David Luebke12/7/2015 CS 551 / 645: Introductory Computer Graphics Review for Midterm.

David Luebke 04/21/23

CS 551 / 645: Introductory Computer Graphics

Review for Midterm


Administrivia

Hand out assignment 2 Hand in assignment 3 Late day policy:

– 1 late day = due tomorrow at noon– Subsequent late days add 24 hours each– Weekends and holidays count

Compiling with C++– UNIX C++ compilers: g++ and /bin/CC– I’ll make sure it works before next assignment


Midterm Examination

Midterm is this Thursday (March 9) Study aids:

– This lecture– Earlier lectures (available on course page)– Last semester’s midterm

See http://www.cs.virginia.edu/~luebke/old_cs551/ But, its not quite the same material

No calculators! (you won’t need them)


Display Technologies

Cathode Ray Tubes– Earliest, still most common graphical display– Understand the basic mechanism

Vacuum tube, phosphors, electron beam

– Pros: bright, fairly high-res, leverages TV tech– Cons: bulky, size-limited, finicky



Vector versus raster display– Vector: traces lines like an oscilloscope

Pros: bright, crisp, uniform lines Cons: wireframe only, flicker for complex scenes

– Raster: fixed scan pattern for electron beam, intensity controlled by scan-out from frame buffer

Pros: display solid objects, image complexity limited only by framebuffer resolution

Cons: discreet sampling (aliasing), memory cost



LCDs– Understand the basic mechanism

Polarized light, crystals twist 90º unless excited Basically a light valve: reflective or transmissive

– Pros: light-weight and thin– Cons: expensive, high-power (when backlit),

limited in size



Also know:– Plasma display panels– Digital Micromirror Devices


Framebuffers

Memory array storing image in pixels Issues: memory speed, size, bus contention Different types in common use, motivated

mainly by memory cost– True-Color: 24 bits, 8 per RGB (or 32 bits with )– Hi-Color: 16 bits (R = 6, G = 6, B = 4)– Pseudo-Color: 8 bits index into 256-entiry color

lookup table (entries typically 24-bits)


Mathematical Foundations

Geometry (2-D, 3-D) Trigonometry Vector spaces

– Elements: scalars and vectors– Operations:

Addition (identity & inverse) Scalar multiplication (distributive rule)

– Linear combinations, dimension, basis sets– Inner (dot) product, vector (cross) product


Mathematical Foundations

Affine spaces– Elements: points– Operations:

Subtraction (point - point = vector) Addition (vector + point = point)

Matrices– Linear transforms, vector-matrix multiplication– Matrix-matrix multiplication– Composition of linear transforms = matrix

concatenation


The Rendering Pipeline

Transform

Illuminate

Transform

Clip

Project

Rasterize

Model & CameraParameters

Rendering Pipeline Framebuffer Display


The Rendering Pipeline: 3-D

ModelingTransforms

Scene graphObject geometry

LightingCalculations

ViewingTransform

Clipping

ProjectionTransform

Result:

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

• Vertices shaded according to lighting model

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

• Exactly those vertices & portions of polygons in view frustum

• 2-D screen coordinates of clipped vertices


Geometric Transforms

Modeling transforms: object coordinates world coordinates

Viewing transform: world coordinates eye coordinates– eye coordinates == camera coordinates == view

coordinates

Projection transform: eye coordinates 2-D screen coordinates


Geometric Transforms

Understand homogeneous coordinates– [x, y, z, w]T == (x/w, y/w, z/w)– Allows us to capture translation and projection as

matrices

Know your 4x4 Euclidean transform matrices:– Translation, scale, rotation about X, Y, Z

Understand rotation about an arbitrary axis Understand order of composition for matrices

– In OpenGL: using column vectors as points order from right to left


Perspective Projection

Geometry of the perspective projection:

P (x, y, z)X

Z

Viewplane

d

(0,0,0) x’ = ?



Desired result:

dzdz

y

z

ydy

dz

x

z

xdx

z

y

d

y

z

x

d

x

,','

',

'



A matrix that accomplishes this:

0100

0100

0010

0001

d

M eperspectiv


Rasterizing Lines

Review McMillan’s great java-enabled lecture First stab: slope-intercept + symmetry A case study in optimization

– Special case boundary conditions if necessary– Optimize inner loops

Incremental update using DDA (biggest win) Low-level tricks: integer arithmetic, compare to 0, etc. Culmination: Bresenham’s algorithm

– Be aware of diminishing returns and readability/portability tradeoffs


Rasterizing Triangles

Triangles are nice to deal with because they are always planar and always convex

Triangle rasterization techniques:– REYES: recursive subdivision of primitive– Warnock: recursive subdivision of screen– Edge walking– Edge equations



Edge walking: – Draw edges vertically

– Fill in horizontal spans for each scanline

– Interpolate colors down edges

– At each scanline, interpolate edge colors across span

– Pros: Fast: touch only lit pixels,

touch pixels only once

– Cons: Finicky: lots of special cases,

hard to get just right



Edge Equations– Equation of a line defines two half-spaces– Triangle can be represented as intersection of

three half-spaces:

A1x + B1y + C1 < 0

A2 x + B

2 y + C2 < 0

A 3x

+ B 3

y +

C 3 <

0

A1x + B1y + C1 > 0

A 3x

+ B 3

y +

C 3 >

0 A2 x + B

2 y + C2 > 0



Basic algorithm: – Walk pixels in bounding box– Evaluate three edge equations– If all are greater than zero, shade pixel

Issues:– Computing edge equations: numerical precision– Interpolating parameters (i.e., color): just like

another edge equation (why?)

Optimizing the algorithm– Like line rasterization: DDA, early termination, etc.


Rasterizing General Polygons

Parity test:– Starting outside polygon,

count edges crossed.

– Odd = inside,even = outside

Big cost: testing every edgeagainst every pixel

Solution: the active edge table algorithm– Sort edges by Y

– Keep a list of edges that intersect current scanline, sorted by their X-intersection w/ scanline

A

B

C

D

E

F

G I

H


Clipping Lines

Cohen-Sutherland Algorithm– Clip 2-D line segments to rectangular viewport– Designed for rapid trivial accept & trivial reject ops

4-bit outcodes divide screen into 9 regions Bitwise operations determine whether to accept, reject,

or intersect with a viewport edge & recurse May require multiple iterations


Clipping Polygons

Clipping polygons fundamentally more difficult– Polygons can gain or lose edges– Concave polygons can even multiply

Sutherland-Hodgman Algorithm– Simplify by divide-and-conquer: consider each

clipping plane individually Input: polygon as ordered list of vertices Output: polygon as ordered list of vertices Lends itself to pipelined hardware implementation


Clipping Polygons

Sutherland-Hodgman Algorithm– Know the details:

Point-plane test Line-plane intersection Rules:

inside outside

s

p

p output

inside outside

s

p

no output

inside outside

sp

i output

inside outside

sp

i outputp output


Clipping in 3-D

Problem: clipping under perspective must happen before homogeneous divide– Solution 1: clip to hither plane in eye coordinates, then

multiply by projection matrix, then do homogeneous divide

Better: transform to canonical perspective coordinates to simplify clipping

– Solution 2: clip after projection (must clip all 4 homogeneous coordinates)

– Solution 3 (ugly but common): clip to hither & yon before projection, clip to 2-D viewport after projection and divide


Color

Rods and cones Cones and color perception

– Metamers– 3-D color: X, Y, and Z; CIE color space

Gamma correction


Lighting

Definitions: illumination, lighting, shading Illumination:

– Direct versus indirect– Light properties: geometry, spectrum

Common simplifications: ambient, directional, and point

– Surface material: geometry, reflectance, microstructure

Common simplification: Phong lighting Diffuse (Lambertian) reflection: incoming light reflected

equally in all directions, proportional to N • L Specular reflection: approximate falloff with (V • R)nshiny


Lighting

Putting it all together: the Phong lighting model

Note: evaluate per light, per color component Common simplification: constant V (viewer

infinitely far away)

lights

i

n

sdiambientatotal

shiny

RVkLNkIIkI#

1

ˆˆˆˆ


Shading

Where to apply lighting calculations?– Once per face: flat shading– Once per vertex, interpolate resulting color:

Gouraud shading– Once per pixel, interpolating normal vectors from

vertices: Phong shading

Flat shading Phong Shading

David Luebke12/7/2015 CS 551 / 645: Introductory Computer Graphics Review for Midterm.

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