Correspondence. Imagine have 2 photos of the same object from a slightly different position. Then the point \(p_1\) and \(p_2\) on the respective images are corresponding if they are projecting the same physical point. Sign for correspondence is \(p_1\ \hat{=}\ p_2\).

Homogeneous coordinates. This is a coordinate system used in projective geometry and will be used here from now on as well. 2D vector \(\begin{bmatrix} x\\ y \end{bmatrix}\) in cartesian coordinates is expressed as 3D vector \(\begin{bmatrix} wx\\ wy\\ w \end{bmatrix}, \forall w

eq 0\) in homogeneous coordinates. Similarly, 3D vectors in cartesian coordinates are 4D vectors in homogeneous coordinates. Also, note that \(\begin{bmatrix} w_{1}x\\ w_{1}y\\ w_1 \end{bmatrix}=\begin{bmatrix} w_{2}x\\ w_{2}y\\ w_2 \end{bmatrix}, \forall w_1

eq 0,\ w_2

eq 0\) in homogeneous coordinates.

Matrices are used to represent certain geometric transformations in the homogeneous coordinates. Transformation of the point \(p\) is realized by a simple multiplication so that \(p’=Mp\). In addition, transformations can merged into a single one by the standard matrix multiplication.