Blog

Marc Alexa, TU Berlin
CG
1
Computer Graphics I
Lighting, Shading
Marc Alexa, TU Berlin
CG
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3D Graphics Pipeline
§ Application
§ Geometry
Perspective Transformation, canonical view volume
Clipping
Culling
Approximation of light transport
Projection
§ Rasterization
§ Output
Marc Alexa, TU Berlin
CG
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Local shading model
§ Shading
Calculating the lightness (gray-value) of primitives
Model typically considers
– Light source (position, direction, intensity, spectral properties), and
– Surface properties (geometry, reflection behavior) of objects.
Simulation of light transport
Real time computation requires simplified models
Shading models are based on physical laws of optics and human
perception
Marc Alexa, TU Berlin
CG
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Light / object interaction
§ Light in CG is modeled with 3 quantities
§ Interaction with object surface = change of the three
quantities
Material properties = type of change and dependence on geometry
of situation
§ We limit interaction to reflection, only
Marc Alexa, TU Berlin
CG
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Light sources
§ Directed / parallel light source
– Position is point at infinity: (x,y,z,0)T
§ Point light source
– Finite position: (x,y,z,1)T
§ Spot
– Position, direction, opening angle
§ Ambient light
Marc Alexa, TU Berlin
CG
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Ambient light
§ Scene with only ambient light
Marc Alexa, TU Berlin
CG
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Directed light
§ Scene with ambient light and parallel light source
Marc Alexa, TU Berlin
CG
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Point light
§ Scene with ambient light and point light source
Marc Alexa, TU Berlin
CG
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Types of reflection
§ Specular (mirror-like) = reflection without
diffusion in different directions
§ Diffuse reflection equally distributed light
energy in all directions
§ Mixed diffuse and specular reflection is a
combination of the two
Marc Alexa, TU Berlin
CG
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Reflection distribution
§ Most surfaces have complex reflection behavior
Varying with incident and reflection angle of light
Modelled as combination of three basic types
+ + =
specular + glossy + diffuse = distribution
Marc Alexa, TU Berlin
CG
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Representing light intensity
§ Light power is called flux and measured in Watts (W)
§ Flux per solid angle = intensity (W/sr)
§ Flux per area
Light received by surface = irradiance (W/m2)
Light leaving a surface = radiosity (W/m2)
§ Photometry = units related to human vision by integrating over the
human spectral sensitivity
Luminous flux (power) measured in Lumen (lm)
Luminous flux per solid angle: luminous intensity, measured in Candela (cd = lm/sr)
Luminous flux per area: illuminance, measured in Lux (lx = lm/m2)
Marc Alexa, TU Berlin
CG
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Light rays in computer graphics
§ In CG we model light travelling along rays
§ Intensity of light is constant along ray
§ Appropriate physical unit: Flux per solid angle per area
Radiometry: Radiance (W/sr m2)
Photometry: Luminance (lm/sr m2 = cd/m2 = lx/sr)
Marc Alexa, TU Berlin
CG
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Bidrectional reflectance distribution
function (BRDF)
§ Description of reflection behavior
§ Proportion of reflected radiance L to irradiance E
§ Incoming light: index i
§ Reflected light: Index r
( ) ( )( )
( )
( ) ( )ò W
==
iiiii
rrr
iii
rrr
iirr dL
L
E
L
qqfl
qfl
qfl
qfl
qfqflr
l
l
l
l
cos,,
,,
,,
,,
,,,,
,
,
,
,
Marc Alexa, TU Berlin
CG
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BRDF
Properties
1. Reciprocity
ρλ remains unchanged, if incidence and reflection angle are swapped
2. ρλ is generally anisotropic
Rotating the surface around the surface normal changes the reflection
Prominent example would be cloth.
3. Superposition
Light from multiple directions behaves linearly
Integration over all incident directions yields
( )ò
W
W=
i
iiir dLL qr ll cos,,
Marc Alexa, TU Berlin
CG
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BRDF
Simplification in CG
§ Reflection is always positive
§ In Computer Graphics we mostly use another quantity
§ Proportion of reflected and incoming irradiance
This quantity is dimensionless and between 0 and 1
We apply it to radiance (which is physically implausible)
r
E
E
rr
r
l
l
l
l= ￡ ￡
,
,
, 0 1
Marc Alexa, TU Berlin
CG
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BRDF
Surfaces
§ On a microscopic scale all surface are
rough
§ Shadowing
§ “Masking” of reflected light
Shadow Shadow
Masked light
Marc Alexa, TU Berlin
CG
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Ideal diffuse reflection
§ Simplest case: reflected radiance is independent of reflection
direction
Lambert reflection
§ Incident light (irradiance) depends on angle θ of light source
to normal vector of surface
§ This angle expresses the illuminated surface in direction of
the light source
Falls off as the cosine of the angle
Lambert’s cosine law
dI LdA= 1 1cosq
Marc Alexa, TU Berlin
CG
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Lambert’s cosine law
Marc Alexa, TU Berlin
CG
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Ideal Diffuse Reflection
Computation
§ Angle between light source and surface normal is incidence
angle:
Idiffuse = kd Ilight cos θ
§ using vectors
Idiffuse = kd Ilight (n l)
nl
θ
kd : diffuse component
”color of surface”
Marc Alexa, TU Berlin
CG
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Ideal specular reflection
§ Ideal specular reflection is described by the following rules:
Incident and reflected ray subtend the same angle with the surface
normal
Incident and reflected ray, and the surface normal are in a common
plane
In polar coordinates:
q qr i= and f f pr i= +
Marc Alexa, TU Berlin
CG
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Ideal specular reflection
Geometry of mirror reflection
qi qr
L
P
P
R
S
L
N
θ
S
P = N ( N · L )
2 P = R + L
2 P – L = R
2 (N ( N · L )) – L = R
Marc Alexa, TU Berlin
CG
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Ideal specular reflection
Geometry of mirror reflection Geometry of refraction
qi qr
q1
q2
Medium 1
Medium 2
Marc Alexa, TU Berlin
CG
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Refraction
§ Incident ray, surface normal, and refracted ray are in one
plane
§ Sine of incident angle is proportional to sine of refraction
angle
§ The proportion depends on the two media
§ n1 resp. n2 are the indices of refraction
The index of refraction is defined as the speed of light in the medium
relative to the speed of light in vacuum.
n n n
n
const1 1 2 2 1
2
2
1
sin sin sin
sin
.q q q
q
= = =
Marc Alexa, TU Berlin
CG
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Refraction
Total internal reflection
§ Light travels to medium with lower index of refraction n2 < n1 Angle to normal increases There is an incidence angle θT so that refraction angle is 90o Law of refraction gives: § For angles larger than θT Light cannot propagate into other medium All light is reflected at the interface Total internal reflection sin .q T n n = 2 1 qT Medium 1 Medium 2 Marc Alexa, TU Berlin CG 25 Specular-diffuse reflection § Ideal reflectors (Lambert or specular) are rare § One really should determine ρλ(λ, r,θr, i,θi) § But we observe in many cases A maximum in the direction of the specular reflection Fall-off towards increasing angles from this direction § In CG we often treat this part, which is dependent on the viewer, independent of the purely diffuse reflection Marc Alexa, TU Berlin CG 26 Specular-diffuse reflection Specular (mirror-like) Mixed Diffuse Marc Alexa, TU Berlin CG 27 Phong model § The view-dependent part is modelled as § rs,0 is a constant between 0 and 1 § g is the angle between viewing direction and the ideally reflected illumination § The exponent (shininess) models how quickly the reflection decreases with larger angles r rs s m= , cos0 g g ! L ! N ! R ! E Marc Alexa, TU Berlin CG 28 Phong model § diffusely reflected radiance § diffuse reflection § radiance of incident light § angle between viewer and ideally reflected illumination § shininess Lspec rs L g m g ! L ! N ! R ! E( ) ( ) í ì >××××=
sonst
ERfallsERLr
m
s
0
0,
!!!!
( )