Matrix Transformation

Perspective Transformation

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

{\mathbf{T}_P} = \left[ {\begin{array}{*{20}{c}} \mathbf{A} & \mathbf{t}\\ {{\mathbf{a}^T}} & v \end{array}} \right]. \end{equation}

Its upper left corner is the invertible matrix $\mathbf{A}$. The upper right corner is the translation $\mathbf{t}$, and 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 lower 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. Perspective transformation is the most general transformation that has been said so far.

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.

In Eigen

Eigen::Perspective3d T;