MATLAB Implementation of Affine Transformation

Affine transformation is one of the fundamental operations in computer vision and image processing, capable of performing linear transformations such as translation, rotation, scaling, and shearing on images. In MATLAB, affine transformations can be implemented using built-in functions or through manual computation of transformation matrices.

### Method 1: Using MATLAB Built-in Functions MATLAB provides the `affine2d` and `imwarp` functions to simplify affine transformation implementation. First, define an affine transformation matrix (typically a 3x3 matrix for 2D transformations), then use `affine2d` to convert it into a MATLAB-recognizable transformation object. Finally, apply the transformation to the target image using the `imwarp` function. This approach is suitable for quickly implementing common affine transformations like translation, rotation, or scaling. The `affine2d` function creates a geometric transformation object that stores the transformation matrix, while `imwarp` handles the actual image warping with automatic interpolation.

### Method 2: Manual Transformation Matrix Computation For more flexible control (such as custom transformation parameters), you can manually construct the affine transformation matrix and apply it to image pixel coordinates through matrix multiplication. The basic steps include: Defining the transformation matrix containing translation, rotation, scaling, and other parameters. Calculating the new coordinate positions for each pixel point. Using interpolation methods (such as bilinear interpolation) to fill the transformed image. This method requires manual implementation of coordinate transformation using matrix operations like: new_coords = transform_matrix * [x; y; 1], followed by interpolation to handle non-integer pixel positions.

Although more complex, this approach is suitable for scenarios requiring special transformations (such as simultaneous rotation and shearing) or when you need precise control over the transformation process.

### Extended Applications Affine transformations are not only used in image processing but also applied in coordinate system conversions, such as in Geographic Information Systems (GIS) or robot localization. Understanding the principles of affine transformations helps in learning more complex transformations, such as perspective transformations or nonlinear deformations. The transformation matrix format can be extended to 3D applications using 4x4 matrices for spatial transformations.