david luebke 1 12/2/2015 cs 551/651: advanced computer graphics advanced ray tracing radiosity

David Luebke 1 04/21/23

CS 551/651: Advanced Computer Graphics

Advanced Ray Tracing

Radiosity


Administrivia

My penance: Ray tracing homeworks very slow to grade Many people didn’t include READMEs Many (most) people didn’t include

workspace/project files or Makefiles Some people’s don’t work Nobody’s works perfectly

Quiz 1: Tuesday, Feb 20


Recap: Stochastic Sampling

Sampling theory tells us that with a regular sampling grid, frequencies higher than the Nyquist limit will alias

Q: What about irregular sampling? A: High frequencies appear as noise, not

aliases This turns out to bother our visual system less!



Poisson distribution: Completely random Add points at random until area is full. Uniform distribution: some neighboring samples

close together, some distant



Poisson disc distribution: Poisson distribution, with minimum-distance

constraint between samples Add points at random, removing again if they are

too close to any previous points Jittered distribution

Start with regular grid of samples Perturb each sample slightly in a random direction More “clumpy” or granular in appearance



Spectral characteristics of these distributions: Poisson: completely uniform (white noise). High

and low frequencies equally present Poisson disc: Pulse at origin (DC component of

image), surrounded by empty ring (no low frequencies), surrounded by white noise

Jitter: Approximates Poisson disc spectrum, but with a smaller empty disc.


Recap: Nonuniform Supersampling

To be correct, need to modify filtering step:


Recap: Nonuniform Supersampling

Approximate answer: weighted average filter

Correct answer: multistage filtering Real-world answer: ignore the problem

I(i, j) h(x-i, y-j) h(x-i, y-j)I’(x,y) =


Recap: Antialiasing and Texture Mapping

Texture mapping is uniquely harder Potential for infinite frequencies

Texture mapping is uniquely easier Textures can be prefiltered


Recap: Antialiasing and Texture Mapping

Issue in prefiltering texture is how much texture a pixel filter covers

Simplest prefiltering scheme: MIP-mapping Idea: approximate filter size, ignore filter shape Create a pyramid of texture maps Each level doubles filter size


Recap: Mip-Mapping

Tri-linear mip-mapping Pixel size depth d Linearly interpolate colors within 2 levels

closest to d Linearly interpolate color between levels

according to d


Distributed Ray Tracing

Distributed ray tracing is an elegant technique that tackles many problems at once Stochastic ray tracing: distribute rays

stochastically across pixel Distributed ray tracing: distribute rays

stochastically across everything



Distribute rays stochastically across: Pixel for antialiasing Light source for soft shadows Reflection function for soft (glossy) reflections Time for motion blur Lens for depth of field

Cook: 16 rays suffice for all of these I done told you wrong: 4x4, not 8x8



Distributed ray tracing is basically a Monte Carlo estimation technique

Practical details: Use lookup tables (LUTs) for distributing rays

across functions See W&W Figure 10.14 p 263


Backwards Ray Tracing

Traditional ray tracing traces rays from the eye, through the pixel, off of objects, to the light source

Backwards ray tracing traces rays from the light source, into the scene, into the eye

Why might this be better?



Backwards ray tracing can capture: Indirect illumination Color bleeding Caustics

Remember what a caustic is? Give some examples Remember where caustics get the name?



Usually implies two passes: Rays are cast from light into scene Rays are cast from the eye into scene, picking up

illumination showered on the scene from the first pass



Q: How might these two passes “meet in the middle?”



Arvo: illumination maps tile surfaces with regular grids, like texture maps Shoot rays outward from lights Every ray hit deposits some of its energy into

surface’s illumination map Ignore first generation hits that directly illuminate

surface (Why?) Eye rays look up indirect illumination using

bilinear interpolation



Watt & Watt, Plate 34 Illustrates Arvo’s method of backwards ray tracing

using illumination maps Illustrates a scene with caustics

Related idea: photon maps


Advanced Ray Tracing Wrapup

Backwards ray tracing accounts for indirect illumination by considering more general paths from light to eye

Distributed ray tracing uses a Monte Carlo sampling approach to solve many ray-tracing aliasing problems


Next Up: Radiosity

Ray tracing: Models specular reflection easily Diffuse lighting is more difficult

Radiosity methods explicitly model light as an energy-transfer problem Models diffuse interreflection easily Shiny, specular surfaces more difficult


Radiosity Introduction

First lighting model: Phong Still used in interactive graphics Major shortcoming: local illumination!

After Phong, two major approaches: Ray tracing Radiosity



Ray tracing: ad hoc approach to simulating optics Deals well with specular reflection Trouble with diffuse illumination

Radiosity: theoretically rigorous simulation of light transfer Very realistic images But makes simplifying assumption: only diffuse

interaction!



Ray-tracing: Computes a view-dependent solution End result: a picture

Radiosity: Models only diffuse interaction, so can compute a

view-independent solution End result: a 3-D model


Radiosity

Basic idea: represent surfaces in environment as many discrete patches

A patch, or element, is a polygon over which light intensity is constant


Radiosity

Model light transfer between patches as a system of linear equations

Solving this system gives the intensity at each patch

Solve for R, G, B intensities and get color at each patch

Render patches as colored polygons in OpenGL. Voila!


Fundamentals

Theoretical foundation: heat transfer Need system of equations that describes

surface interreflections Simplifying assumptions:

Environment is closed All surfaces are Lambertian reflectors


Radiosity

The radiosity of a surface is the rate at which energy leaves the surface

Radiosity = rate at which the surface emits energy + rate at which the surface reflects energy Notice: previous methods distinguish light sources

from surfaces In radiosity all surfaces can emit light Thus: all emitters inherently have area


Radiosity

Break environment up into a finite number n of discrete patches Patches are opaque Lambertian surfaces of finite

size Patches emit and reflect light uniformly over their

entire surface Q: What’s wrong with this model? Q: What can we do about it?


Radiosity

Then for each surface i:

Bi = Ei + i Bj Fji (Aj / Ai)

where

Bi, Bj= radiosity of patch i, j

Ai, Aj= area of patch i, j

Ei = energy/area/time emitted by i

i = reflectivity of patch i

Fji = Form factor from j to i


Form Factors

Form factor: fraction of energy leaving the entirety of patch i that arrives at patch j, accounting for: The shape of both patches The relative orientation of both patches Occlusion by other patches


Form Factors

Some examples…

Form factor: nearly 100%


Form Factors

Some examples…

Form factor: roughly 50%


Form Factors

Some examples…



Form Factors

In diffuse environments, form factors obey a simple reciprocity relationship:

Ai Fij = Ai Fji

Which simplifies our equation:

Bi = Ei + i Bj Fij

Rearranging to:

Bi - i Bj Fij = Ei


Form Factors

So…light exchange between all patches becomes a matrix:

What do the various terms mean?

nnnnnnnnn

n

n

E

E

E

B

B

B

FFF

FFF

FFF

2

1

2

1

21

22222212

11121111

1

1

1


Form Factors

1 - 1F11 - 1F12 … - 1F1n B1 E1

- 2F21 1 - 2F22 … - 2F2n B2 E2

. . … . . .

. . … . . .

. . … . . .

- pnFn1 - nFn2 … 1 - nFnn Bn En

Note: Ei values zero except at emitters Note: Fii is zero for convex or planar patches Note: sum of form factors in any row = 1 (Why?) Note: n equations, n unknowns!


Radiosity

Now “just” need to solve the matrix! W&W: matrix is “diagonally dominant” Thus Guass-Siedel must converge

End result: radiosities for all patches Solve RGB radiosities separately, color each

patch, and render! Caveat: actually, color vertices, not patches

(see F&vD p 795)


Radiosity

Q: How many form factors must be computed? A: O(n2) Q: What primarily limits the accuracy of the

solution? A: The number of patches


Radiosity

Where we go from here: Evaluating form factors Progressive radiosity: viewing an approximate

solution early Hierarchical radiosity: increasing patch resolution

on an as-needed basis


The End

david luebke 1 12/2/2015 cs 551/651: advanced computer graphics advanced ray tracing radiosity

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