# Transport

## Scattering

Fog

Rainbow

#### Wavelength Dependence Mediated by Selective Absorption

Beer

• Beer's Law: I' = I exp( -\alpha (lambda) t )
• \alpha(\lambda) is the absorption coefficient
• \alpha(\lambda) t is the optical density
• zero density is transparent
• infinite density is opaque

# Surface Reflectances

According to Jim Kajiya, Bidirectional Reflectance Distribution Function should tell the whole story. But where does it come from?

• R( incoming angle (2 dof), outgoing angle (2 dof), wavelength (1 dof) )
• incoming and outgoing angles: 0 < \theta < \pi/2, 0 < \phi < 2\pi
• wavelength 400 nm < \lambda < 700nm
• we won't even talk about sampling density (3.14 million values at 10 degree, 10 nm sampling)
• potentially also a function of distance along the surface (2 dof) and difference in surface normal (1 dof )
• if the surface varies, then also parameters that control variation

Maybe it could come from measurement

• even if you could measure it how would you encode the result

Maybe you could simulate the system

• just how complex is the system

Maybe you could find a simple model

• capturing only the important aspects of the system

## Smooth Surfaces

#### Body reflectance

• just like selective absorption
• doesn't come out at the same place: Why doesn't it matter?
• edges of a surface

#### Lambertian reflectance

Lambert 1728-1777

#### The Moon

• What would the full moon look like if its surface were Lambertian?
• What does the full moon actually look like?

## Colour Mixing

Paint is the usual example, but there are other examples

1. Fruit ripening
2. Leaves turning colour in the fall
3. Colour of human skin

#### Kubelka-Munk colour mixing model

1. Infinitesimally thin layers, thickness proportional to the density of the pigment
2. Use reflection/refraction plus Beer's law
3. Solve in equilibrium at the boundaries between layers
• Infinite depth
• White underlayer
• Black underlayer

Engineering model

• We don't use densities, reflection/refraction coefficients, etc calculated from physics.
• We get humans to say whether colours look the same or not, and by how much they differ, and find the best coefficients.
• Recall difference between
• blind surgery -- calculate like paint -- and
• computer-animated movies -- let the artist play with coefficients