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

• 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 ?

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)

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

array of pixels

(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)

array of pixels

(0, 0)

• Irrelevant to z

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

• Irrelevant to z

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

• Viewport transform matrix:

Mviewport =

width2 0 0 width

0 height2 0 height

20 0 1 00 0 0 1

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Slide courtesy of Prof. Ren Ng, UC Berkeley

Today: Rasterizing Triangles into Pixels

Life of Pi (2012)

Drawing Machines

CNC Sharpie Drawing Machine

Aaron Panone with Matt W. Moore

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

Laser Cutters

Oscilloscope

Cathode Ray Tube

Oscilloscope Art

Jerobeam Fenderson!15

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

Television - Raster Display CRT

Cathode Ray Tube Raster Scan(modulate intensity)

Frame Buffer: Memory for a Raster Display

A Sampling of Different Raster Displays

Flat Panel Displays

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

LED Array Display

Light emitting diode array

DMD Projection Display

itz; E

DMD Projection Display

Array of micro-mirror pixels DMD = Digital Micromirror Device

Electrophoretic (Electronic Ink) Display

Drawing to Raster Displays

Polygon Meshes

Life of Pi (2012)

Triangle Meshes

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)

Drawing a Triangle To The Framebuffer

(“Rasterization”)

What Pixel Values Approximate a Triangle?

Input: position of triangle vertices projected on screen

Output: set of pixel valuesapproximating triangle

?(2.2, 1.3)

(4.4, 11.0)(15.3, 8.6)

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

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) …

Let’s Try Rasterization As 2D Sampling

Sample If Each Pixel Center Is Inside Triangle

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

inside(t,x,y) = 1

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)

Recall: Sample Locations

(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)

Triangle = Intersection of Three Half Planes

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

Line Equation Derivation

T = P1 � P0 = (x1 � x0, y1 � y0)

Line Tangent Vector

(-y,x)

Perp(x, y) = (�y, x)

General Perpendicular Vector in 2D

N = Perp(T ) = (�(y1 � y0), x1 � x0)

Line Normal Vector

P = (x, y)

V = P � P0 = (x� x0, y � y0)

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

Line Equation

P = (x, y)

V = P � P0 = (x� x0, y � y0)

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

Line Equation Tests

L(x, y) = V ·N > 0

L(x, y) = V ·N = 0

Line Equation Tests

L(x, y) = V ·N < 0

Line Equation Tests

Point-in-Triangle Test: Three Line Tests

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

L0(x, y) > 0

Pi = (Xi, Yi)

dXi = Xi+1 - Xi

dYi = Yi+1 - Yi

= Ai x + Bi y + Ci

L1(x, y) > 0

Pi = (Xi, Yi)

dXi = Xi+1 - Xi

dYi = Yi+1 - Yi

= Ai x + Bi y + Ci

L2(x, y) > 0

Pi = (Xi, Yi)

dXi = Xi+1 - Xi

dYi = Yi+1 - Yi

= Ai x + Bi y + Ci

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;

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?

Incremental Triangle Traversal (Faster?)

Modern Approach: Tiled Triangle Traversal

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

Real LCD Screen Pixels (Closeup)

iPhone 6S Galaxy S5

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?

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

The Display Physically Emits This Signal

Compare: The Continuous Triangle Function

What’s Wrong With This Picture?

Jaggies!

Aliasing (Jaggies)

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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