cse3431-lecture14-color
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Experiment 1
Experiment 2
Experiment 3
Experiment 4
Experiment 4: the proof
Experiment 4
Experiment 4
Another interesting phenomenon
Fourier Series (advanced)
Any periodic and integrable function f(x) can be
approximated with a series
where
f(x) = f(x + T ), ⇥x
f(x) �
a0
2
+
⇥⇤
n=1
�
an cos
�
2�nx
T
⇥
+ bn sin
�
2�nx
T
⇥⇥
,
a0 =
1
T
⌅ �
��
f(x)dx,
an =
1
T
⌅ �
��
f(x) cos
�
2�nx
T
⇥
dx,
bn =
1
T
⌅ �
��
f(x) sin
�
2�nx
T
⇥
dx,
n = 1, 2, 3, . . .
Fourier series
• In some sense the function is analyzed in “function
coordinates” that happen to be cosines and sines.
Note:
• It is an orthogonal basis!
π
−π
cos(nx) sin(mx)dx = 0, for m ̸= n.
Example 1
0 2.5 5
y = sin(x)
f
Time of Spatial Domain Frequency Domain
Example 2: sum of two sinusoidals
0 2.5 5 7.5
0 2.5 5
-0.8
0.8
y = sin(x) + 0.5 sin(2x+1.5)
f 2f
FOURIER
SERIES
y1 = sin(x)
y2 = 0.5 sin(2x + 1.5)
Time or Spatial Domain Frequency Domain
y = f(x)
Example 3
Visible Spectrum
We perceive electromagnetic energy having wavelengths in the
range of 400-700 nms as visible light
Text
From:
http://www.yorku.ca/eye/
http://www.yorku.ca/eye/
http://www.yorku.ca/eye/
(0,1,1) = (1,1,1) – (1,0,0)
(1,0,1) = (1,1,1) – (0,1,0)
(1,1,0) = (1,1,1) – (0,0,1)
Question
• Ok, our color perception can be modeled with a 3D
linear space.
• But what are the basis vectors of this space?
• In other words, how do we compute a single frame of
reference for color? We all have different eyes!
Standard Color Space
To answer this question we will need to look
first at the pure spectral colors
What are the rgb coordinates of the
pure spectral colors?
• Perception of color is largely a result of a psycho-
physical process.
• The question can only be answered experimentally.
• This was done long time ago through an experiment of
“Color Matching”.
Weight functions, not spectrums.
Given a spectrum I(λ), we can use these
functions to compute its color:
where R,G,B are the unit vectors (the
standardized color of spectral red,green, and blue
light sources used in the experiment)
R =
Z
�
r(�)I(�)d�
G =
Z
�
g(�)I(�)d�
B =
Z
�
b(�)I(�)d�
C = RR + GG + BB
Color Matching functions
I(λ)
λ
Problem
Negative coefficients
This will happen with any choice of visible primaries.
Adding colors creates a less saturated color.
Solution: affine transformation of (r,g,b)
The RGB cube in XYZ
(e.g. brown, pink)
From XYZ to (x,y,Y)
From (x,y,Y) to XYZ
Working in XYZ
Text
RGB vs CMY
RGB CMY
Affine transformation
Equivalent colors between monitors
(color conversion)
Monitor 1 has phosphors with colors:
R1 = (X1r,Y1r,Z1r)
G1 = (X1g,Y1g,Z1g)
B1 = (X1b,Y1b,Z1b)
Monitor 2 has phosphors with colors :
R2 = (X2r,Y2r,Z2r)
G2 = (X2g,Y2g,Z2g)
B2 = (X2b,Y2b,Z2b)
Given color C1= (R1c,G1c,B1c) in monitor 1 what is the
equivalent color C2 = (R2c,G2c,B2c) in monitor 2 ?
Color in monitor one
Given color C1= (R1c,G1c,B1c) in monitor one
its coordinates C =(Xc,Yc,Zc) in XYZ-space
are:
Equivalent color in monitor 2
Similarly for monitor 2:
C=M2C2
Putting both together:
Other Color Spaces
• NTSC YIQ (TV)
• Y : luminance, I,Q color information
• Relation to RGB
2
6666
4
Y
I
Q
3
7777
5
=
2
6666
4
0.30 0.59 0.11
0.60 �0.28 �0.32
0.21 �0.52 0.31
3
7777
5
2
6666
4
R
G
B
3
7777
5
Visualization of the HSV
Tone Mapping
Simplest definition
• Map a set of colours to another set of colours
Appears in many areas that deal with visual
information (e.g. images)
• Computer graphics
• Digital photography
• Printing
Often associated with High Dynamic Range
images
Computer graphics
What does color intensity > MAX_DISP_INT
means?
• How do you map the following color (10,1,1) to display
intensities in [0,1]
• Clamp –> (10,1,1) –> (1,1,1)
• Uniform Scale –> (10,1,1) –> (1,0.1,0.1)
How do you map the colours of an entire
image to the colours of a monitor?
• Color histogram normalization
• Perception-based techniques
Digital Photography (HDR Images)
Same scene 6 different exposures (i.e. tones)
Courtesy Dean S. Pemberton, Wikipedia
Digital Photography (HDR Images)
mixed –>
Courtesy
Dean S. Pemberton,
Wikipedia
Computer vision
• Often the problem relates to balancing colour
histograms
• Object recognition under varying illumination
conditions
Printing
The most complex application where tone
mapping is crucial