 Image Characteristics
 Image Digitization
Spatial domain
Intensity domain
1
What is an Image ?
• An image is a projection of a 3D scene into a 2D
projection plane.
• An image can be defined as a 2 variable function
f(x,y): R2R , where for each position (x,y) in the
projection plane, f(x,y) defines the light intensity at this
point.
2
Image as a function
3
Image Acquisition
g(i,j)
i(x,y)
f(x,y)=i(x,y)r(x,y)
r(x,y)
pixel=picture element
4
Acquisition System
World
Camera
Digitizer
CMOS sensor
Digital
Image
5
Image Types
Three types of images:
– Binary images
g(x,y)  {0 , 1}
– Gray-scale images
g(x,y)  C
typically c={0,…,255}
– Color Images
three channels:
gR(x,y)C gG(x,y)C gB(x,y)C
6
Gray Scale Image
x = 58
y=
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
210
206
201
216
221
209
204
214
209
208
207
208
204
200
205
59
60
61
62
63
64
65
66
67
68
69
70
71
72
209
196
207
206
206
214
212
215
205
209
210
205
206
203
210
204
203
192
211
211
224
213
215
214
205
211
209
203
199
202
202
197
201
193
194
199
208
207
205
203
199
209
209
236
203
197
195
198
202
196
194
191
208
204
202
217
197
195
188
199
247
210
213
207
197
193
190
180
196
186
194
194
203
197
197
143
207
156
208
220
204
191
172
187
174
183
183
188
183
196
71
56
69
57
56
173
214
188
196
185
177
187
185
190
181
64
63
65
69
63
64
60
69
86
149
209
187
183
183
173
80 84 54 54 57 58
58 53 53 61 62 51
57 55 52 53 60 50
60 55 77 49 62 61
60 55 46 97 58 106
60 59 51 62 56 48
62 66 76 51 49 55
72 55 49 56 52 56
62 66 87 57 60 48
71 63 55 55 45 56
90 62 64 52 93 52
239 58 68 61 51 56
221 75 61 58 60 60
196 122 63 58 64 66
186 105 62 57 64 63
7
Color Image
8
Notations
• Image Intensity – Light energy emitted from a unit area in the image
– Device dependence
• Image Brightness – The subjective appearance of a unit area in the image
– Context dependence
– Subjective
• Image Gray-Level – The relative intensity at each unit area
– Between the lowest intensity (Black value) and the highest
intensity (White value)
– Device independent
9
Intensity vs. Brightness
10
f1 < f2, Df1 = Df2
Intensity
Intensity vs. Brightness
Df2
Df1
f2
f1
Equal intensity steps:
Equal brightness steps:
11
Weber Law
• Describe the relationship between the physical magnitudes
of stimuli and the perceived intensity of the stimuli.
• In general, Df needed for just noticeable difference (JND)
over background f was found to satisfy:
Df
 const
f
Brightness  log(f)
12
What about Color Space?
• JND in XYZ color space
was measured by Wright
and Pitt, and MacAdam in
the thirties
• MacAdam ellipses: JND
plotted at the CIE-xy
diagram
• Conclusion: measuring
perceptual distances in the
cie-XYZ space is not a
good idea
Perceptually Uniform Color Space
• Most common: CIE-L*a*b* (CIELAB) color space.
• L* represents luminance.
• a* represents the difference between green and red, and
b* represents the difference between yellow and blue.
Perceptually Uniform Color Space
• XYZ to CIELAB conversion:

 200 X
a *  500  X X 0 
b
*
1/ 3
X0
1/ 3
 Y Y0 
1/ 3
 Z Z 0 
1/ 3

116Y Y0   16
*
L 
903Y Y0 


1/ 3
for Y Y0  0.01
otherwise
• where (X0,Y0,Z0) are the XYZ values of a reference white
point
Digitization
• Two stages in the digitization process:
– Spatial sampling: Spatial domain
– Quantization: Gray level
f
j
x
1
2
3
4
5
1 100 100 100 100 100
y
i
2 100 0
0
0
100
3 100 0
0
0
100
4 100 0
0
0
100
5 100 100 100 100 100
Continuous Image
f(x,y)
Digital Image
g(i,j)  C
16
Spatial Sampling
• When a continuous scene is imaged on
the sensor, the continuous image is
divided into discrete elements - picture
elements (pixels)
Spatial Sampling
Sampling
• The density of the sampling denotes the
separation capability of the resulting image
• Image resolution defines the finest
details that are still visible by the image
• We use a cyclic pattern to test the
separation capability of an image
numberof cycles
Frequency 
unit length
1
Wavelength
frequency
0
x
Sampling Rate
Nyquist Frequency
•
Nyquist Rule: To observe details at frequency f
(wavelength d) one must sample at frequency > 2f
(sampling intervals < d/2)
•
The Frequency 2f is the Nyquist Frequency.
• Aliasing: If the pattern wavelength is less than 2d
erroneous patterns may be produced.
1D Example:
0
Aliasing - Moiré Patterns
Temporal Aliasing
Temporal Aliasing Example
Image De-mosaicing
• Can we do better than Nyquist?
Image De-mosaicing
• Basic idea: use correlations between color
bands
A joint Histogram of rx v.s. gx
500
450
Green derivative
400
350
300
250
200
150
100
50
100
200
300
Red derivative
400
500
Quantization
• Choose number of gray levels (according
to number of assigned bits)
• Divide continuous range of intensity values
Quantization
Quantization
• Low freq. areas are more sensitive to
quantization
8 bits image
4 bits image
How should we quantize an image?
• Simplest approach: uniform quantization
10
Zk  Z0
Z i 1  Z i 
K
8
Gray-Level
Z i 1  Z i
qi 
2
6
4
2
0
0
10
20
30
40
50
60
70
80
90
Sensor Voltage
q0
Z0
q1
Z1
q2
Z2
.... ....
q3
Z3
Z4
....
Zk-1
quantization
level
qk-1
Zk
sensor
voltage
100
Non-uniform Quantization
• Quantize according to visual sensitivity
(Weber’s Law)
• Non uniform sensor voltage distribution
q 0 q1 q2
q3
Z0 Z1 Z2 Z3
High Visual
Sensitivity
q4
Z4
q5
Z5
q6
Z6
Z7
Low Visual
Sensitivity
Optimal Quantization (Lloyd-Max)
• Content dependant
• Minimize quantization error
q0
q1
q2
q3
quantization
level
sensor
voltage
Z0
Z1
Z2
Z3
Z4
Optimal Quantization (Lloyd-Max)
• Also known as Loyd-Max quantizer
• Denote P(z) the probability of sensor voltage
• The quantization error is :
k 1 zi 1
E    P  z  z  qi  dz
2
i  0 zi
zi1
• Solution:
qi 
 zPz dz
zi
zi1
 Pz dz
qi 1  qi
zi 
2
zi
• Iterate until convergence (but optimal minimum is not
guaranteed).
Example
8 bits image
4 bits image
Uniform quantization
4 bits image
Optimal quantization
Color Quantization
• Common color resolution for high quality images is
256 levels for each Red, Greed, Blue channels, or
2563 = 16777216 colors.
• How can an image be displayed with fewer colors
than it contains?
• Select a subset of colors (the colormap or pallet) and
map the rest of the colors to them.
from: Daniel Cohen-Or
Color Quantization
• With 8 bits per pixel and color look up table we can
display at most 256 distinct colors at a time.
• To do that we need to choose an appropriate set of
representative colors and map the image
into these colors
111
14
126
5
12
36
36
111
36
12
17
111
17
111 200
36
12
36
14
36
12
36
14
36
17
111
14
126 17
111
36
36
12 126 200
from: Daniel Cohen-Or
12
Color Quantization
2
colors
16
colors
4
from: Danielcolors
Cohen-Or
256
colors
Color Quantization
Naïve (uniform) Color
Quantization 24 bit to 8 bit:
Retaining 3-3-2 most
significant bits of the R,G
and B components.
false contours
from: Daniel Cohen-Or
Median Cut
R
G
B
Median Cut
from: Daniel Cohen-Or
Median Cut
from: Daniel Cohen-Or
Median Cut
from: Daniel Cohen-Or
Median Cut
from: Daniel Cohen-Or
Median Cut
from: Daniel Cohen-Or
The median cut algorithm
Color_MedCut (Image, n){
For each pixel in Image with color C, map C in RGB space;
B = {RGB space};
While (n-- > 0) {
L = Heaviest (B);
Split L into L1 and L2;
Remove L from B, and add L1 and L2 instead;
}
For all boxes in B do
assign a representative (color centroid);
For each pixel in Image do
map to one of the representatives;
}
from: Daniel Cohen-Or
Better Solution
from: Daniel Cohen-Or
Generalized Lloyed Algorithm (GLA)
pi
0 
qi
qi1
from: Daniel Cohen-Or
Generalized Lloyed Algorithm (GLA)
pi
0 
qi
qi1
from: Daniel Cohen-Or
Generalized Lloyed Algorithm (GLA)
pi
2 
qi
qi1
from: Daniel Cohen-Or
• The GLA algorithms aims at minimizing the quantization
error:
K
E     p j  qi 
2
i 1 jCi
Color_GLloyd(Image, K) {
- Guess K cluster centre locations
- Repeat until convergence {
- For each data point finds out which centre it’s closest to
- For each centre finds the centroid of the points it owns
- Set a new set of cluster centre locations
- optional: split clusters with high variance
}
}
24
8 bit
4
from: Daniel Cohen-Or
More on Color Quantization
• Observation 1: Distances and quantization errors
measured in RGB space, do not relate to human
perception.
• Solution: Apply quantization in perceptually uniform
color space (such as CIELAB).
More on Color Quantization
Original
RGB Quantization
Lab Quantization
More on Color Quantization
Sensitivity
• Observation 2: Quantization errors are spatially
dependent: we are more sensitive to errors at lower
spatial frequencies.
1
3
10
30 100
Spatial Frequency
More on Color Quantization
• Solution: Assign weight for each pixel color
• Using this scheme we minimize:
W
k 1
E    w j  p j  qi 
i 0 jCi
W
W
2
250
200
w
150
100
50
0
300
200
100
W
0 0
50
w
100
150
200
250
Original
Standard quantization
Weighted quantization
THE
END
57