David Luebke 04/20/23

CS 551 / 645: Introductory Computer Graphics

David Luebke

cs551@cs.virginia.edu

http://www.cs.virginia.edu/~cs551

David Luebke 04/20/23

Administrivia

Course evaluations on the web– Help make this class better next semester

Looking into alternative location for final– Who wants to move?

David Luebke 04/20/23

The Plan

Today– Finish up radiosity– Recap up till midterm

Tomorrow– Recap since midterm

David Luebke 04/20/23

Recap: Radiosity Fundamentals

Model light transfer between patches as a system of linear equations

Solving this system gives the intensity at each patch– Simplifying assumptions:

Environment is closed All surfaces are Lambertian (perfectly diffuse) reflectors

– Solving for R, G, B intensities produces color at each patch

Render patches as colored polygons

David Luebke 04/20/23

Recap: Radiosity Fundamentals

Radiosity is the rate at which energy leaves a 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

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Recap: The Radiosity Equation

For each patch 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

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Recap: 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

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Recap: The (Expanded) Radiosity Equation

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!

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Recap: Evaluating Form Factors (Hemicubes) 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 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?)

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Recap: Hemicubes

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

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Form Factors: Ray Casting

Idea: shoot rays from center of patch in hemispherical pattern

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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

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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 What kind of scene might ray tracing actually be faster

on?

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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

– What are the problems with this approach?

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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

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Radiosity Continued

Lots more to know about radiosity:– Progressive radiosity: viewing an approximate

solution early– Hierarchical radiosity: increasing patch resolution

on an as-needed basis

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Review for Exam

Quick recap of lecture topics for exam… – Display technologies

Vector versus raster: what’s a pixel? CRT (black & white, color): shadow mask, phosphors,

electron guns LCD: polarizing crystals that line up under an E-field,

losing their polarization and acting as light valves Pros and cons of different technologies

– Framebuffers Fast, dual-ported memory bank for holding pixels True-color (24-bit) vs Pseudocolor (8-bit indexed) vs hi-

color (16-bit, 6-6-4 ?)

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Review For Exam

Color– Basic physiology: retina, rods, cones

– Different types of cones: L, M, S

– Metamers: perceptually identical color senstions caused by different spectra

– CIE Color Space (X, Y, Z) Color mix-and-match experiment Hypothetical light sources X, Y, Z (why?) Three dimensional shape, often simplified to 2-D gamut

– Other color spaces (RGB, HSV)

– Gamma correction: linearize non-linear response of display device

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Review For Exam

Rasterizing lines– A methodology in optimizing code– Simplest way: solve slope-intercept equation

Check slope and step in X or Y

– DDA: find incremental change per inner loop– Bresenham: take advantage of rational slope,

endpoints to optimize with integer arithmetic

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Review For Exam

Rasterizing polygons– Can break into triangles for convenience

Trivial for convex polys, difficult for complex polys

– Edge equations: Evaluate equation of edge (linear expression) per pixel If all three edges are positive, light pixel Can evaulate R,G,B, Z as linear expressions too Hardware: SIMD array Software: bounding box Issues: ensuring consistent edge equations, numerical

stability when evaluating edge equations

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Review For exam

Rasterizing triangles: edge walking– Find edges (DDA)– Interpolate colors down edges– Interpolate colors across scanlines– Fast: touches only pixels necessary– Difficult: lots of special cases, fractional offsets,

precision worries

Rasterizing triangles: general issues– Need consistent rules about pixels right on edge

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Review For Test

Can also rasterize general polygons– Active edge table: sort edges by ymin and ymax,

then by x intersection w/ scanline– March up scanline by scanline, filling pixels

according to parity rule– Hard to interpolate color, etc. correctly -- not

planar in color space, in general

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Review For Test

Clipping– Want only portions of lines, polygons in viewport– Cohen-Sutherland line clipping: binary outcodes– Polygon clipping is harder

Triangle in, 7-gon out Concave polyon in, 2 polygons out

– Sutherland-Hodgman: clip against view planes in succession

In-out rules Line-plane intersection

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Review For Test

Clipping in 3-D– Sutherland-Hodgman extends easily to clipping

against six view-frustum planes– Issue: when in the pipeline to clip?

World coordinates: arbitrary planes expensive Camera coordinates: two planes easy, four hard Canonical perspective coordinates: two easy, four okay Canonical orthographic coordinates/screen coordinates:

reduces matrix multiplies, requires clipping in homogeneous coordinates

Common shortcut: clip near-far in camera coordinates, multiply by perspective matrix, clip left-right-top-bottom

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Review For Test

3-D graphics:

Transform

Illuminate

Transform

Clip

Project

Rasterize

Model & CameraModel & CameraParametersParameters Rendering PipelineRendering Pipeline FramebufferFramebuffer DisplayDisplay

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Review For Test Rendering pipeline:

ModelingTransforms

Scene graphObject geometry

LightingCalculations

ViewingTransform

Clipping

ProjectionTransform

Result:Result:

• All vertices of scene in shared 3-D “world” coordinate All vertices of scene in shared 3-D “world” coordinate systemsystem

• Vertices shaded according to lighting modelVertices shaded according to lighting model

• Scene vertices in 3-D “view” or “camera” coordinate Scene vertices in 3-D “view” or “camera” coordinate systemsystem

• Exactly those vertices & portions of polygons in view Exactly those vertices & portions of polygons in view frustumfrustum

• 2-D screen coordinates of clipped vertices2-D screen coordinates of clipped vertices

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Review For Test

Transformations– Shift in coordinate systems (basis sets)– Accomplished via matrix multiplication– Modeling transforms: object->world– Viewing transform: world->view– Projection transform: view->screen

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Review For Test

Rigid-body transforms– Rotation: 2-D. 3-D (canonical & arbitrary axis)– Scaling– Translation: homogeneous coords, 4x4 matrices– Composiing transforms

Matrix multiplication Order from right to left

Projection transforms– Geometry of perspective projection– Derive perspective projection matrix