quantization and sampling presentation ppt

4
Digital Image
Digital image = a multidimensional
array of numbers (such as intensity image)
or vectors (such as color image)
Each component in the image
called pixel associates with
the pixel value (a single number in
the case of intensity images or a
vector in the case of color images).



6 26 37
25 13 22

2 15 87 39
10 10 16 28


54 96 67
54 47 42

65 65 39
 9 65 70 56 43



 65 87 99
32
43 92
5485
15 60
3
21
3299 70 56 78
90 96 67
85

2.3 IMAGE SENSING AND ACQUISITION
Image Sensing
o Scene
o Molecules
o Human Brains
o …
o Illumination
o Radar
o Infrared
o X-ray
o Sun
o …
o Reflection
5

CONT’D
6
Single imaging
Sensor
Line Sensor
Array Sensor

IMAGE ACQUISITION USING A SINGLE SENSOR
7
The most familiar sensor of
this type is the photodiode
It is constructed of silicon
materials and whose output
voltage waveform is
proportional to light.
The use of a filter in front of a
sensor improves selectivity.
For example, a green (pass)
filter in front of a light sensor
favors light in the green band
of the color spectrum.
As a consequence, the sensor
output will be stronger for
green light than for other
components in the visible
spectrum.

IMAGE ACQUISITION USING SENSOR STRIPS
5

Image Acquisition Using Sensor Strip
6

IMAGE ACQUISITION USING SENSOR ARRAYS
7

 Used for convert a continuous
image into a digital image
 Contains an array of light sensors
 Converts photon into electric charges
accumulated in each sensor unit
CCD KAF-3200E from Kodak.
(2184 x 1472 pixels,
Pixel size 6.8 microns2)
8
Image Sensors : Array Sensor
Charge-Coupled Device (CCD)

A SIMPLE IMAGE FORMATION
MODEL
f(x,y) = i(x,y)  r(x,y)
0 < i(x,y) < 
0 < r(x,y) < 1
(from total absorption to
total reflectance)
Sample values of
r(x,y):
0.01: black velvet
0.93: snow
Intensity of a monochrome
image f at (x0,y0): gray level l
of the image at that point
l=f(x0, y0)
Lmin ≤ l ≤ Lmax
Where:
Lmin: Positive
Lmax: Finite
In practice:
Lmin = Imin rmin
Lmax = Imax rmax
e.g. for indoor image processing:
Lmin ≈ 10 Lmax ≈ 1000
[L , L
min max] : gray scale
Often shifted to [0,L-1] 
l=0: black
l=L-1: white
All intermediate values are shades
of gray
12

2.4 Image Sampling and Quantization
11
• The output of most sensors is continuous in
amplitude and spatial coordinates.
• Converting an analog image to a digital image
require sampling and quantization
• Sampling: is digitizing the coordinate values
• Quantization: is digitizing the amplitude values

SAMPLING & QUANTIZATION
12
 The spatial and amplitude digitization of f(x,y) is
called:
⚫ image sampling when it refers to spatial coordinates
(x,y) and
⚫ gray-level quantization when it refers to the amplitude.

Image Sampling and Quantization
13
Spatial sampling is
accomplished by sensor
arrangement and
mechanical movement.

IMAGE SAMPLING AND
QUANTISATION
A digital sensor can only measure a limited number of
samples at a discrete set of energy levels
Quantisation is the process of converting a continuous
analogue signal into a digital representation of this signal
14

IMAGE SAMPLING AND QUANTISATION
15

(CONT…)
Remember that a digital image is always only an
approximation of a real world scene
20

21
original Sampled by 2 Sampled by 4
Sampled by 8 Sampled by 16

UNIFORM QUANTIZATION
 Digitized in amplitude (or pixel value)
 PGM – 256 levels  4 levels
0
255
64
128
192
0
3
1
2

23
original 128 levels (7 bits) 16 levels (4 bits)
4 levels (2 bits) 2 levels (1 bit)

Representing Digital Images
21

Image “After snow storm”
Fundamentals of Digital Images
f(x,y)
x
y
 An image: a multidimensional function of spatial coordinates.
 Spatial coordinate: (x,y) for 2D case such as photograph,
(x,y,z) for 3D case such as CT scan images
(x,y,t) for movies
 The function f may represent intensity (for monochrome images)
or color (for color images) or other associated values.
Origin
22

REPRESENTING DIGITAL IMAGES
0  ai,j  L-1 Where L = 2k
The dynamic range of an image is the range of values
spanned by the gray scale.
The number, b, of bits required to store a digitized image of
size M by N is
b = M  N  k
The pixel intensity levels (gray scale levels) are in the
interval of [0, L-1].
26

77 66 68 67 98 93 79 81
79 61 61 71 61 78 88 94
79 93 84 64 72 76 95 94
97 65 71 75 75 72 95 111
120 81 82 76 72 77 78 83
150 146 112 83 78 62 9127 85
156 145 158 125 107 121 95 86
158 166 147 146 153 149 129 107
Elaine image of size 512
by 512 pixels (5 by 5
inches), The dynamic
range is [0, 255].
Find the following:
• The number of bits
required to represent a
pixel
• The size of the image in
bits?

25

Digital Image Types : Intensity Image
Intensity image or monochrome image
each pixel corresponds to light intensity
normally represented in gray scale (gray
level).


 9
10 10 16 28
 6 26 37

15 25 13 22
32 15 87 39
Gray scale values
26




6 26 37
5 25 13 22

2 15 87 39
3



54 96 67
54 47 42

65 65 39


 65 87 99
32
43 92
5485
 99 70 56 78
1 32
 
21 60 90 96 67
85
RGB components
10 10 16 28
 9 65 70 56 43
30
Digital Image Types : RGB Image
Color image or RGB image:
each pixel contains a vector
representing red, green and
blue components.

Image Types : Binary Image
Binary image or black and white image
Each pixel contains one bit :
1 represent white
0 represents black

 1
1
0
0
0 0 0 0
 0 0 
1 1 1 1
1 1
Binary data
31

Image Types : Index Image
 

2
7
6
1 4 9
 4
6 5
Index value
Index
No.
Red
component
Green
component
Blue
component
1 0.1 0.5 0.3
2 1.0 0.0 0.0
3 0.0 1.0 0.0
4 0.5 0.5 0.5
5 0.2 0.8 0.9
… … … …
Index image
Each pixel contains index number
pointing to a color in a color table
Color Table
32

Effect of Spatial Resolution
256x256 pixels
64x64 pixels
128x128 pixels
32x32 pixels
33

SPATIAL RESOLUTION
The spatial resolution of an image is determined by
how sampling was carried out
Spatial resolution simply refers to the smallest
discernable detail in an image
⚫ Vision specialists will
often talk about pixel
size
⚫ Graphic designers will
talk about dots per
inch (DPI)

Effect of Spatial Resolution
32

SPATIAL RESOLUTION (CONT…)
33

34

35

36

37

38

INTENSITY LEVEL RESOLUTION
39
Intensity level resolution refers to the number of
intensity levels used to represent the image
⚫ The more intensity levels used, the finer the level
of detail discernable in an image
⚫ Intensity level resolution is usually given in terms
of the number of bits used to store each intensity
level
Number of Bits
Number of Intensity
Levels
Examples
1 2 0, 1
2 4 00, 01, 10, 11
4 16 0000, 0101, 1111
8 256 00110011, 01010101
16 65,536 1010101010101010

INTENSITY LEVEL RESOLUTION (CONT…)
128 grey levels (7 bpp) 64 grey levels (6 bpp) 32 grey levels (5 bpp)
16 grey levels (4 bpp) 8 grey levels (3 bpp) 4 grey levels (2 bpp) 2 grey levels (1 bpp)
256 grey levels (8 bits per pixel)
43

41

42

43

44

45

46

47

48

RESOLUTION: HOW MUCH IS
ENOUGH?
50
The big question with resolution is always how much is
enough?
⚫ This all depends on what is in the image and what you would
like to do with it
⚫ Key questions include
 Does the image look aesthetically pleasing?
 Can you see what you need to see within the image?

RESOLUTION: HOW MUCH IS ENOUGH?
(CONT…)
The picture on the right is fine for counting the number of
cars, but not for reading the number plate
51

52
Low Detail Medium Detail High Detail

53

54

55

Basic Relationship of Pixels
x
Conventional indexing method 59
y
(0,0)
(x-1,y-1) (x,y-1) (x+1,y-1)
(x-1,y) (x,y) (x+1,y)
(x-1,y+1) (x,y+1) (x+1,y+1)

(x,y-1)
(x-1,y) p (x+1,y)
(x,y+1)
Neighbors of a Pixel
Neighborhood relation is used to tell adjacent pixels. It is
useful for analyzing regions.
4-neighbors of p:
N4(p) =
(x1,y)
(x+1,y)
(x,y1)
(x,y+1)
Note: q  N4(p) implies p  N4(q)
4-neighborhood relation considers only vertical and
horizontal neighbors.
60

(x-1,y-1) (x,y-1) (x+1,y-1)
(x-1,y) p (x+1,y)
(x-1,y+1) (x,y+1) (x+1,y+1)
CONT’D
8-neighbors of p:
(x1,y1)
(x,y1)
(x+1,y1)
(x1,y)
(x+1,y)
(x1,y+1)
(x,y+1)
(x+1,y+1)
N8(p) =
8-neighborhood relation considers all neighbor pixels.
61

(x-1,y-1) (x+1,y-1)
p
(x-1,y+1) (x+1,y+1)
Diagonal neighbors of p:
ND(p) =
(x1,y1)
(x+1,y1)
(x1,y1)
(x+1,y+1)
Diagonal -neighborhood relation considers only diagonal
neighbor pixels.
CONT’D
62

If pixel p at location (x,y) then its neighbors are:
• 4-neighbors N4(p)
(x-1 , y), (x+1 , y), (x , y-1), (x , y+1)
• 4-diagonal neighbors ND(p)
(x-1 , y-1), (x-1 , y+1), (x+1 , y+1), (x+1 , y-1)
• 8-neighbors N8(p)
All pixels in N4(p) and in ND(p)
2.5 Some Basic Relationship Between Pixels
63

Three type of adjacency:
(a) 4-adjacency. Two pixels p and q with values from V
are 4-adjacent if q is in the set N4(p).
(b) 8-adjacency. Two pixels p and q with values from V
are 8-adjacent if q is in the set N8(p).
(c) m-adjacency (mixed adjacency). Two pixels p and q
with values from V are m-adjacent if
(i) q is in N4(p), or
(ii) q is in ND(p) and the set N4(p)  N4(q) has
no pixels whose values are from V
V: The set of gray-level values used to define adjacency 64
Adjacency

Adjacency
A pixel p is adjacent to pixel q is they are connected.
Two image subsets S1 and S2 are adjacent if some pixel
in S1 is adjacent to some pixel in S2
S1
S2
We can define type of adjacency: 4-adjacency, 8-adjacency
or m-adjacency depending on type of connectivity. 65

0 1 0
0 1 0
0 0 1
4-adjacency
q
p
0 0
0 1 0
0 0 1
8-adjacency
q
1
m-adjacency
0 1
0 1 0
0 0 1
q
1
0 1
0 1 0
0 0 1
1 q
8-adjacency
!?
Adjacency
66

CONT’D
 Subset adjacency
⚫ S1 and S2 are adjacent if some pixel in S1 is adjacent to
some pixel in S2
A path (curve) from pixel p with coordinates (x,y) to pixel
q with coordinates (s,t) is a sequence of distinct pixels:
(x0,y0), (x1,y1), …, (xn,yn)
where (x0,y0)=(x,y), (xn,yn)=(s,t), and (xi,yi) is
adjacent to (xi-1,yi-1), for 1≤i ≤n ; n is the length of the
path.
If (x0, y0) = (xn, yn): A closed path.
64

 Region
⚫ We call R a region of the image if R is a connected set
 Boundary
⚫ The boundary of a region R is the set of pixels in the
region that have one or more neighbors that are not in
R
 Edge
⚫ Pixels with derivative values that exceed a preset
threshold
65
CONT’D

 Distance measures
⚫ Euclidean distance
⚫ City-block distance
D4 ( p,q) | (x  s) |  | (y t) |
⚫ Chessboard distance
D8 ( p,q)  max(| (x  s) |,| (y t) |)
1
 (y t)2
]2
66
e
D ( p,q)  [(x  s)2
CONT’D

 Dm distance: The shortest m-path between
the points
67
CONT’D

For pixels p, q, and z, with coordinates (x,y), (s,t), and
(v,w), respectively, D is a distance function or metric if
(a) D(p,q)  0 ,
(b) D(p,q) = D(q,p), (symmetry)
(c) D(p,z)  D(p,q) + D(q,z) (triangular inequality)
Euclidean distance between p and q is
De(p,q) = [ (x - s)2 + (y - t)2 ]1/2
For this distance measure, the pixels
having a distance less than or equal
to some value r from (x,y) are the
points contained in a disk of radius
r centered at (x,y).
Distance Measure (Euclidean)
71

The D4 distance (city-block) between p and q i
D4(p,q) = |x – s | + |y – t |
Diamond shape
The D8 distance (chessboard) between p and q is
D8(p,q) = max ( |x – s | , |y – t | )
2
s 2 1 2
2 1 0 1 2
2 1 2
2
2 2 2 2 2
2 1 1 1 2
2 1 0 1 2
2 1 1 1 2
Distance Measure (City block, Chessboard)
2 2 2 2 7
2
2

If distance depend on the path between two pixels such
as m-adjacency then the Dm distance between two pixels
is defined as the shortest m-path between the pixels.
0 0 1
0 1 0
1 0 0
q
Dm( p , q ) = 2
0 0 1
1 1 0
1 0 0
Dm( p , q ) = 3
p
0 1 1
1 1 0
1 0 0
Dm( p , q ) = 4
Distance Measure of Path
70

Distance
For pixel p, q, and z with coordinates (x,y), (s,t) and (u,v),
D is a distance function or metric if
 D(p,q)  0 (D(p,q) = 0 if and only if p = q)
 D(p,q) = D(q,p)
 D(p,z)  D(p,q) + D(q,z)
Example: Euclidean distance
D (p,q)  (x  s)2
 (y  t)2
e
71

D4-distance (city-block distance) is defined as
D4(p,q)  x  s  y  t
2
2 1 2
2 1 0 1 2
2 1 2
2
Pixels with D4(p) = 1 is 4-neighbors of p. 72

D8-distance (chessboard distance) is defined as
D8(p,q)  max( x  s, y  t )
Pixels with D4(p) = 1 is 4-neighbors of p. 73
2 2 2 2 2
2 1 1 1 2
2 1 0 1 2
2 1 1 1 2
2 2 2 2 2

Path
A path from pixel p at (x,y) to pixel q at (s,t) is a sequence
of distinct pixels:
(x0,y0), (x1,y1), (x2,y2),…, (xn,yn)
such that
(x0,y0) = (x,y) and (xn,yn) = (s,t)
and
(xi,yi) is adjacent to (xi-1,yi-1), i = 1,…,n
p
q
We can define type of path: 4-path, 8-path or m-path
depending on type of adjacency. 77

p
q
p
q
p
q
8-path from p to q
results in some ambiguity
75
m-path from p to q
solves this ambiguity
8-path m-path

Find the shortest 4-, 8-, m-path
between p and q for
V= {0, 1} and V={1, 2}
3
2
1
2
2
0
(q)
1
2
1 2 1 1
1
(p)
0 1 2
76
Path Length

Images are represented by Matrices, and matrix division
is not defined. The following image division
C = A/B
means that the division is carried out between
corresponding pixels in the two images A and B to form
image C.
77
Image Operation on a Pixel Basis

Linear operation
H is said to be a linear operator if, for any two
images f and g and any two scalars a and b,
H(af + bg) = a H( f ) + b H( g )
Linear and Nonlinear Operation
78

IMAGE MULTIPLICATION (DIVISION)
82

IMAGE MULTIPLICATION (DIVISION)
83

REFERENCES
 “Digital Image Processing”, 2/ E, Rafael C. Gonzalez & Richard
E. Woods, www.prenhall.com/gonzalezwoods.
 Only Original Owner has full rights reserved for copied
images.
 This PPT is only for fair academic use.
84

Questions?
85
Think Smarter,
Work Harder

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