Computer Graphics - Light Transport - Philipp Slusallek & Arsne - - PowerPoint PPT Presentation
Computer Graphics - Light Transport - Philipp Slusallek & Arsne Prard-Gayot Overview So far Nuts and bolts of ray tracing Today Light Physics behind ray tracing Physical light quantities Perception of light
– Nuts and bolts of ray tracing
– Light
– Light transport simulation
– Reflectance properties – Shading
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[Wikipedia]
– 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
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– 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
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– Macroscopic geometry ( Reflection Models) – Tristimulus color model ( Human Visual System) – Psycho-physics: tone mapping, compression, … ( RIS course)
– Macroscopic objects – Incoherent light – Light: scalar, real-valued quantity – Linear propagation – Superposition principle: light contributions add, do not interact – No attenuation in free space
– No microscopic structures (≈ λ): diffraction, interference – No polarization – No dispersion, …
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– Units for measuring solid angle: steradian [sr] (dimensionless)
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– 𝑒𝑣 = 𝑠 𝑒𝜄 – 𝑒𝑤 = 𝑠´ 𝑒Φ = 𝑠 sin 𝜄 𝑒Φ – 𝑒𝐵 = 𝑒𝑣 𝑒𝑤 = 𝑠2 sin𝜄 𝑒𝜄𝑒Φ – 𝑒𝜕 = Τ 𝑒𝐵 𝑠2 = sin 𝜄 𝑒𝜄𝑒Φ
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du r dθ r’ dΦ dA dv θ Φ dω 1
and (ii) dividing by the squared distance to the origin: d𝜕 = d𝐵 cos 𝜄
𝑠2
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– 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 [J] Q (#Photons x Energy = 𝑜 ⋅ ℎ𝜉) – Radiant power [watt = J/s] Φ (Total Flux) – Intensity [watt/sr] I (Flux from a point per s.angle) – Irradiance [watt/m2] E (Incoming flux per area) – Radiosity [watt/m2] B (Outgoing flux per area) – Radiance [watt/(m2 sr)] L (Flux per area & proj. s. angle)
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– The power (flux) traveling through some point x – In a specified direction ω = (θ, φ) – Per unit area perpendicular to the direction of travel – Per unit solid angle
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ω
dA
𝑒𝐵
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𝑒𝐵
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– Light is composed of electromagnetic waves – These waves have different frequencies and wavelengths – Most transfer quantities are continuous functions of wavelength
– Each measurement L(x,ω) is for a discrete band of wavelength
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– The human eye is sensitive to a limited range of wavelengths
– Our visual system responds differently to different wavelengths
– Photometric quantities are derived from radiometric quantities by integrating them against this function
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The eye detects radiance f
rod sensitive to flux
angular extent of rod = resolution ( 1 arcminute2)
r
2 2 /
' l r
angular extent of pupil aperture (r 4 mm) = solid angle
l
projected rod size = area
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l A
radiance = flux per unit area per unit solid angle
A L '
' A L flux proportional to area and solid angle As l increases: const
2 2 2
L l r l L
photons / second = flux = energy / time = power (𝚾) (1 arcminute = 1/60 degrees)
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f r l A
Where does the Sun turn into a star ? Depends on apparent S un disc size on retina Photon flux per rod stays the same on Mercury, Earth or Neptune Photon flux per rod decreases when ’ < 1 arcminute2 (beyond Neptune)
' A
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1
L
1
d
1
dA
2
L
2
d
2
dA
l
The radiance in the direction of a light ray remains constant as it propagates along the ray Flux leaving surface 1 must be equal to flux arriving on surface 2
2 2 1
l dA d
2 1 2
l dA d
From geometry follows
2 2 1 2 2 1 1
l dA dA dA d dA d T
Ray throughput 𝑈:
𝑀1𝑒Ω1𝑒𝐵1 = 𝑀2𝑒Ω2𝑒𝐵2 𝑀1 = 𝑀2
– Power (total flux) of a point light source
– Intensity of a light source (radiance cannot be defined, no area)
– Irradiance on a sphere with radius r around light source:
– Irradiance on some other surface A
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dA r d 𝐹 𝑦 = 𝑒Φ 𝑒𝐵 = 𝑒Φ 𝑒𝜕 𝑒𝜕 𝑒𝐵 = 𝐽 𝑒𝜕 𝑒𝐵 = Φ 4𝜌 ⋅ 𝑒𝐵 cos 𝜄 𝑠2𝑒𝐵 = Φ 4𝜌 ⋅ cos 𝜄 𝑠2
– Illuminating quantity
– Double distance from emitter: area of sphere is four times bigger
– For point light sources (!)
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1 2 2 2 1 2
Irradiance E: E2 E1 d1 d2
– Emitted energy / time
– Point, line, area, volume
– Thermal, line spectrum
– Goniometric diagram
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Black body radiation (see later)
– Point source (approx.) – White light (by def.)
– Area source – Scattering: blue
– Brighter – Haze: whitish
– Multiple scattering in clouds – Uniform grey
26 Courtesy Lynch & Livingston
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– Lights (emitters) – Object surfaces (partially absorbing)
– Radiosity = Irradiance minus absorbed photons flux density
– No absorption in-between objects
– Emitted photons = absorbed photons (+ escaping photons) → Global Illumination, discussed in RIS lecture
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– Surface position – Outgoing direction
– Incoming radiance from all directions – Direction-dependent reflectance (BRDF: bidirectional reflectance distribution function)
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𝜕𝑝 𝜕𝑗
𝑠 𝜕𝑗, 𝑦, 𝜕𝑝
i
x
i
– Expresses energy equilibrium in scene
– Non-zero only for light sources
– Integral over all possible incoming directions of radiance times angle-dependent surface reflection function
– Unknown radiance appears both on the left-hand side and inside the integral – Numerical methods necessary to compute approximate solution
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x
i
– An example: polygon rendering in OpenGL
– Follows roughly the eye’s sensitivity
– Simplifies integration to summation
– Parameterized function
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– Illumination from discrete point light sources only – direct illumination only
– Evaluates angle-dependent reflectance function (BRDF) – shading process
– Recursive ray tracing
specular surfaces)
– Ray tracing for global illumination
(Monte Carlo methods)
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– Incoming radiance at x is outgoing radiance at y
– Ray-Tracing operator: y = 𝑆𝑈 𝑦, 𝜕𝑗
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y
i
y
y
i
i
𝑀 𝑦, 𝜕𝑝 = 𝑀𝑓 𝑦, 𝜕𝑝
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𝑀 𝑦, 𝜕𝑝 = 𝑀𝑓 𝑦, 𝜕𝑝
– Radiosity equation: simplified form of the rendering equation
– Assumes local constancy: diffuse reflection, radiosity, visibility
– Form factor: percentage of light flowing between 2 patches – Form system of linear equations – Iterative solution – Discussed in details in RIS course
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– Reflectance factor or albedo: between [0,1]
– Defines percentage of light leaving dAy arriving at dA
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𝑔
𝑠 𝜕 𝑦, 𝑧 , 𝑦, 𝜕𝑝 = 𝑔 𝑠 𝑦 ⇒
𝑠 𝑦 𝐹 𝑦 = 𝑀𝑝 𝑦
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𝑠 𝑦 𝐹(𝑦) = 𝐶𝑓 𝑦 + 𝜍 𝑦 𝐹(𝑦)
– Fredholm equation 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
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𝐶 𝑦 = 𝐶𝑓 𝑦 + 𝜍(𝑦) න
𝑧∈𝑇
𝐺 𝑦, 𝑧 𝐶 𝑧 𝑒𝐵𝑧 𝑔 𝑦 = 𝑦 + 𝐿[𝑔 𝑦 ] 𝐿 𝑔 𝑦 = ∫ 𝑙 𝑦, 𝑧 𝑔 𝑧 𝑒𝑧 𝐿 𝑏𝑔 + 𝑐 = 𝑏𝐿 𝑔 + 𝑐𝐿[]
– Converges only if |K| < 1 which is true in all physical settings
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𝐶 𝑦 = 𝐶𝑓 𝑦 + 𝜍(𝑦) න
𝑧∈𝑇
𝐺 𝑦, 𝑧 𝐶 𝑧 𝑒𝐵𝑧 𝐶 ⋅ = 𝐶𝑓 ⋅ + 𝐿 𝐶 ⋅ ⇒ 𝐽 − 𝐿 𝐶 ⋅ = 𝐶𝑓 ⋅ 𝐶(⋅) = 𝐽 − 𝐿 −1 𝐶𝑓 ⋅ 1 1 − 𝑦 = 1 + 𝑦 + 𝑦2 + ⋯ 1 𝐽 − 𝐿 = 𝐽 + 𝐿 + 𝐿2 + ⋯ 𝐽 − 𝐿 1 𝐽 − 𝐿 = 𝐽 − 𝐿 𝐽 + 𝐿 + 𝐿2 + ⋯ = 𝐽 + 𝐿 + 𝐿2 + ⋯− 𝐿 + 𝐿2 + ⋯ = 𝐽 ( )
– Direct light from the light source – Light which is reflected and transported at most once – Light which is reflected and transported up to n times
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𝐶1 ⋅ = 𝐶𝑓 ⋅ 𝐶2 ⋅ = 𝐶𝑓 ⋅ + 𝐿[𝐶𝑓 ⋅ ] 𝐶𝑜 ⋅ = 𝐶𝑓 ⋅ + 𝐿[𝐶𝑜−1 ⋅ ]
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– Radiance – Radiosity – Irradiance – Intensity
– Key equation in graphics (!) – Integral equation – Describes global balance of radiance
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