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CS 551/651: Advanced Computer GraphicsPowerPoint Presentation

CS 551/651: Advanced Computer Graphics

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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

David Luebke 211/17/2014

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!

David Luebke 311/17/2014

Recap: Stochastic Sampling

- Poisson distribution:
- Completely random
- Add points at random until area is full.
- Uniform distribution: some neighboring samples close together, some distant

David Luebke 411/17/2014

Recap: Stochastic Sampling

- 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

David Luebke 511/17/2014

Recap: Stochastic Sampling

- 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.

David Luebke 611/17/2014

Recap: Nonuniform Supersampling

- To be correct, need to modify filtering step:

David Luebke 711/17/2014

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)

I’(x,y)=

h(x-i, y-j)

David Luebke 811/17/2014

Recap: Antialiasing and Texture Mapping

- Texture mapping is uniquely harder
- Potential for infinite frequencies

- Texture mapping is uniquely easier
- Textures can be prefiltered

David Luebke 911/17/2014

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

David Luebke 1011/17/2014

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

David Luebke 1111/17/2014

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

David Luebke 1211/17/2014

Distributed Ray Tracing

- Distribute rays stochastically across:
- Pixel for antialiasing
- Light sourcefor 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

David Luebke 1311/17/2014

Distributed Ray Tracing

- 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

David Luebke 1411/17/2014

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?

David Luebke 1511/17/2014

Backwards Ray Tracing

- Backwards ray tracing can capture:
- Indirect illumination
- Color bleeding
- Caustics
- Remember what a caustic is?
- Give some examples
- Remember where caustics get the name?

David Luebke 1611/17/2014

Backwards Ray Tracing

- 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

David Luebke 1711/17/2014

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 1911/17/2014

Backwards Ray Tracing

- Watt & Watt, Plate 34
- Illustrates Arvo’s method of backwards ray tracing using illumination maps
- Illustrates a scene with caustics

- Related idea: photon maps

David Luebke 2011/17/2014

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

David Luebke 2111/17/2014

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

David Luebke 2211/17/2014

Radiosity Introduction

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

- After Phong, two major approaches:
- Ray tracing
- Radiosity

David Luebke 2311/17/2014

Radiosity Introduction

- 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!

David Luebke 2411/17/2014

Radiosity Introduction

- 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

David Luebke 2511/17/2014

Radiosity

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

David Luebke 2611/17/2014

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!

David Luebke 2711/17/2014

Fundamentals

- Theoretical foundation: heat transfer
- Need system of equations that describes surface interreflections
- Simplifying assumptions:
- Environment is closed
- All surfaces are Lambertian reflectors

David Luebke 2811/17/2014

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

David Luebke 2911/17/2014

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?

David Luebke 3011/17/2014

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

David Luebke 3111/17/2014

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

David Luebke 3211/17/2014

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 3911/17/2014

Form Factors

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

David Luebke 4011/17/2014

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 4111/17/2014

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)

David Luebke 4211/17/2014

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 4311/17/2014

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

David Luebke 4411/17/2014

The End

David Luebke 4511/17/2014

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