04 - Viewing Transformations, Rasterization

CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

04 - Viewing Transformations, Rasterization


Viewing transformations

World Coordinates

Screen Space (pixels)

Modeling Camera Projection Viewport


Coordinate Systems

object coordinates

world coordinates

camera coordinates

screen coordinates

Image Copyright: Mark Pauly


Viewing Transformationobject space

model camera projection viewport

canonical view volumeworld space

camera space screen space


Viewport transformation

viewport

canonical view volume

screen space1-1

nx

ny

2

4xscreen

yscreen1

3

5 =

2

4nx/2 0 nx�1

2

0 ny/2 ny�12

0 0 1

3

5

2

4xcanonical

ycanonical1

3

5

How does it look in 3D?


Orthographic Projection

projection

canonical view volume

camera space

xz

y(l,b,n)

(r,t,f)

Morth =

2

664

2r�l 0 0 � r+l

r�l

0 2t�b 0 � t+b

t�b

0 0 2n�f �n+f

n�f

0 0 0 1

3

775


Camera Transformation

camera

world space

camera space

1. Construct the camera reference system given: 1. The eye position e2. The gaze direction g3. The view-up vector t

uv w

e

w = � g

||g||

u =t⇥w

||t⇥w||

v = w ⇥ u


Change of frame

o x

y

euv

p

p = (px, py) = o+ pxx+ pyyp = (pu, pv) = e+ puu+ pvv

2

4pxpy1

3

5 =

2

41 0 ex0 1 ey0 0 1

3

5

2

4ux vx 0uy vy 00 0 1

3

5

2

4pupv1

3

5 =

2

4ux vx exuy vy ey0 0 1

3

5

2

4pupv1

3

5

pxy =

u v e0 0 1

�puv puv =

u v e0 0 1

��1

pxy

Can you write it directly without the inverse?


Camera Transformation

camera

world space

camera space

1. Construct the camera reference system given: 1. The eye position e2. The gaze direction g3. The view-up vector t

uv w

e

w = � g

||g||

u =t⇥w

||t⇥w||

v = w ⇥ u

2. Construct the unique transformations that converts world coordinates into camera coordinates

Mcam =

u v w e0 0 0 1

��1


Viewing Transformationobject space

model camera projection viewport

canonical view volumeworld space

camera space screen space


Algorithm• Construct Viewport Matrix

• Construct Projection Matrix

• Construct Camera Matrix

•

• For each model

• Construct Model Matrix

•

• For every point p in each primitive of the model

•

• Rasterize the model

Morth

Mvp

Mcam

Mmodel

M = MvpMorthMcam

Mfinal = MMmodel

pfinal = Mfinalp

Mvp

Morth

Mcam

Mmodel

Mfinal


References

Fundamentals of Computer Graphics, Fourth Edition 4th Edition by Steve Marschner, Peter Shirley

Chapter 7

https://www.amazon.com/s/ref=dp_byline_sr_book_1?ie=UTF8&text=Steve+Marschner&search-alias=books&field-author=Steve+Marschner&sort=relevancerank

https://www.amazon.com/Peter-Shirley/e/B00IZTGJ9O/ref=dp_byline_cont_book_2


Rasterization

CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Image Copyright: Andrea Tagliasacchi


2D Canvas

(0, 0)

(width-1, height-1)

pixel gridcanvas(-1.0, -1.0)

(1.0, 1.0)

Image Copyright: Andrea Tagliasacchi


2D Canvas

(0, 0)

(width-1, height-1)

pixel gridcanvas(-1.0, -1.0)

(1.0, 1.0)



Implicit Geometry Representation• Define a curve as zero set of 2D implicit function

• F(x,y) = 0 → on curve

• F(x,y) < 0 → inside curve

• F(x,y) > 0 → outside curve

• Example: Circle with center (cx, cy) and radius r

F (x, y) = (x� cx)2 + (y � cy)

2 � r2


Implicit Geometry Representation• Define a curve as zero set of 2D implicit function

• F(x,y) = 0 → on curve

• F(x,y) < 0 → inside curve

• F(x,y) > 0 → outside curve

By Ag2gaeh - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=45004240


Implicit Rasterization

for all pixels (i,j)

(x,y) = map_to_canvas (i,j)

if F(x,y) < 0

set_pixel (i,j, color)


Implicit Geometry Representation

• Example: Triangle



Barycentric Interpolation• Barycentric coordinates:

• p = αa + βb + γc with α + β + γ = 1

ab

c

p



• p = αa + βb + γc with α + β + γ = 1

• Unique for non-collinear a,b,c

2

4ax bx cxay by cy1 1 1

3

5 ·

2

4↵��

3

5 =

2

4px

py

1

3

5

ab

c

p



• p = αa + βb + γc with α + β + γ = 1

• Unique for non-collinear a,b,c

• Ratio of triangle areas

ab

c

p↵(p) =

area(p,b, c)

area(a,b, c)

�(p) =area(p, c,a)

area(a,b, c)

�(p) =area(p,a,b)

area(a,b, c)



• p = αa + βb + γc with α + β + γ = 1 • Unique for non-collinear a,b,c• Ratio of triangle areas • α(p), β(p), γ(p) are linear functions

a

b

c a

b

c a

b

c

1

1

1


γ <0

β<0 α<0


• p = αa + βb + γc with α + β + γ = 1 • Unique for non-collinear a,b,c• Ratio of triangle areas • α(p), β(p), γ(p) are linear functions • Gives inside/outside information

ab

c

α,β,γ > 0



• p = αa + βb + γc with α + β + γ = 1 • Unique for non-collinear a,b,c• Ratio of triangle areas • α(p), β(p), γ(p) are linear functions • Gives inside/outside information • Use barycentric coordinates to interpolate vertex normals (or other data, e.g. colors)

AB

C

P

n(P) = ↵ · n(A) + � · n(B) + � · n(C)


Color Interpolation

http://www.cgchannel.com/2010/11/cg-science-for-artists-part-2-the-real-time-rendering-pipeline/

AB

C

P

Per-vertex Per-pixel

Evaluate color on vertices, then interpolates it

Interpolates positions and normals, then evaluate color on each pixel


Triangle Rasterization• Each triangle is represented as three 2D points (x0, y0), (x1, y1), (x2, y2)

• Rasterization using barycentric coordinates

x = α · x0 + β · x1 + γ · x2

y = α · y0 + β · y1 + γ · y2

α + β + γ = 1

γ < 0 β < 0

α < 0(x2, y2)

(x1, y1)

(x0, y0)

(x, y)


• Each triangle is represented as three 2D points (x0, y0), (x1, y1), (x2, y2)

• Rasterization using barycentric coordinates

Triangle Rasterization

for all y do for all x do compute (α,β,γ) for (x,y)

if (α ∈ [0,1] and β ∈ [0,1] and γ ∈ [0,1]

set_pixel (x,y)


Clipping

• Ok if you do it brute force

• Care is required if you are explicitly tracing the boundaries


Objects Depth Sorting• To handle occlusion, you

can sort all the objects in a scene by depth

• This is not always possible!


z-buffering

Image Depth (z)

• You render the image both in the Image and in the depth buffer, where you store only the depth

• When a new fragment comes in, you draw it in the image only if it is closer

• This always work and it is cheap to evaluate! It is the default in all graphics hardware

• You still have to sort for transparency…


z-buffer quantization and “z-fighting”• The z-buffer is quantized (the

number of bits is heavily dependent on the hardware platform)

• Two close object might be quantized differently, leading to strange artifacts, usually called “z-fighting”


• Render nxn pixels instead of one

• Assign the average to the pixel

Super Sampling Anti-Aliasing

Image Copyright: Fritz Kessler

c1 c2

c3 c4

c1 + c2 + c3 + c44


Many different names and variants• SSAA (FSAA)

• MSAA

• CSAA

• EQAA

• FXAA

• TX AA

Copyright: tested.com (http://www.tested.com/tech/pcs/1194-how-to-choose-the-right-anti-aliasing-mode-for-your-gpu/#)

MSAA

http://tested.com

http://www.tested.com/tech/pcs/1194-how-to-choose-the-right-anti-aliasing-mode-for-your-gpu/#




References

Fundamentals of Computer Graphics, Fourth Edition 4th Edition by Steve Marschner, Peter Shirley

Chapter 8

https://www.amazon.com/s/ref=dp_byline_sr_book_1?ie=UTF8&text=Steve+Marschner&search-alias=books&field-author=Steve+Marschner&sort=relevancerank

https://www.amazon.com/Peter-Shirley/e/B00IZTGJ9O/ref=dp_byline_cont_book_2

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04 - Viewing Transformations, Rasterization