8.C] Explain various geometric operations on images.

Answer:

Geometric Operations on Images

Geometric operations involve transforming images through various manipulations like translation, scaling, rotation, reflection, and shearing. These operations are fundamental in image processing and computer graphics.

1. Translation

Translation moves an image from one position to another. If a point in an image at coordinate (x, y) is moved to a new position (x', y'), the translation can be represented as:

\[ x’ = x + \delta x \]
\[ y’ = y + \delta y \]

In matrix form, the translation of a point (x, y) in homogeneous coordinates can be expressed as:

\[ \begin{bmatrix} x’ \\ y’ \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & \delta x \\ 0 & 1 & \delta y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]

2. Scaling

Scaling changes the size of an image. Scaling can either enlarge or shrink the image. For a point (x, y) scaled to (x', y'), the scaling is represented as:

\[ x’ = x \times S_x \]
\[ y’ = y \times S_y \]

In matrix form, scaling in homogeneous coordinates is represented as:

\[ \begin{bmatrix} x’ \\ y’ \\ 1 \end{bmatrix} = \begin{bmatrix} S_x & 0 & 0 \\ 0 & S_y & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]

where \(S_x\) and \(S_y\) are the scaling factors along the x and y axes, respectively.

3. Rotation

Rotation involves turning the image around a specific point, usually the origin. If an image is rotated by an angle \(\theta\), the new coordinates (x', y') are given by:

\[ x’ = x \cos \theta – y \sin \theta \]
\[ y’ = x \sin \theta + y \cos \theta \]

In matrix form, rotation in homogeneous coordinates is represented as:

\[ \begin{bmatrix} x’ \\ y’ \\ 1 \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]

4. Reflection

Reflection produces a mirror image of the image. Reflection across the x-axis, y-axis, and lines can be represented as:

• • Reflection across the x-axis:

\[ \begin{bmatrix} x’ \\ y’ \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]

• • Reflection across the y-axis:

\[ \begin{bmatrix} x’ \\ y’ \\ 1 \end{bmatrix} = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]

• • Reflection across the line \(y = x\):

\[ \begin{bmatrix} x’ \\ y’ \\ 1 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]

• • Reflection across the line \(y = -x\):

\[ \begin{bmatrix} x’ \\ y’ \\ 1 \end{bmatrix} = \begin{bmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]

5. Shearing

Shearing transforms an image by shifting its rows or columns. For shearing in the x-direction and y-direction, the transformations are:

• • Shearing in the x-direction:

\[ x’ = x + a \cdot y \]
\[ y’ = y \]

\[ \begin{bmatrix} x’ \\ y’ \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & a & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]

• • Shearing in the y-direction:

\[ x’ = x \]
\[ y’ = y + b \cdot x \]

\[ \begin{bmatrix} x’ \\ y’ \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ b & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]

where \(a\) and \(b\) are shear factors in the x and y directions, respectively.