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CS 551 / 645:
Introductory Computer Graphics
Geometric transforms: perspective projection
Color
David Luebke
7/27/2016
Administrivia



Problems submitting assignment 2 late
Hand out assignment 3….or not
Graphics Lunch (Glunch)…Fridays at noon,
typically in Olsson 236D (this week in 228E)
– uva.cs.graphics
David Luebke
7/27/2016
Perspective Projection

We talked about geometric transforms,
focusing on modeling transforms
– Ex: translation, rotation, scale, gluLookAt()
– These are encapsulated in the OpenGL
modelview matrix
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Can also express perspective projection (and
other projections) as a matrix
Next few slides: representing perspective
projection with the projection matrix
David Luebke
7/27/2016
Recap: Transformation Matrices

Represent these transformation using
matrices
– Rotation, scale, shear: 3x3 matrices suffice
– Would be nice to work translation and projection
into the same system

Solution: homogeneous coordinates
– A point in homogeneous coordinates: [x, y, z, w]T
– Corresponding point in 3-D: [x/w, y/w, z/w]T
– Now transformation matrices are 4x4
David Luebke
7/27/2016
Recap: Homogeneous
Coordinates

4x4 matrix for rotation about the X axis:
0
0
1
0 cos()  sin( )
Rx  
0 sin( ) cos()

0
0
0
David Luebke
0
0
0

1
7/27/2016
Recap: Homogeneous
Coordinates

4x4 matrix for scaling by Sx, Sy, Sz:
 Sx 0
 0 Sy
S
0 0

0 0
David Luebke
0 0
0 0
Sz 0 

0 1
7/27/2016
Recap: Homogeneous
Coordinates

4x4 matrix for translating by Tx, Ty, Tz:
1
0
T
0

0
David Luebke
0 0 Tx 
1 0 Ty 
0 1 Tz 

0 0 1
7/27/2016
Recap: Compositing Transforms
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We can composite the effect of multiple
transforms by multiplying their matrices:
Ex: rotate 90° about X, then 10 units down Z:
 x'  1
 y '  0
 
 z '  0
  
 w' 0
David Luebke
0 0 0  1
0
0
1 0 0  0 cos(90)  sin( 90)
0 1 10 0 sin( 90) cos(90)

0 0 1  0
0
0
0  x 
0  y 
0  z 
 
1   w
7/27/2016
Recap: Compositing Transforms

These transforms, in general, do not
commute
– Translate then rotate very different from rotate
then translate

Write transforms down from right to left in the
order in which they take place
– Example: p’ = Ry-1 Rx  -1 Rz Rx  Ry p
David Luebke
7/27/2016
More On Homogeneous Coords
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The w coordinate of a homogeneous point is
typically 1
Decreasing w makes the point “bigger”,
meaning further from the origin
Homogeneous points with w = 0 are thus
“points at infinity”, meaning infinitely far away
in some direction. (What direction?)
To help illustrate this, imagine subtracting two
homogeneous points
David Luebke
7/27/2016
Perspective Projection
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In the real world, objects exhibit perspective
foreshortening: distant objects appear smaller
The basic situation:
David Luebke
7/27/2016
Perspective Projection
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When we do 3-D graphics, we think of the
screen as a 2-D window onto the 3-D world:
How tall should
this bunny be?
David Luebke
7/27/2016
Perspective Projection
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The geometry of the situation is that of
similar triangles. View from above:
View
plane
X
P (x, y, z)
x’ = ?
(0,0,0)
Z
d

What is x’?
David Luebke
7/27/2016
Perspective Projection
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Desired result for a point [x, y, z, 1]T projected
onto the view plane:
x' x
 ,
d z
dx
x
x' 

,
z
z d

y' y

d z
dy
y
y' 

, zd
z
z d
What could a matrix look like to do this?
David Luebke
7/27/2016
A Perspective Projection Matrix
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Answer:
1
0
Mperspective  
0

0
David Luebke
0
1
0
0
0
1
0 1d
0

0
0

0
7/27/2016
A Perspective Projection Matrix
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Example:
 x  1
 y  0


 z  0

 
 z d  0

0
1
0
0
0
1
0 1d
0  x 



0  y 
0  z 
 
0  1 
Or, in 3-D coordinates:
 x

,
z d
David Luebke

y
, d 
zd

7/27/2016
A Perspective Projection Matrix

OpenGL’s gluPerspective() command
generates a slightly more complicated matrix:
 f
 aspect

 0

 0

 0
where
0
0
f
0
 Ζ far  Z near 


Z Z 
far 
 near
1
0
0
0
0
 2  Z far  Z near

 Z Z
near
far

0









 fov y 

f  cot 
 2 
– Can you figure out what this matrix does?
David Luebke
7/27/2016
Projection Matrices
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Now that we can express perspective
foreshortening as a matrix, we can composite
it onto our other matrices with the usual
matrix multiplication
End result: a single matrix encapsulating
modeling, viewing, and projection transforms
David Luebke
7/27/2016
Matrix Operations In OpenGL

Certain commands affect the current matrix
in OpenGL
– glMatrixMode() sets the current matrix
– glLoadIdentity() replaces the current matrix
with an identity matrix
– glTranslate() postmultiplies the current matrix
with a translation matrix
– gluPerspective() postmultiplies the current
matrix with a perspective projection matrix

It is important that you understand the order
in which OpenGL concatenates matrices
David Luebke
7/27/2016
Matrix Operations In OpenGL
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In OpenGL:
– Vertices are multiplied by the modelview matrix
– The resulting vertices are multiplied by the
projection matrix

Example:
– Suppose you want to scale an object, translate it,
apply a lookat transformation, and view it under
perspective projection. What order should you
make calls?
David Luebke
7/27/2016
Matrix Operations in OpenGL
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Problem: scale an object, translate it, apply a lookat
transformation, and view it under perspective

A correct code fragment:
glMatrixMode(GL_PERSPECTIVE);
glLoadIdentity();
gluPerspective(…);
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
gluLookAt(…);
glTranslate(…);
glScale(…);
/* Draw the object... */
David Luebke
7/27/2016
Matrix Operations in OpenGL

Problem: scale an object, translate it, apply a lookat
transformation, and view it under perspective

An incorrect code fragment:
glMatrixMode(GL_PERSPECTIVE);
glLoadIdentity();
glTranslate(…);
glScale(…);
gluPerspective(…);
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
gluLookAt(…);
/* Draw the object... */
David Luebke
7/27/2016
Color

Next topic: Color
To understand how to make realistic images, we
need a basic understanding of the physics and
physiology of vision. Here we step away from the
code and math for a bit to talk about basic
principles.
David Luebke
7/27/2016
Basics Of Color
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Elements of color:
David Luebke
7/27/2016
Basics of Color
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Physics:
– Illumination

Electromagnetic spectra
– Reflection
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
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Material properties (i.e., conductance)
Surface geometry and microgeometry (i.e., polished
versus matte versus brushed)
Perception
– Physiology and neurophysiology
– Perceptual psychology
David Luebke
7/27/2016
Physiology of Vision
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The eye:
The retina
– Rods
– Cones

David Luebke
Color!
7/27/2016
Physiology of Vision
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The center of the retina is a densely packed
region called the fovea.
– Cones much denser here than the periphery
David Luebke
7/27/2016
Physiology of Vision: Cones
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Three types of cones:
– L or R, most sensitive to red light (610 nm)
– M or G, most sensitive to blue light (560 nm)
– S or B, most sensitive to blue light (430 nm)
– Color blindness results from missing cone type(s)
David Luebke
7/27/2016
Physiology of Vision: The Retina

Weirdly, rods and
cones are at the back
of the retina, behind a
mostly-transparent
neural structure that
collects their
response.
David Luebke
7/27/2016
Perception: Metamers
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A given perceptual sensation of color derives
from the stimulus of all three cone types
Identical perceptions of color can thus be
caused by very different spectra
David Luebke
7/27/2016
Perception: Other Gotchas
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Color perception is also difficult because:
– It varies from person to person (thus std observers)
– It is affected by adaptation (transparency demo)
– It is affected by surrounding color:
David Luebke
7/27/2016
Color Spaces
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Three types of cones suggests color is a 3D
quantity. How to define 3D color space?
Idea: shine given wavelength () on a
screen, and mix three other wavelengths
(R,G,B) on same screen. Have user adjust
intensity of RGB until colors are identical:


David Luebke
How closely does this
correspond to a color CRT?
Problem: sometimes need
to “subtract” R to match 
7/27/2016
CIE Color Space
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The CIE (Commission Internationale
d’Eclairage) came up with three hypothetical
lights X, Y, and Z with these spectra:

Note that:
X~R
Y~G
Z~B
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Idea: any wavelength  can be matched
perceptually by positive combinations of X,Y,Z
David Luebke
7/27/2016
CIE Color Space
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The gamut of all colors perceivable is thus a
three-dimensional shape in X,Y,Z:

For simplicity, we
often project to the
2D plane X+Y+Z=1
X = X / (X+Y+Z)
Y = Y / (X+Y+Z)
Z=1-X-Y
David Luebke
7/27/2016
CIE Chromaticity Diagram (1931)
David Luebke
7/27/2016
Device Color Gamuts
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Since X, Y, and Z are hypothetical light
sources, no real device can produce the
entire gamut of perceivable color
Example: CRT monitor
David Luebke
7/27/2016
Device Color Gamuts

The RGB color cube sits within CIE color
space something like this:
David Luebke
7/27/2016
Device Color Gamuts
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We can use the CIE chromaticity diagram to
compare the gamuts of various devices:
Note, for example,
that a color printer
cannot reproduce
all shades available
on a color monitor
David Luebke
7/27/2016
Converting Color Spaces

Simple matrix operation:
 R'  XR
G '   YR
  
 B'  ZR

XG
YG
ZG
XB   R 
YB  G 
ZB   B 
The transformation C2 = M-12 M1 C1 yields
RGB on monitor 2 that is equivalent to a
given RGB on monitor 1
David Luebke
7/27/2016
Converting Color Spaces

Converting between color models can also
be expressed as such a matrix transform:
0.11   R 
Y  0.30 0.59
 I   0.60  0.28  0.32 G 
  
 
Q  0.21  0.52 0.31   B 

YIQ is the color model used for color TV in
America. Y is luminance, I & Q are color
– Note: Y is the same as CIE’s Y
– Result: backwards compatibility with B/W TV!
David Luebke
7/27/2016
Gamma Correction

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We generally assume colors are linear
But most display devices are inherently
nonlinear
– I.e., brightness(voltage) != 2*brightness(voltage/2)

Common solution: gamma correction
– Post-transformation on RGB values to map them
to linear range on display device:
1

– Can have separate  for R, G, B
yx
David Luebke
7/27/2016