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 N_{4}(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 ** N_{8}(p)**.

N_{8}(p) = N_{4}(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:

**4-adjacency:**Two pixels p and q with values from V are 4-adjacent if q is in the set N_{4}(p).**8-adjacency:**Two pixels p and q with values from V are 8-adjacent if q is in the set N_{8}(p).**m-adjacency (Mixed Adjaceny):**Two pixels p and q with values from V are m-adjacent if- q is in N
_{4}(p), or - q is in N
_{D}(p) and the set N_{4}(p)∩N_{4}(q) has no pixels whose values are from V.

- q is in N

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:

**4-connectivity:**If two or more pixels are 4-adjacent to each other, they are said to be 4-connected.**8-connectivity:**If two or more pixels are 8-adjacent to each other, they are said to be 8-connected.**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*are said to be

_{j,}**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*.

## FAQ

### What is adjacency in Image Processing

Two pixels are connected if they are neighbors and their gray levels satisfy some specified criterion of similarity.