Neighbors of a Pixel – Relationships between Pixels

In this article, we will look at some of the most essential relationships between pixels in a digital image. As previously stated, an image is denoted by f. (x, y). This tutorial uses lowercase letters like p and q to refer to specific pixels.

Neighbors of a Pixel

A pixel p at coordinates (x, y) has four horizontal and vertical neighbors whose coordinates are given by

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

This set of pixels, called the 4-neighbors of p, is denoted by N₄(p).

The four diagonal neighbors of p have coordinates

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

and are denoted by N_D(p).

Diagonal neighbors together with the 4-neighbors are called the 8-neighbors of p, denoted by N₈(p).

N₈(p) = N₄(p) + N_D(p)

The set of image locations of the neighbors of a point p is called the neighborhood of p. The neighborhood is said to be closed if it contains p. Otherwise, the neighborhood is said to be open.

Adjacency between pixels

Let V be the set of intensity values used to define adjacency. In a binary image, V ={1} if we are referring to the adjacency of pixels with the value 1. In a gray-scale image, the idea is the same but set V typically contains more elements.

For example, in the adjacency of pixels with intensity values ranging from 0 to 255, set V might be any subset of these 256 values.

There are three types of adjacency:

1. 4-adjacency: Two pixels p and q with values from V are 4-adjacent if q is in the set N₄(p).
2. 8-adjacency: Two pixels p and q with values from V are 8-adjacent if q is in the set N₈(p).
3. m-adjacency (Mixed Adjaceny): Two pixels p and q with values from V are m-adjacent if
   1. q is in N₄(p), or
   2. q is in N_D(p) and the set N₄(p)∩N₄(q) has no pixels whose values are from V.

Mixed adjacency is a modification of 8-adjacency, and is introduced to eliminate the ambiguities that may result from using 8-adjacency.

Connectivity between pixels

It is a vital topic in digital image processing.

It is used to define the boundaries of objects and region components in an image.

Let S represent 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 is S, the set of pixels that are connected to it in S is called a connected component of S. If it only has one component, and that component is connected, then S is called a connected set.

On the basis of adjacency, there are three forms of connectivity. They are as follows:

1. 4-connectivity: If two or more pixels are 4-adjacent to each other, they are said to be 4-connected.
2. 8-connectivity: If two or more pixels are 8-adjacent to each other, they are said to be 8-connected.
3. M-connectivity: If two or more pixels are m-adjacent to each other, they are said to be m-connected.

Region

Let R represent a subset of pixels in an image. We call R a region f the image if R is a connected set.

Two regions, R_i and R_j, are said to be adjacent if their union forms a connected set.

Regions that are not adjacent are said to be disjoint. We consider 4 and 8-adjacency when referring to regions.

Boundary

The boundary is also known as the border or contour. The boundary of a region R is the set of pixels in R that are adjacent to pixels in the complement of R.

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