Explain Geometric transformations and Parametric transformations
Answer:-
In computer vision and image processing, geometric transformations refer to operations that change the spatial arrangement of pixels in an image. These transformations allow tasks such as: aligning images, registering different views, simulating camera changes, or manipulating images through rotation, scaling, and warping.
Geometric Transformations: Definition and Role
A geometric transformation is any function that maps input image coordinates (x, y) to output image coordinates (x', y'):
(x', y') = T(x, y)
Transformations can be linear or non-linear, global or local. Common examples include:
- Translation: shifts image in space
- Rotation: rotates around a point
- Scaling: enlarges or shrinks image
- Affine: combines linear transform with translation
- Projective (Homography): applies perspective changes
Parametric Transformations
Parametric transformations are a subset of geometric transformations defined using a fixed number of parameters. They follow the model:
(x', y') = T(x, y; θ)
Where θ is a parameter vector.
| Transformation | Parameters | Use Case |
|---|---|---|
| Translation | tx, ty | Object tracking |
| Rotation | θ | Image alignment |
| Scaling | sx, sy | Zooming |
| Affine | 6 params | Image rectification |
| Homography | 8 params | Panorama stitching |
Matrix Representation (Homogeneous Coordinates)
Geometric and parametric transformations can be expressed as matrix multiplication using homogeneous coordinates:
Where HHH is a 3×3 transformation matrix:
- For affine transformations, the last row is fixed as [0 0 1]
- For projective transformations (homographies), the matrix is general and can represent perspective effects
Properties of Parametric Transformations
- Compact and low-dimensional
- Global: same transform applies to entire image
- Invertible
- Differentiable: useful in optimization
Applications
- Image stitching – using homographies
- Motion tracking – estimating object transformation
- Camera calibration – correcting lens distortion
- Image registration – aligning different image sources
Difference Between Geometric and Parametric Transformations
| Feature | Geometric Transformation | Parametric Transformation |
|---|---|---|
| Scope | General (any mapping) | Specific, defined by parameters |
| Defined by | Function (can be complex) | Finite parameter vector |
| Representation | May not be matrix-based | Often matrix-based |
| Examples | Free-form warping | Affine, homography |
| Applications | General warps | Alignment, tracking |