ComputerVision_week_2_Wednesday_2023.pdf

ComputerVision_week_2_Thursday_2023.pdf

Projective geometry cont.

Summary

Projective geometry is an alternative algebraic representation of geometric objects and transformations.

Homogeneous coordinates are a system of coordinates used in projective geometry

• Use a n+1 dimensional vector to represent an n dimensional euclidean point
• Points at infinity can be represented using finite coordinates
• A single matrix can represent affine and projective transformations

2D Transformations

original shape

• Scaling

  Each component multiplied by a scalar (uniform scaling: same scalar for each component)

  $x'=ax\\ y'=by$

  Matrix representation (euclidean):

  $$ \begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$

• Shear

  Matrix representation (euclidean):

  $$ \begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} 1 & a \\ b & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$

• Rotation

  Matrix representation (euclidean):

  $$ \begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$

• Translation

  $x' = x + t_x$, $y'= y+t_y$

  → can’t be represented by a 2x2 matrix multiplication (because it’s not a linear transofrmation, the origin’s image is not the origin…)

  But it can be with 3x3 matrix multiplication, using homogenous coordinates:

  $$ \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} $$