Download Path Tracing and More

Path Tracing
and
More
Notation again
• L(P, w): light leaving P in direction w
• To estimate light arriving at the eye, we trace to a point P, and
compute L(P, w), where w points from P to the eye.
• Five terms:
1.
2.
3.
4.
5.
Emitted light E(P,w)
diffusely scattered direct light
specularly scattered direct light
diffusely scattered indirect light
specularly scattered indirect light
1. diffusely scattered direct light:
1.
2.
Single-sample estimate, samples taken over light sources
include visibility term, squared distance, dot product with light-normal
2. specularly scattered direct light
1.
Integral reduces to a sum over "impulse directions"
3. diffusely scattered indirect light
1.
Single sample estimate
4. specularly scattered indirect light
1.
Integral reduces to a sum over "impulse directions"
• Can combine 2 and 4 to specular scattering of total light from specularreflection direction
• Case 3 requires computing radiance leaving some point Q, but only scattered,
not emitted.
• Our "estimateL" procedure should take an extra parameter saying whether to include
emitted radiance or not
• Cases 3 and 4 involve recursion: control the probability of recurring (in 3) by
the overall reflectance of the material at P or (in 4) by the "impulse factor".
Classes of light paths (Heckbert, Shirley, ???)
• L = light source (emitter)
• S = specular
• D = diffuse
• E = eye
• Regular-expression syntax
•
•
•
•
*: zero or more
?: 0 or 1
+: 1 or more
|: or
• Read left-to-right
Light path examples
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LE: a lamp in the scene that's visible to the eye.
LDE: directly lit surface
L(D|S)E: one bounce path from either a diffuse or specular surface
Quiz: leaves light, then one or more diffuse bounces, then a specular
bounce, then eye
• Quiz 2: What light paths did your raytracer handle?
• Note: 'light path' is a mistaken terminology
• really describes a scattering sequence
• LDE and LSE could both describe a path taken by a photon from a light point, to a
surface point P, to the eye.
• Note 2: sometimes extend notation to G (glossy) and T (transmission)
Rendering algorithms and the paths they
handle
• Appel's ray-casting: L(D|G)E
• Whitted ray-tracing: E [S*](D|G)L
• Kajiya path-tracing: E [(D|G|S)+](D|G)]L
• Radiosity: E D* L
Transmitted light, reflected light, waves
• The particle model isn't enough to explain the world
• Week 1 phenomena:
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double rainbows
mirages
crepuscular rays
water vapor: why is water different colors in different phases?
color when wet: wet dirt and wet wood both get darker
shimmering oil on water
hair color
fuzzy things are lighter around edges
Bokeh effect on cameras
xmas light haloes/spikes
nighttime red sky
prisms and splitting into rainbow colors
moon looks bigger near horizon
transparent skim milk
distant mountains look blue-grey
Wave Theory of light explains many of these
• Light is an electromagnetic wave
• The electric field due to a beam of light is orthogonal to the direction of
propagation; magnetic field orthogonal to both
• Field is a planar wave: electric field is the same at all points of a plane
perpendicular to direction of propagation.
• Wave is sinusoidal
• Moves with speed 𝑐 (speed of light) in empty space.
• For a light-wave moving in the x direction, write 𝐸(𝑥, 𝑦, 𝑧, 𝑡) for electric
field vector at point (𝑥, 𝑦, 𝑧) at time 𝑡.
• Independent of 𝑦, 𝑧; so just write 𝐸 𝑥, 𝑡
• The vector points only in the 𝑦𝑧 plane, so 𝐸 𝑥, 𝑡 = 0, 𝐸𝑦 𝑥, 𝑡 , 𝐸𝑧 𝑥, 𝑡
Sinusoidal wave
• 𝐸 𝑥, 𝑡 = 0, 𝐸𝑦 𝑥, 𝑡 , 𝐸𝑧 𝑥, 𝑡
• 𝐸𝑦 𝑥, 𝑡 =
𝑥
𝐴𝑦 sin(2𝜋
𝜆
− 2𝜋𝑓𝑡 + Δ𝑦 )
• 𝜆 = wavelength. Fix 𝑡, and this looks like a sine function that completes one period as 𝑥
ranges over a length 𝜆
• 𝑓= frequency. Fix 𝑥, and this looks like
a sine function that varies up and down as 𝑡
1
ranges over a time period of length .
𝑓
• Δ𝑦 = phase offset: perhaps the sine wave isn't zero at 𝑥 = 𝑡 = 0, because the axes are
shifted.
• 𝐴𝑦 , 𝐴𝑧 are "amplitudes" that tell how bright the light is.
• Total amplitude 𝐴 =
𝐴2𝑦 + 𝐴𝑧2
Sinusoidal wave
• 𝐸 𝑥, 𝑡 = 0, 𝐸𝑦 𝑥, 𝑡 , 𝐸𝑧 𝑥, 𝑡
• 𝐸𝑦 𝑥, 𝑡
• 𝐸𝑧 𝑥, 𝑡
𝑥
= 𝐴𝑦 sin(2𝜋
𝜆
𝑥
= 𝐴𝑧 sin(2𝜋
𝜆
− 2𝜋𝑓𝑡 + Δ𝑧 )
𝑥
= 𝐴𝑦 sin(2𝜋
𝜆
𝑥
= 𝐴𝑧 sin(2𝜋
𝜆
𝑐
− 2𝜋 𝑡 + Δ𝑦 )
𝜆
𝑐
− 2𝜋 𝑡 + Δ𝑧 )
𝜆
− 2𝜋𝑓𝑡 + Δ𝑦 )
• Because speed of light is 𝑐 (in empty space), this becomes
• 𝐸𝑦 𝑥, 𝑡
• 𝐸𝑧 𝑥, 𝑡
i.e., 𝑓 =
𝑐
.
𝜆
Sinusoidal wave
• 𝐸 𝑥, 𝑡 = 0, 𝐸𝑦 𝑥, 𝑡 , 𝐸𝑧 𝑥, 𝑡
• 𝐸𝑦 𝑥, 𝑡
• 𝐸𝑧 𝑥, 𝑡
𝑥
= 𝐴𝑦 sin(2𝜋
𝜆
𝑥
= 𝐴𝑧 sin(2𝜋
𝜆
𝑐
− 2𝜋 𝑡 + Δ𝑦 )
𝜆
𝑐
− 2𝜋 𝑡 + Δ𝑧 )
𝜆
• Relationship between 𝐴𝑦 and 𝐴𝑧 ? None.
• Relationship between Δ𝑦 , Δ𝑧 ? None.
• Difference Δ𝑦 − Δ𝑧 has interesting effect on shape of field.
Δ𝑦 − Δ𝑧 = 0
• So Δ𝑦 = Δ𝑧 ; by adjusting position of axes, assume both are zero
𝑥
𝑐
𝐸𝑦 𝑥, 𝑡 = 𝐴𝑦 sin(2𝜋 − 2𝜋 𝑡) = 𝐴𝑦 sin 𝑄𝑥 + 𝑅𝑡
𝑥𝜆
𝑐𝜆
𝐸𝑧 𝑥, 𝑡 = 𝐴𝑧 sin(2𝜋 − 2𝜋 𝑡) = 𝐴𝑧 sin 𝑄𝑥 + 𝑅𝑡
𝜆
𝜆
• These are multiples of one another.
• "Linearly polarized"
• Sketch!
𝜋
Δ𝑦 − Δ𝑧 =
2
• So Δ𝑦 = Δ𝑧 ; by adjusting position of axes, assume are zero
𝑥
𝑐
𝐸𝑦 𝑥, 𝑡 = 𝐴𝑦 sin(2𝜋 − 2𝜋 𝑡) = 𝐴𝑦 sin 𝑄𝑥 + 𝑅𝑡
𝑥𝜆
𝑐𝜆
𝐸𝑧 𝑥, 𝑡 = 𝐴𝑧 sin(2𝜋 − 2𝜋 𝑡) = 𝐴𝑧 sin 𝑄𝑥 + 𝑅𝑡
𝜆
𝜆
• These are multiples of one another.
• "Linearly polarized"
• Sketch!
𝜋
Δ𝑦 − Δ𝑧 =
2
• So Δ𝑦 = Δ𝑧 +
𝜋
;
2
by adjusting position of axes, assume Δ𝑧 = 0
𝑥
𝑐
𝜋
𝐸𝑦 𝑥, 𝑡 = 𝐴𝑦 sin(2𝜋 − 2𝜋 𝑡) = 𝐴𝑦 sin 𝑄𝑥 + 𝑅𝑡 +
𝜆𝑥
𝜆𝑐
2
𝐸𝑧 𝑥, 𝑡 = 𝐴𝑧 sin(2𝜋 − 2𝜋 𝑡) = 𝐴𝑧 sin 𝑄𝑥 + 𝑅𝑡
𝜆
𝜆
• Rewrite slightly
𝐸𝑦 𝑥, 𝑡 = 𝐴𝑦 cos 𝑄𝑥 + 𝑅𝑡
𝐸𝑧 𝑥, 𝑡 = 𝐴𝑧 sin 𝑄𝑥 + 𝑅𝑡
(where I've absorbed a sign-change into 𝐴𝑦 )
• "Circularly polarized"
Other cases
𝜋
:
2
• 𝐴𝑦 and 𝐴𝑧 are different, or Δ𝑦 − Δ𝑧 isn't a multiple of elliptical
polarization
• Theorem: Every wave can be written as a sum of a linearly and a
circularly polarized wave.
• Corollary: Understanding these two cases lets you know the whole
story.
What happens when light is in a conductive
or semi-conductive material?
• Electric potential is constant along conductors
• So electric field, in direction of conductors, is zero (or near zero, for
slightly resistive conductors)
• As light passes through a material with many embedded wires…
• the electric field in the horizontal direction gets killed off
• …and the wires heat up
Wires in a transparent material???
• Yep!
• You stretch plastic in one direction as it cools, aligning long, slightly
conductive, molecule chains with the stretch-axis
• Get "polarized plastic"
• Make sunglasses from it and sell them for $$$
Polarization of everyday light
• Total mix of polarizations and amplitudes
• When such light passes through a linear polarizer, it's attenuated by
about a factor of 0.7 ≈
2
.
2
• But it comes out linearly polarized!
Wave-like light at non-conducting interfaces
(like water)
•
•
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•
Sum of waves at the interface must be zero
Light moves more slowly in the water
Light has same frequency in the water, hence different wavelength
𝑐
𝜈 = . 𝜈 called "index of refraction". Often written 𝑛 instead of 𝜈.
v
Always at least 1.
Continuity of electric field at interface
• To make the arriving and refracted fields "line up", need
this picture:
𝜃1
𝑛1
𝑛2
𝜃2
𝑛2
𝑛1
=
sin 𝜃1
sin 𝜃2
in
Refraction at interface
•
𝑛2
𝑛1
=
sin 𝜃1
sin 𝜃2
• "Snell's Law" (Willebrord Snellius, 1580 – 1626 CE)
• Ibn Sahl (984 CE)
At interface, energy changes
• Energy is related to 𝐴𝑓 … but 𝐴 gets altered
• For energy balance, and not just continuity, there must be a reflected
wave as well
•
•
•
•
Again, assumes no heating losses in material!
Continuity says 𝜃r = 𝜃i
𝜃t determined by Snell/Sahl
What about magnitudes?
𝜃𝑖
𝜃𝑟
𝑛1
𝑛2
𝜃𝑡
Reflected/transmitted magnitudes
• For light whose electric field is in the plane of the scattering surface
(p-polarized light) the fraction reflected 𝑅𝑝 given by
𝑛2 cos 𝜃𝑖 − 𝑛1 cos 𝜃𝑡
𝑟𝑝 =
; 𝑅𝑝 = 𝑟𝑝2
𝑛2 cos 𝜃𝑖 + 𝑛1 cos 𝜃𝑡
• Fraction transmitted 𝑇𝑝 = 1 − 𝑅𝑝
• For light whose electric field is in the plane determined by the normal
and the propagation direction (s-polarized),
𝑛1 cos 𝜃𝑖 − 𝑛2 cos 𝜃𝑡
𝑟𝑠 =
; 𝑅𝑠 = 𝑟𝑠2
𝑛1 cos 𝜃𝑖 + 𝑛2 cos 𝜃𝑡
Consequences
• Incoming equal mix of s- and p-polarized waves is reflected/transmitted as
an unequal mix of s- and p-polarized waves
• Common to assume arriving light is equal-mix polarized
• In that case, reflected energy fraction is
1
𝑅𝐹 = (𝑅𝑠 + 𝑅𝑝 )
2
• "F" is for Fresnel
• Radiance changes at interfaces because of speed-of-light change, affecting
wavelength.
𝐿 𝑃, 𝜔𝑟 = 𝑅𝐹 𝐿 𝑃, −𝜔𝑖
𝑛22
𝐿 𝑃, 𝜔𝑡 = (1 − 𝑅𝐹 ) 2 𝐿 𝑃, −𝜔𝑖
𝑛1
Getting Fresnel effects right
• Surprisingly difficult
• When we move from a dense medium (e.g., glass) to a less-dense one
(e.g., air), with 𝑛1 > 𝑛2 , there's an angle at which 𝑅𝑠 = 𝑅𝑝 = 1, i.e.,
all light is reflected. Approx 41 degrees for glass in air.
• Called "total internal reflection"; essential for some touch interfaces.