Explain Geometric transformations and Parametric transformations

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.

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