lecture 6: rasterization 1 (triangles)lingqi/teaching/... · cs180, spring 2019 lingqi yan, uc...

CS180: Introduction to Computer GraphicsSpring 2019, Lingqi Yan, UC Santa Barbara

Rasterization 1 (Triangles)Lecture 6:

http://www.cs.ucsb.edu/~lingqi/teaching/cs180.html

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Last Lectures

• Transformation- Modeling + Viewing

- Model + View + Projection

- Linear, Affine, Projection transformations

- Homogeneous coordinates

!2


Today

!3

• Finishing up Viewing

- Viewport transformation

• Rasterization

- Different raster displays

- Rasterizing a triangle


What’s after MVP?

• Model transformation (placing objects)

• View transformation (placing camera)

• Projection transformation- Orthographic projection (cuboid to “canonical” cube [-1, 1]3)

- Perspective projection (frustum to “canonical” cube)

• Canonical cube to ?

!4


Canonical Cube to Screen

• What is a screen?

- A typical kind of raster display

• Raster == discretize (personally)

- Usually to an array of pixels

- Size of the array: resolution

• Pixel (FYI, short for “picture element”)

- For this course only: A pixel is a little square with uniform color

- Color is a mixture of (red, green, blue)

!5



• Defining the screen space- Slightly different from the “tiger book”

!6

array of pixels

X

Y

(0, 0)

Pixel (x, y) is centered at(x + 0.5, y + 0.5)

Pixels’ indices are from (0, 0) to (width - 1, height - 1)

Pixels’ indices are in the form of (x, y), where both x and y are integers

The screen covers range (0, 0) to (width, height)



!7

array of pixels

X

Y

(0, 0)

• Irrelevant to z

• Transform in xy plane: [-1, 1]2 to [0, width] x [0, height]

x

z

y



!8

• Irrelevant to z

• Transform in xy plane: [-1, 1]2 to [0, width] x [0, height]

• Viewport transform matrix:

Mviewport =

0

BB@

width2 0 0 width

2

0 height2 0 height

20 0 1 00 0 0 1

1

CCA

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Slide courtesy of Prof. Ren Ng, UC Berkeley


Today: Rasterizing Triangles into Pixels

!9

Life of Pi (2012)



Drawing Machines

!10



CNC Sharpie Drawing Machine

Aaron Panone with Matt W. Moore

!11

http://44rn.com/projects/numerically-controlled-poster-series-with-matt-w-moore/



Laser Cutters

!12



Oscilloscope

!13



Cathode Ray Tube

!14



Oscilloscope Art

Jerobeam Fenderson!15

https://www.youtube.com/watch?v=rtR63-ecUNo



Television - Raster Display CRT

Cathode Ray Tube Raster Scan(modulate intensity)

!16



Frame Buffer: Memory for a Raster Display

!17



A Sampling of Different Raster Displays

!18



Flat Panel Displays

!19

B.Woods, Android Pit

Low-Res LCD Display

Color LCD, OLED, …



LCD (Liquid Crystal Display) Pixel

Principle: block or transmit light by twisting polarization Illumination from backlight(e.g. fluorescent or LED) Intermediate intensity levels by partial twist

!20

[H&

B fig

. 2-1

6]



LED Array Display

!21

Light emitting diode array



DMD Projection Display

!22

[Y.K

. Rab

inow

itz; E

KB

Tec

hnol

ogie

s



DMD Projection Display

!23

[Tex

as In

stru

men

ts]

Array of micro-mirror pixels DMD = Digital Micromirror Device



Electrophoretic (Electronic Ink) Display

!24

[Wik

imed

ia C

omm

ons

—Se

narc

lens

]

http://commons.wikimedia.org/wiki/User:Senarclens



Drawing to Raster Displays

!25



Polygon Meshes

!26

Life of Pi (2012)



Triangle Meshes

!27



Triangle Meshes

!28



Triangles - Fundamental Shape Primitives

Why triangles?

• Most basic polygon

• Break up other polygons

• Optimize one implementation

• Triangles have unique properties

• Guaranteed to be planar

• Well-defined interior

• Well-defined method for interpolating values at vertices over triangle (barycentric interpolation)

!29



Drawing a Triangle To The Framebuffer

(“Rasterization”)

!30



What Pixel Values Approximate a Triangle?

Input: position of triangle vertices projected on screen

Output: set of pixel valuesapproximating triangle

!31

?(2.2, 1.3)

(4.4, 11.0)(15.3, 8.6)



Today, Let’s Start With A Simple Approach: Sampling

!32



Sampling a Function

Evaluating a function at a point is sampling. We can discretize a function by sampling. for( int x = 0; x < xmax; x++ ) output[x] = f(x);

Sampling is a core idea in graphics.We sample time (1D), area (2D), direction (2D), volume (3D) …

!33



Let’s Try Rasterization As 2D Sampling

!34



Sample If Each Pixel Center Is Inside Triangle

!35



Sample If Each Pixel Center Is Inside Triangle

!36

Define Binary Function: inside(tri,x,y)

inside(t,x,y) = 1

0

pixel (x,y) in triangle t

otherwise



Rasterization = Sampling A 2D Indicator Function

for( int x = 0; x < xmax; x++ ) for( int y = 0; y < ymax; y++ ) Image[x][y] = f(x + 0.5, y + 0.5);

Rasterize triangle tri by sampling the function f(x,y) = inside(tri,x,y)

!38



Recall: Sample Locations

!39

(0,0) (w,0)

(0,h) (w,h)

Sample location for pixel (x,y)

(x+1/2,y+1/2)



Evaluating inside(tri,x,y)

!40



Triangle = Intersection of Three Half Planes

!41

P0

P1

P2



Each Line Defines Two Half-Planes

Implicit line equation • L(x,y) = Ax + By + C

• On line: L(x,y) = 0 • Above line: L(x,y) > 0 • Below line: L(x,y) < 0

!42

> 0

< 0

= 0



Line Equation Derivation

!43

P0

P1

T

T = P1 � P0 = (x1 � x0, y1 � y0)

Line Tangent Vector




!44

(x,y)

(-y,x)

Perp(x, y) = (�y, x)

General Perpendicular Vector in 2D




!45

P0

P1

T

N

N = Perp(T ) = (�(y1 � y0), x1 � x0)

Line Normal Vector




!46

P0

P1

N

P = (x, y)

V

V = P � P0 = (x� x0, y � y0)

L(x, y) = V ·N = �(x� x0)(y1 � y0) + (y � y0)(x1 � x0)



Line Equation

!47

P0

P1

N

P = (x, y)

V

V = P � P0 = (x� x0, y � y0)

L(x, y) = V ·N = �(x� x0)(y1 � y0) + (y � y0)(x1 � x0)



Line Equation Tests

!48

P0

P1

N

P

V

L(x, y) = V ·N > 0



L(x, y) = V ·N = 0

Line Equation Tests

!49

P0

P1

N

PV



L(x, y) = V ·N < 0

Line Equation Tests

!50

P0

P1

N

PV



Point-in-Triangle Test: Three Line Tests

!51

P0

P1

P2

Compute line equations from pairs of vertices

Pi = (Xi, Yi)

dXi = Xi+1 - Xi

dYi = Yi+1 - Yi

Li (x, y) = –(x - Xi) dYi + (y - Yi) dXi

= Ai x + Bi y + Ci

Li (x, y) = 0 : point on edge < 0 : outside edge > 0 : inside edge




!52

P0

P1

P2

L0(x, y) > 0

Pi = (Xi, Yi)

dXi = Xi+1 - Xi

dYi = Yi+1 - Yi


= Ai x + Bi y + Ci





!53

P0

P1

P2

L1(x, y) > 0

Pi = (Xi, Yi)

dXi = Xi+1 - Xi

dYi = Yi+1 - Yi


= Ai x + Bi y + Ci





!54

P0

P1

P2

L2(x, y) > 0

Pi = (Xi, Yi)

dXi = Xi+1 - Xi

dYi = Yi+1 - Yi


= Ai x + Bi y + Ci





!55

Sample point s = (sx, sy) is inside the triangle if it is inside all three lines.

inside(sx, sy) =L0 (sx, sy) > 0 &&L1 (sx, sy) > 0 &&L2 (sx, sy) > 0;

P0

P1

P2

Note: actual implementation of inside(sx,sy) involves ≤ checks based on edge rules



Edge Cases (Literally)

Is this sample point covered by triangle 1, triangle 2, or both?

!56

1

2



Incremental Triangle Traversal (Faster?)

!57

P0

P1

P2



Modern Approach: Tiled Triangle Traversal

!58

P0

P1

P2Traverse triangle in blocks

Test all samples in block in parallel

All modern GPUs have special-purpose hardware for efficient point-in-triangle tests

Advantages: - Simplicity of wide parallel execution

overcomes cost of extra point-in-triangle tests (most triangles cover many samples, especially when super-sampling)

- Can skip sample testing work: entire block not in triangle (“early out”), entire block entirely within triangle (“early in”)



Signal Reconstruction on Real Displays

!59



Real LCD Screen Pixels (Closeup)

iPhone 6S Galaxy S5

!60

iphonearena.com iphonearena.com

Notice R,G,B pixel geometry! But in this class, we will assume a colored square full-color pixel.



Aside: What About Other Display Methods?

!61

Color print: observe half-tone pattern

Assume Display Pixels Emit Square of Light

LCD pixelon laptop

Each image sample sent to the display is converted into a little square of light of the appropriate color: (a pixel = picture element)

* LCD pixels do not actually emit light in a square of uniform color, but this approximation suffices for our current discussion



So, If We Send The Display This Sampled Signal

!63



The Display Physically Emits This Signal

!64



Compare: The Continuous Triangle Function

!65



What’s Wrong With This Picture?

!66

Jaggies!



Aliasing (Jaggies)

!67

Is this the best we can do?

Thank you!

lecture 6: rasterization 1 (triangles)lingqi/teaching/... · cs180, spring 2019 lingqi yan, uc...

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