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```									      Image Processing
Chapter 2
Digital Image Fundamentals
Reflected Light
The colours that we perceive are determined by the nature
of the light reflected from an object
For example, if white light is shone onto a green object
most wavelengths are absorbed, while green light is
reflected from the object

Colours
Absorbed
Simple Image Model
Monochrome(Gray) Image : f(x,y)
f(x,y) = intensity value at coordinates (x,y)
0 < f(x,y) < ∞ ; f(x,y) is energy
Simple Image Model
Simple image formation
f(x,y) = i(x,y)r(x,y)
i(x,y): illumination (determined
by ill. Source)
0 < i(x,y) < ∞
r(x,y) reflectance (determined
by imaged object)
0 < r(x,y) < 1
In real situation
Lmin ≤ l (=f(x,y)) ≤ Lmax
L : gray level
Sampling & Quantization
Image sampling
Digitization of spatial coordinates (x,y)
A digital sensor can only measure a limited number
of samples at a discrete set of energy levels
Quantization
Amplitude digitization
In all types of sensors, quantization of the sensor output
completes the process of generating digital image.
The quality of a digital image is determined to a large
degree by the number of samples and discrete gray
levels used in sampling and quantization
Sampling & Quantization
Sampling & Quantization
Remember that a digital image is always only an approximation
of a real world scene
Representing Digital Image
a digital image is composed of M rows and N columns of
pixels each storing a value

Pixel values are most
often grey levels in the
range 0-255(black-white)

Images can easily
be represented as
matrices
Representing Digital Image

 f (0,0)          f (0,1)       ... f (0, N  1) 
 f (1,0)            f (1,1)  ...     f (1, N  1) 
f ( x, y )                                                    
      .                .      .            .      
Continuous                                                      
image        f ( M  1,0)   f ( M  1,1) ... f ( M  1, N  1)

Digital Image

The notation (0,1), is used to signify the the second sample
along the first row, not the actual physical coordinates when
the image was sampled
Storage
For M x N image with L(=2k) discrete gray level
The number, b, of bits required to store the image is
b = MNk
ex1) 1024 x 1024 x 8bit = 1Mbytes
What is Digital Image?
Digital image : x,y,f(x,y), three values are all
discretized
Pixel : Image elements, Picture elements, Pels
What is Digital Image?
Comparison
f(x,y) : 2D still image
f(x,y,z) : 3D object
f(x,y,t) : Video or Image Sequence
f(x,y,z,t) : moving 3D object
Spatial Resolution (x,y)
Spatial resolution is the smallest discernible detail in
an image
A line pair : a line and its adjacent space
A widely used definition of resolution is the smallest
number of discernible line pairs per unit distance
ex) 100 line pairs/mm
But, unit distance or unit area is omitted in most cases
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
size
– Graphic designers will
inch (DPI)
Spatial Resolution
Spatial Resolution
Gray-level Resolution
Gray-level resolution is the smallest discernible
change in gray level (but, highly subjective!)
Due to hardware considerations, we only consider
quantization level
Usually an integer power of 2. The most common
level is 28=256
However, we can find some systems that can
digitize the gray levels of an image with 10 to 12
bits of accuracy.
Gray-level Resolution
Gray-level Resolution
Isopreference curves
Effects produced on image quality by varying N and K
Relation between subjective image quality and resolution
Tested by images with low/medium/high detail
Result
A few gray levels may be needed for high detailed image
Perceived quality in the other two image categories remained the same in
some intervals in which the spatial resolution was increased, but the number
of gray levels actually decrease
Aliasing
Shannon sampling theorem
if the function is sampled at a rate equal to or greater
than twice its highest frequency, it is possible to
recover completely the original function from its
samples
if the function is undersampled, then a phenomenon
called aliasing corrupts the sampled image
The corruption is in the form of additional frequency
components being introduced into the sampled
function. These are called aliased frequencies
Nyquist freq. = 0.5 x sampling rate
Aliasing
Aliasing
Except for a special case, it is impossible to satisfy the sampling
theorem in practice
The principal approach for reducing the aliasing effects on an
image is to reduce its high-frequency components by blurring the
image prior to sampling
However, aliasing is always present in a sampled image
The effect of aliased frequencies can be seen under the right
conditions in the form of so-called Moiré patterns
Zooming and Shrinking
Zooming : requires two steps
Creation of a new pixel location
Assignment of a gray level to those new locations
Nearest neighbour interpolation (ex 500 x 500 image)
Laying an imaginary 750 x 750 grid over the original image
Spacing in the grid will be less than one pixel
Look for the closest pixel in the original image and assign its
gray level to the new pixel in the grid.
Fast but produces checkerboard effect that is particularly
objectionable at high factor of magnification.
Pixel replacement
Applicable when we want to increase with integer number of
times
We can duplicate each column
Then we duplicate each row of the enlarged image.
Zooming and Shrinking
Bilinear interpolation (four neighbours of point)
Let x`, y` a point in the zoomed image
v (x`, y`) a gray level assigned to it
v is given by
v (x`, y`) = ax` + by` + cx`y` +d
The coefficients are determined from the four equations using
the four neighbours.

Shrinking is done in the same manner as just described
for zooming
Zooming and Shrinking
Neighbors of a Pixel
N4(p) : 4-neighbors of pixel p(x, y)
{ p(x+1, y), p(x-1, y), p(x, y+1), p(x, y-1)}

ND(p) : diagonal neighbors of pixel p(x, y)
{ p(x+1, y+1), p(x-1, y-1), p(x-1, y+1), p(x+1, y-1)}

N8(p) : 8-neighbors of pixel p(y,x)
N8(p) = N4(p) U ND(p)

Some of the neighbors of pixel p lie outside the
digital image if the pixel p is on the border of the
image
Neighbors of a Pixel
Connectivity
Def: Two pixels are connected, if they
are adjacent in some sense and if their
gray levels are similar.

Let V be the set of gray levels used to
define connectivity; e.g. V={1} for the
connectivity of pixels with value 1 in a
binary image with 0 and 1.
Connectivity (cont.)
In a gray-scale image, for the connectivity of pixels
with a range of intensity values of , say, 100 to 120,
it follows that V={100,101,102,…,120}.

We consider three types of connectivity:
4-connectivity
Two pixels p and q with values
from V are 4-connected
if q is in the set N4(p) .
Connectivity (cont.)
8-connectivity
Two pixels p and q with values from V are 8-
connected if q is in the set N8(p) .
Connectivity (cont.)
m-connectivity (mixed connectivity)
Two pixels p and q with values from V are m-
connected if
(i) q is in N4(p) , or
(ii) q is in ND ( p) and ND( p) ∩ N4 (q) is empty
Path
A (digital) path(or curve) from pixel p at (x,y) to
pixel q at (s,t) is a sequence of distinct pixels with
coordinates
(x0,y0), (x1,y1), …,, (xn,yn)
where (x0,y0) =(x,y), (xn,yn)=(s,t), and pixel (xi,yi)
and (xi-1,yi-1) are adjacent for 1≤ i ≤ n
n is the length of the path
If (x0,y0) =(xn,yn), the path is a closed path
The path can be defined 4-,8-,m-paths depending
Connectivity
Let S be a subset of pixels in an image. Two pixels p and q are
said to be connected in S if there exists a path between
them consisting entirely of pixels in S
For any pixel p in S, the set of pixels that are connected to
it in S is called a connected component of S.
If it only has one connected component, then set S is
called a connected set.
Let R be a subset of pixels in an image. We call R a
region of the image if R is a connected set
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
Distance Measures
Let pixels be
p=p(x,y), q=q(s,t), z=z(u,v)
D() is a distance function or metric if
(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)
Distance Measures
Euclidean distance
De(p,q) = [(x-s)2 + (y-t)2]1/2
D4 distance (city-block distance)
D4(p,q) = |x-s| + |y-t|
2
Distance from (x,y) less than                212
or equal some value r                       21012
from a diamond centered at (x,y)             212
The pixels with D4 = 1 are the 4-neighbor
2
Distance Measures
D8 distance (chessboard distance)
D8(p,q) = max(|x-s|, |y-t|)

22222
21112
21012
21112
22222
Arithmetic/Logic Operations
Arithmetic operation
Subtraction: p-q
Multiplication: pxq
Division: p ÷ q
Logic Operation
AND: p AND q (p. q)
OR: p OR q (p + q)
COMPLEMENT: NOT q ( q )
Logic Operations
Logic Operations

 Try the following problems
 1, 5, 7, 9, 11, 13, 15

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