href="#MJX-500-TEX-I-1D465"></use></g></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><msub><mo data-mjx-texclass="OP">∫</mo><mi mathvariant="normal">Ω</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>x</mi></mrow><mo>≈</mo><munder><mrow data-mjx-texclass="OP"><munder><mfrac><mrow><msub><mo data-mjx-texclass="OP">∫</mo><mi mathvariant="normal">Ω</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>x</mi></mrow></mrow><mrow><msub><mo data-mjx-texclass="OP">∫</mo><mi mathvariant="normal">Ω</mi></msub><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>x</mi></mrow></mrow></mfrac><mo>⏟</mo></munder></mrow><mrow data-mjx-texclass="ORD"><mtable data-mjx-smallmatrix="true" columnspacing="0em" rowspacing="0.1em"><mtr><mtd><mtext>The average value</mtext></mtd></mtr><mtr><mtd><mrow><mtext>of </mtext><mrow data-mjx-texclass="ORD"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mtext> over the interval</mtext></mrow></mtd></mtr></mtable></mrow></munder><mo>⋅</mo><msub><mo data-mjx-texclass="OP">∫</mo><mi mathvariant="normal">Ω</mi></msub><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>x</mi></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><ul><li><p><span>Approximate the result of integration by averaging </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.299ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1900 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-389-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-389-TEX-N-28" d="M94 250Q94 319 104 381T127 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-389-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-389-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="mi"><use data-c="1D453" xlink:href="#MJX-389-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(550,0)"><use data-c="28" xlink:href="#MJX-389-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(939,0)"><use data-c="1D465" xlink:href="#MJX-389-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1511,0)"><use data-c="29" xlink:href="#MJX-389-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">f(x)</script><span> over the interval of integration</span></p></li><li><p><span>When is it (more) accurate?</span></p><ul><li><p><span>When the </span><strong><span>support</span></strong><span> of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.133ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1827 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-390-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path id="MJX-390-TEX-N-28" d="M94 250Q94 319 104 381T127 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-390-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-390-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="mi"><use data-c="1D454" xlink:href="#MJX-390-TEX-I-1D454"></use></g><g data-mml-node="mo" transform="translate(477,0)"><use data-c="28" xlink:href="#MJX-390-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(866,0)"><use data-c="1D465" xlink:href="#MJX-390-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1438,0)"><use data-c="29" xlink:href="#MJX-390-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">g(x)</script><span> is small:</span></p><ul><li><p><span>The support of a real-valued function </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.244ex" height="2.059ex" role="img" focusable="false" viewBox="0 -705 550 910" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-387-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></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="1D453" xlink:href="#MJX-387-TEX-I-1D453"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">f</script><span> is the subset of the function domain containing the elements which are not mapped to zero.</span></p></li></ul></li><li><p><span>When </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.133ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1827 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-390-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path id="MJX-390-TEX-N-28" d="M94 250Q94 319 104 381T127 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-390-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-390-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="mi"><use data-c="1D454" xlink:href="#MJX-390-TEX-I-1D454"></use></g><g data-mml-node="mo" transform="translate(477,0)"><use data-c="28" xlink:href="#MJX-390-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(866,0)"><use data-c="1D465" xlink:href="#MJX-390-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1438,0)"><use data-c="29" xlink:href="#MJX-390-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">g(x)</script><span> is </span><strong><span>smooth</span></strong><span>.</span></p></li><li><p><span>When </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.299ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1900 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-389-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-389-TEX-N-28" d="M94 250Q94 319 104 381T127 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-389-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-389-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="mi"><use data-c="1D453" xlink:href="#MJX-389-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(550,0)"><use data-c="28" xlink:href="#MJX-389-TEX-N-28"></use></g><g 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xlink:href="#MJX-502-TEX-N-6C" transform="translate(19763,0)"></use><use data-c="69" xlink:href="#MJX-502-TEX-N-69" transform="translate(20041,0)"></use><use data-c="74" xlink:href="#MJX-502-TEX-N-74" transform="translate(20319,0)"></use><use data-c="79" xlink:href="#MJX-502-TEX-N-79" transform="translate(20708,0)"></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"><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><mfrac><mrow><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">Ω</mi><mo>+</mo></mrow></msub><mi>V</mi><mo stretchy="false">(</mo><mtext>p</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></mrow><mrow><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">Ω</mi><mo>+</mo></mrow></msub><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></mrow></mfrac><mo>⋅</mo><munder><mrow data-mjx-texclass="OP"><munder><mrow><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">Ω</mi><mo>+</mo></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><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>i</mi></msub><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></mrow><mo>⏟</mo></munder></mrow><mrow data-mjx-texclass="ORD"><mtext>Do shading directly without considering visibility</mtext></mrow></munder></math></mjx-assistive-mml></mjx-container></div></div><ul><li><p><strong><span>Separate visibility from the integration</span></strong><span>. The right part now remains only the direct shading.</span></p></li><li><p><span>This is the basic concept behind shadow mapping.</span></p></li><li><p><span>When is it accurate?</span></p><ul><li><p><span>Small </span><strong><span>support</span></strong><span>: Point/directional lighting</span></p><ul><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.28ex" height="1.902ex" role="img" focusable="false" viewBox="0 -683 1008 840.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-394-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-394-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="1D43F" xlink:href="#MJX-394-TEX-I-1D43F"></use></g><g data-mml-node="mi" transform="translate(714,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-394-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>L</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">L_i</script><span> has small support.</span></p></li></ul></li><li><p><span>Smooth </span><strong><span>integrand</span></strong><span>: Diffuse BSDF/Constant radiance area lighting.</span></p><ul><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.018ex" height="2.059ex" role="img" focusable="false" viewBox="0 -705 891.9 910" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-393-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-393-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 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="1D453" xlink:href="#MJX-393-TEX-I-1D453"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-393-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>f</mi><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">f_r</script><span> is smooth, or </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.28ex" height="1.902ex" role="img" focusable="false" viewBox="0 -683 1008 840.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-394-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-394-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="1D43F" xlink:href="#MJX-394-TEX-I-1D43F"></use></g><g data-mml-node="mi" transform="translate(714,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-394-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>L</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">L_i</script><span> is smooth.</span></p></li></ul></li></ul></li></ul><p><span>Will be reviewed again when discussing Ambient Occlusions.</span></p><p> </p><h2 id='iii-percentage-closer-filtering-pcf-and-percentage-closer-soft-shadows-pcss'><span>III. Percentage Closer Filtering (PCF) and Percentage Closer Soft Shadows (PCSS)</span></h2><p><img src="../images/Lecture03-img-11.png" alt="image-20230728180618799" style="zoom:50%;" /></p><ul><li><p><span>Upper: Point Light, </span><strong><span>umbra</span></strong><span> only</span></p><ul><li><p><span>Inside the umbra of a light source, the viewer cannot see that light source.</span></p></li></ul></li><li><p><span>Lower: Area Light, with </span><strong><span>penumbra</span></strong></p><ul><li><p><span>Inside the penumbra of a light source, the viewer can see only part of that light source.</span></p></li></ul></li></ul><p> </p><h3 id='percentage-closer-filtering'><span>Percentage Closer Filtering</span></h3><ul><li><p><span>Provides </span><strong><span>anti-aliasing</span></strong><span> at shadows' edges</span></p><ul><li><p><strong><span>Not for soft shadows</span></strong><span> (PCSS, which is for soft shadows, will be introduced later)</span></p></li><li><p><strong><span>Filtering the results of shadow comparisons</span></strong></p><ul><li><p><span>Averaging the results of shadow comparisons</span></p></li></ul></li></ul></li><li><p><span>Why not filter the shadow map?</span></p><ul><li><p><span>Texture filtering just averages color components:</span></p><ul><li><p><span>You'll get </span><strong><span>blurred</span></strong><span> shadow map first</span></p></li></ul></li><li><p><span>Averaging depth values and comparing then will still lead to a binary visibility.</span></p></li></ul></li></ul><p> </p><h4 id='the-algorithm'><span>The Algorithm</span></h4><p><span>Compute the </span><strong><span>visibility term</span></strong><span> in the rendering equation:</span></p><ol><li><p><span>Instead of performing a single query, perform multiple (e.g., </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.028ex" height="1.579ex" role="img" focusable="false" viewBox="0 -676 2222.4 698" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-395-TEX-N-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path><path id="MJX-395-TEX-N-D7" d="M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="37" xlink:href="#MJX-395-TEX-N-37"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="D7" xlink:href="#MJX-395-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="37" xlink:href="#MJX-395-TEX-N-37"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>×</mo><mn>7</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">7 \times 7</script><span>) depth comparisons for each </span><strong><span>fragment</span></strong><span>:</span></p><ul><li><p><strong><span>Method</span></strong><span>: In the shadow map (figure below), query the </span><strong><span>neighboring region</span></strong><span> of point </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.699ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 751 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-397-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></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="1D443" xlink:href="#MJX-397-TEX-I-1D443"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">P</script><span> on the shadow map</span></p></li></ul><ul><li><p><strong><span>Querying the fragment</span></strong><span>: the farther the fragment is to the light source, the blurrier its corresponding shadow will be.</span></p></li></ul></li></ol><ol start='2' ><li><p><span>Then, averages the </span><strong><span>results of</span></strong><span> comparisons.</span></p></li></ol><p><span>For example:</span></p><ul><li><p><span>For point </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.699ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 751 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-397-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></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="1D443" xlink:href="#MJX-397-TEX-I-1D443"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">P</script><span> on the floor,</span></p><ol start='' ><li><p><span>Compare its depth </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.873ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 828 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-398-TEX-I-1D437" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z"></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="1D437" xlink:href="#MJX-398-TEX-I-1D437"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">D</script><span> with all texels in the red box, e.g., </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.028ex" height="1.554ex" role="img" focusable="false" viewBox="0 -665 2222.4 687" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-399-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><path id="MJX-399-TEX-N-D7" d="M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="33" xlink:href="#MJX-399-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="D7" xlink:href="#MJX-399-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="33" xlink:href="#MJX-399-TEX-N-33"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><mn>3</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">3 \times 3</script><span>.</span></p></li><li><p><span>Get the compared results, e.g., </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="10.937ex" height="8.597ex" role="img" focusable="false" viewBox="0 -2150 4834 3800" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -3.733ex;"><defs><path id="MJX-400-TEX-S4-23A1" d="M319 -645V1154H666V1070H403V-645H319Z"></path><path id="MJX-400-TEX-S4-23A3" d="M319 -644V1155H403V-560H666V-644H319Z"></path><path id="MJX-400-TEX-S4-23A2" d="M319 0V602H403V0H319Z"></path><path id="MJX-400-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><path id="MJX-400-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-400-TEX-S4-23A4" d="M0 1070V1154H347V-645H263V1070H0Z"></path><path id="MJX-400-TEX-S4-23A6" d="M263 -560V1155H347V-644H0V-560H263Z"></path><path id="MJX-400-TEX-S4-23A5" d="M263 0V602H347V0H263Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><use data-c="23A1" xlink:href="#MJX-400-TEX-S4-23A1" transform="translate(0,996)"></use><use data-c="23A3" xlink:href="#MJX-400-TEX-S4-23A3" transform="translate(0,-1006)"></use><svg width="667" height="402" y="49" x="0" viewBox="0 100.5 667 402"><use data-c="23A2" xlink:href="#MJX-400-TEX-S4-23A2" transform="scale(1,1.002)"></use></svg></g><g data-mml-node="mtable" transform="translate(667,0)"><g data-mml-node="mtr" transform="translate(0,1400)"><g data-mml-node="mtd"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-400-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(1500,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-400-TEX-N-30"></use></g></g><g data-mml-node="mtd" transform="translate(3000,0)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-400-TEX-N-31"></use></g></g></g><g data-mml-node="mtr"><g data-mml-node="mtd"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-400-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(1500,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-400-TEX-N-30"></use></g></g><g data-mml-node="mtd" transform="translate(3000,0)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-400-TEX-N-31"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-1400)"><g data-mml-node="mtd"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-400-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(1500,0)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-400-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(3000,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-400-TEX-N-30"></use></g></g></g></g><g data-mml-node="mo" transform="translate(4167,0)"><use data-c="23A4" xlink:href="#MJX-400-TEX-S4-23A4" transform="translate(0,996)"></use><use data-c="23A6" xlink:href="#MJX-400-TEX-S4-23A6" transform="translate(0,-1006)"></use><svg width="667" height="402" y="49" x="0" viewBox="0 100.5 667 402"><use data-c="23A5" xlink:href="#MJX-400-TEX-S4-23A5" transform="scale(1,1.002)"></use></svg></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="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\begin{bmatrix} 1 & 0 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}</script></p></li><li><p><span>Take the average to get visibility, e.g., </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.154ex" height="1.579ex" role="img" focusable="false" viewBox="0 -676 2278 698" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-401-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-401-TEX-N-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path id="MJX-401-TEX-N-36" d="M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z"></path><path id="MJX-401-TEX-N-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-401-TEX-N-30"></use><use data-c="2E" xlink:href="#MJX-401-TEX-N-2E" transform="translate(500,0)"></use><use data-c="36" xlink:href="#MJX-401-TEX-N-36" transform="translate(778,0)"></use><use data-c="36" xlink:href="#MJX-401-TEX-N-36" transform="translate(1278,0)"></use><use data-c="37" xlink:href="#MJX-401-TEX-N-37" transform="translate(1778,0)"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.667</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">0.667</script></p></li></ol></li></ul><p><img src="../images/Lecture03-img-12.png" alt="image-20230728181322513" style="zoom: 50%;" /></p><p><img src="../images/Lecture03-img-13.png" alt="image-20230728181503347" style="zoom:50%;" /></p><p align="center">Wrong example: filtering on the result</p><p><img src="../images/Lecture03-img-17.png" alt="image-20230731170653958" style="zoom: 50%;" /></p><h4 id='problems-1'><span>Problems</span></h4><ul><li><p><span>Does filtering size matter?</span></p><ul><li><p><span>Smaller -> Sharper</span></p></li><li><p><span>Larger -> Softer</span></p></li></ul></li><li><p><span>Can we use PCF to achieve soft shadow effects?</span></p><ul><li><p><span>Using a larger filter size leads to visual </span><strong><span>approximation</span></strong><span> of soft shadows</span></p></li></ul></li><li><p><strong><span>Key thoughts</span></strong><span>:</span></p><ul><li><p><span>From hard shadows to soft shadows</span></p></li><li><p><span>What's the </span><strong><span>correct size</span></strong><span> to filter?</span></p></li><li><p><span>Is the produced shadow uniform?</span></p><ul><li><p><span>No. The closer the object is to the projecting surface, the sharper the projected shadow will be.</span></p></li></ul></li></ul></li></ul><p> </p><h3 id='percentage-closer-soft-shadows'><span>Percentage Closer Soft Shadows</span></h3><p><img src="../images/Lecture03-img-15.png" alt="image-20230728182327005" style="zoom:50%;" /></p><ul><li><p><span>The shadow becomes sharper as the tip gets closer to the plane. 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xlink:href="#MJX-403-TEX-I-1D464"></use></g><g data-mml-node="mtext" transform="translate(749,-150) scale(0.707)"><use data-c="4C" xlink:href="#MJX-403-TEX-N-4C"></use><use data-c="69" xlink:href="#MJX-403-TEX-N-69" transform="translate(625,0)"></use><use data-c="67" xlink:href="#MJX-403-TEX-N-67" transform="translate(903,0)"></use><use data-c="68" xlink:href="#MJX-403-TEX-N-68" transform="translate(1403,0)"></use><use data-c="74" xlink:href="#MJX-403-TEX-N-74" transform="translate(1959,0)"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mtext>Light</mtext></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">w_\text{Light}</script><span>: the width (dimension) of light</span></p><ul><li><p><span>Shadow map cannot be produced with an </span><strong><span>area light</span></strong><span>.</span></p></li><li><p><span>The soft shadow created by the area light, is simulated by:</span></p><ol start='' ><li><p><span>First using a point light to generate the shadow map, </span></p></li><li><p><span>Then specify the dimension of that light when doing calculation, i.e. </span><strong><span>post-processing</span></strong></p></li></ol></li></ul></li><li><p><span>The ratio in the formula corresponds to word </span><em><span>percentage</span></em><span> in its name</span></p></li></ul><p> </p><h4 id='pcss---the-complete-algorithm'><span>PCSS - The Complete Algorithm</span></h4><ol start='' ><li><p><span>Blocker Search:</span></p><ul><li><p><span>Get the </span><strong><span>average</span></strong><span> blocker depth in a certain region inside the shadow map:</span></p><ul><li><p><span>In a region, what is the average depth of the area that covers the shading point?</span></p></li></ul></li></ul></li><li><p><span>Penumbra Estimation:</span></p><ul><li><p><span>Use the average blocker depth to determine </span><strong><span>filter size</span></strong></p></li></ul></li><li><p><span>Percentage Closer Filtering</span></p></li></ol><p> </p><h4 id='problems-2'><span>Problems</span></h4><ul><li><p><span>Which region to perform blocker search? What </span><strong><span>size</span></strong><span>?</span></p><ul><li><p><span>Can be set constant (e.g., </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.028ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 2222.4 688" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-404-TEX-N-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path id="MJX-404-TEX-N-D7" d="M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="35" xlink:href="#MJX-404-TEX-N-35"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="D7" xlink:href="#MJX-404-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="35" xlink:href="#MJX-404-TEX-N-35"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>×</mo><mn>5</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">5 \times 5</script><span>), but can be </span><strong><span>better with heuristics</span></strong></p></li><li><p><span>Depends on the size of light, and distance between receiver and the light:</span></p><p><img src="../images/Lecture03-img-16.png" alt="image-20230731172224524" /></p><p><span>When estimating the blocker depth, create a cone as the figure shows, and use the projected area on the shadow map covered by the cone.</span></p></li></ul></li></ul><p><img src="../images/Lecture03-img-18.png" alt="image-20230731172655220" /></p><p align="center">Video Game: Dying Light</p><p> </p></div></div>
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