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CS 551/651: Advanced Computer GraphicsPowerPoint Presentation

CS 551/651: Advanced Computer Graphics

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Administrivia

- Quiz 1: Tuesday, Feb 20
- Yes, I’ll have your homework graded by then (somehow)
- Normal written exam (oral later)

David Luebke 211/17/2014

Recap: Distributed Ray Tracing

- Distributed ray tracing: an elegant stochastic approach that distributes rays across:
- Pixel for antialiasing
- Light sourcefor 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

David Luebke 311/17/2014

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

David Luebke 411/17/2014

Recap: Backwards Ray Tracing

- Arvo: illumination mapstile 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

David Luebke 511/17/2014

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

David Luebke 611/17/2014

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

David Luebke 711/17/2014

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

- The radiosity of a surface is the rate at which energy leaves the surface
- Simplifying assumptions
- Environment is closed
- All surfaces have Lambertian reflectance
- Surface patches emit and reflect light uniformly over their entire surface

David Luebke 811/17/2014

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

David Luebke 911/17/2014

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

David Luebke 1011/17/2014

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

David Luebke 1711/17/2014

Form Factors

- So…light exchange between all patches becomes a matrix:
- What do the various terms mean?

David Luebke 1811/17/2014

Form Factors

1 - 1F11 - 1F12 … - 1F1n B1E1

- 2F21 1 - 2F22 … - 2F2n B2E2

. . … . . .

. . … . . .

. . … . . .

- 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!

David Luebke 1911/17/2014

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)

David Luebke 2011/17/2014

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

David Luebke 2111/17/2014

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

David Luebke 2211/17/2014

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”?

David Luebke 2311/17/2014

Form Factors: Hemicube

- Hemicube algorithm: Think Z-buffer
- Render the model onto a hemicubeas 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?

David Luebke 2411/17/2014

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?)

David Luebke 2511/17/2014

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?)

- Aliasing! Low resolution buffer can’t capture actual polygon contributions very exactly

David Luebke 2611/17/2014

Form Factors: Ray Casting

- Idea: shoot rays from center of patch in hemispherical pattern

David Luebke 2711/17/2014

Form Factors: Ray Casting

- Advantages:
- Hemisphere better approximation than hemicube
- More even sampling reduces aliasing

- Don’t need to keep item buffer
- Slightly simpler to calculate coverage

- Hemisphere better approximation than hemicube

David Luebke 2811/17/2014

Form Factors: Ray Casting

- 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?

David Luebke 2911/17/2014

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?

- Calculating form factors at the patch vertices helps address some problems:

David Luebke 3011/17/2014

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

- Analytical:

David Luebke 3111/17/2014

Substructuring

- More patches better results
- Problem: # form factors grows quadratically with # patches
- Substructuring: adaptively subdivide patches into elements where high radiosity gradient is found

David Luebke 3211/17/2014

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

David Luebke 3311/17/2014

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

David Luebke 3411/17/2014

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

David Luebke 3511/17/2014

Progressive Radiosity

- Good news: iterative solver of radiosity matrix will converge
- Bad news: can take a longtime
- Progressive radiosity: reorder computation to allow viewing of partial results

David Luebke 3611/17/2014

Progressive Radiosity

- 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

David Luebke 3711/17/2014

Progressive Radiosity

1 - 1F11 - 1F12 … - 1F1n B1E1

- 2F21 1 - 2F22 … - 2F2n B2E2

. . … . . .

. . … . . .

. . … . . .

- 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

David Luebke 3811/17/2014

Progressive Radiosity

- Progressive radiosity shoots light from a patch into the environment:
Bjdue to Bi = j Bj Fjij rather thanBidue to Bj = i Bj Fijj

- Given an estimate of Bi, evaluating this equation estimates patch i’s contribution to the rest of the scene

David Luebke 3911/17/2014

Progressive Radiosity

- A problem: evaluating the equationBjdue to Bi = j Bj Fjij requires knowing Fji for each patch j
- Determining these values requires a hemicube computation per patch
- Use reciprocity relationship to getBjdue to Bi = j Bj Fij(Ai/Aj)j

David Luebke 4011/17/2014

Progressive Radiosity

- 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

- Pick patch w/ highest estimated radiosity

David Luebke 4111/17/2014

Progressive Radiosity

- 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?

David Luebke 4211/17/2014

Progressive Radiosity

- 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

- Pick patch with most energy to shoot

David Luebke 4311/17/2014

The End

David Luebke 4411/17/2014

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