4-5 Basic Relationship Between Pixels

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Basic Relationship between Pixels

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• Different Neighbours• Connectivity• Region• Connectivity – pathways• Distance Measures• Arithmetic/Logical Operations• Logical Operations

Objectives

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Terminology:

f(x,y) = image

S = subset of pixels f(x,y)

p,q,etc = particular

pixels in image

Basic Relationship between Pixels

p

S

f(x,y)

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4-Neighbours

A pixel p at coordinates (x,y) has 4 horizontal and vertical neighbours, each being a unit distance from (x,y)

Note: one or more of these points might lie outside image

(x,y)

(x-1,y)

(x+1, y)

(x, y-1)

(x, y+1)

y

x

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Example

P1 P2 P3

P4 P5 P6

P7 P8 P9

y0 1 2

x

0

1

2

N4(P5) = p4,p6 ---- Horizontal Neighbors

p2,p8 ---- Vertical Neighbors

N4 (1,1) = (1,0) (1,2)

(0,1) (2,1)

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Diagonal Neighbours ND(p)

• A pixel p at coordinates (x,y) has 4 diagonal neighbours, each being a unit distance from (x,y)

p(x,y)

(x-1, y-1)

(x-1, y+1)

(x+1, y-1)

(x+1, y-1)

Y

x

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P1 P2 P3

P4 P5 P6

P7 P8 P9

y

ND(P5) = p1,p3,p9,p7

ND (1,1) = (0,0) (0,2)

(2,2) (2,0)

0 1 2

x

0

1

2

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8 Neighbours N8(p) • 8-neighbours of p= 4 diagonal neighbours of p and 4-

neighbours of p

N8(p) = ND(p) + N4(p) p(x,y)

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Problems

P1 P2 P3

P4 P5 P6

P7 P8 P9

yX Y co-ordinate of pixel P2 = (0,1)

‘’ ‘’ ‘’ P5 =

‘’ ‘’ P9 =

0 1 2

x

0

1

2

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Problems

P1 P2 P3

P4 P5 P6

P7 P8 P9

N4 (P5)=(L,T,R,B) = (P4,P2,P6,P8)

ND(P5)=(LT,RT,RB,LB) = (P1,P3,P9,P7)

N8(P5) = (L,-----LB) = (P4,P1,P2,P3, P6,P9,P8,P7)

N4 (P1)= (L,T,R,B) = (0,0,P2,P4)

ND(P1)=(LT,RT,RB,LB) = (0,0,P5,0)

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Connectivity

Connectivity is an important concept for establishing:-

- boundaries of objects - components of regions in an image

Region = set of pixels in which there is a path between any pair of its pixels, all of whose pixels also belong to the set.

i.e. adjacent pixels

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Connectivity

Two pixels are connected if:-

- they are adjacent (neighbours)

- and colour or grey levels are similar

Region 1

Region 2

Region 3

Region 4

Region 5

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Connectivity - types

V = set of grey-levels used to define intensity

V(31,32) = in a grey-scale image, the sub-set of

grey values whose intensity value is 31, 32.

3 types of connectivity:- · 4-connectivity/ 4-adjacency

· 8-connectivity

· m-connectivity

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4-Connectivity

Two pixels p and q with values from V are 4-connected if q is in the set N4(p)

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Problem

1 1 0

0 1 1

0 0 1

1 1 0

0 1 1

0 0 1

p q

V = (1)


p

q

1 1 0

0 1 1

0 0 1

p

q1 1 0

0 1 1

0 0 1

p

q

A) B)

C)D)

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Problems

1 2 3 4

0 1 3 0

1 3 4 4

V1 = (1,2)

1 2 3 4

0 1 3 0

1 3 4 4

(1,2)

(3)

(4)

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8-Connectivity


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Problems

1 2 3 4

0 1 3 0

1 3 4 4

1 2 3 4

0 1 3 0

1 3 4 4

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m-Connectivity

Two pixels p and q with values from V are m-

connected if:-

1) q is in the set N4(p),

or

2) q is in the set ND(P)

and the set N4(p) N4(q) is empty

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Problem

1 1 0

0 1 1

0 0 1

1 1 0

0 1 1

0 0 1

p q

V = (1)

p

q

1 0 0

0 1 1

0 0 1

p

q1 1 0

0 1 1

0 0 1

p

q

q is in the set N4(p), or q is in the set ND(P) and the set N4(p) N4(q) is empty

B)

C)D)

A)

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Basic Relationships Between Pixels• 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 .

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Connectivity - pathways

A pixel p is adjacent to a pixel q if they are connected.

A path from pixel to pixel q will be a sequence of distinct pixels with their own coordinates.

0 1 1 0 1 1 0 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1

4 neighbours 8 neighbours m neighbours

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Examples: Adjacency and Path

01,1 11,2 11,3 0 1 1 0 1 102,1 22,2 02,3 0 2 0 0 2 003,1 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)

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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.

1 1 1 0 1 1 1 0 1 1 1 0 0 1 0 0

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Boundary (or border)

• 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.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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Problem• 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

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• 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

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• 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

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• 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

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• 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

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Distance function D• 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)

p (i,j)

q (h,k)

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Distance Measures

Euclidean: DE(p,q) = [(i-h)2 + (j-k)2]1/2

City Block: D4(p,q) = |i-h| + |j-k|

Chessboard: D8(p,q) = max{|i-h| , |j-k|}

p (i,j)

q (h,k)

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Problems

Euclidean: DE(p,q) = [(i-h)2 + (j-k)2]1/2

City Block: D4(p,q) = |i-h| + |j-k|

Chessboard: D8(p,q) = max{|i-h| , |j-k|}

p (i,j)

q (h,k)

If p(I,j) = (2,2) and q(h,k) = (4,4) find Euclidean, City Block, Chessboard distances

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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|)

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distance function D• 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

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Arithmetic/Logical Operations

Arithmetic and logical operations between pixels are used in most branches of image processing.

•Arithmetic operations apply to multivalued pixels.

• Logic operations apply only to binary images

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Arithmetic OperationsFor entire images, Arithmetic operations are carried out pixel by pixel.

Addition: p + q used for image averaging to reduce noise

Subtraction: p – q remove background information

Multiplication: p * q used to correct grey level pq shadingp x q

Division: p q “ “ “ “··

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Logical Operations

Basic tools of binary image processing where they are used for such tasks as masking, feature detection, and shape analysis

And: p AND qp • q

Or: p OR qp + q

Complement: NOT qq–

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Logical Operations - Examples

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Arithmetic/Logical Operations - Footnote

In addition to pixel-by-pixel operations on entire images, arithmetic and logical operations are used in neighbourhood-oriented operations.

i.e. mask/filter operations

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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

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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.

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• 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.

4-5 Basic Relationship Between Pixels

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