Rasterization (03) RNDr. Martin Madaras, PhD. - - PowerPoint PPT Presentation

Principles of Computer Graphics and Image Processing

Rasterization (03)

• RNDr. Martin Madaras, PhD.

martin.madaras@stuba.sk

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Last lessons summary

CG reference model

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Computer Vision/ Computer Graphics

Computer Graphics

CG reference model

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 Geometry space

 continuous  3Dimensional

 Screen space

 discrete  2Dimensional

3D Scene vs. 2D image

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Geometry vs. screen space

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3D Continuous Parametric Models 2D Discrete Non-parametric Pixels

3D polygon rendering

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 Many applications use rendering of 3D polygons with

direct illumination

3D polygon rendering

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 Many applications use rendering of 3D polygons with

direct illumination

Quake 3, ID software

3D polygon rendering

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 Many applications use rendering of 3D polygons with

direct illumination

CATIA, Dassault Systemes

3D polygon rendering

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 What steps are necessary to produce an image of a 3D

scene?

Ray Casting

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 One approach is to cast rays from the camera…

Ray Casting

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 And find intersections with the scene…  We are going to describe different approach this lesson

3D polygon rendering

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 Second approach is called Rasterization  Way how to efficiently draw primitives into screen space

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How the lectures should look like #1

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Rasterization

3D rendering pipeline

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3D polygons Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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3D rendering pipeline

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Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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array of vertex positions x,y,z { 0,1,0, 1,1,0, 1,0,0, 0,0,0}

OpenGL executes steps of the 3D rendering pipeline for each polygon

3D rendering pipeline

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3D polygons Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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Transform into 3D world coordinate system

3D rendering pipeline

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3D polygons Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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Transform into 3D world coordinate system Illuminate according to light

3D rendering pipeline

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3D polygons Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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Transform into 3D world coordinate system Illuminate according to light Transform into 3D camera coordinate system

3D rendering pipeline

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3D polygons Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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Transform into 3D world coordinate system Illuminate according to light Transform into 3D camera coordinate system Transform into 2D camera coordinate system Clip polygons outside of camera’s view Draw pixels

3D rendering pipeline

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3D polygons Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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Transform into 3D world coordinate system Illuminate according to light Transform into 3D camera coordinate system Transform into 2D camera coordinate system Clip polygons outside of camera’s view Draw pixels

3D rendering pipeline

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3D polygons Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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Transform into 3D world coordinate system Illuminate according to light Transform into 3D camera coordinate system Transform into 2D camera coordinate system Clip polygons outside of camera’s view Draw pixels

3D rendering pipeline

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3D polygons Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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 Model transformation

 local → global coordinates

 View transformation

 global → camera

 Projection transformation

 camera → screen

 Clipping, rasterization,

texturing & Lighting

 might take place earlier

Transformations

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3D polygons Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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Transform into 3D world coordinate system Illuminate according to light Transform into 3D camera coordinate system Transform into 2D camera coordinate system Clip polygons outside of camera’s view Draw pixels

Transformations

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3D Object coordinates

Modeling Transformation

3D World coordinates

Viewing Transformation

3D Camera coordinates

Projection Transformation

2D Camera coordinates

Window to Viewport Transformation

2D Image coordinates

P’(x’, y’)

Transformations map points from one coordinate system to another

Camera coordinates

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 Canonical coordinate system

 Convention is right-handed (looking down -z)  Convenient for projection, clipping etc.

Coordinate systems

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 DirectX <= 9, left handed only

Local coordinates

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 Each object has its own coordinate system

Global coordinates

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 One system for the whole scene

Local → Global coordinates

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 Translation

            = 1 1 1 ) 1 , , ( ) 1 , ' , ' ( y t x t y x y x

Local → Global coordinates

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 Rotation

          − = 1 cos sin sin cos ) 1 , , ( ) 1 , ' , ' (     y x y x

Local → Global coordinates

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 All transformations combined

Transformations

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 Transformation from one coordinate system to another

• ne is a composition of partial transformations:

 Translation  Rotation  Scaling

All transformations

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 Model transformation

 Unify coordinates by transforming local

to global coordinates

 View transformation

 Transform global coordinates so that they are

aligned with camera coordinates

 T

• make projection computable

Model transformation

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 Transformation local → global  Combination of rotate, translate, scale  Matrix multiplication

Model transformation

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 Translation, rotation, scaling

          − 1 cos sin sin cos               1 y s x s

            1 1 1 y t x t

Camera coordinates

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 XY of screen + Z as direction of view

Global→camera coordinates

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 T * Ry * Rx

 Translation, rotation, rotation

 T * Ry * Rx * Rz

 if the camera is rolled

 Projection P

 orthogonal, perspective, isometric ...

Viewing Transformation

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 Mapping from world to camera coordinates

 Eye position maps to origin  Right vector maps to X axis  Up vector maps to

Y axis

 Back vector maps to Z axis

Finding the Viewing Transformation

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 We have the camera (in world coordinates)  We want T taking objects from world to camera

𝑞𝐷 = 𝑈𝑞𝑋

 Trick: find T taking objects in camera to world

𝑞𝑋 = 𝑈−1𝑞𝐷 𝑦′ 𝑧′ 𝑨′ 𝑥′ = 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 𝑕 ℎ 𝑗 𝑘 𝑙 𝑚 𝑛 𝑜 𝑝 𝑞 𝑦 𝑧 𝑨 𝑥

Finding the Viewing Transformation

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 Trick: Map from camera coordinates to world

 Origin maps to eye position  z axis maps to Back vector  y axis maps to Up vector  x axis maps to Right vector  T

• get 𝑈−1we just need to invert 𝑈

𝑦′ 𝑧′ 𝑨′ 𝑥′ =

𝑠

𝑦

𝑣𝑦 𝑐𝑦 𝑓𝑌 𝑠

𝑧

𝑣𝑧 𝑐𝑧 𝑓𝑧 𝑠

𝑨

𝑣𝑨 𝑐𝑨 𝑓𝑨 𝑠

𝑥

𝑣𝑥 𝑐𝑥 𝑓𝑥

𝑦 𝑧 𝑨 𝑥

Finding the Viewing Transformation

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 Trick: Map from camera coordinates to world

 Origin maps to eye position  z axis maps to Back vector  y axis maps to Up vector  x axis maps to Right vector  T

• get 𝑈−1we just need to invert 𝑈

𝑦′ 𝑧′ 𝑨′ 𝑥′ =

𝑠

𝑦

𝑣𝑦 𝑐𝑦 𝑓𝑌 𝑠

𝑧

𝑣𝑧 𝑐𝑧 𝑓𝑧 𝑠

𝑨

𝑣𝑨 𝑐𝑨 𝑓𝑨 𝑠

𝑥

𝑣𝑥 𝑐𝑥 𝑓𝑥

𝑦 𝑧 𝑨 𝑥

Vectors vs Positions

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 There is a fundamental difference between vectors and

positions in homogeneous coordinates!

 Position

 In homogeneous coordinates p = {x, y, z, 1}  Can be moved so translation will apply

 Vector

 In homogenous coordinates v = {x, y, z, 0}  Cannot be moved, its just direction

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Projections summary

Projection types

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 Orthogonal

Projection types

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 Parallel

Projection types

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 Isometric (parallel but not orthogonal)

Projection types

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 Perspective

Projection types

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 Perspective

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Viewport transformation

Viewport transformation

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 Global coordinates



e.g. (-50..50 cm, -50..50 cm, -50..50 cm)

 Camera coordinates

 e.g. (-1..1, -1..1, -1..1)

 Viewport (window)

 e.g. (0..1200 px, 0..800 px)

Viewport transformation

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          + − + − = 1 ) 1 , , ( ) 1 , , (

min min min min

yv yc s xv xc s s s y x y x

y x y x p p v v

min max min max

xc xc xv xv sx − − =

min max min max

yc yc yv yv sy − − =

3D rendering pipeline

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3D polygons Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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Transform into 3D world coordinate system Illuminate according to light Transform into 3D camera coordinate system Transform into 2D camera coordinate system Clip polygons outside of camera’s view Draw pixels

3D rendering pipeline

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3D polygons Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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Transform into 3D world coordinate system Illuminate according to light Transform into 3D camera coordinate system Transform into 2D camera coordinate system Clip polygons outside of camera’s view Draw pixels

3D polygon rendering

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 Closed sequence of lines

General problem

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 Given a continuous

geometric representation

• f an object

 Decide which pixels

are occupied by the object

General problem

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Digital Differential Analyzer

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 dd = (y2 – y1) / (x2 – x1)

: float

Digital Differential Analyzer

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Pseudocode:

y = y1 for x = x1 to x2 begin setpixel (x, round(y)) y = y + dd end

Digital Differential Analyzer

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Watch for line slope

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 if abs(dd) > 1  exchange x↔y

in algorithm

Bresenham algorithm

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 DDA requires floating point  Bresenham works with integers only  main idea: for each x there are only 2 possible y values,

pick the one with the smaller error. accumulate error

• ver iterations.

 modify for other slopes and

• rientations

Circle, ellipse rasterization

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 Bresenham for circles (midpoint algorithm)  Can be modified for ellipses

Polygon rasterization

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Scanline algorithm: For each scan line:

• 1. Find the intersections of polygon and the scan line
• 2. Sort the intersections by x coordinate
• 3. Fill the pixels between subsequent pairs of intersections

Scan-line algorithm

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Scan-line algorithm

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 (works also for non-convex polygons)

Filled polygon

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 How to draw all pixels inside a polygon?

Filled polygon

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 We need to determine INSIDE / OUTSIDE

Filled polygon

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 We need to determine INSIDE / OUTSIDE

Filled polygon

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 We need to determine INSIDE / OUTSIDE

Filled polygon

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 We need to determine INSIDE / OUTSIDE

Filled polygon

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 We need to determine INSIDE / OUTSIDE

Filled polygon

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 We need to determine INSIDE / OUTSIDE

Filled polygon

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 We need to determine INSIDE / OUTSIDE

Filled polygon

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 If convex / concave vertices are handled correctly

Filled triangle

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 Polygons defined using triangles  Lets draw triangles instead

Filled triangle

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 Split triangle horizontally into two parts  Use linear interpolation to draw lines

Filled triangle

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 Fill using horizontal lines

A B C

Filled triangle

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 Fill using horizontal lines

𝑌 = 𝑚𝑓𝑠𝑞(𝐵, 𝐷, 𝑢1) 𝑍 = 𝑚𝑓𝑠𝑞(𝐶, 𝐷, 𝑢1) 𝑎 = 𝑚𝑓𝑠𝑞(𝑌, 𝑍, 𝑢2)

A B C X Y Z 𝑢1 𝑢2

Rasterization alias

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Aliasing

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 continuous → discrete, artifacts might appear  rasterization alias – jagged edges  sampling

 creating

• bservation of

continuous phenomenon in discrete intervals

 sampling frequency

 pixel density

Forms of alias

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 spatial alias

 jaggy edges  moiré  texture

distortion

 temporal

 “wagon wheel”

Anti-aliasing

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 general (global) anti-aliasing - supersampling

 works on all objects

 object (local) anti-aliasing

 line anti-aliasing  silhouette anti-aliasing  texture anti-aliasing

Super-sampling

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 For each pixel perform multiple

sub-pixel observations and combine the results

Regular (grid) Random (stochastic) Poisson Jitter

Super-sampling

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Super-sampling

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Super-sampling

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Next lessons

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3D polygons Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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Shading and Lighting (06)

Rest of rendering pipeline - next lessons

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3D polygons Modeling Transformation Lighting Viewing Transformation Projection Transformation Clipping Scan Conversion 2D Image

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Visibility, Culling, Cropping (07)

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Shading and Lighting

Next Week

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Acknowledgements

 Thanks to all the people, whose work is shown here and whose

slides were used as a material for creation of these slides:

Matej Novotný, GSVM lectures at FMFI UK Peter Drahoš, PPGSO lectures at FIIT STU

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www.slido.com #PPGSO03 martin.madaras@stuba.sk

Questions ?!