Post on 07-Jul-2020
Chapter 8.
The Rendering Equation
Graphics Programming, 29th Sep.
Graphics and Media Lab.
Seoul National University 2011 Fall
3D Rendering by David Keegan
Goal of This Chapter
• Understand the basic radiometry
• Understand BRDF and reflection models
• Understand the rendering equation
Solid Angle
• A quantity that measures the relative coverage of a hemisphere.
– Unit: sr – steradian
• The whole hemisphere is 2π sr.
(Radiant) Flux F
• Time rate of the flow of radiant energy Q
– Amount of energy arriving at a surface during 1 sec.
– Unit: Watt
Irradiance E
• Flux arriving at a surface location x
– Unit: W/m2
– Positional distribution of flux
surface
(Radiant) Intensity I
• Radiant flux per unit solid angle
– Unit: W/sr
– Directional distribution of flux
Radiance L
• Radiant intensity per unit projected area
– Unit: W/m2sr
– Flux density per unit projected area per unit solid angle
surface
Radiance L
• Radiant intensity per unit projected area
– Unit: W/m2sr
– Irradiance in terms of radiance:
dxLxE cos),()(
Radiance L
• Radiant intensity per unit projected area
– Unit: W/m2sr
– Flux in terms of radiance:
Light Emission Example
• Assume a point light source with power
• What is the irradiance E at a surface?
• What is the radiance L at a surface?
Light Scattering
• Materials interact with light in different ways.
• Different materials have different appearances given the same lighting conditions.
• The reflectance properties of a surface are described by a reflectance function, which models the interaction of light reflecting at a surface.
• The bi-directional reflectance distribution function (BRDF) is the most general expression of reflectance of a material
The BRDF
• The BRDF is defined as the ratio between differential radiance reflected in an exitant direction, and incident irradiance through a differential solid angle
• The BRDF is defined as the ratio between differential radiance reflected in an exitant direction, and incident irradiance through a differential solid angle
The BRDF
incoming
outgoing
Why BRDF is Complicated? • How about
• Differential irradiance dE(p,w’) – dEi(x,w’) Li(x,w’)cosi dw’ – Why do we put d in front of E?
• dLr(x,w) differential exitance radiance due to dEi(x,w’) • dLr(x,w) dEi(x,w’)
– Why? – The relationship is called the “linearity assumption”.
• fr(x,w,w’) fr(x,w’,w) – Reciprocity: the value of the BRDF will remain unchanged if the
incident and exitant directions are interchanged.
ww
w
w
www
dpL
xdL
xdE
xdLxf
ii
r
i
rr
cos),(
),(
),(
),(),,(
),(
),(),,(
w
www
xL
xLxf
i
rr
The Scattering Equation
wwwww
wwwww
dxLpf
dxLpfxL
ir
iirr
)(n ),(),,(
cos ),(),,(),(
• Calculation of the exitant radiance from x toward w, by summing contributions made by all the incident radiances.
Reflectance r
• Why dF instead of F?
• BRDF vs. Reflectance – Both are about a surface point x.
– Reflectance: General (or average) tendency of reflection
– BRDF: Bi-directional reflection distribution
Object Color and Reflectance
• How do we see the color? – The surface absorbs photons of certain range of l
– Is it general or direction specific behavior of the surface?
• Object color is represented with the reflectance! – For red, use the vector reflectance r(1,0,0).
– For yellow, use r(0.5,0.5,0).
– …
Diffuse Reflection
• A diffuse surface reflects light in all directions
– A special case of diffuse reflection is Lambertian: the reflection is uniform over the whole hemisphere.
Rendered using Mitsuba
Lambertian
• For a Lambertian surface, the reflected radiance is constant in all directions regardless of the irradiance.
• This gives a constant BRDF:
)(
)(
)(
),()(,
xE
xL
xE
xLxf
i
r
i
rdr
w
Constant wrt w
Specular Reflection
• Specular reflection happens when light strikes a smooth surface – e.g. Metal, glass, and water
Specular Reflection
• Specular reflection happens when light strikes a smooth surface – e.g. Metal, glass, and water
• The reflected radiance due to specular reflection is:
and the reflection direction is:
Relationship Between Reflectance and Index of Refraction
• From left-to-right, only the index of refraction is varied. – We can observe the reflectance is related with the IOR.
– Why such thing happens?
Image from Maya 2009 Mental Ray Document
The Fresnel Equations
• For smooth homogeneous metals and dielectrics, specular reflectance is given by the following formulae for polarized lights:
Specular Reflectance for Unpolarized Light
• For unpolarized light the specular reflectance becomes:
• The amount of refracted ray is computed as:
)(2
1)( 22
|| rrr rs F
More General Reflection Models
• Phong
• Microfacet models
– Torrance-Sparrow
– Oren-Nayar
• Lafortune
• BSSDF
What’s Next?
• Now, we know how to describe light.
• We also know how it gets reflected.
• From now on, let’s talk about how light travels around!
The Rendering Equation
• Introduced by David Immel et al. and James Kajiya in 1986.
• The rendering equation describes the total amount of light coming from a point x along a particular viewing direction. – Based on the law of conservation of energy.
Image from Wikipedia
The Rendering Equation
• In short, the outgoing radiance Lo is the sum of the emitted radiance Le and the reflected radiance Lr:
• Using BRDF, we can rewrite about eq. as:
• How can we solve the above eq?
– Difficult!
Understanding the Situation
E
D
S
L
D
E: Eye, D: Diffuse, S: Specular
Light -> Eye
E
D
S
L
D
E: Eye, D: Diffuse, S: Specular
Light -> Diffuse -> Eye
E
D
S
L
D
E: Eye, D: Diffuse, S: Specular
Light -> Specular -> Eye
E
D
S
L
D
E: Eye, D: Diffuse, S: Specular
Light -> Specular -> Specular -> Diffuse -> Eye
E
D
S
L
D
E: Eye, D: Diffuse, S: Specular
Light -> Specular -> Specular -> Diffuse -> Diffuse -> Specular -> Diffuse -> Specular -> … -> Eye
E
D
S
L
D
E: Eye, D: Diffuse, S: Specular
Computing Rendering Equation
• You might have felt that solving the rendering eq won’t be a trivial job.
– How can we evaluate this integral numerically?
• Let’s rewrite the eq in more readable (and computable) form.
1. Neumann series expansion
2. Light transport notation
Neumann Series Expansion
• The rendering eq can be represented using the following compact form:
where the integral operator T is:
Neumann Series Expansion
• Then, recursive evaluation of L=Le+TL gives:
E
D
S
L
D
Light Transport Notation
• We can also represent light traversals in general form. – L: a light source
– E: the eye
– S: a specular reflection/refraction
– D: a diffuse reflection
• Using regular expression: – (k)+: one or more of k events
– (k)*: zero or more of k events
– (k)?: zero or one k events
– (k|k’): a k or a k’ events
LE
E
D
S
L
D
LSE
E
D
S
L
D
LSSDE
E
D
S
L
D
L(S|D)+DE
E
D
S
L
D
Ray Tracing Models
• Appel’s Method (1968) – E(D|G)L – aka. Ray casting – Only computes local illuminations
• Whitted’s Method (1980) – ES*(D|G)L – aka. (Recursive) ray tracing – Only computes local illuminations
• Kajiya’s Method (1986) – E[(D|G|S)+(D|G)]L – aka. Path tracing – Fully computes local and global illuminations
* G is glossy reflection
How to Solve the Rendering Equation
• Trace all possible paths, and integrates every contribution
– Called the “path tracing”
– Extremely expensive.
• The next two lectures will teach how to integrate the equation numerically.
– Monte-Carlo integration
E
D
S
L
D