Geometric transformation is an image processing technique that changes the position of pixels in an image by mapping their coordinates from one location to another. It consists of two main steps:
- Spatial Transformation: Changes the coordinates of pixels from
(x, y) to(x', y') - Intensity Interpolation: Assigns intensity values to the new pixel locations after transformation.
Mathematical Representation
A pixel at coordinate
\begin{bmatrix}\\x'\\y'\end{bmatrix}= T\begin{bmatrix}\\x\\y \end{bmatrix}= \begin{bmatrix}\\a &b \\c& d\end{bmatrix}\begin{bmatrix}\\x\\y \end{bmatrix}
where:
- (x, y) is the input coordinate
- (x', y') is the output coordinate
- T is the geometric transformation matrix
Types of Geometric Transformations
Geometric transformations modify the position, orientation, size or shape of an image by changing the coordinates of its pixels.
1. Translation
Translation shifts an image from one location to another without changing its size, shape, or orientation. Every pixel is moved by the same distance along the x-axis and y-axis.
Applications: Image alignment, object tracking and image registration.
\begin{bmatrix}x' \\y'\end{bmatrix}=\begin{bmatrix}x + \Delta x \\y + \Delta y\end{bmatrix}
2. Rotation
Rotation turns an image around a fixed point, usually the image center, by a specified angle. Positive angles rotate the image counterclockwise, while negative angles rotate it clockwise.
Applications: Image correction, object recognition and computer vision systems.
\begin{bmatrix}x' \\y'\end{bmatrix}=\begin{bmatrix}\cos\theta & -\sin\theta \\\sin\theta & \cos\theta\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}
3. Scaling
Scaling changes the size of an image by enlarging or shrinking it. Uniform scaling changes both dimensions equally, while non-uniform scaling changes them independently.
Applications: Image resizing, zooming and display optimization.
\begin{bmatrix}x' \\y'\end{bmatrix}=\begin{bmatrix}s_x & 0 \\0 & s_y\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}
4. Shearing (Skewing)
Shearing shifts pixels in one direction while keeping the other direction fixed, causing the image to appear slanted or tilted. Unlike rotation, it changes the angles between lines in the image.
Applications: Perspective correction, image distortion and graphics design.
\begin{bmatrix}x' \\y'\end{bmatrix}=\begin{bmatrix}1 & k_y \\k_x & 1\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}
5. Affine Transformation
Affine transformation combines multiple transformations such as translation, rotation, scaling, and shearing into a single operation. It preserves straight lines and parallel lines, making it one of the most widely used geometric transformations.
Applications: Image registration, object tracking, and image warping.
\begin{bmatrix}\\x'\\y'\\1 \end{bmatrix}=A\begin{bmatrix}\\x\\ y\\1\end{bmatrix}= \begin{bmatrix}a &b &c \\ d&e &f \\ 0& 0 &1 \end{bmatrix}\begin{bmatrix}\\x\\y\\1\end{bmatrix}
where:
- (x, y) is the input coordinate
- (x', y') is the output coordinate
- A is the affine transformation matrix
6. Perspective Transformation
Perspective transformation changes the viewpoint of an image by mapping points from one plane to another. It creates the effect of depth and distance, making objects appear closer or farther away.
Applications: Image stitching, camera calibration, document scanning, and 3D reconstruction.
m
\begin{bmatrix}\\x'\\ y'\\w'\end{bmatrix}=H\begin{bmatrix}\\x \\y\\1\end{bmatrix}=\begin{bmatrix}a &b &c \\ d &e &f \\ g &h &i \end{bmatrix}\begin{bmatrix}\\x\\y\\1\end{bmatrix}
Where:
- w'x' = ax + by + c
- w'y' = dx + ey + f
- w' = gx + hy + i
To find x' and y' we need to normalize them with respect to w'. Thus the coordinate equations for perspective transformation are:
x'=\frac{ax+by+c}{gx+hy+i_{}}y'=\frac{dx+ey+f}{gx+hy+i}
Applications
- Image Registration: Registering images from various views or from multiple sensors.
- Computer Vision: Improving media related to computer vision.
- Medical Imaging: Understanding and comparing images of MRI, X ray or CT scan.
- Remote Sensing: Geographical information system analyse satellite or aerial images using this.
- Robotics: Navigation of robots is done using image analysis of images captured by their camera sensor.