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Lecture19.html
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</style><title>Lecture19</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="#games101-lecture-19---cameras-lenses-and-light-fields">GAMES101 Lecture 19 - Cameras, Lenses and Light Fields</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-cameras-lenses">I. Cameras, Lenses</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="#pinhole-image-formation">Pinhole Image Formation</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="#field-of-view-fov-and-focal-length">Field of View (FOV) and Focal Length</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="#field-of-view">Field of View</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="#focal-length">Focal Length</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="#sensor-sizes">Sensor Sizes</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="#exposure">Exposure</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="#exposure-control-in-photography">Exposure Control in Photography</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="#iso-gain">ISO (Gain)</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="#f-number-f-stop-exposure-levels">F-Number (F-Stop): Exposure Levels</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="#shutter-speed">Shutter Speed</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="#constant-exposure-f-stop-vs-shutter-speed">Constant Exposure: F-Stop vs Shutter Speed</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h4"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#fastslow-photography---applications">Fast/Slow Photography - Applications</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="#high-speed-photography">High-Speed Photography</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="#long-exposure-photography">Long-Exposure Photography</a></div><ul class="outline-children"></ul></li></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="#thin-lens-approximation">Thin Lens Approximation</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="#real-lens---aberrations">Real Lens - Aberrations</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h4"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#ideal-thin-lens">Ideal Thin Lens</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="#the-thin-lens-equation">The Thin Lens Equation</a></div><ul class="outline-children"></ul></li></ul></li><li class="outline-item-wrapper outline-h4"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#defocus-blur">Defocus Blur</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="#f-numbers">F-Numbers</a></div><ul class="outline-children"></ul></li></ul></li><li class="outline-item-wrapper outline-h4"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#ray-tracing-ideal-thin-lenses">Ray Tracing Ideal Thin Lenses</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="#setup">Setup</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="#rendering">Rendering</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="#depth-of-field">Depth of Field</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-light-fieldlumigraph">II. Light Field/Lumigraph</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-plenoptic-function">The Plenoptic Function</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="#the-plenoptic-surface">The Plenoptic Surface</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="#view-synthesis">View Synthesis</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="#lumigraph">Lumigraph</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="#parameterization">Parameterization</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h4"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#recording-the-lumigraph">Recording the Lumigraph</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="#camera-array">Camera array</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="#integral-imaging">Integral Imaging</a></div><ul class="outline-children"></ul></li></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="#light-field-camera">Light Field Camera</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="#getting-a-photo-from-the-light-field-camera">Getting a Photo from the Light Field Camera</a></div><ul class="outline-children"></ul></li></ul></li></ul></li></ul></li></div></div><div id='write' class=''><h1 id='games101-lecture-19---cameras-lenses-and-light-fields'><span>GAMES101 Lecture 19 - Cameras, Lenses and Light Fields</span></h1><p><a href='https://sites.cs.ucsb.edu/~lingqi/teaching/resources/GAMES101_Lecture_19.pdf'><span>GAMES101_Lecture_19.pdf</span></a></p><p align="center">Imaging = Synthesis + Capture</p><h2 id='i-cameras-lenses'><span>I. Cameras, Lenses</span></h2><p><img src="../images/Lecture19-img-1.png" alt="img-1" style="zoom: 50%;" /></p><p align="center">Cross-section of Nikon D3, 14-24mm F2.8 lens</p><ul><li><p><strong><span>Pinholes & Lenses</span></strong><span> form images</span></p></li><li><p><span>Shutter exposes sensor for precise duration</span></p></li><li><p><span>Sensor accumulates irradiance during exposure</span></p><ul><li><p><span>The sensor records irradiance, and therefore all pixel values would be similar</span></p></li><li><p><span>If the sensor records radiance, then we can form an image without lenses/pinholes by capturing light from programmed direction</span></p></li></ul></li></ul><p> </p><h3 id='pinhole-image-formation'><span>Pinhole Image Formation</span></h3><ul><li><p><span>No depth of focus - captures the entire 3D scene sharply and without blur</span></p></li></ul><p> </p><h3 id='field-of-view-fov-and-focal-length'><span>Field of View (FOV) and Focal Length</span></h3><h4 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display="block"><mtext>FOV</mtext><mo>=</mo><mn>2</mn><mi>arctan</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mi>h</mi><mrow><mn>2</mn><mi>f</mi></mrow></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container></div></div></li></ul><p> </p><h4 id='focal-length'><span>Focal Length</span></h4><ul><li><p><span>For historical reasons, it is common to refer to the angular field of view by focal length of a lens used on a 35mm-format film (36x24 mm)</span></p><ul><li><p><span>When we say current </span><strong><span>cell phones</span></strong><span> have approximately 28mm "equivalent" focal length, this uses the above convention.</span></p></li></ul></li><li><p><span>Examples:</span></p><ul><li><p><span>Focal length -> </span><em><strong><span>Diagonal FOV</span></strong></em></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg 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552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path><path id="MJX-720-TEX-N-B0" d="M147 628Q147 669 179 692T244 715Q298 715 325 689T352 629Q352 592 323 567T249 542Q202 542 175 567T147 628ZM313 628Q313 660 300 669T259 678H253Q248 678 242 678T234 679Q217 679 207 674T192 659T188 644T187 629Q187 600 198 590Q210 579 250 579H265Q279 579 288 581T305 595T313 628Z"></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-720-TEX-N-35"></use><use data-c="30" xlink:href="#MJX-720-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mtext" transform="translate(1000,0)"><use data-c="6D" xlink:href="#MJX-720-TEX-N-6D"></use><use data-c="6D" xlink:href="#MJX-720-TEX-N-6D" transform="translate(833,0)"></use></g><g data-mml-node="mo" transform="translate(2943.8,0)"><use data-c="2192" xlink:href="#MJX-720-TEX-N-2192"></use></g><g data-mml-node="mn" transform="translate(4221.6,0)"><use data-c="34" xlink:href="#MJX-720-TEX-N-34"></use><use data-c="37" xlink:href="#MJX-720-TEX-N-37" transform="translate(500,0)"></use></g><g data-mml-node="mi" class="" transform="translate(5221.6,0)"><use data-c="B0" xlink:href="#MJX-720-TEX-N-B0"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn><mtext>mm</mtext><mo accent="false" stretchy="false">→</mo><mn>47</mn><mi mathvariant="normal" class="MathML-Unit">°</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">50\text{mm} \to 47 \degree</script><span>, normal lens</span></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="14.076ex" height="1.667ex" role="img" focusable="false" viewBox="0 -715 6221.6 737" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-721-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><path id="MJX-721-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-721-TEX-N-6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path id="MJX-721-TEX-N-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path><path id="MJX-721-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-721-TEX-N-B0" d="M147 628Q147 669 179 692T244 715Q298 715 325 689T352 629Q352 592 323 567T249 542Q202 542 175 567T147 628ZM313 628Q313 660 300 669T259 678H253Q248 678 242 678T234 679Q217 679 207 674T192 659T188 644T187 629Q187 600 198 590Q210 579 250 579H265Q279 579 288 581T305 595T313 628Z"></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="32" xlink:href="#MJX-721-TEX-N-32"></use><use data-c="30" xlink:href="#MJX-721-TEX-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-721-TEX-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mtext" transform="translate(1500,0)"><use data-c="6D" xlink:href="#MJX-721-TEX-N-6D"></use><use data-c="6D" xlink:href="#MJX-721-TEX-N-6D" transform="translate(833,0)"></use></g><g data-mml-node="mo" transform="translate(3443.8,0)"><use data-c="2192" xlink:href="#MJX-721-TEX-N-2192"></use></g><g data-mml-node="mn" transform="translate(4721.6,0)"><use data-c="31" xlink:href="#MJX-721-TEX-N-31"></use><use data-c="32" xlink:href="#MJX-721-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="mi" class="" transform="translate(5721.6,0)"><use data-c="B0" xlink:href="#MJX-721-TEX-N-B0"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>200</mn><mtext>mm</mtext><mo accent="false" stretchy="false">→</mo><mn>12</mn><mi mathvariant="normal" class="MathML-Unit">°</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">200\text{mm} \to 12 \degree</script><span>, telephoto lens</span></p></li></ul></li></ul><p><img src="../images/Lecture19-img-3.png" referrerpolicy="no-referrer" alt="img-3"></p><p><em><span>Normally we fix the size of sensor for convenience, but in fact they should all be taken into consideration.</span></em></p><p> </p><h4 id='sensor-sizes'><span>Sensor Sizes</span></h4><p><img src="../images/Lecture19-img-4.png" referrerpolicy="no-referrer" alt="img-4"></p><p align="center"> Credit: <a>lensvid.com </a> </p><p> </p><h3 id='exposure'><span>Exposure</span></h3><p><em><strong><span>Definition</span></strong></em><span>: Exposure is the product of time and irradiance.</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block 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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="1D43B" xlink:href="#MJX-668-TEX-I-1D43B"></use></g><g data-mml-node="mo" transform="translate(1165.8,0)"><use data-c="3D" xlink:href="#MJX-668-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2221.6,0)"><use data-c="1D447" xlink:href="#MJX-668-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(3147.8,0)"><use data-c="D7" xlink:href="#MJX-668-TEX-N-D7"></use></g><g data-mml-node="mi" transform="translate(4148,0)"><use data-c="1D438" xlink:href="#MJX-668-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>H</mi><mo>=</mo><mi>T</mi><mo>×</mo><mi>E</mi></math></mjx-assistive-mml></mjx-container></div></div><ul><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.593ex" height="1.532ex" role="img" focusable="false" viewBox="0 -677 704 677" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-722-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></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="1D447" xlink:href="#MJX-722-TEX-I-1D447"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">T</script><span> - the exposure time</span></p><ul><li><p><span>Controlled by shutter</span></p></li></ul></li><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-723-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 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opening/closing the aperture (if camera has </span><em><span>iris control</span></em><span>)</span></p></li></ul></li><li><p><strong><span>Shutter speed</span></strong></p><ul><li><p><span>Change the duration the sensor pixels integrate light</span></p></li></ul></li><li><p><strong><span>ISO gain</span></strong></p><ul><li><p><span>Change the amplification (analog and/or digital) between sensor values and digital image values</span></p></li></ul></li></ul><p><img src="../images/Lecture19-img-5.png" alt="img-5" style="zoom:50%;" /></p><ul><li><p><span>From top to bottom: aperture size, shutter speed, ISO gain</span></p></li></ul><p> </p><h4 id='iso-gain'><span>ISO (Gain)</span></h4><p><span>Third variable for exposure</span></p><p><strong><span>Film</span></strong><span>: trade </span><strong><span>sensitivity</span></strong><span> for </span><strong><span>grain</span></strong></p><p><strong><span>Digital</span></strong><span>: trade </span><strong><span>sensitivity</span></strong><span> for </span><strong><span>noise</span></strong></p><ul><li><p><span>Multiply signal before analog-to-digital conversion</span></p></li><li><p><span>Linear effect (ISO 200 needs half the light as ISO 100)</span></p></li></ul><p><img src="../images/Lecture19-img-6.png" alt="img-6" style="zoom:67%;" /></p><p> </p><h4 id='f-number-f-stop-exposure-levels'><span>F-Number (F-Stop): Exposure Levels</span></h4><p><img src="../images/Lecture19-img-7.png" referrerpolicy="no-referrer" alt="img-7"></p><ul><li><p><span>Written as F</span><strong><span>N</span></strong><span> or F/</span><strong><span>N</span></strong><span>, where </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.009ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 888 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-730-TEX-I-1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 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="1D441" xlink:href="#MJX-730-TEX-I-1D441"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">N</script><span> is the f-number.</span></p></li></ul><p><em><strong><span>Definition</span></strong></em><span>: The f-number of a lens is defined as</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n115" cid="n115" 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="7.718ex" height="4.676ex" role="img" focusable="false" viewBox="0 -1381 3411.6 2067" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -1.552ex;"><defs><path id="MJX-669-TEX-I-1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z"></path><path id="MJX-669-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-669-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-669-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 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="mi"><use data-c="1D441" xlink:href="#MJX-669-TEX-I-1D441"></use></g><g data-mml-node="mo" transform="translate(1165.8,0)"><use data-c="3D" xlink:href="#MJX-669-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2221.6,0)"><g data-mml-node="mi" transform="translate(320,676)"><use data-c="1D453" xlink:href="#MJX-669-TEX-I-1D453"></use></g><g data-mml-node="mi" transform="translate(220,-686)"><use data-c="1D434" xlink:href="#MJX-669-TEX-I-1D434"></use></g><rect width="950" height="60" x="120" y="220"></rect></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>N</mi><mo>=</mo><mfrac><mi>f</mi><mi>A</mi></mfrac></math></mjx-assistive-mml></mjx-container></div></div><p><span>which is the </span><strong><span>focal length</span></strong><span> </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-734-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-734-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> divided by the </span><strong><span>diameter of the aperture</span></strong><span>.</span></p><ul><li><p><span>An f-stop of </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-726-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"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-726-TEX-N-32"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">2</script><span> is sometimes written </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.507ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1550 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-727-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-727-TEX-N-2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z"></path><path id="MJX-727-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"><g data-mml-node="mi"><use data-c="1D453" xlink:href="#MJX-727-TEX-I-1D453"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(550,0)"><g data-mml-node="mo"><use data-c="2F" xlink:href="#MJX-727-TEX-N-2F"></use></g></g><g data-mml-node="mn" transform="translate(1050,0)"><use data-c="32" xlink:href="#MJX-727-TEX-N-32"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">f/2</script><span>, reflecting the fact that the absolute aperture diameter </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-735-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 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="mi"><use data-c="1D434" xlink:href="#MJX-735-TEX-I-1D434"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">A</script><span> can be computed by dividing the focal length </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-734-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-734-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> by the relative aperture </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.009ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 888 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-730-TEX-I-1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 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="1D441" xlink:href="#MJX-730-TEX-I-1D441"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">N</script><span>.</span></p></li></ul><p><img src="../images/Lecture19-img-18.png" alt="img-18" style="zoom:50%;" /></p><p> </p><h4 id='shutter-speed'><span>Shutter Speed</span></h4><p><img src="../images/Lecture19-img-8.png" referrerpolicy="no-referrer" alt="img-8"></p><ul><li><p><strong><span>Motion Blur</span></strong><span>: handshake, subject movement</span></p><ul><li><p><span>Doubling shutter time doubles motion blur</span></p></li></ul></li><li><p><strong><span>Rolling shutter</span></strong><span>: while the shutter is moving, different parts of photo is actually taken at different times</span></p><ul><li><p><span>May also be caused during the imaging process, where the CMOS stores data linearly (doesn't capture every pixel simultaneously)</span></p></li></ul><p><img src="../images/Lecture19-img-9.png" referrerpolicy="no-referrer" alt="img-9"></p></li></ul><p> </p><h4 id='constant-exposure-f-stop-vs-shutter-speed'><span>Constant Exposure: F-Stop vs Shutter Speed</span></h4><figure><table><thead><tr><th><span>F-Stop</span></th><th><span>1.4</span></th><th><span>2.0</span></th><th><span>2.8</span></th><th><span>4.0</span></th><th><span>5.6</span></th><th><span>8.0</span></th><th><span>11.0</span></th><th><span>16.0</span></th><th><span>22.0</span></th><th><span>32.0</span></th></tr></thead><tbody><tr><td><strong><span>Shutter</span></strong></td><td><span>1/500</span></td><td><strong><span>1/250</span></strong></td><td><span>1/125</span></td><td><strong><span>1/60</span></strong></td><td><span>1/30</span></td><td><strong><span>1/15</span></strong></td><td><span>1/8</span></td><td><strong><span>1/4</span></strong></td><td><span>1/2</span></td><td><strong><span>1</span></strong></td></tr></tbody></table></figure><p><span>These combinations gives equivalent exposure.</span></p><ul><li><p><span>For moving objects, photographers must trade off </span><strong><span>depth of field</span></strong><span> and </span><strong><span>motion blur</span></strong></p></li></ul><p> </p><p> </p><h4 id='fastslow-photography---applications'><span>Fast/Slow Photography - Applications</span></h4><h5 id='high-speed-photography'><span>High-Speed Photography</span></h5><p><span>Normal exposure = </span></p><ul><li><p><span>extremely fast shutter speed, times</span></p></li><li><p><span>large aperture and/or high ISO</span></p></li></ul><p><img src="../images/Lecture19-img-10.png" alt="img-10" style="zoom:33%;" /></p><p> </p><h5 id='long-exposure-photography'><span>Long-Exposure Photography</span></h5><p><img src="../images/Lecture19-img-11.png" referrerpolicy="no-referrer" alt="img-11"></p><p> </p><h3 id='thin-lens-approximation'><span>Thin Lens Approximation</span></h3><p><span>Real lens designs are highly complex.</span></p><h4 id='real-lens---aberrations'><span>Real Lens - Aberrations</span></h4><p><img src="../images/Lecture19-img-12.png" alt="img-12" style="zoom:33%;" /></p><p> </p><h4 id='ideal-thin-lens'><span>Ideal Thin Lens</span></h4><p><img src="../images/Lecture19-img-13.png" alt="img-13" style="zoom: 50%;" /></p><ul><li><p><span>All parallel rays entering a lens pass through the focal point of that lens</span></p></li><li><p><span>All rays through a focal point will be in parallel after passing the lens</span></p></li><li><p><span>Focal length can be arbitrarily changed (using a </span><strong><span>zoom lens</span></strong><span>)</span></p></li></ul><p> </p><h5 id='the-thin-lens-equation'><span>The Thin Lens Equation</span></h5><p><img src="../images/Lecture19-img-14.png" alt="img-14" style="zoom:50%;" /></p><p><strong><span>Gaussian Thin Lens Equation</span></strong><span>:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n200" cid="n200" 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="13.821ex" height="5.052ex" role="img" focusable="false" viewBox="0 -1342 6108.9 2233" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.016ex;"><defs><path id="MJX-670-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-670-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-670-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-670-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 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transform="translate(220,-686)"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-670-TEX-I-1D467"></use></g><g data-mml-node="mi" transform="translate(498,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-670-TEX-I-1D456"></use></g></g><rect width="992" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(3777.7,0)"><use data-c="2B" xlink:href="#MJX-670-TEX-N-2B"></use></g><g data-mml-node="mfrac" transform="translate(4778,0)"><g data-mml-node="mn" transform="translate(415.5,676)"><use data-c="31" xlink:href="#MJX-670-TEX-N-31"></use></g><g data-mml-node="msub" transform="translate(220,-686)"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-670-TEX-I-1D467"></use></g><g data-mml-node="mi" transform="translate(498,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-670-TEX-I-1D45C"></use></g></g><rect width="1090.9" height="60" x="120" y="220"></rect></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mn>1</mn><mi>f</mi></mfrac><mo>=</mo><mfrac><mn>1</mn><msub><mi>z</mi><mi>i</mi></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>z</mi><mi>o</mi></msub></mfrac></math></mjx-assistive-mml></mjx-container></div></div><ul><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.016ex" height="1.357ex" role="img" focusable="false" viewBox="0 -442 890.9 599.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-736-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><path id="MJX-736-TEX-I-1D45C" d="M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></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="1D467" xlink:href="#MJX-736-TEX-I-1D467"></use></g><g data-mml-node="mi" transform="translate(498,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-736-TEX-I-1D45C"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">z_o</script><span> - object distance</span></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.792ex" height="1.357ex" role="img" focusable="false" viewBox="0 -442 792 599.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-737-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><path id="MJX-737-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="1D467" xlink:href="#MJX-737-TEX-I-1D467"></use></g><g data-mml-node="mi" transform="translate(498,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-737-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>z</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">z_i</script><span> - image distance</span></p></li><li><p><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-734-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 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id='defocus-blur'><span>Defocus Blur</span></h4><p><strong><span>Circle of Confusion, CoC</span></strong><span>: an optical spot caused by a cone of light rays from a lens not coming to a perfect focus when imaging a point source.</span></p><ul><li><p><strong><span>Proportional</span></strong><span> to the size of the </span><strong><span>aperture</span></strong></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n216" cid="n216" 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="37.571ex" height="5.21ex" role="img" focusable="false" viewBox="0 -1459 16606.2 2302.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -1.909ex;"><defs><path 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data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-671-TEX-I-1D467"></use></g><g data-mml-node="mi" transform="translate(498,-150) scale(0.707)"><use data-c="1D460" xlink:href="#MJX-671-TEX-I-1D460"></use></g></g><g data-mml-node="mo" transform="translate(1379.9,0)"><use data-c="2212" xlink:href="#MJX-671-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(2380.1,0)"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-671-TEX-I-1D467"></use></g><g data-mml-node="mi" transform="translate(498,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-671-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(3172,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-671-TEX-N-7C"></use></g></g><g data-mml-node="msub" transform="translate(1549,-686)"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-671-TEX-I-1D467"></use></g><g data-mml-node="mi" transform="translate(498,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-671-TEX-I-1D456"></use></g></g><rect width="3650" height="60" x="120" y="220"></rect></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>C</mi><mo>=</mo><mi>A</mi><mfrac><msup><mi>d</mi><mo data-mjx-alternate="1">′</mo></msup><msub><mi>z</mi><mi>i</mi></msub></mfrac><mo>=</mo><mi>A</mi><mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>z</mi><mi>s</mi></msub><mo>−</mo><msub><mi>z</mi><mi>i</mi></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mi>z</mi><mi>i</mi></msub></mfrac><mo>=</mo><mfrac><mi>f</mi><mi>N</mi></mfrac><mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>z</mi><mi>s</mi></msub><mo>−</mo><msub><mi>z</mi><mi>i</mi></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mi>z</mi><mi>i</mi></msub></mfrac></math></mjx-assistive-mml></mjx-container></div></div></li></ul><p><img src="../images/Lecture19-img-16.png" alt="img-16" style="zoom:50%;" /></p><p><img src="../images/Lecture19-img-17.png" alt="img-17" style="zoom: 50%;" /></p><p> </p><h5 id='f-numbers'><span>F-Numbers</span></h5><p><span>View the </span><strong><span>Exposure</span></strong><span> section.</span></p><p> </p><h4 id='ray-tracing-ideal-thin-lenses'><span>Ray Tracing Ideal Thin Lenses</span></h4><h5 id='setup'><span>Setup</span></h5><ul><li><p><span>Choose sensor size, lens focal length </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-734-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-734-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> and aperture size </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-735-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 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="mi"><use data-c="1D434" 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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><path id="MJX-736-TEX-I-1D45C" d="M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></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="1D467" xlink:href="#MJX-736-TEX-I-1D467"></use></g><g data-mml-node="mi" transform="translate(498,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-736-TEX-I-1D45C"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">z_o</script></p><ul><li><p><span>Compute corresponding depth of sensor </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.792ex" height="1.357ex" role="img" focusable="false" viewBox="0 -442 792 599.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-737-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><path id="MJX-737-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="1D467" xlink:href="#MJX-737-TEX-I-1D467"></use></g><g data-mml-node="mi" transform="translate(498,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-737-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>z</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">z_i</script><span> from the equation (</span><strong><span>focusing</span></strong><span>)</span></p></li></ul></li></ul><p> </p><p><img src="../images/Lecture19-img-19.png" alt="img-19" style="zoom:50%;" /></p><h5 id='rendering'><span>Rendering</span></h5><ul><li><p><span>For each pixel </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.922ex" height="1.742ex" role="img" focusable="false" viewBox="0 -759 849.5 770" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-741-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-741-TEX-V-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></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="1D465" xlink:href="#MJX-741-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(605,363) scale(0.707)"><use data-c="2032" xlink:href="#MJX-741-TEX-V-2032"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">x'</script><span> on the sensor (</span><strong><span>film</span></strong><span> actually)</span></p></li><li><p><span>Sample random points </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.362ex" height="1.742ex" role="img" focusable="false" viewBox="0 -759 1043.9 770" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-739-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 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transform="translate(605,363) scale(0.707)"><g data-c="2033"><use data-c="2032" xlink:href="#MJX-739-TEX-V-2032"></use><use data-c="2032" xlink:href="#MJX-739-TEX-V-2032" transform="translate(275,0)"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mo data-mjx-alternate="1">″</mo></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">x''</script><span> on the lens plane</span></p></li><li><p><span>Since the ray passing through the lens will hit </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.802ex" height="1.742ex" role="img" focusable="false" viewBox="0 -759 1238.4 770" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-740-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-740-TEX-V-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></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="1D465" xlink:href="#MJX-740-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(605,363) scale(0.707)"><g data-c="2034"><use data-c="2032" xlink:href="#MJX-740-TEX-V-2032"></use><use data-c="2032" xlink:href="#MJX-740-TEX-V-2032" transform="translate(275,0)"></use><use data-c="2032" xlink:href="#MJX-740-TEX-V-2032" transform="translate(550,0)"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mo data-mjx-alternate="1">‴</mo></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">x'''</script></p><ul><li><p><span>Consider the virtual ray </span><strong><span>connecting </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.922ex" height="1.742ex" role="img" focusable="false" viewBox="0 -759 849.5 770" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-741-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-741-TEX-V-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></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="1D465" xlink:href="#MJX-741-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(605,363) scale(0.707)"><use data-c="2032" xlink:href="#MJX-741-TEX-V-2032"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">x'</script><span> and the center of lens</span></strong></p></li></ul></li><li><p><span>Estimate radiance on ray </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.683ex" height="1.742ex" role="img" focusable="false" viewBox="0 -759 3837.8 770" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-742-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-742-TEX-V-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path><path id="MJX-742-TEX-N-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 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="msup"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-742-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(605,363) scale(0.707)"><g data-c="2033"><use data-c="2032" xlink:href="#MJX-742-TEX-V-2032"></use><use data-c="2032" xlink:href="#MJX-742-TEX-V-2032" transform="translate(275,0)"></use></g></g></g><g data-mml-node="mo" transform="translate(1321.7,0)"><use data-c="2192" xlink:href="#MJX-742-TEX-N-2192"></use></g><g data-mml-node="msup" transform="translate(2599.5,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-742-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(605,363) scale(0.707)"><g data-c="2034"><use data-c="2032" xlink:href="#MJX-742-TEX-V-2032"></use><use data-c="2032" xlink:href="#MJX-742-TEX-V-2032" transform="translate(275,0)"></use><use data-c="2032" xlink:href="#MJX-742-TEX-V-2032" transform="translate(550,0)"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mo data-mjx-alternate="1">″</mo></msup><mo accent="false" stretchy="false">→</mo><msup><mi>x</mi><mo data-mjx-alternate="1">‴</mo></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">x'' \to x'''</script></p></li></ul><p> </p><h4 id='depth-of-field'><span>Depth of Field</span></h4><p><img src="../images/Lecture19-img-20.png" referrerpolicy="no-referrer" alt="img-20"></p><p><span>Set the CoC as the maximum permissible blur spot on the image plane</span></p><ul><li><p><span>Such that they will appear as a single pixel finally</span></p></li></ul><p><img src="../images/Lecture19-img-21.png" alt="img-21" style="zoom: 50%;" /></p><p><strong><span>Depth of Field</span></strong><span>: Depth range in a scene where the corresponding CoC is considered </span><strong><span>small enough</span></strong></p><p><img src="../images/Lecture19-img-22.png" alt="img-22" style="zoom:50%;" /></p><p align="center"><a>http://graphics.stanford.edu/courses/cs178/applets/dof.html</a></p><p> </p><h2 id='ii-light-fieldlumigraph'><span>II. Light Field/Lumigraph</span></h2><h3 id='the-plenoptic-function'><span>The Plenoptic Function</span></h3><p><strong><span>Definition:</span></strong><span> The </span><strong><span>Plenoptic function</span></strong><span> describes the intensity of light viewed from </span><strong><span>any</span></strong><span> point, to any direction, at any time:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n263" cid="n263" 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="13.952ex" height="2.477ex" role="img" focusable="false" viewBox="0 -845 6166.7 1095" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-672-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><path id="MJX-672-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-672-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><path id="MJX-672-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-672-TEX-I-1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z"></path><path id="MJX-672-TEX-I-1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path id="MJX-672-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path id="MJX-672-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-672-TEX-N-20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path><path id="MJX-672-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="1D443" xlink:href="#MJX-672-TEX-I-1D443"></use></g><g data-mml-node="mo" transform="translate(751,0)"><use data-c="28" xlink:href="#MJX-672-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1140,0)"><use data-c="1D703" xlink:href="#MJX-672-TEX-I-1D703"></use></g><g data-mml-node="mo" transform="translate(1609,0)"><use data-c="2C" xlink:href="#MJX-672-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2053.7,0)"><use data-c="1D719" xlink:href="#MJX-672-TEX-I-1D719"></use></g><g data-mml-node="mo" transform="translate(2649.7,0)"><use data-c="2C" xlink:href="#MJX-672-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(3094.3,0)"><use data-c="1D706" xlink:href="#MJX-672-TEX-I-1D706"></use></g><g data-mml-node="mo" transform="translate(3677.3,0)"><use data-c="2C" xlink:href="#MJX-672-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(4122,0)"><use data-c="1D461" xlink:href="#MJX-672-TEX-I-1D461"></use></g><g data-mml-node="mo" transform="translate(4483,0)"><use data-c="2C" xlink:href="#MJX-672-TEX-N-2C"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(4927.7,0)"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-672-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(313.8,31) translate(-250 0)"><use data-c="20D7" xlink:href="#MJX-672-TEX-N-20D7"></use></g></g></g><g data-mml-node="mo" transform="translate(5499.7,0)"><use data-c="29" xlink:href="#MJX-672-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(5888.7,0)"><use data-c="2C" xlink:href="#MJX-672-TEX-N-2C"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>ϕ</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>t</mi><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>,</mo></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="1.061ex" height="1.618ex" role="img" focusable="false" viewBox="0 -705 469 715" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.023ex;"><defs><path id="MJX-755-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="mi"><use data-c="1D703" xlink:href="#MJX-755-TEX-I-1D703"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\theta</script><span> and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.348ex" height="2.034ex" role="img" focusable="false" viewBox="0 -694 596 899" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-756-TEX-I-1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z"></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="1D719" xlink:href="#MJX-756-TEX-I-1D719"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\phi</script><span> describes the spherical position, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.597ex" role="img" focusable="false" viewBox="0 -694 583 706" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.027ex;"><defs><path id="MJX-745-TEX-I-1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></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="1D706" xlink:href="#MJX-745-TEX-I-1D706"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\lambda</script><span> is the wavelength of light, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.817ex" height="1.441ex" role="img" focusable="false" viewBox="0 -626 361 637" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-746-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></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="1D461" xlink:href="#MJX-746-TEX-I-1D461"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">t</script><span> is the time and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.937ex" role="img" focusable="false" viewBox="0 -845 572 856" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-754-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-754-TEX-N-20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></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="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-754-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(313.8,31) translate(-250 0)"><use data-c="20D7" xlink:href="#MJX-754-TEX-N-20D7"></use></g></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="ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\vec{x}</script><span> is the viewing position.</span></p><ul><li><p><strong><span>Grayscale snapshot</span></strong><span>: </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.875ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3038.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-748-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><path id="MJX-748-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-748-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><path id="MJX-748-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-748-TEX-I-1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z"></path><path id="MJX-748-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="1D443" xlink:href="#MJX-748-TEX-I-1D443"></use></g><g data-mml-node="mo" transform="translate(751,0)"><use data-c="28" xlink:href="#MJX-748-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1140,0)"><use data-c="1D703" xlink:href="#MJX-748-TEX-I-1D703"></use></g><g data-mml-node="mo" transform="translate(1609,0)"><use data-c="2C" xlink:href="#MJX-748-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2053.7,0)"><use data-c="1D719" xlink:href="#MJX-748-TEX-I-1D719"></use></g><g data-mml-node="mo" transform="translate(2649.7,0)"><use data-c="29" xlink:href="#MJX-748-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>P</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">P(\theta, \phi)</script></p></li><li><p><strong><span>Color snapshot</span></strong><span>: </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="9.2ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4066.3 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-749-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><path id="MJX-749-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-749-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><path id="MJX-749-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-749-TEX-I-1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z"></path><path id="MJX-749-TEX-I-1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path id="MJX-749-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="1D443" xlink:href="#MJX-749-TEX-I-1D443"></use></g><g data-mml-node="mo" transform="translate(751,0)"><use data-c="28" xlink:href="#MJX-749-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1140,0)"><use data-c="1D703" xlink:href="#MJX-749-TEX-I-1D703"></use></g><g data-mml-node="mo" transform="translate(1609,0)"><use data-c="2C" xlink:href="#MJX-749-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2053.7,0)"><use data-c="1D719" xlink:href="#MJX-749-TEX-I-1D719"></use></g><g data-mml-node="mo" transform="translate(2649.7,0)"><use data-c="2C" xlink:href="#MJX-749-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(3094.3,0)"><use data-c="1D706" xlink:href="#MJX-749-TEX-I-1D706"></use></g><g data-mml-node="mo" transform="translate(3677.3,0)"><use data-c="29" xlink:href="#MJX-749-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>P</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>ϕ</mi><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">P(\theta, \phi, \lambda)</script></p></li><li><p><strong><span>Movie</span></strong><span>: </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="11.023ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4872 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-750-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><path id="MJX-750-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-750-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><path id="MJX-750-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-750-TEX-I-1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z"></path><path id="MJX-750-TEX-I-1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path id="MJX-750-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path id="MJX-750-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="1D443" xlink:href="#MJX-750-TEX-I-1D443"></use></g><g data-mml-node="mo" transform="translate(751,0)"><use data-c="28" xlink:href="#MJX-750-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1140,0)"><use data-c="1D703" xlink:href="#MJX-750-TEX-I-1D703"></use></g><g data-mml-node="mo" transform="translate(1609,0)"><use data-c="2C" xlink:href="#MJX-750-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2053.7,0)"><use data-c="1D719" xlink:href="#MJX-750-TEX-I-1D719"></use></g><g data-mml-node="mo" transform="translate(2649.7,0)"><use data-c="2C" xlink:href="#MJX-750-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(3094.3,0)"><use data-c="1D706" xlink:href="#MJX-750-TEX-I-1D706"></use></g><g data-mml-node="mo" transform="translate(3677.3,0)"><use data-c="2C" xlink:href="#MJX-750-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(4122,0)"><use data-c="1D461" xlink:href="#MJX-750-TEX-I-1D461"></use></g><g data-mml-node="mo" transform="translate(4483,0)"><use data-c="29" xlink:href="#MJX-750-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>P</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>ϕ</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">P(\theta, \phi, \lambda, t)</script></p></li><li><p><strong><span>Holographic movie</span></strong><span>: </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="13.323ex" height="2.477ex" role="img" focusable="false" viewBox="0 -845 5888.7 1095" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-751-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><path id="MJX-751-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-751-TEX-I-1D703" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 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441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z"></path><path id="MJX-751-TEX-I-1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path id="MJX-751-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path id="MJX-751-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-751-TEX-N-20D7" d="M377 694Q377 702 382 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data-mml-node="mo" transform="translate(1609,0)"><use data-c="2C" xlink:href="#MJX-751-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2053.7,0)"><use data-c="1D719" xlink:href="#MJX-751-TEX-I-1D719"></use></g><g data-mml-node="mo" transform="translate(2649.7,0)"><use data-c="2C" xlink:href="#MJX-751-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(3094.3,0)"><use data-c="1D706" xlink:href="#MJX-751-TEX-I-1D706"></use></g><g data-mml-node="mo" transform="translate(3677.3,0)"><use data-c="2C" xlink:href="#MJX-751-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(4122,0)"><use data-c="1D461" xlink:href="#MJX-751-TEX-I-1D461"></use></g><g data-mml-node="mo" transform="translate(4483,0)"><use data-c="2C" xlink:href="#MJX-751-TEX-N-2C"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(4927.7,0)"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-751-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(313.8,31) translate(-250 0)"><use data-c="20D7" xlink:href="#MJX-751-TEX-N-20D7"></use></g></g></g><g data-mml-node="mo" transform="translate(5499.7,0)"><use data-c="29" xlink:href="#MJX-751-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>P</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>ϕ</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>t</mi><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">P(\theta, \phi, \lambda, t, \vec{x})</script></p></li></ul><p> </p><p><strong><span>Definition</span></strong><span>: A </span><strong><span>ray</span></strong><span> is defined by</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n276" cid="n276" 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="9.175ex" height="2.477ex" role="img" focusable="false" viewBox="0 -845 4055.3 1095" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-673-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><path id="MJX-673-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-673-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><path id="MJX-673-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 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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-673-TEX-N-20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path><path id="MJX-673-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="1D443" xlink:href="#MJX-673-TEX-I-1D443"></use></g><g data-mml-node="mo" transform="translate(751,0)"><use data-c="28" xlink:href="#MJX-673-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1140,0)"><use data-c="1D703" xlink:href="#MJX-673-TEX-I-1D703"></use></g><g data-mml-node="mo" transform="translate(1609,0)"><use data-c="2C" xlink:href="#MJX-673-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2053.7,0)"><use data-c="1D719" xlink:href="#MJX-673-TEX-I-1D719"></use></g><g data-mml-node="mo" transform="translate(2649.7,0)"><use data-c="2C" xlink:href="#MJX-673-TEX-N-2C"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(3094.3,0)"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-673-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(313.8,31) translate(-250 0)"><use data-c="20D7" xlink:href="#MJX-673-TEX-N-20D7"></use></g></g></g><g data-mml-node="mo" transform="translate(3666.3,0)"><use data-c="29" xlink:href="#MJX-673-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>ϕ</mi><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></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="1.061ex" height="1.618ex" role="img" focusable="false" viewBox="0 -705 469 715" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.023ex;"><defs><path id="MJX-755-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="mi"><use data-c="1D703" xlink:href="#MJX-755-TEX-I-1D703"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\theta</script><span> and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.348ex" height="2.034ex" role="img" focusable="false" viewBox="0 -694 596 899" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-756-TEX-I-1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z"></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="1D719" xlink:href="#MJX-756-TEX-I-1D719"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\phi</script><span> describes the orientation, and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.937ex" role="img" focusable="false" viewBox="0 -845 572 856" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-754-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-754-TEX-N-20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></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="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-754-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(313.8,31) translate(-250 0)"><use data-c="20D7" xlink:href="#MJX-754-TEX-N-20D7"></use></g></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="ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\vec{x}</script><span> describes the origin of that ray.</span></p><p> </p><h3 id='the-plenoptic-surface'><span>The Plenoptic Surface</span></h3><p><span>Describe the </span><strong><span>radiance information</span></strong><span> of an object by 4D rays, which is represented by:</span></p><ul><li><p><span>a 2D </span><strong><span>position</span></strong><span> (surface coordinate), and</span></p></li><li><p><span>a 2D </span><strong><span>direction</span></strong><span> (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.061ex" height="1.618ex" role="img" focusable="false" viewBox="0 -705 469 715" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.023ex;"><defs><path id="MJX-755-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="mi"><use data-c="1D703" xlink:href="#MJX-755-TEX-I-1D703"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\theta</script><span> and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.348ex" height="2.034ex" role="img" focusable="false" viewBox="0 -694 596 899" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-756-TEX-I-1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z"></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="1D719" xlink:href="#MJX-756-TEX-I-1D719"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\phi</script><span>)</span></p></li></ul><p> </p><h4 id='view-synthesis'><span>View Synthesis</span></h4><p><img src="../images/Lecture19-img-23.png" alt="image-20230724191725683" style="zoom:33%;" /></p><p><span>Place a camera at a certain direction, When looking to an object, we know all the radiance information of each ray we have casted, and thus we can </span><strong><span>synthesize the view simply by</span></strong><span>:</span></p><ul><li><p><span>Looking up the radiance information </span><strong><span>through the plenoptic function</span></strong><span>.</span></p></li></ul><p> </p><p><span>Furthermore, we may completely </span><strong><span>ignore the shape</span></strong><span> of the object, and solely record the light field.</span></p><p><img src="../images/Lecture19-img-24.png" alt="image-20230724192309037" style="zoom: 33%;" /></p><p align="center">Outside the convex space</p><p> </p><h3 id='lumigraph'><span>Lumigraph</span></h3><h4 id='parameterization'><span>Parameterization</span></h4><ul><li><p><span>2D </span><strong><span>Position</span></strong><span>, 2D </span><strong><span>Direction</span></strong><span>:</span></p><p><img src="../images/Lecture19-img-25.png" alt="image-20230724193147214" style="zoom:33%;" /></p></li><li><p><strong><span>2-Plane Parameterization</span></strong><span>: 2D </span><strong><span>Position</span></strong><span>, 2D </span><strong><span>Position</span></strong><span>: </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.644ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2052.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-762-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-762-TEX-I-1D460" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 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385H26Z"></path><path id="MJX-762-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="mo"><use data-c="28" xlink:href="#MJX-762-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D460" xlink:href="#MJX-762-TEX-I-1D460"></use></g><g data-mml-node="mo" transform="translate(858,0)"><use data-c="2C" xlink:href="#MJX-762-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(1302.7,0)"><use data-c="1D461" xlink:href="#MJX-762-TEX-I-1D461"></use></g><g data-mml-node="mo" transform="translate(1663.7,0)"><use data-c="29" xlink:href="#MJX-762-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(s, t)</script><span> and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.158ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2279.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-763-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-763-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-763-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-763-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path id="MJX-763-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="mo"><use data-c="28" xlink:href="#MJX-763-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D462" xlink:href="#MJX-763-TEX-I-1D462"></use></g><g data-mml-node="mo" transform="translate(961,0)"><use data-c="2C" xlink:href="#MJX-763-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(1405.7,0)"><use data-c="1D463" xlink:href="#MJX-763-TEX-I-1D463"></use></g><g data-mml-node="mo" transform="translate(1890.7,0)"><use data-c="29" xlink:href="#MJX-763-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(u, v)</script></p><p><img src="../images/Lecture19-img-26.png" alt="image-20230724193250559" style="zoom:33%;" /></p></li></ul><h4 id='recording-the-lumigraph'><span>Recording the Lumigraph</span></h4><p><img src="../images/Lecture19-img-27.png" alt="image-20230724193642148" style="zoom:50%;" /></p><p><em><span>Assume we are viewing from left of the </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.391ex" height="1.027ex" 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xmlns="http://www.w3.org/2000/svg" width="5.158ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2279.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-763-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-763-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 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xlink:href="#MJX-762-TEX-I-1D461"></use></g><g data-mml-node="mo" transform="translate(1663.7,0)"><use data-c="29" xlink:href="#MJX-762-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(s, t)</script><span>: move the viewing position.</span></p></li><li><p><span>Fix </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.644ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2052.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-762-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 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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="mo"><use data-c="28" xlink:href="#MJX-763-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D462" xlink:href="#MJX-763-TEX-I-1D462"></use></g><g data-mml-node="mo" transform="translate(961,0)"><use data-c="2C" xlink:href="#MJX-763-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(1405.7,0)"><use data-c="1D463" xlink:href="#MJX-763-TEX-I-1D463"></use></g><g data-mml-node="mo" transform="translate(1890.7,0)"><use data-c="29" xlink:href="#MJX-763-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(u, v)</script><span>: view the </span><strong><span>same object</span></strong><span> from </span><strong><span>different directions</span></strong><span>.</span></p></li></ul><h5 id='camera-array'><span>Camera array</span></h5><p><img src="../images/Lecture19-img-28.png" alt="image-20230724193942414" style="zoom:50%;" /></p><p> </p><h5 id='integral-imaging'><span>Integral Imaging</span></h5><p><img src="../images/Lecture19-img-29.png" alt="image-20230724194252279" style="zoom:67%;" /></p><p><span>Flies record lumigraph of the scene, or </span><strong><span>radiance</span></strong><span>.</span></p><ul><li><p><strong><span>Spatially-multiplexed</span></strong><span> light field capture using lenslets.</span></p><ul><li><p><span>Trade-off between spatial and angular resolution</span></p></li></ul></li></ul><p> </p><h3 id='light-field-camera'><span>Light Field Camera</span></h3><ul><li><p><strong><span>Prof. Ren Ng</span></strong><span>: Founder of the company Lytro, who makes light field cameras.</span></p></li><li><p><strong><span>Computational Refocusing</span></strong><span>: virtually changing focal length, aperture, size, etc., </span><strong><span>after</span></strong><span> taking</span></p></li></ul><p><img src="../images/Lecture19-img-30.png" alt="image-20230724195214794" style="zoom:50%;" /></p><p align="center">Picture taken by a light field camera</p><p><img src="../images/Lecture19-img-31.png" alt="image-20230724195237553" style="zoom: 50%;" /></p><ul><li><p><span>Each pixel (</span><strong><span>irradiance</span></strong><span>) is now stored as a block of pixels (</span><strong><span>radiance</span></strong><span>, or irradiance at different directions)</span></p><ul><li><p><span>If each disk of "recorded radiance "is averaged as a single pixel, then the resulting picture is the same as what a normal camera would have taken.</span></p></li></ul></li></ul><h4 id='getting-a-photo-from-the-light-field-camera'><span>Getting a Photo from the Light Field Camera</span></h4><p><img src="../images/Lecture19-img-32.png" alt="Lecture19-img-32" style="zoom: 50%;" /></p><ul><li><p><strong><span>Moving the camera around</span></strong><span>: Always choose the pixel at a fixed position in each disk</span></p></li><li><p><strong><span>Computational/Digital Refocusing</span></strong><span>: Changing focal length, and pick the refocused rays accordingly</span></p></li></ul><p> </p><p><strong><span>Why does it have these functions?</span></strong></p><ul><li><p><span>The light field contains everything</span></p></li></ul><p> </p><p><strong><span>Pros & Cons</span></strong></p><ul><li><p><span>Insufficient </span><strong><span>spatial</span></strong><span> resolution: Same film used for both spatial and directional information</span></p></li><li><p><span>High cost</span></p></li></ul><p> </p><p> </p></div></div>
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