David Luebke 1 10/12/2015 CS 551/651: Advanced Computer Graphics Advanced Ray Tracing Radiosity.

David Luebke 1 04/19/23

CS 551/651: Advanced Computer Graphics

Advanced Ray Tracing

Radiosity


Administrivia

Quiz 1: Tuesday, Feb 20 Yes, I’ll have your homework graded by then

(somehow) Normal written exam (oral later)


Recap: Distributed Ray Tracing

Distributed ray tracing: an elegant stochastic approach that distributes rays across: Pixel for antialiasing Light source for soft shadows Reflection function for soft (glossy) reflections Time for motion blur Lens elements for depth of field

Cook: 16 rays suffice for all of these


Recap: Backwards Ray Tracing

Two-pass algorithm: Rays are cast from light into scene Rays are cast from the eye into scene, picking up

illumination showered on the scene in the first pass Backwards ray tracing can capture:

Indirect illumination Color bleeding Caustics


Recap: Backwards Ray Tracing

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


Recap: Radiosity

Ray tracing: Models specular reflection easily Diffuse lighting is more difficult View-dependent, generates a picture

Radiosity methods explicitly model light as an energy-transfer problem Models diffuse interreflection easily But only diffuse; no shiny (specular) surfaces View-independent, generates a 3-D model


Recap: Radiosity

Basic idea: represent surfaces in environment as many discrete patches A patch, or element, is a polygon over which light

intensity is constant Model light transfer between patches as a system

of linear equations Solve this system for the intensity at each patch Solve for R,G,B intensities; get color at each patch Render patches as colored polygons in OpenGL


Recap: Fundamentals

Definition: 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

Simplifying assumptions Environment is closed All surfaces have Lambertian reflectance Surface patches emit and reflect light uniformly over

their entire surface


Radiosity

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

We’ll return later to the calculation of form factors


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 (what’s that?)

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

patch, and render! Caveat: for rendering, we 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


Roadmap

So, we know the basic radiosity algorithm Represent light transfer as a matrix Solve the matrix to get radiosity (=color) per patch

Next topics: Evaluating form factors Progressive radiosity: viewing an approximate

solution early Hierarchical radiosity: increasing patch resolution

on an as-needed basis


Form Factors

Calculating form factors is hard Analytic form factor between two polygons in

general case: open problem till last few years Q: So how might we go about it? Hint: Clearly form factors are related to

visibility: how much of patch j can patch i “see”?


Form Factors: Hemicube

Hemicube algorithm: Think Z-buffer Render the model onto a hemicube as seen from the center of patch

i Store item IDs instead of color Use Z-buffer to resolve visibility See W&W p 278

Q: Why hemicube, not hemisphere?


Form Factors: Hemicubes

Advantages of hemicubes Solves shape, size, orientation, and occlusion

problems in one framework Can use hardware Z-buffers to speed up form

factor determination (How?)


Form Factors: Hemicubes

Q: What are some disadvantages of hemicubes? Aliasing! Low resolution buffer can’t capture

actual polygon contributions very exactly Causes “banding” near lights (plate 41)

Actual form factor is over area of patch; hemicube samples visibility at only center point on patch (So?)


Form Factors: Ray Casting

Idea: shoot rays from center of patch in hemispherical pattern



Advantages: Hemisphere better approximation than hemicube

More even sampling reduces aliasing Don’t need to keep item buffer Slightly simpler to calculate coverage



Disadvantages: Regular sampling still invites aliasing Visibility at patch center still isn’t quite the same

as form factor Ray tracing is generally slower than

Z-buffer-like hemicube algorithms Depends on scene, though Q: What kind of scene might ray tracing actually be

faster on?


Form Factors

Source-to-vertex form factors Calculating form factors at the patch vertices

helps address some problems:for every patch vertex

for every source patch

sample source evenly with rays

visibility = % rays that hit Q: What are the problems with this

approach?


Form Factors

Summary of form factor computation Analytical:

Expensive or impossible (in general case) Hemicube

Fast, especially using graphics hardware Not very accurate; aliasing problems

Ray casting Conceptually cleaner than hemicube Usually slower; aliasing still possible


Substructuring

More patches better results Problem: # form factors grows quadratically

with # patches Substructuring: adaptively subdivide patches

into elements where high radiosity gradient is found


Substructuring

Elements are second-class patches: When a patch is subdivided, form factors are

computed from the elements to other patches But form factors from the other patches to the

elements are not computed However, the form factors from other patches to the

subdivided patch are updated using more accurate area-weighted average of elements


Substructuring

Elements vs. patches, cont. Elements “gather” radiosity from other patches But those other patches only gather radiosity from

the “parent” patch, not the individual elements So an element’s contribution to other patches is

approximated coarsely by it’s patch’s radiosity


Substructuring

Bottom line: Substructuring allows subpatch radiosities to be

computed without changing the size of the form-factor matrix

Show examples: W&W plate 38, F&vD plate III.21

Note: texts aren’t clear about adaptive subdivision vs substructuring


Progressive Radiosity

Good news: iterative solver of radiosity matrix will converge

Bad news: can take a long time Progressive radiosity: reorder computation to

allow viewing of partial results



Radiosity as described uses Gauss-Seidel iterative solver Must do an entire iteration to get an estimate of

patch radiosities Must precompute and store all O(n2) form factors



1 - 1F11 - 1F12 … - 1F1n B1 E1

- 2F21 1 - 2F22 … - 2F2n B2 E2

. . … . . .

. . … . . .

. . … . . .

- pnFn1 - nFn2 … 1 - nFnn Bn En

Evaluating row i estimates radiosity of patch i based on all other patches

We say the patch gathers light from the environment



Progressive radiosity shoots light from a patch into the environment:

Bj due to Bi = j Bj Fji j rather than

Bi due to Bj = i Bj Fij j Given an estimate of Bi, evaluating this

equation estimates patch i’s contribution to the rest of the scene



A problem: evaluating the equation

Bj due to Bi = j Bj Fji j requires knowing Fji for each patch j

Determining these values requires a hemicube computation per patch

Use reciprocity relationship to get

Bj due to Bi = j Bj Fij (Ai/Aj) j



Now evaluation requires only a single hemicube about patch i Compute, use, and discard form factors Drastically reduces total storage!

Reorder radiosity computation: Pick patch w/ highest estimated radiosity

Shoot to all other patches Update their estimates

Pick new “brightest” patch and repeat



We can look at the scene after every iteration through this loop

Q: How will it look after 1 loop? Q: 2 loops? Q: If m = # of light sources, how will it

look after m loops? After 2m loops?



Subtleties: Pick patch with most energy to shoot

Energy = radiosity * area = Bj Ai

A patch may be selected to shoot again after new light has been shot to it

So don’t shoot Bj , shoot Bj, the amount of radiosity patch i has received since it was last shot


The End

David Luebke 1 10/12/2015 CS 551/651: Advanced Computer Graphics Advanced Ray Tracing Radiosity.

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