A Simple Method for Extracting the Natural Beauty of Hair

1
A Simple Method for Extracting the Natural Beauty
of Hair
How many roads must a man walk down

• Ken-ichi Anjyo, Yoshiaki Usami, Tsuneya Kurihara
• Presented by Chris Lutz

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Presentation Overview

• The Dr. B presentation style
• Ill just read the first sentence of each
  paragraph and try to wing it.
• Next, Ill write some vocab on the board.
• Finally, Ill scribble down some incomplete
  matrices and run out of time.
• (Im assuming Dave can take a joke.)
• Man, its late. Wow.

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Real Presentation Overview
(Its not all that much better, really)

• How hair is actually represented and how it is
  allowed to move
• WHAM! Whole lotta math comin atchya
• Dynamic behaviors
• Coordinate system
• Inertia and force fields
• Rendering Techniques Results

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Philosophy and Caveats

• This paper is concerned (oddly enough) with the
  natural beauty of hair.
• To that end, the authors sacrifice some realism
• Physically-based simulation
• Collision detection between hair body as well
  as hair other hair
• Shadowing of hair onto itself
• In a sense, theyre after pretty pictures

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Modeling Process Overview

• 1. Create a model of a head
• 2. Define an ellipsoidal hull
• A rough approximation, yes, but not too bad
• Its faster for collisions pore placement
• 3. Calculate hair bending (cantilever beam)
• 4. Cut and style as desired
• Essentially, blow-dry the hair by applying
  various directional forces

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Not So Touchably Soft

• Hair is modeled as a series of segments connected
  at bending points
• Apply some force to the beam and watch it deform
• d2y/dx2 -M/(EI)
• E Youngs Modulus
• I 2nd momentum of area

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Get Volume Through Math

• Bending by sequentially averaged concentrated
  loads
• Mi -gd(?1k-i1p ?1k-ip)/2
  -gd(k-i1)2/2
• g force on the hair
• 1..k number of segments in the hair
• d segment length
• Displacement of node yi (-1/2)(Mi/EI)d2
• 2D define a new vector ei of magnitude yi
• ei pi-2?pi-1 yi
• pi (d/ei)ei pi-1
• New node pi (d/ei)ei pi-1

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3D Bending

• Set up coordinate system
• Use 2D deformation formulas along both a0 (x) and
  a2 (z) axes
• The deflectional vector is is then just y1a1
  y2a2

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Avoiding the Issue of Hair Piercing the Skull

• Since collision detection using the actual head
  model is hard, revert to using an ellipsoidal
  representation
• Check every new pi for collision
• If the new pi collides with the head, move it to
  a close point on the plane defined by pi-2,
  pi-1, and pi
• Which way do you move it?

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The Taming of the Do

• (a) Initial conditions zero-g bed-head
• (b) Gravity kicks in
• (c) Apply external forces (blow-dry) the hair
  cut (define y-threshold and pore location)
• (d) Paul Mitchel would be proud

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Other Hip Styles
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The Answer, My Friend...

• So we want to add wind that means keeping track
  of inertia and applied forces
• Again, some realism issues
• Hair is modeled as rigid segments connected with
  flexible joints
• Hair-to-hair collisions friction is not modeled
  in a physically correct manner
• Use a pseudo force-field and solve differential
  equations

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Single Hair Dynamics

• Set up an initial polar coordinate system like
  the one to the right
• Track the projection of the hair onto the ? and ?
  planes

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Plane Definitions
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More Math Stuff

• ?i(t) d2?i/dt2 ci ui F?
• ?i(t) d2?i/dt2 ci vi F?
• Variables
• ci reciprocal number of the inertia moment of si
• ui (1/2)si
• vi half of segment si projected onto ? plane
• F? F? the respective force components

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Still More Math Stuff

• Given ?in-1 and ?in
• ?in1 ?in1-2?in?in-1 (?t)2ciuiF?
• Similar thing for ?in1
• Outer loop is segment number I
• Inner loop is time loop n
• By selectively manipulating the cis, you can
  simulate frictional effects
• You can also manipulate joint stiffness

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Hey Look! A Dab of Reality!

• We should try and keep track of the inertia
  moment of hair
• kd length of hair (k segments of length d)
• ? line density
• IS (1/3)?(kd)2
• Ii (?/3i)k3d2
• Putting this in for (?t)2ciui results in
  (3(?t)2i) / (2k3?d)

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A Bouncy Hold All Day Long

• (a) 10,000 hairs of length lt 18
• (b) F (-200, 0, 0), 10th frame
• (c) F (-20, -250, 0), 15th frame
• (d) 20th frame shows the effects of the force
  (wind) shift

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Pseudo-Force Fields

• Again, hairs could pierce the skull (bad)
• Define a pseudo-force field to replace the
  specified force for hairs close to the head
• Make this PsFF ellipsoidal to make it easier

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Pseudo-Force Field Definition

• F original user-specified force (wind)
• Di segment direction of si defined in terms of
  the ellipsoid
• E(p) Di (Ex(pi), Ey(pi), Ez(pi))
• Ex, Ey, Ez partial derivatives of polynomial
• For some value ?(? lt 1)
• If (Di, F) lt ?Di?F then si is facing
  into F
• If si is near the head, replace F with ?iF
• (near pore) 0 lt ?i lt 1 (end of hair)
• Ellipsoidal representation wont always look good

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Lighting Silky Shiny

• Diffuse term is neglected
• Instead of polylines, assume the hair is
  cylindrical
• ?s(P) ks(1-(T,H)2)n/2
• n specular exponent
• ks reflection coeff.
• Close to Phong shading

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Rendering Time Measures

• Use z-buffer algorithm to render the hair
  linearly interpolate colors across segments
  (oversample if hairs are thin)
• 20,000 hairs with lt 20 segments 50 sec for
  modeling, several for rendering on a SG Iris
  Power Series w/ VGX gfx board
• The image with a different camera angle was
  obtained almost in real-time

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Split Ends (Closing Issues)

• If you define (complicated) regional forces, you
  could create almost any hairstyle, right?
• It seems hard to add any human touches, e.g.
  ponytails or dreadlocks.
• In all honesty, when would anyone really care
  about the exact physics of hair?
• It would be nice to have a shot of
  lighter-colored hair, just to see what it looked
  like.