Visual_Computing/Asset/hair-survey.html at main · PeterHUistyping/Visual_Computing · GitHub

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
<!DOCTYPE html>
        <html>
        <head>
            <meta charset="UTF-8">
            <title>Hair - Computer Graphics</title>
            <style>
/* From extension vscode.github */
/*---------------------------------------------------------------------------------------------
 *  Copyright (c) Microsoft Corporation. All rights reserved.
 *  Licensed under the MIT License. See License.txt in the project root for license information.
 *--------------------------------------------------------------------------------------------*/

.vscode-dark img[src$=\#gh-light-mode-only],
.vscode-light img[src$=\#gh-dark-mode-only],
.vscode-high-contrast:not(.vscode-high-contrast-light) img[src$=\#gh-light-mode-only],
.vscode-high-contrast-light img[src$=\#gh-dark-mode-only] {
	display: none;
}

</style>
            <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex/dist/katex.min.css">
<link href="https://cdn.jsdelivr.net/npm/katex-copytex@latest/dist/katex-copytex.min.css" rel="stylesheet" type="text/css">
        <link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/Microsoft/vscode/extensions/markdown-language-features/media/markdown.css">
<link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/Microsoft/vscode/extensions/markdown-language-features/media/highlight.css">
<style>
            body {
                font-family: -apple-system, BlinkMacSystemFont, 'Segoe WPC', 'Segoe UI', system-ui, 'Ubuntu', 'Droid Sans', sans-serif;
                font-size: 14px;
                line-height: 1.6;
            }
        </style>
        <style>
.task-list-item {
    list-style-type: none;
}

.task-list-item-checkbox {
    margin-left: -20px;
    vertical-align: middle;
    pointer-events: none;
}
</style>
<style>
:root {
  --color-note: #0969da;
  --color-tip: #1a7f37;
  --color-warning: #9a6700;
  --color-severe: #bc4c00;
  --color-caution: #d1242f;
  --color-important: #8250df;
}

</style>
<style>
@media (prefers-color-scheme: dark) {
  :root {
    --color-note: #2f81f7;
    --color-tip: #3fb950;
    --color-warning: #d29922;
    --color-severe: #db6d28;
    --color-caution: #f85149;
    --color-important: #a371f7;
  }
}

</style>
<style>
.markdown-alert {
  padding: 0.5rem 1rem;
  margin-bottom: 16px;
  color: inherit;
  border-left: .25em solid #888;
}

.markdown-alert>:first-child {
  margin-top: 0
}

.markdown-alert>:last-child {
  margin-bottom: 0
}

.markdown-alert .markdown-alert-title {
  display: flex;
  font-weight: 500;
  align-items: center;
  line-height: 1
}

.markdown-alert .markdown-alert-title .octicon {
  margin-right: 0.5rem;
  display: inline-block;
  overflow: visible !important;
  vertical-align: text-bottom;
  fill: currentColor;
}

.markdown-alert.markdown-alert-note {
  border-left-color: var(--color-note);
}

.markdown-alert.markdown-alert-note .markdown-alert-title {
  color: var(--color-note);
}

.markdown-alert.markdown-alert-important {
  border-left-color: var(--color-important);
}

.markdown-alert.markdown-alert-important .markdown-alert-title {
  color: var(--color-important);
}

.markdown-alert.markdown-alert-warning {
  border-left-color: var(--color-warning);
}

.markdown-alert.markdown-alert-warning .markdown-alert-title {
  color: var(--color-warning);
}

.markdown-alert.markdown-alert-tip {
  border-left-color: var(--color-tip);
}

.markdown-alert.markdown-alert-tip .markdown-alert-title {
  color: var(--color-tip);
}

.markdown-alert.markdown-alert-caution {
  border-left-color: var(--color-caution);
}

.markdown-alert.markdown-alert-caution .markdown-alert-title {
  color: var(--color-caution);
}

</style>

        </head>
        <body class="vscode-body vscode-light">
            <h1 id="hair---computer-graphics">Hair - Computer Graphics</h1>
<p>Peter HU, Nov 2024.</p>
<h2 id="dataset">Dataset</h2>
<ul>
<li><a href="https://avatarneo.com/">head mesh: avatar online</a></li>
<li><a href="https://huliwenkidkid.github.io/liwenhu.github.io/">USC-HairSalon (2015)</a>
<ul>
<li>online game assets</li>
<li><a href="http://www.thesimsresource.com/">ELECTRONIC ARTS, 2014</a></li>
<li><a href="http://www.newseasims.com/">Newsea SIMS</a></li>
</ul>
</li>
<li><a href="https://www.hao-li.com/Hao_Li/Hao_Li_-_publications_%5BSingle-View_Hair_Modeling_Using_A_Hairstyle_Database%5D.html">Single-view-hair (2015)</a></li>
</ul>
<h2 id="codebase">Codebase</h2>
<ul>
<li><a href="https://cs.yale.edu/homes/che/projects/perm/">PERM (2024)</a>
<ul>
<li>augment USC-HairSalon</li>
</ul>
</li>
<li><a href="https://github.com/facebookresearch/CT2Hair">CT2Hair (2023)</a></li>
<li><a href="https://haar.is.tue.mpg.de/">HAAR (2023)</a>
<ul>
<li><a href="https://github.com/SamsungLabs/NeuralHaircut">NeuralHaircut (2023)</a> for strand-based prior model;</li>
</ul>
</li>
<li><a href="https://github.com/GAP-LAB-CUHK-SZ/HairStep">Hairstep (2023)</a></li>
</ul>
<p>others: <a href="https://arxiv.org/pdf/2201.02610">Embodied Hands (MANO)</a> &amp; <a href="https://dl.acm.org/doi/pdf/10.1145/2816795.2818013">Body (SMPL)</a></p>
<h2 id="representation">Representation</h2>
<p>TODO: more recent papers, relationship with renderer, conversion between reps.</p>
<ul>
<li>hairstyling (modeling) + hair simulation + hair rendering</li>
<li>Shape category: long, curly or wavy hair / short, straight</li>
<li><a href="https://ieeexplore.ieee.org/document/4069232">A Survey on Hair Modeling: Styling, Simulation, and Rendering, K. Ward et al., 2007</a></li>
<li><a href="https://ieeexplore.ieee.org/abstract/document/8323371">A Survey of Image-Based Techniques for Hair Modeling, Y. Bao, 2018</a></li>
<li><strong>Tutorial</strong>
<ul>
<li><a href="https://jacob-lopez.github.io/SairHimulator/report.html">Kajiya Kay Model, Modified Marschner Model</a></li>
<li><a href="https://developer.download.nvidia.com/presentations/2010/SIGGRAPH/HairCourse_SIGGRAPH2010.pdf">Advanced Techniques in Real-time Hair Rendering and Simulation, Siggraph Course 2010</a></li>
</ul>
</li>
</ul>
<h3 id="geometric-based">Geometric based</h3>
<!-- curve (line), surface, cylinder, trigonal prism. -->
<table>
<thead>
<tr>
<th>reps</th>
<th>curve</th>
<th>surface</th>
<th>cylinder</th>
<th>trigonal<br />prism</th>
</tr>
</thead>
<tbody>
<tr>
<td>ex</td>
<td>line segment<br />sine waves<br />B-spline<br />Catmull-Rom</td>
<td>N-U R B-Spline<br />(strips)</td>
<td>generalized<br />cylinder</td>
<td>wisp based</td>
</tr>
<tr>
<td>pros</td>
<td>realistic</td>
<td>fewer geometry</td>
<td>by cluster</td>
<td></td>
</tr>
<tr>
<td>cons</td>
<td>large size</td>
<td>unrealistic</td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
<p><strong>poly line</strong>: a set of connected line segments</p>
<p>To reduce the large number, key/guide hairs + population</p>
<ul>
<li>a portion of all hair strands (key hairs) and</li>
<li>populating the rest based on the explicitly modeled ones.</li>
</ul>
<p><strong>Single Strand Interpolation</strong> creates hair strands around each explicitly modeled curve based on the shape of the curve. <em>Wisps and generalized cylinder</em> based techniques use this approach for generating a complete hair model.</p>
<p><strong>Multi Strand Interpolation</strong> creates hair strands in-between a number of explicitly modeled curves by interpolating their shapes.</p>
<p>cons: limited hairstyles, lack details</p>
<h4 id="curve">curve</h4>
<p>curve + thickness, e.g. sine waves, B-spline, Catmull-Rom spline</p>
<p>It ignores the details of the tubular shape of hair fibers, which is often unimportant, because, in most cases, the projected thickness of a hair strand is less than the size of a pixel on the
screen.</p>
<p>Rendering (line segment and cylinder)</p>
<blockquote>
<p>reference: <a href="https://dl.acm.org/doi/10.1145/74333.74361">Rendering fur with three dimensional textures, J. T. Kajiya, 1989</a></p>
</blockquote>
<p>the first shading model for hair rendering, widely cited.</p>
<ul>
<li>the diffuse component: the Lambert shading model w.r.t small cylinder by integration.</li>
<li>the specular component: an ad hoc model for for cylindrical surfaces similar to the Phong light reflection model.
<ul>
<li>or (unnecessarily) convert the surface micro-geometry to be represented in the volume directly to lighting models.</li>
</ul>
</li>
</ul>
<h4 id="u-shaped-strips">u-shaped strips</h4>
<blockquote>
<p>reference:  <a href="https://ieeexplore.ieee.org/document/1238296">An Enhanced Framework for Real-time Hair Animation, W. Liang et al., 2003</a>.</p>
</blockquote>
<p><em>Real-time conversion</em>  that project each vertex onto the scalp (spheres) and connect the vertices with their projections.</p>
<h4 id="thin-shell-volume">thin shell volume</h4>
<blockquote>
<p>reference:  <a href="https://ieeexplore.ieee.org/abstract/document/889055">A Thin Shell Volume for Modeling Human Hair, T. Kim et al., 2000</a>.</p>
</blockquote>
<p>surface + cylinder + particles (under constraints for hair-hair interaction)</p>
<p>Rendering</p>
<ul>
<li>shading model (line segment, see above).</li>
<li>antialiasing (line segment) via super-sampling, where each time shifting the image plane slightly.</li>
</ul>
<h4 id="wisp">wisp</h4>
<blockquote>
<p>reference:  <a href="https://link.springer.com/article/10.1007/s003710050169">A system of 3D hair style synthesis based on the wisp model, L. Chen et al., 1999</a>.</p>
</blockquote>
<p>cylinder or trigonal prism</p>
<p>Rendering</p>
<p>The <em>illumination</em> is given by ambient, diffuse and specular terms, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> is the angle between incoming light and normal, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span></span></span></span> is the angle between normal and view point,</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>C</mi><mi>h</mi></msub><mo>=</mo><msub><mi>L</mi><mi>a</mi></msub><msub><mi>K</mi><mi>a</mi></msub><mo>+</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">[</mo><msub><mi>K</mi><mi>d</mi></msub><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>+</mo><msub><mi>K</mi><mi>s</mi></msub><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mi>n</mi></msup><mo stretchy="false">(</mo><mi>θ</mi><mo>+</mo><mi>ϕ</mi><mo>−</mo><mi>π</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">C_h = L_a K_a + \sum_i L_i [K_d \cos\theta + K_s \sin^n(\theta+\phi-\pi)]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">h</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.3277em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7222em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mclose">)]</span></span></span></span></span></p>
<ul>
<li>for dry and black hair, reduce the weight of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">K_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mi>a</mi></msub></mrow><annotation encoding="application/x-tex">K_a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> but emphasize <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">K_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</li>
<li>This lighting model still has its limitations, however, such as the lack of a back light visual effect.</li>
<li>compute for three control lines of each wisp + interpolation for those strands within one wisp.</li>
</ul>
<p>Modified <em>shadow</em> Z-buffering method (shadow mask).</p>
<ul>
<li>3D data <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">x,y,z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span> instead of depth.  Assigning an effective radius to every point, we union these balls, forming a mask-like.</li>
<li>Complexity <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi mathvariant="normal">#</mi><mtext> objects</mtext><mo stretchy="false">)</mo><mo>→</mo><mi>O</mi><mo stretchy="false">(</mo><mi mathvariant="normal">#</mi><mtext> 2D array</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\# \text{ objects}) \rightarrow O(\# \text{ 2D array})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">#</span><span class="mord text"><span class="mord"> objects</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">#</span><span class="mord text"><span class="mord"> 2D array</span></span><span class="mclose">)</span></span></span></span>.</li>
</ul>
<p><em>Artificial objects</em> (e.g. hair pins, hair bands, ponytails, and braids)</p>
<ul>
<li>give some wisps a certain restriction. These wisps are required to pass through a region of 3D space.
<ul>
<li>For example, we can get a ponytail if we let the control lines of a group of wisps pass through a little circle behind the CG character's head.</li>
<li>Handling a braid is a little tricky. A braid is heavier and stiffer than a single hair strand or wisp. It basically consists three wisps. A repeated loop changes the positions of these three wisps. (However, they do not really have to be three individual wisps. They may be just three branches of one wisp. They may also consist of several wisps separately.)</li>
</ul>
</li>
</ul>
<h4 id="generalized-cylinder">generalized cylinder</h4>
<blockquote>
<p>reference: <a href="https://dl.acm.org/doi/10.1145/566654.566627">Interactive Multiresolution Hair Modeling and Editing T. Kim et al., 2002</a></p>
</blockquote>
<p>With a space curve <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">C</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{C}(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">C</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> and series of contours <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">R</mi><mo stretchy="false">(</mo><mi>θ</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{R}(\theta, t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">R</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> along the curve, GC defines the boundary of a hair cluster when <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">r=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>,</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi>r</mi><mo separator="true">,</mo><mi>θ</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold">C</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>r</mi><mi mathvariant="bold">R</mi><mo stretchy="false">(</mo><mi>θ</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi mathvariant="bold">N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>+</mo><mi mathvariant="bold">B</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>sin</mi><mo>⁡</mo><mi>θ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf(r, \theta, t) = \mathbf{C}(t) + r \mathbf{R}(\theta, t) [\mathbf{N}(t) \cos\theta  +  \mathbf{B}(t) \sin\theta]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">C</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathbf">R</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mopen">[</span><span class="mord mathbf">N</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">B</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">]</span></span></span></span></span></p>
<p>, where we define coordinate frame of the curve <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> to be the tangent vector, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> to be the principal normal vector, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> to be the bi-normal vector. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> as the angle around the tangent vector <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>, starting from the principal normal <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span>.</p>
<ul>
<li>auxiliary scale: add scale <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>N</mi></msub><mo separator="true">,</mo><msub><mi>s</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">s_N, s_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to two orthogonal directions.</li>
<li>auxiliary twist: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mo>=</mo><mi>θ</mi><mo>+</mo><mrow><mi mathvariant="bold">W</mi><mo stretchy="false">(</mo><mi mathvariant="bold">t</mi><mo stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex">\hat{\theta} = \theta + \mathbf{W(t)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9579em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mopen">(</span><span class="mord mathbf">t</span><span class="mclose">)</span></span></span></span></span>, updating the rotating angles of contours along the curve.</li>
</ul>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold">R</mi><mo stretchy="false">(</mo><mi>θ</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msub><mi>s</mi><mi>N</mi></msub><mo>⋅</mo><mi mathvariant="bold">N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>cos</mi><mo>⁡</mo><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mo>+</mo><msub><mi>s</mi><mi>B</mi></msub><mo>⋅</mo><mi mathvariant="bold">B</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>sin</mi><mo>⁡</mo><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf{R}(\theta, t) [    s_N \cdot \mathbf{N}(t) \cos\hat{\theta}  +  s_B \cdot \mathbf{B}(t) \sin\hat{\theta}  ]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">R</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2079em;vertical-align:-0.25em;"></span><span class="mord mathbf">N</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5945em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2079em;vertical-align:-0.25em;"></span><span class="mord mathbf">B</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></p>
<p>When <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">t=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>, the hair strand is on the scalp (head model). Catmull-Rom spline curve for smoothness.</p>
<p>multi-resolution: after GC, he/she subdivides each hair cluster, adding more detail until the desired appearance is achieved.</p>
<p>Rendering:</p>
<ul>
<li>OpenGL <strong>poly line</strong></li>
<li>shading: use the model for line segment (see above) and disable the lighting calculation in OpenGL.
<ul>
<li>The shaded color is computed at each point of the line segments and colors are interpolated with OpenGL.</li>
<li>Other shading models such as that in [Goldman 1997] could be equally applicable.</li>
</ul>
</li>
<li>self-shadowing: use opacity shadow maps algorithm [Kim and Neumann 2001], a fast approximation of deep shadow maps [Lokovic and Veach 2000].
<ul>
<li>since shadows are view-independent, they can be computed once and cached for reuse while the user interactively changes views.</li>
</ul>
</li>
<li>antialiasing: drawing order based on the distance from the camera, inspired by [Levoy and Whitted 1985].
<ul>
<li>OpenGL antialiased line drawing option alone is not sufficient.</li>
</ul>
</li>
</ul>
<h4 id="mesh">mesh</h4>
<blockquote>
<p>reference: <a href="http://www.cemyuksel.com/research/hairmesh_rendering/">Hair mesh, C. Yuksel et al, 2009</a></p>
</blockquote>
<p>Layers of polygonal meshes set <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi>N</mi></msubsup><msup><mi>F</mi><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\sum_{k=0}^N F^{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2809em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9812em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span></span>, which typically contain quadrilateral or triangular faces, with no restrictions
on the types of polygons used. Layer at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> has one-to-one correspondence to a face at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>.</p>
<ul>
<li>support Face Delete, Layer Insert / Remove,  Edge/ Vertex Separate.</li>
</ul>
<h3 id="volumetric-based">Volumetric based</h3>
<p>basic element: voxels grid or bounding box.</p>
<p>pros: efficient spatial-aware intermediate representation to encode hair, for all sorts of downstream tasks.</p>
<p>cons: when hair animates, such volumes should be updated for every frame, making pre-filtering inefficient.</p>
<blockquote>
<p>reference:  <a href="https://www.sciencedirect.com/science/article/pii/S2468502X18300652">Hair-GAN: Recovering 3D hair structure from a single image..., M. Zhang, 2019</a>, etc. [full lists at <a href="https://cs.yale.edu/homes/che/projects/perm/">PERM (2024)</a> §2. related work.]</p>
</blockquote>
<p>3D volumetric field, grid of shape <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mi>x</mi></msub><mo>×</mo><msub><mi>n</mi><mi>y</mi></msub><mo>×</mo><msub><mi>n</mi><mi>z</mi></msub><mo>=</mo><mn>128</mn><mo>×</mo><mn>192</mn><mo>×</mo><mn>128</mn></mrow><annotation encoding="application/x-tex">n_x \times n_y \times n_z= 128 \times 192 \times 128</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">128</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">192</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">128</span></span></span></span>, aligned each hair model to a unified bust model within a volumetric bounding box.</p>
<ul>
<li>occupancy field (binary True and False), confidence map for reverse work</li>
<li>flow vector field / orientation field
<ul>
<li>see above</li>
</ul>
</li>
<li>hierarchical (top-down approach)
<ul>
<li>see above multi-resolution</li>
</ul>
</li>
</ul>
<h4 id="vector-field-points">Vector field (points)</h4>
<p>For each grid vertex <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">v</mi><mi mathvariant="bold">i</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{v_i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, Laplace operator <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">v</mi><mi mathvariant="bold">i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∑</mo><mrow><mi>j</mi><mo>∈</mo><mi>N</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold">t</mi><mi mathvariant="bold">j</mi></msub><mo>−</mo><msub><mi mathvariant="bold">t</mi><mi mathvariant="bold">i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta(\mathbf{v_i}) = \sum_{j \in N(i)} (\mathbf{t_j}-\mathbf{t_i})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2247em;vertical-align:-0.4747em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2253em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">i</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4747em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> is adopted in mesh editing for smooth deformation. Minimize with constraint index set <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>,</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">t</mi><mn mathvariant="bold">1</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi mathvariant="bold">t</mi><mi mathvariant="bold">n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">t</mi><mi mathvariant="bold">i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><msup><mi mathvariant="normal">∣</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><munder><mo>∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>C</mi></mrow></munder><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><msub><mi mathvariant="bold">t</mi><mi mathvariant="bold">i</mi></msub><mo>−</mo><msub><mi mathvariant="bold">c</mi><mi mathvariant="bold">i</mi></msub><mi mathvariant="normal">∣</mi><msup><mi mathvariant="normal">∣</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E(\mathbf{t_1}, ..., \mathbf{t_n}) = \sum_i || \Delta (\mathbf{t_i}) ||^2 + \omega \sum_{i \in C} || \mathbf{t_i} - \mathbf{c_i} ||^2
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1611em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3277em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣∣Δ</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">∣</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.3717em;vertical-align:-1.3217em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8557em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.3217em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣∣</span><span class="mord"><span class="mord mathbf">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p>
<ul>
<li>weighted tangent vectors, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mi>r</mi><mi>a</mi><mi>d</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">∇</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">grad(f) = \nabla f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span></li>
<li>iteratively re-weighted least square of a linear system</li>
</ul>
<blockquote>
<p>reference:  <a href="http://www.kunzhou.net/2009/hairSyn.pdf">Example-Based Hair Geometry Synthesis, L. Wang, 2009</a></p>
</blockquote>
<ul>
<li>(achieved) First, when hair roots are close on the scalp surface, corresponding strands tend to form wisps with similar geometry.</li>
<li>(aim to solve) Second, every hair strand has a curved trajectory in the hair volume and ( those from <em><strong>different</strong></em> hair roots, but) spatially close portions of the trajectories tend to share similar <strong>tangent vectors</strong>. &lt;- added</li>
</ul>
<h3 id="volume-density-function">Volume density function</h3>
<blockquote>
<p>reference:  <a href="https://dl.acm.org/doi/10.1145/74334.74359">Hypertexture, K. Perlin, 1989</a>.</p>
</blockquote>
<p>Rendering</p>
<p><em>Volume rendering</em> with</p>
<ul>
<li>object density functions <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">]</mo><mo separator="true">,</mo><mi>x</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mn mathvariant="double-struck">3</mn></msup></mrow><annotation encoding="application/x-tex">D(x) \in [0,1], x \in \mathbb{R^3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></li>
<li>density modulation functions for soft regions <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(x) \in (0,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span>.
<ul>
<li>bias to push up/down, variance gain (gradient flatter or steeper), noise, turbulence, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|x|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sin(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">in</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>.</li>
</ul>
</li>
</ul>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>Hypertexture</mtext><mo>=</mo><msub><mi>f</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><msub><mi>f</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>f</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Hypertexture} = f_n ( . . . . f_2 (f_1 (f_0 (D(x)))))
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">Hypertexture</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">....</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)))))</span></span></span></span></span></p>
<p>Pros: a significant reduction of storage requirements + naturalness of the transparent property of hair regions.</p>
<p>Cons: relatively slow rendering performance <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, typically a few hours as originally reported (1999); despite its power, one does <strong>not</strong> have an explicit control over its shape, but</p>
<blockquote>
<p>reference:  <a href="https://www.sciencedirect.com/science/article/pii/S1524070399905180">The Cluster Hair Model, X. Yang et al., 2000</a>.</p>
</blockquote>
<p>Embed a volume density model into a generalized cylinder.</p>
<ul>
<li>Each cluster is first approximated by a polygonal boundary (mask).</li>
<li>When a ray hits the polygonal surface, predefined density functions are used to accumulate density. By approximating the high frequency detail with volume density functions, the method produces antialiased images of hair clusters.</li>
<li>Cons: this method does not allowchanges in the density functions, making hairs appear as if they always stay together.</li>
</ul>
<h3 id="complex-fluid-flow">Complex fluid flow</h3>
<blockquote>
<p>reference:  <a href="https://vdoc.pub/download/computer-animation-and-simulation-2000-proceedings-of-the-eurographics-workshop-in-interlaken-switzerland-august-2122-2000-sf9i3fdjd9c0">Interactive Hair Styler based on Fluid Flow, S. Hadap et al., 2000, P87</a>.</p>
</blockquote>
<p>explicit control + reduce complexity + volumetric</p>
<p>Ideal flow with elements: stream + source / vortex</p>
<ul>
<li>incompressible (density won't change)</li>
<li>stable, inviscid (no viscosity)</li>
<li>ir-rotational</li>
</ul>
<p>Laplace equation (PDE) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>ϕ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\nabla^2 \phi = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal">ϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>, where potential <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span></span></span></span> and velocity <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>V</mi><mo>⃗</mo></mover><mo>=</mo><mi mathvariant="normal">∇</mi><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\vec{V} = \nabla \phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∇</span><span class="mord mathnormal">ϕ</span></span></span></span>.</p>
<ul>
<li>complex flow, i.e. linear combination <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mo>∑</mo><mi>i</mi></msub><mover accent="true"><msub><mi>V</mi><mi>i</mi></msub><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\sum_i \vec{V_i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.266em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.162em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9663em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span> defined by user (as the sum of sols to PDE is also a valid sol.)</li>
<li>small sources along the the boundary panel <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, each having <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> contributions, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mo>∑</mo><mi>j</mi></msub><msub><mi>λ</mi><mi>j</mi></msub><mover accent="true"><msub><mi>S</mi><mi>j</mi></msub><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\sum_j \lambda_j \vec{S_j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4021em;vertical-align:-0.4358em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.162em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9663em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span> (obstacle avoidance)
<ul>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>V</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo>⃗</mo></mover><mo>⋅</mo><mover accent="true"><msub><mi>n</mi><mi>i</mi></msub><mo>^</mo></mover><mo>=</mo><msub><mi>b</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\vec{V(p_i)} \cdot \hat{n_i} = b_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.283em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.033em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.319em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.25em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, streamline will be parallel to the boundary surface, except the starting point.</li>
<li>solve <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>, given the other variables are const.</li>
<li>compute for the coarse grid and subdivision flow for points within (cheaper computation) and combine with <em>volume density function (noise / perturbations).</em></li>
</ul>
</li>
</ul>
<p>Rendering</p>
<p>volume rendering / visualization techniques</p>
<h3 id="texture-based">Texture based</h3>
<p>Texture mapping the surfaces with hair images and using alpha mapping to create fake illusion of hair strands.</p>
<h3 id="conversion">Conversion</h3>
<blockquote>
<p>reference: <a href="https://dl.acm.org/doi/10.1145/2766931">Single-view hair modeling using a hairstyle database, Hu et al., 2015</a></p>
</blockquote>
<p>Pose transformation (fixed standard head model) via translating, rotating and/or scaling. (flipping symmetrically in the end)</p>
<p>Mesh ---- <em>uniform sampling</em> -----&gt; hair strands ----&gt; 3D orientation field -----&gt; hair strands</p>
<p>Each strand is represented as a set of equally spaced sample points. Guided by the diffused 3D orientation field, smoothly diffuse the field into the entire 3D volume as proposed by Paris et al. [2008].</p>
<!-- TODO: add more -->
<blockquote>
<p>reference: <a href="https://www.sciencedirect.com/science/article/pii/S2468502X18300652">Hair-GAN: Recovering 3D hair structure from a single image..., M. Zhang, 2019</a></p>
</blockquote>
<p>polygon-strips --------&gt; dense 3D orientation volume</p>
<h3 id="low-quality-hair">Low-quality hair</h3>
<ul>
<li>idea1: texture based (IQOO 13 hardware)</li>
<li>idea2: downsample hair + boundary (guide strands)</li>
<li>idea3: sketch from users.</li>
<li>idea4: fitting to surface (with somehow good quality)
<ul>
<li>it is very difficult to get realistic results with surface approximations of hair for most hair models.</li>
</ul>
</li>
</ul>
<h2 id="renderer">Renderer</h2>
<p>High-quality hair</p>
<ul>
<li>Blender (ok, mesh 1GB...)</li>
<li>Mitsuba (doesn't support)</li>
<li>poly line in OpenGL (TODO)</li>
</ul>
<h3 id="rendering-techniques">Rendering techniques</h3>
<ul>
<li>shading model (diffuse + specular)
<ul>
<li><a href="https://jacob-lopez.github.io/SairHimulator/report.html">Kajiya Kay Model (1989), Modified Marschner (2003) Model</a></li>
<li>modified illumination for line segment, cylinder</li>
</ul>
</li>
<li>multiple scattering
<ul>
<li>path tracing ~ 100 hours</li>
<li>photon mapping ~ 1.5 hours</li>
<li>spherical harmonics in a grid ~ 30 minutes</li>
<li>dual scattering ~ 20 fps</li>
</ul>
</li>
<li>shadow
<ul>
<li>depth test or z-buffer (with modification)</li>
<li>deep shadow maps [Lokovic and Veach 2000]</li>
<li>opacity shadow maps algorithm [Kim and Neumann 2001]</li>
<li>density clustering [Mertens et al. 2004]</li>
<li>deep opacity maps [Yuksel et al. 2008]</li>
<li><strong>self-shadowing</strong> is essential for volumetric hair.</li>
</ul>
</li>
<li>antialiasing
<ul>
<li>since hair strands are very thin, it is important to draw them smoothly with correct filtering.</li>
<li>depend on the drawing order [McReynolds 1997]</li>
</ul>
</li>
<li>color and texture
<ul>
<li><em>artificial objects</em> (e.g. hair pins, hair bands, ponytails, and braids)</li>
<li>interpolated colors</li>
</ul>
</li>
<li>level of detail (LOD)</li>
</ul>
<h3 id="blender-conversion-steps">Blender conversion steps</h3>
<ol>
<li>
<p>Verify the Converted Hair</p>
<p>After converting hair (via Object &gt; Convert &gt; Mesh or similar), inspect the result in Edit Mode.
You will see vertices along the strands but no faces or thickness.</p>
</li>
<li>
<p>Convert to Curve</p>
<p>Select the hair mesh in Object Mode.
Go to the Object menu: Object &gt; Convert To &gt; Curve from Mesh/Text.
This converts the vertex strands into curves, making them easier to manipulate.</p>
</li>
<li>
<p>Add Thickness to the Curves</p>
<p>Go to the Object Data Properties (the green curve icon in the Properties Panel).
Under the Geometry section, locate the Bevel subsection.
Adjust the Depth value to add thickness to the strands.
Use the Resolution slider to refine the shape.</p>
</li>
<li>
<p>Convert Back to Mesh (Optional)</p>
<p>Once the curves have the desired thickness, convert them back to a mesh:
Object &gt; Convert To &gt; Mesh from Curve/Meta/Surf/Text.</p>
</li>
</ol>

            <script async src="https://cdn.jsdelivr.net/npm/katex-copytex@latest/dist/katex-copytex.min.js"></script>

        </body>
        </html>