[PPT] - CS-184: Computer Graphics Lecture #8: Projection Prof. James PowerPoint Presentation

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CS-184: Computer Graphics

Lecture #8: Projection

Prof. James O’Brien

University of California, Berkeley

V2016-F-08-1.0

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Today

Windowing and Viewing Transformations
Windows and viewports
Orthographic projection
Perspective projection

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Screen Space

Monitor has some number of pixels
e.g. 1024 x 768
Some sub-region used for given program
You call it a window
Let’s call it a viewport instead

[0,0] [1024,768] [60,350] [690,705] [0,0] [1024,768]

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Screen Space

May not really be a “screen”
Image file
Printer
Other
Little pixel details
Sometimes odd
Upside down
Hexagonal

From Shirley textbook.

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Screen Space

Viewport is somewhere on screen
You probably don’t care where
Window System likely manages this detail
Sometimes you care exactly where
Viewport has a size in pixels
Sometimes you care (images, text, etc.)
Sometimes you don’t (using high-level library)

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Screen Space

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Integer Pixel Addresses

i=3 j=5 10 × 10 Image Resolution

0.5,-0.5

nx-0.5,ny-0.5

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Screen Space

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Float Pixel Coordinates

u= 0.35 = (i + 0.5)/nx

0,0 1,1

v= 0.55 = (j + 0.5)/ny

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Canonical View Space

Canonical view region
2D: [-1,-1] to [+1,+1]

From Shirley textbook.

1,-1

+1,+1

x=0.0, y=0.0

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Canonical View Space

Canonical view region
2D: [-1,-1] to [+1,+1]

x y y (1,1) (-1,-1) x (nx-0.5, -0.5) (-0.5, ny-0.5) x y (1,-1) (-1,1) (nx/2,-ny/2) (-nx/2,ny/2) y x reflect-y translate scale

From Shirley textbook. (Image coordinates are up-side-down.)

  x0 y0 1   =   

nx 2 0 nx1 2

0 ny

2 ny1 2

0 0 1      x y 1  

−

Remove minus for up-side-up

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Projection

Process of going from 3D to 2D
Studies throughout history (e.g. painters)
Different types of projection
Linear
Orthographic
Perspective
Nonlinear

Orthographic is special case of perspective... Many special cases in books just

ne of these two...

}

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Perspective Projections

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Ray Generation vs. Projection

Viewing in ray tracing

start with image point
compute ray that projects to that point
do this using geometry

Viewing by projection

start with 3D point
compute image point that it projects to
do this using transforms

Inverse processes

ray gen. computes the preimage of projection

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Linear Projection

Projection onto a planar surface
Projection directions either
Converge to a point
Are parallel (converge at infinity)

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Linear Projection

A 2D view

Orthographic Perspective

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Linear Projection

Orthographic Perspective

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Linear Projection

Orthographic Perspective

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Orthographic Perspective

Note how different things can be seen Parallel lines “meet” at infinity

Linear Projection

A 2D view

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Orthographic Projection

No foreshortening
Parallel lines stay parallel
Poor depth cues

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Orthographic Projection

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Canonical View Space

Canonical view region
2D: [-1,-1] to [+1,+1]
Define arbitrary window and define objects
Transform window to canonical region
Do other things (we’ll see clipping latter)
Transform canonical to screen space
Draw it.

From Shirley textbook.

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Canonical View Space

World Coordinates Canonical Screen Space

(Meters) (Pixels) Note distortion issues...

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Canonical View Space

Canonical view region
3D: [-1,-1,-1] to [+1,+1,+1]
Assume looking down -Z axis
Recall that “Z is in your face”

[1,1,1] [-1,-1,-1]

Z

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Orthographic Projection

Convert arbitrary view volume to canonical

[1,1,1] [-1,-1,-1]

Z

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Orthographic Projection

View vector Up vector Right = view X up Origin Center near,top,right far,bottom,left *Assume up is perpendicular to view.

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Orthographic Projection

Step 1: translate center to origin

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Orthographic Projection

Step 1: translate center to origin
Step 2: rotate view to -Z and up to +Y

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Orthographic Projection

Step 1: translate center to origin
Step 2: rotate view to -Z and up to +Y
Step 3: center view volume

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Orthographic Projection

Step 1: translate center to origin
Step 2: rotate view to -Z and up to +Y
Step 3: center view volume
Step 4: scale to canonical size

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Orthographic Projection

Step 1: translate center to origin
Step 2: rotate view to -Z and up to +Y
Step 3: center view volume
Step 4: scale to canonical size

M = S·T2·R·T1 M = Mo·Mv

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Perspective Projection

Foreshortening: further objects appear smaller
Some parallel line stay parallel, most don’t
Lines still look like lines
Z ordering preserved (where we care)

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Perspective Projection

Pinhole a.k.a center of projection

Image from D. Forsyth

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Perspective Projection

Foreshortening: distant objects appear smaller

Image from D. Forsyth

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Perspective Projection

Vanishing points
Depend on the scene
Not intrinsic to camera

“One point perspective”

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Perspective Projection

Vanishing points
Depend on the scene
Nor intrinsic to camera

“Two point perspective”

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Perspective Projection

Vanishing points
Depend on the scene
Not intrinsic to camera

“Three point perspective”

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Perspective Projection

u v n

View Frustum

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Perspective Projection

View Up Distance to image plane i Y

Z

Top t Bottom b Near n Far f Center

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Perspective Projection

Step 1: Translate center to origin

Y

Z

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Perspective Projection

Step 1: Translate center to origin
Step 2: Rotate view to -Z, up to +Y

Y

Z

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Perspective Projection

Step 1: Translate center to origin
Step 2: Rotate view to -Z, up to +Y
Step 3: Shear center-line to -Z axis

Y

Z

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Perspective Projection

Step 1: Translate center to origin
Step 2: Rotate view to -Z, up to +Y
Step 3: Shear center-line to -Z axis
Step 4: Perspective
Z

! ! ! ! ! ! " # $ $ $ $ $ $ % & − + 1 1 1 i f i f i

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Perspective Projection

Step 4: Perspective
Points at z=-i stay at z=-i
Points at z=-f stay at z=-f
Points at z=0 goto z=±∞
Points at z=-∞ goto z=-(i+f)
x and y values divided by -z/i
Straight lines stay straight
Depth ordering preserved in [-i,-f ]
Movement along lines distorted
Z

! ! ! ! ! ! " # $ $ $ $ $ $ % & − + 1 1 1 i f i f i

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From Shirley textbook.

Perspective Projection

WRONG!

! ! ! ! ! ! " # $ $ $ $ $ $ % & − + 1 1 1 i f i f i

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Perspective Projection

ˆ z

“Eye” plane Top Near Far S

m

e h

r

i z

n

t a l l i n e s View vector

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Perspective Projection

ˆ z

Visualizing division of x and y but not z

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Perspective Projection

ˆ z

Motion in x,y

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Perspective Projection

ˆ z

Note that points on near plane fixed

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Perspective Projection

ˆ z

Motion in z

−∞

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Perspective Projection

ˆ z

Total motion

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Perspective Projection

Step 1: Translate center to orange
Step 2: Rotate view to -Z, up to +Y
Step 3: Shear center-line to -Z axis
Step 4: Perspective
Step 5: center view volume
Step 6: scale to canonical size
Z

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Perspective Projection

Step 1: Translate center to orange
Step 2: Rotate view to -Z, up to +Y
Step 3: Shear center-line to -Z axis
Step 4: Perspective
Step 5: center view volume
Step 6: scale to canonical size
Z

M = Mo ·Mp ·Mv Mo Mp Mv

} } }

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Perspective Projection

There are other ways to set up the projection matrix
View plane at z=0 zero
Looking down another axis
etc...
Functionally equivalent

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r(t) = p+t d

Vanishing Points

Consider a ray:

d p

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Vanishing Points

Ignore Z part of matrix
X and Y will give location in image plane
Assume image plane at z=-i

! ! ! ! " # $ $ $ $ % & − 1 1 1 ! ! ! " # $ $ $ % & ! ! ! " # $ $ $ % & − = ! ! ! " # $ $ $ % & z y x I I I

w y x

1 1 1

whatever

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Vanishing Points

! ! ! " # $ $ $ % & − = ! ! ! " # $ $ $ % & ! ! ! " # $ $ $ % & − = ! ! ! " # $ $ $ % & z y x z y x I I I

w y x

1 1 1 ! " # $ % & − − = ! " # $ % & z y z x I I I I

w y w x

/ / / /

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Vanishing Points

Assume

dz = −1

! ! ! ! " # $ $ $ $ % & + − + + − + = ! " # $ % & − − = ! " # $ % & t p td p t p td p z y z x I I I I

z y y z x x w y w x

/ / / / ! " # $ % & = ±∞ →

y x

d d t Lim 61

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Vanishing Points

All lines in direction d converge to same point in the image

plane -- the vanishing point

Every point in plane is a v.p. for some set of lines
Lines parallel to image plane ( ) vanish at infinity

! " # $ % & = ±∞ →

y x

d d t Lim

dz = 0 What’s a horizon?

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Perspective Tricks

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Right Looks Wrong (Sometimes)

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. .

From Correction of Geometric Perceptual Distortions in Pictures, Zorin and Barr SIGGRAPH 1995

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Right Looks Wrong (Sometimes)

65 From WIRED Magazine

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Strangeness

66 The Ambassadors by Hans Holbein the Younger

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Strangeness

67 The Ambassadors by Hans Holbein the Younger

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Ray Picking

Pick object by picking point on screen
Compute ray from pixel coordinates.

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Ray Picking

Transform from World to Screen is:
Inverse:
What Z value?

! ! ! ! " # $ $ $ $ % & = ! ! ! ! " # $ $ $ $ % &

w z y x w z y x

W W W W I I I I M

! ! ! ! " # $ $ $ $ % & = ! ! ! ! " # $ $ $ $ % &

− w z y x w z y x

I I I I W W W W

1

M

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r(t) = aw +t(bw −aw) bs = [sx,sy,−f] as = [sx,sy,−i]

Ray Picking

Recall that:
Points at z=-i stay at z=-i
Points at z=-f stay at z=-f

r(t) = p+t d

Depends on screen details, YMMV General idea should translate...

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Depth Distortion

Recall depth distortion from perspective
Interpolating in screen space different than in world
Ok, for shading (mostly)
Bad for texture

Screen World Half way in screen space Half way in world space

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Depth Distortion

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P1 P2 P3 P4 S1 = P1/h1 S2 = P2/h2 S3 = P3/h3 S4 = P4/h4

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Depth Distortion

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P1 P2 P3 P4 S1 = P1/h1 S2 = P2/h2 S3 = P3/h3 S4 = P4/h4 X = X i Sibi Q = X i Piai

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X i Pibi/hi = @X i Piai 1 A / B @ X j hjaj 1 C A

Depth Distortion

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Independent of given vertex locations.

bi/hi = ai/ B @ X j hjaj 1 C A ∀i

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Depth Distortion

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Linear equations in the .

bi/hi = ai/ B @ X j hjaj 1 C A ∀i B @ X j hjaj 1 C A bi/hi − ai = 0 ∀i ai

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Depth Distortion

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B @ X j hjaj 1 C A bi/hi − ai = 0 ∀i ai X i ai = X i bi = 1

Not invertible so add some extra constraints. Linear equations in the .

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Depth Distortion

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For a line:

a1 = h2bi/(b1h2 + h1b2) a1 = h2h3b1/(h2h3b1 + h1h3b2 + h1h2b3)