Solution:

Shrink Operators:

Shrink operators are iterative operators which convert 1 into 0 i.e. converts a foreground pixel that is present in the background of the object into a background pixel.

If $p(k, j)$ is the pixel function evaluated at $(k, j)$ and $1($.$) is a unit step function, then the$ shrink operator acting on the pixel $p(k, j)$ of a digital image is defined as

$\operatorname{Shrink}(i) .1(k, j)=I(k, j)$ AND $1(i-1-[8-p(k, j)]) ; 0 \leq i \leq 8$

$p(k, j)$ returns the number of foreground pixels

$[8-p(k, J)]$ returns the number of background pixels.

Swell operators:

Swell operators are iterative operators which convert 0 into 1 i.e. convert the background pixel which is present in the foreground of an object into a foreground pixel and are dual of shrink operators.

If p( k, j) is the pixel function evaluated at (k, j) and 1(.) is a unit step function, then the swell operator acting on the pixel p( k,j) of a digital image is defined as,

Swell (i). I $(k, j)=I(k, j)$ OR $1(p(k, j)-1]) ; 0 \leq i \leq 8$

$p(k, j)$ returns the number of foreground pixels

Connectedness:

A pixel is 4-connectedness to its neighbors if and only if at least one of the four pixels to the east, north, west, and south have the same value.

A pixel is 8-connected to its neighbors if and only if at least one of its eight neighbors has the same value.

Foreground region: 4-connected

Background region: 8-connected

Skeleton operator:

An iterative operator which shrinks the image as much as possible while preserving the Euler the number is called a skeleton operator.

When the skeleton operator is applied to a simple region, it reduces to a point and whereas if it is applied to a region with a single hole, the final result is a thin ring.

Bulk operator:

An iterative operator which swells the image as much as possible while preserving the Euler the number is called bulk operator.

It is the dual skeleton operator

They are used to highlight defects in parts. However, they are not effective in removing salt and-pepper noise from an image.

Euler number:

Euler number is defined as the number of parts minus the number of holes in a given image. The part is a connected foreground region and a hole is an isolated background region enclosed by a part.

Euler number of A = 1 -1 =0; B = 1 – 2 = -1; C= 1- 0= 1