Computer Graphics - Light Transport - Hendrik Lensch Computer - - PowerPoint PPT Presentation
Computer Graphics - Light Transport - Hendrik Lensch Computer Graphics WS07/08 Light Transport Overview So far simple shading Today Physics behind ray tracing Physical light quantities Perception of light
Computer Graphics WS07/08 – Light Transport
Computer Graphics WS07/08 – Light Transport
– simple shading
– Physics behind ray tracing – Physical light quantities – Perception of light – Light sources – Light transport simulation
– Light-matter interaction – Reflectance function – Reflection models
Computer Graphics WS07/08 – Light Transport
– Linear propagation ⇒Geometrical optics
– Polarization ⇒Jones Calculus: matrix representation
– Diffraction, Interference ⇒Maxwell equations: propagation of light
– Light comes in discrete energy quanta: photons ⇒Quantum theory: interaction of light with matter
– Electromagnetic force: exchange of virtual photons ⇒Quantum Electrodynamics (QED): interaction between particles
Computer Graphics WS07/08 – Light Transport
– Linear propagation ⇒Geometrical optics
– Polarization ⇒Jones Calculus: matrix representation
– Diffraction, Interference ⇒Maxwell equations: propagation of light
– Light comes in discrete energy quanta: photons ⇒Quantum theory: interaction of light with matter
– Electromagnetic force: exchange of virtual photons ⇒Quantum Electrodynamics (QED): interaction between particles
Computer Graphics WS07/08 – Light Transport
– Macroscopic geometry – Tristimulus color model – Psycho-physics: tone mapping, compression, …
– Light: scalar, real-valued quantity – Linear propagation – Macroscopic objects – Incoherent light – Superposition principle: light contributions add up linearly – No attenuation in free space
– Microscopic structures (≈λ) – Diffraction, Interference – Polarization – Dispersion
Computer Graphics WS07/08 – Light Transport
the angle subtended by a curve in the plane, is the length of the corresponding arc on the unit circle. the solid angle subtended by an object, is the surface area of its projection onto the unit sphere, Units for measuring solid angle: steradians [sr]
Computer Graphics WS07/08 – Light Transport
2 2
( ) ( )
1 1
φ φ φ θ φ θ
Computer Graphics WS07/08 – Light Transport
The solid angle subtended by a small surface patch S with area ΔA is
and (ii) dividing by the square of the distance to the origin:
2
Computer Graphics WS07/08 – Light Transport
– Radiometry is the science of measuring radiant energy transfers. Radiometric quantities have physical meaning and can be directly measured using proper equipment such as spectral photometers.
– energy [watt second] n · hν (Photon Energy) – radiant power (total flux) [watt] Φ – radiance [watt/(m2 sr)] L – irradiance [watt/m2] E – radiosity [watt/m2] B – intensity [watt/sr] I
Computer Graphics WS07/08 – Light Transport
ω θ ω d dA x L d cos ) , ( 2 = Φ– the power (flux) traveling at some point x – in a specified direction ω = (θ,φ), – per unit area perpendicular to the direction of travel, – per unit solid angle.
angle dω, from the projected differential area dA cosθ is: ω
dA
2
Computer Graphics WS07/08 – Light Transport
– Since light is composed of electromagnetic waves of different frequencies and wavelengths, most of the energy transfer quantities are continuous functions of wavelength. – In graphics each measurement L(x,ω) is for a discrete band of wavelength only (often some abstract R, B, G)
Computer Graphics WS07/08 – Light Transport
Irradiance E is defined as the total power per unit area (flux density) incident onto a surface. To obtain the total flux incident to dA, the incoming radiance Li is integrated over the upper hemisphere Ω+ above the surface:
+ +
Ω Ω 2 / 2
π π
i i i
Computer Graphics WS07/08 – Light Transport
Radiosity B is defined as the total power per unit area (flux density) leaving a surface. To obtain the total flux radiated from dA, the
above the surface.
Ω Ω 2 / 2
π π
Computer Graphics WS07/08 – Light Transport
– The human eye is sensitive to a limited range of radiation wavelengths (roughly from 380nm to 770nm). – The response of our visual system is not the same for all wavelengths, and can be characterized by the luminuous efficiency function V(λ), which represents the average human spectral response. – A set of photometric quantities can be derived from radiometric quantities by integrating them against the luminuous efficiency function V(λ). – Separate curves exist for light and dark adaptation of the eye.
Computer Graphics WS07/08 – Light Transport
Computer Graphics WS07/08 – Light Transport
The eye detects radiance f radiance = flux per unit area per unit solid angle
dA d L ⋅ Ω Φ = '
photons / second = flux = energy / time = power
Φ
rod sensitive to flux
A ' d d ⋅ Ω ∝ Φ
flux proportional to area and solid angle r
2 2 /
' l r d ⋅ ≈ Ω π
Angular extend of pupil aperture (r ≤ 4 mm) = solid angle
' Ω d
angular extend of rod = resolution (≈ 1 arc minute2)
Ω d Ω d
l
dA
projected rod size = area
Ω ⋅ ≈ d l dA
2
Computer Graphics WS07/08 – Light Transport
r f l
' Ω d dA Ω d ' dA
Where does the Sun turn into a star ? − Depends on apparent Sun disc size on retina ⇒ Photon flux per rod stays the same on Mercury, Earth or Neptune ⇒ Photon flux per rod decreases when dΩ’ < 1 arc minute (beyond Neptune) As l increases:
const '
2 2 2
= ⋅ Ω = Ω ⋅ ∝ Φ l r d l d dA π
Computer Graphics WS07/08 – Light Transport
2
l dΩ r ⇒ Flux does not depend on distance l ⇒ Nebulae always appear b/w
Computer Graphics WS07/08 – Light Transport
The radiance in the direction of a light ray remains constant as it propagates along the ray
2 2 2 1 1 1
dA d L dA d L ⋅ Ω ⋅ = ⋅ Ω ⋅
Flux leaving surface 1 must be equal to flux arriving on surface 2
1
L
1
Ω d
1
dA
2
L
2
Ω d
2
dA
l
2 1
L L =
2 2 1
l dA d = Ω
2 1 2
l dA d = Ω
From geometry follows
2 2 1 2 2 1 1
l dA dA dA d dA d T ⋅ = ⋅ Ω = ⋅ Ω =
Ray throughput
Computer Graphics WS07/08 – Light Transport
– Power (total flux) of a point light source
∀ Φg= Power of the light source [watt]
– Intensity of a light source
– Irradiance on a sphere with radius r around light source:
– Irradiance on some other surface A
2 2
cos 4 cos 4 ) ( r dA r dA dA d I dA d x E
g g g
θ π θ π ω ⋅ Φ = ⋅ Φ = = Φ =
dA r dω θ
Computer Graphics WS07/08 – Light Transport
1 2 2 2 1 2
Irradiance E: E2 E1
d1 d2
– Illuminating quantity
– Double distance from emitter: sphere area four times bigger
– For point light sources
Computer Graphics WS07/08 – Light Transport
– Emitted energy / time
– Point, area, volume
– Thermal, line spectrum
– Goniometric diagram
Computer Graphics WS07/08 – Light Transport
Courtesy Lynch & Livingston
– Point source (approx.) – White light (by def.)
– Area source – Scattering: blue
– Brighter – Haze: whitish
– Multiple scattering in clouds – Uniform grey
Computer Graphics WS07/08 – Light Transport
– neon advertisements – sodium vapor lamps
– CRT, LCD display – (Overcast) sky
– Clear light bulb, filament
– Xenon lamp – Arc lamp – Laser diode
– Spot-lights – Beamers – Distant sources
– Torchieres – Frosted glass lamps
– “Photons everywhere”
Computer Graphics WS07/08 – Light Transport
– Surface position – Outgoing direction – Incoming illumination direction
– Incoming radiance from all directions – Direction-dependent reflectance (BRDF: bidirectional reflectance distribution function)
i i i i
r
Ω
i
ω
x
i
θ ) , (
L ω
x
i
ω
) , , (
r
x f ω ω
) , (
i i x
L ω ) , (
L ω
Computer Graphics WS07/08 – Light Transport
– Illumination from light sources only - local illumination (integral sum) – Evaluates angle-dependent reflectance function - shading
– Recursive ray tracing
specular surfaces)
– Forward ray tracing
methods)
– Combination of both
i i i i
r
Ω
Computer Graphics WS07/08 – Light Transport
– Lights (emitters) – Object surfaces (partially absorbing)
– Radiosity = Irradiance – absorbed photons flux density
– No absorption in-between objects
– emitted photons = absorbed photons (+ escaping photons)
Global Illumination
Computer Graphics WS07/08 – Light Transport
total radiance = emitted radiance + reflected radiance
– non-zero only for light sources
– integral over all possible incoming directions
reflection function
– unknown radiance appears on lhs and inside the integral – Numerical methods necessary to compute approximate solution
i i i i
r
Ω
i
ω
x
i
θ
Computer Graphics WS07/08 – Light Transport
– Incoming radiance at x is outgoing radiance at y – Ray-Tracing operator
+
Ω
i i i i
r
i i i i i
i
Computer Graphics WS07/08 – Light Transport
+
Ω
i i i i
r
2
y y i
y
i
y
y
i
i
∈
S y y y i i
r
2
Computer Graphics WS07/08 – Light Transport
∈
S y y i
r
2
y i
∈
S y y y i i
r
2
Computer Graphics WS07/08 – Light Transport
– follows roughly the eye’s sensitivity
– Assumes locally constant reflection, visibility, geometry terms
– simplifies integration to summation
– Parameterized function
– Lambertian surface (only diffuse reflection) - Radiosity
An example: polygon rendering in OpenGL
Computer Graphics WS07/08 – Light Transport
r r
r
= =
2 / 2 2 / 2
sin cos ) ( cos ) ( ) (
π π π π
θ φ θ θ ω θ ρ d d x f d x f x
r r
∈
S y y e e
∈
S y y i
r
percentage of light leaving that arrives at
y
diffuse reflectance
Computer Graphics WS07/08 – Light Transport
– Fredholm integral of 2nd kind – Global linking
each other
(occlusions)
– No consideration of volume effects!!
– acts on functions like matrices act on vectors – Superposition principle – Scaling and addition
∈
S y y e
Computer Graphics WS07/08 – Light Transport
e
∈
S y y e
e
e
−
2 2
I K K K K I K K I K I K I K I = + + − + + + = + + + − = − − ...) ( ...) ( ...) )( ( 1 ) (
2 2 2
Computer Graphics WS07/08 – Light Transport
– Direct light from the light source – Light which is reflected and transported one time – Light which is reflected and transported n-times
2
e e e e e e e
1 2 1
−
n e n e e
Computer Graphics WS07/08 – Light Transport
Courtesy Karol Myszkowski, MPII
Computer Graphics WS07/08 – Light Transport
Courtesy Karol Myszkowski, MPII
Computer Graphics WS07/08 – Light Transport
Courtesy Karol Myszkowski, MPII
Computer Graphics WS07/08 – Light Transport
– Radiance – Radiosity – Irradiance – Intensity
– Integral equation – Balance of radiance
– Diffuse reflectance function – Radiative equilibrium between emission and absorption, escape – System of linear equations – Iterative solution