Image Acquisition and Representation

Contents
Image Acquisition and Representation
A Simple Image Formation Model
Image Sampling and Quantization
Image Interpolation
Homework

Light and EM spectrum
Monochromatic light
Intensity is the only attribute, from black to white
Monochromatic images are referred to as gray-
scale images
Chromatic light bands: 0.43 to 0.79 um
The quality of a chromatic light source:
Radiance (W)
Luminance (lm)
Brightness

Acquisition using Sensor Arrays
CCD Array (Charge Couple Devices)
Use in digital cameras
Can be packaged in rugged arrays of 4000x4000

12
A Simple Image Formation Model
0 < ( , ) <
( , ) ( , ) ( , )
where
0 < ( , ) <
and
0 < ( , ) < 1
f x y
f x y i x y r x y
i x y
r x y



( , ) : intensity at the point ( , )
( , ) : illumination at the point ( , )
(the amount of source illumination incident on the scene)
( , ) : reflectance/transmissivity at the point ( , )
(t
f x y x y
i x y x y
r x y x y
he amount of illumination reflected/transmitted by the object)

13
Some Typical Ranges of illumination
Illumination
Lumen — A unit of light flow or luminous flux
Lumen per square meter (lm/m2) — The metric unit of measure for
illuminance of a surface
On a clear day, the sun may produce in excess of 90,000 lm/m2 of illumination
on the surface of the Earth
On a cloudy day, the sun may produce less than 10,000 lm/m2 of illumination
on the surface of the Earth
On a clear evening, the moon yields about 0.1 lm/m2 of illumination
The typical illumination level in a commercial office is about 1000 lm/m2

14
Some Typical Ranges of Reflectance
Reflectance
0.01 for black velvet
0.65 for stainless steel
0.80 for flat-white wall paint
0.90 for silver-plated metal
0.93 for snow

Digital vs. Analog Images
Analog:
Function v = f(x,y): v,x,y are REAL
Digital:
Function v = f(x,y): v,x,y are INTEGER

Sampling means measuring the value of an
image at a finite number of points.
Quantization is the representation of the
measured value at the sampled point by an
integer.

Image Acquisition and Representation

Stepping down from REALity to INTEGER
coordinates x,y: Sampling

Image quantization(example)
256 gray levels (8bits/pixel) 32 gray levels (5 bits/pixel) 16 gray levels (4 bits/pixel)
8 gray levels (3 bits/pixel) 4 gray levels (2 bits/pixel) 2 gray levels (1 bit/pixel)

Image sampling (example)
original image sampled by a factor of 2
sampled by a factor of 4 sampled by a factor of 8

Image downsampling by factor of 2

Color images can be represented by
3D Arrays (e.g. 320 x 240 x 3)

But for the time being we’ll handle 2D
grayvalue images

26
Image Interpolation
Interpolation — Process of using known data to
estimate unknown values
e.g., zooming, shrinking, rotating, and geometric correction
Interpolation (sometimes called resampling) — an
imaging method to increase (or decrease) the number of pixels
in a digital image.
Some digital cameras use interpolation to produce a larger image than the
sensor captured or to create digital zoom

28
Image Interpolation:
Nearest Neighbor Interpolation
f1(x2,y2) =
f(round(x2), round(y2))
=f(x1,y1)
f(x1,y1)
f1(x3,y3) =
f(round(x3), round(y3))
=f(x1,y1)

29
Bilinear Interpolation
(x,y)
The output pixel value is a weighted average of pixels in the nearest 2-by-2
neighborhood
Considers the closest 2x2 neighborhood of known pixel values surrounding the
unknown pixel
It then takes a weighted average of these 4 pixels to arrive at its final interpolated
value
This results in much smoother looking images than nearest neighbor

30
Bilinear Interpolation
f(x,y)=ax + by + cxy + d

31
Bicubic Interpolation
3 3
3
0 0
( , ) i j
ij
i j
f x y a x y
 
 
The intensity value assigned to point (x,y) is obtained by the
following equation
The sixteen coefficients are determined by using the sixteen
nearest neighbors.

Weeks 1 & 2 39
Examples: Interpolation

Homework
Consider the following 4x4 image. Construct the 8x8 image using
nearest neighbor and bilinear interpolation techniques.
9 8 7 6
8 8 4 6
1 1 4 6
0 9 2 3

41
Basic Relationships Between Pixels
Neighborhood
Adjacency
Connectivity
Paths
Regions and boundaries

42
Neighbors of a pixel p at coordinates (x,y)
 4-neighbors of p, denoted by N4(p):
(x-1, y), (x+1, y), (x,y-1), and (x, y+1).
 4 diagonal neighbors of p, denoted by ND(p):
(x-1, y-1), (x+1, y+1), (x+1,y-1), and (x-1, y+1).
 8 neighbors of p, denoted N8(p)
N8(p) = N4(p) U ND(p)

Weeks 1 & 2 43
Adjacency
Let V be the set of intensity values
 4-adjacency: Two pixels p and q with values from V are 4-
adjacent if q is in the set N4(p).
 8-adjacency: Two pixels p and q with values from V are 8-
adjacent if q is in the set N8(p).

44
Adjacency
Let V be the set of intensity values
 m-adjacency: Two pixels p and q with values from V are m-
adjacent if
(i) q is in the set N4(p), or
(ii) q is in the set ND(p) and the set N4(p) ∩ N4(p) has no pixels whose values
are from V.

Weeks 1 & 2 45
Path
 A (digital) path (or curve) from pixel p with coordinates (x0, y0) to pixel q with
coordinates (xn, yn) is a sequence of distinct pixels with coordinates
(x0, y0), (x1, y1), …, (xn, yn)
Where (xi, yi) and (xi-1, yi-1) are adjacent for 1 ≤ i ≤ n.
 Here n is the length of the path.
 If (x0, y0) = (xn, yn), the path is closed path.
 We can define 4-, 8-, and m-paths based on the type of adjacency used.

Weeks 1 & 2 46
Examples: Adjacency and Path
0 1 1 0 1 1 0 1 1
0 2 0 0 2 0 0 2 0
0 0 1 0 0 1 0 0 1
V = {1, 2}

Weeks 1 & 2 47
0 1 1 0 1 1 0 1 1
0 2 0 0 2 0 0 2 0
0 0 1 0 0 1 0 0 1
V = {1, 2}
8-adjacent

Weeks 1 & 2 48
0 1 1 0 1 1 0 1 1
0 2 0 0 2 0 0 2 0
0 0 1 0 0 1 0 0 1
V = {1, 2}
8-adjacent m-adjacent

Weeks 1 & 2 49
01,1 11,2 11,3 0 1 1 0 1 1
02,1 22,2 02,3 0 2 0 0 2 0
03,1 03,2 13,3 0 0 1 0 0 1
V = {1, 2}
8-adjacent m-adjacent
The 8-path from (1,3) to (3,3):
(i) (1,3), (1,2), (2,2), (3,3)
(ii) (1,3), (2,2), (3,3)
The m-path from (1,3) to (3,3):
(1,3), (1,2), (2,2), (3,3)

Weeks 1 & 2 50
Connected in S
Let S represent a subset of pixels in an image. Two pixels p with
coordinates (x0, y0) and q with coordinates (xn, yn) are said to be
connected in S if there exists a path
(x0, y0), (x1, y1), …, (xn, yn)
Where ,0 ,( , )i ii i n x y S   

Weeks 1 & 2 51
Let S represent a subset of pixels in an image
For every pixel p in S, the set of pixels in S that are connected to p is called a
connected component of S.
If S has only one connected component, then S is called Connected Set.
We call R a region of the image if R is a connected set
Two regions, Ri and Rj are said to be adjacent if their union forms a connected
set.
Regions that are not to be adjacent are said to be disjoint.

Weeks 1 & 2 52
Boundary (or border)
 The boundary of the region R is the set of pixels in the region that have one
or more neighbors that are not in R.
 If R happens to be an entire image, then its boundary is defined as the set of
pixels in the first and last rows and columns of the image.
Foreground and background
 An image contains K disjoint regions, Rk, k = 1, 2, …, K. Let Ru denote the
union of all the K regions, and let (Ru)c denote its complement.
All the points in Ru is called foreground;
All the points in (Ru)c is called background.

53
Question
In the following arrangement of pixels, are the two regions (of
1s) adjacent? (if 8-adjacency is used)
1 1 1
1 0 1
0 1 0
0 0 1
1 1 1
1 1 1
Region 1
Region 2

54
Question
In the following arrangement of pixels, are the two parts (of 1s)
adjacent? (if 4-adjacency is used)
1 1 1
1 0 1
0 1 0
0 0 1
1 1 1
1 1 1
Part 1
Part 2

Weeks 1 & 2 55
In the following arrangement of pixels, the two regions (of 1s)
are disjoint (if 4-adjacency is used)
1 1 1
1 0 1
0 1 0
0 0 1
1 1 1
1 1 1
Region 1
Region 2

Weeks 1 & 2 56
In the following arrangement of pixels, the two regions (of 1s)
are disjoint (if 4-adjacency is used)
1 1 1
1 0 1
0 1 0
0 0 1
1 1 1
1 1 1
foreground
background

57
Question
In the following arrangement of pixels, the circled point is part
of the boundary of the 1-valued pixels if 8-adjacency is used,
true or false?
0 0 0 0 0
0 1 1 0 0
0 1 1 0 0
0 1 1 1 0
0 1 1 1 0
0 0 0 0 0

58
Homework: Question 1
In the following arrangement of pixels, the circled point is part
of the boundary of the 1-valued pixels if 4-adjacency is used,
true or false?
0 0 0 0 0
0 1 1 0 0
0 1 1 0 0
0 1 1 1 0
0 1 1 1 0
0 0 0 0 0

59
Distance Measures
Given pixels p, q and z with coordinates (x, y), (s, t), (u, v)
respectively, the distance function D has following
properties:
a. D(p, q) ≥ 0 [D(p, q) = 0, iff p = q]
b. D(p, q) = D(q, p)
c. D(p, z) ≤ D(p, q) + D(q, z)

60
Distance Measures
The following are the different Distance measures:
a. Euclidean Distance :
De(p, q) = [(x-s)2 + (y-t)2]1/2
b. City Block Distance:
D4(p, q) = |x-s| + |y-t|
c. Chess Board Distance:
D8(p, q) = max(|x-s|, |y-t|)

61
Question
In the following arrangement of pixels, what’s the value of the
chessboard distance between the circled two points?
0 0 0 0 0
0 0 1 1 0
0 1 1 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0

62
city-block distance between the circled two points?
0 0 0 0 0
0 0 1 1 0
0 1 1 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0

63
length of the m-path between the circled two points?
0 0 0 0 0
0 0 1 1 0
0 1 1 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0

64
length of the m-path between the circled two points?
0 0 0 0 0
0 0 1 1 0
0 0 1 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0

Image Acquisition and Representation

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