Perspective Transformation

Perspective transformation is the most general transformation. Its matrix form is:

$${\mathbf{T}}_P=\left[\begin{array}{cc}\mathbf{A}& \mathbf{t}\\ {\mathbf{a}}^T& v\end{array}\right]$$

• Its upper left corner is an invertible matrix $\mathbf{A}$.
• The upper right corner is the translation $\mathbf{t}$
• The lower left corner is the scale ${\mathbf{a}}^T$.

Since the homogeneous coordinates are used, when $v\ne 0$, we can divide the entire matrix by $v$ to get a matrix with a bottom right corner of $1$. Otherwise, we get a matrix with a bottom right corner of $0$. Therefore, the 2D perspective transformation has a total of 8 degrees of freedom, and 3D has a total of 15 degrees of freedom.

The transformation from the real world to a camera photo can be seen as a perspective transformation. The reader can imagine what a square tile would look like in a photo. First, it is no longer square. Second, since the close part is larger than the far-away part, it is not even a parallelogram but an irregular quadrilateral.