Find the Euler number of the 8 -connected 4 ×4 image. (i) [ 1 1 1 1 1

(i) The Euler number of the 8-connected $4 times 4$ image is given by the formula: Euler number

Find the Euler number of the 8 -connected 4 ×4 image. (i) [ ...

Find the Euler number of the 8 -connected $4 \times 4$ image. (i) $$ \left[\begin{array}{llll} 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 \end{array}\right] $$ (ii) $$ \left[\begin{array}{llll} 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 \end{array}\right] $$ (iii) $$ \left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{array}\right] $$

Find the Euler number of the 8 -connected $4 \times 4$ image.
(i)
$$
\left[\begin{array}{llll}
1 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 \\
1 & 1 & 0 & 1 \\
1 & 1 & 1 & 1
\end{array}\right]
$$
(ii)
$$
\left[\begin{array}{llll}
0 & 0 & 0 & 1 \\
0 & 1 & 1 & 0 \\
1 & 1 & 1 & 1 \\
0 & 1 & 1 & 0
\end{array}\right]
$$
(iii)
$$
\left[\begin{array}{llll}
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
1 & 0 & 1 & 0 \\
1 & 1 & 1 & 1
\end{array}\right]
$$

Key Concepts

Euler Number

The Euler number in image processing is a topological descriptor calculated as the difference between the number of connected components and the number of holes in a binary image. It provides a summary of an object’s topology and is invariant under continuous deformations, which makes it useful for shape analysis and object recognition.

8-connectedness

8-connectedness refers to a connectivity model where a pixel is considered connected to its eight immediate neighbors (including the diagonals). This connectivity criterion determines how pixels group together into connected components in a digital image and influences the calculation of topological measures such as the Euler number.

Binary Image Representation

Binary image representation involves representing an image as a matrix of values, typically 0s and 1s, where one value indicates the background and the other indicates the object or feature of interest. This simplified model is used for many image processing tasks, including computing connectivity and topological descriptors.

Connected Components and Holes

Connected components are clusters of pixels that are linked according to a specific connectivity rule (such as 8-connectedness), while holes are regions of background pixels that are completely surrounded by pixels of the object. Their count directly influences the Euler number, as it is defined as the number of connected components minus the number of holes.

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