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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
      often talk about pixel
      size
    – Graphic designers will
      talk about dots per
      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
on adjacency type
                 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
  Addition: p+q
  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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