neighbouhood cells

8/6/2019 Neighbouhood cells

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Basic Relationships Between

Pixels

Neighborhood

Adjacency Connectivity

Paths Regions and boundaries


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The four diagonal neighbors of p(x,y) are

given by,(x+1, y+1), (x+1, y-1), (x-1, y+1), (x-1 ,y-1)

This set is denoted by ND(P).

Each of them are at Euclidean distance of

1.414 from P.

Neighbors of a Pixel (Contd..)


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Neighbors of a pixel

a. 4-neighbors of a pixel pare its vertical and

horizontal neighbors

denoted by N4(p)b. 8-neighbors of a pixel p

are its vertical

horizontal and 4diagonal neighbors

denoted by N8(p)

pN4(p)

p

N8(p)



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ND N4 ND

N4 P N4

ND N4 ND

N4 - 4-neighborsND - diagonal neighbors

N8 - 8-neighbors (N4 U ND)


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Adjacency

Two pixels are connected if they are

neighbors and their gray levels satisfysome specified criterion of similarity.

For example, in a binary image two pixels

are connected if they are 4-neighbors and

have same value (0/1).


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Connectivity :

To determine whether the pixelsare adjacent in some sense.

Let V be the set of gray-levelvalues used to define connectivity;

then Two pixels p, q that have values

from the set V are:

a. 4-connected, if q is in the set N4(p)

b. 8-connected, if q is in the set N8(p)

c. m-connected, iff

i. q is in N4(p) or

ii. q is in ND(p) and the set

is empty

0 1 1

0 2 0

0 0 1

0 1 1

0 2 0

0 0 1

a.

b.

c.

V = {1, 2}

)()(44

qNpN I

0 1 1

0 2 0

0 0 1


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Adjacency/Connectivity

0 1 1

0 1 0

0 0 1

8-adjacent

m-adjacent


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Pixel p is adjacentto pixel q if they are connected.

Two image subsets S1 and S2 are adjacent if somepixel in S1 is adjacent to some pixel in S2

Adjacency/Connectivity

S1

S2


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Paths & Path lengths

Apath from pixel p with coordinates(x, y) to pixel q with coordinates (s, t)

is a sequence of distinct pixels withcoordinates:

(x0, y0), (x1, y1), (x2, y2) (xn, yn),

where (x0, y0)=(x, y) and (xn, yn)=(s, t);(xi, yi) is adjacent to (xi-1, yi-1)

Here n is the length of the path.

We can define 4-, 8-, and m-paths basedon type of adjacency used.

ni 1


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Connected Components

Ifp and q are pixels of an image subset S

thenp isconnectedto q in S if there is a

path fromp to q consisting entirely of

pixels in S.

For every pixelp in S, the set of pixels inS that are connected top is called a

connected component of S. If S has only one connected component

then S is called Connected Set.


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Regions and Boundaries

A subset R of pixels in an image is

called aRegion of the image if R is aconnected set.

The boundary of the region R is theset of pixels in the region that haveone or more neighbors that are notin R.

If R happens to be entire Image?


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Distance measures

Given pixelsp, q and zwith coordinates

(x, y), (s, t), (u, v) respectively, the distancefunction 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)


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The follow ing are the different

Distance measures:

Euclidean Distance :

De( p , q ) = [ ( x - s)2

+ ( y - t )2

]

b. City Block Distance: D4( p , q ) = | x -s| + | y - t |

c. Chess Board Distance: D8( p , q ) = m ax ( | x -s| , | y - t | )

22222

21112

21012

21112

22222

2

212

21012212

2


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Relationship between pixels (Contd..)

Arithmetic/Logic Operations:- Addition : p + q

Subtraction: p q

Multiplication: p * q

Division: p / q

AND: p AND q

OR : p OR q

Complement: NOT(q)


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Arithmetic/Logic Operations

Tasks done using neighborhoodprocessing:

Smoothing / averaging

Noise removal / filtering

Edge detection

Contrast enhancement


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END o f Ne ighb or ho od

and Con nect i v i t y


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neighbouhood cells

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