Notable Homographic Transformations

It is possible to provide a brief list of the transformations in computer vision that can be represented through a homography:

1. **Perspective Transformation**: This includes the transformation of images taken from different viewpoints, allowing for the correction of perspective distortions.

2. **Image Rectification**: Homographies can be used to align images taken from different angles to a common plane, making it easier to analyze or stitch images together.

3. **Panorama Stitching**: By applying homographies, multiple images can be combined to create a seamless panoramic view.

4. **Camera Calibration**: Homographies play a crucial role in the calibration process, helping to relate the 2D image plane to the 3D world coordinates.

5. **Object Tracking**: Homographies can assist in tracking objects across frames by mapping their positions in different views.

6. **Augmented Reality**: In augmented reality applications, homographies are used to overlay virtual objects onto real-world scenes accurately.

7. **Image Warping**: Homographies can be employed to warp images for various applications, including artistic effects or correcting lens distortion.

8. **Scene Reconstruction**: By using homographies, it is possible to reconstruct 3D scenes from multiple 2D images taken from different perspectives.

The transformations described in this section allow, given the knowledge of the camera orientation and intrinsic parameters, to derive the matrix $\mathbf{H}$ that determines the transformation. Conversely, by obtaining the homographic matrix through the association of points between the two images, it is possible to derive certain parameters that relate the views to each other.

It is indeed important to note that for all transformations involving a homography (change of viewpoint, perspective projection, IPM, and rectification), when the knowledge of the parameters necessary to generate the transformation is required, the representative matrices can still be implicitly derived by knowing how (at least) 4 points of the image are transformed (see section 8.5.1 for details).

The parameters obtained from the decomposition of the homographic matrix are derived from algebraic minimization. The maximum likelihood solution requires a nonlinear minimization but uses the result obtained from this phase as a starting point.