CS488 - Introduction to Computer Graphics - Lecture 25

Lighting

Bidirectional Ray Tracing

Participating Media

What is fog?

Lots of little water droplets
Light gets scattered

What is beer?

Lots of little colour centres
Light gets absorbed

What they have in common is

The farther light goes the more likely it is to get scattered or absorbed.
The property is described by Beer's Law (named after August Beer, no relation)
- I(x) ~ exp( -k(\lambda) x )

What happens to the light that doesn't make it through?

Shadows

What is a shadow?

Shadows come `for free' in the ray tracer.

Can we make them fast enough to use with OpenGL?

Yes. The methods, in increasing order of cost.

Projective shadows
- Project silhouette of shadowing object onto shadowed object
- Draw a dark area where shadow lies, using alpha blending unless you are trying to get the `deep space' look
- Easy for simple objects onto onto simple geometry, ...
- What about meshed objects?
  - Finding the silhouette of a mesh
    - from a particular direction
    - lots of algorithms: best are linear in the length of the silhouette,
      but with a big constant
    - still more to do
Notice that we know a lot about how to project.
Shadow maps
- Project as if each light is an eye
  - Scene behind every point that appears in the virtual frame buffer is in shadow
  - Store the distance to each point in the z-buffer
- Project towards the eye
  For each point that is visible
  - Transform the point to the light's coordinate frame
  - Is the distance stored in the light's virtual z-buffer greater than the distance to the light
    - If `yes' then the point is shadowed from that light
    - If `no' then it is illuminated by that light
How does this interact with scan conversion?

What if the light is inside the view frustrum?
- Remember that the eye ray is not the same as the axis of the view frustrum.
Shadow volumes
- Project from light as for shadow maps
- Define a set of polygons that are the boundaries of the volume that is in shadow.
  - Front-facing wrt eye +1
  - Back-facing -1
- Count along the ray from eye to point, staring with zero
  - Staring with:
    - zero if eye is not in shadow
    - number of times it's shadowed otherwise
  - If > 0 in shadow
  - If 0 in light

Currently (2009) the preferred technique is shadow maps

Global Illumination

Comment on global illumination. If you are doing a walk-through, you can calculate the illumination on each polygon once, then re-render (re-project) the scene from different viewpoints as the user moves around.

Radiosity

Calculating illumination

Each small bit of surface in the scene

receives some amount of light (possibly none)
- from other bits of surface: \sum_bits (light emitted in the direction of this bit) * (fraction occluded)
- B(y, <y-x>, \lambda) = \sum_surfaces (I(x, <y-x>, \lambda) + L(x, <y-x>, \lambda) * F(x,y) * dx.dy
emits some amount of light (possibly none)
- I(x, <z>, \lambda )
re-emits some amount of light (possibly none)
- sum_directions (received light from ...) * (BRDF to ...)
- L(x, <y-x>, \lambda) = \sum_<z> B(x, <z>, \lambda) * R(<z>, <y-x>, \lambda)

Solve the resulting equations.

F(x, y)dx.dy is known from the geometry
I(x, <z>, \lambda) and R(<z-in>, <z-out>, \lambda) are surface properties in the model
B(x, <z>, \lambda) and L(x, <z>, \lambda) are unknown.
Substitute B into the third equation.
The result is a set of linear equations that can be solved for L

Once L is known,

B is easily calculated.
The light field is easily calculated at point P
- LF(P, <z>, \lambda) = sum_x L(x, <P-x>, \lambda) \delta(<z>, <P-x>)

The Light Field

Let's turn our attention away from the surfaces of objects and onto the volume between objects

At every point in this volume there is a light density

for every possible direction
for every visible wavelength

This quantity LF(P, <z>, \lambda ) is the light field. If we knew it we could

evaluate it at the eye position
at the angle heading for each pixel
to get RGB for that pixel

The evaluation is, in fact, just a projective transformation of the light field.

How do we get the light field?

by measurement
by calculation
- Radiosity is the obvious method

How is the light field used in 2009?

routine applications for backdrops
- Think about a window in a dark room
- Light passes only one direction
- What's wrong with treating a window like a 2D scene on the wall?
  - Easy to do by texture mapping
- How would we get the necessary data?
  - calculation
    - maybe radiosity
  - measurement
    - remote controlled digital camera
    - still the problems of storage and reconstruction
yesterday's excitement

But tomorrow!!

Plenoptic Function

Think about what the viewer can do.

The seriously handicapped viewer can
- not move in position
- not move the direction of gaze
Ray tracing is perfect.
The mildly handicapped viewer can
- not move in position
- gaze in any direction
Ray trace onto a sphere surrounding the viewer and reproject from the sphere to a view plane whenever the direction of gaze changes.
The unhandicapped viewer can
- move around
- gaze in any direction
Ray trace onto a sphere at each accessible point.

The third is the light field, also called the plenoptic function, and it has to be recalculated every time something in the scene moves.

`Backdrop' Applications

Imagine making a game or a movie

There is an area accessible to the players (actors, camera), and
there is an area inaccessible to the players (actors, camera).

An easy backdrop

Surround the accessible volume with a sphere (actually a hemi-sphere)
Ray trace the scene outside the accessible volume onto the sphere
Put the re-projected portion of the sphere into the frame buffer, depth buffer set to infinity
Where is the eye point?
- The centre of the sphere works for the mildly handicapped viewer.
- What is missing for the unhandicapped viewer?

A more difficult backdrop

Photography
Perhaps a window