Finding the linear mapping between homogeneous coordinates

[stephchia]

Homework Statement

If I have an affine camera with a projection relationship governed by:

\begin{equation}
\begin{bmatrix}
x & y
\end{bmatrix}^T = A
\begin{bmatrix}
X & Y & Z
\end{bmatrix}^T + b
\end{equation}
where A is a 2x3 matrix and b is a 2x1 vector. How can I form a matrix representing the linear mapping between the world point (X,Y,Z) and image point (x,y) if they are represented by homogeneous vectors?

Homework Equations

NIL

The Attempt at a Solution

I understand that for a camera perspective projection, the linear mapping between homogeneous coordinates where the equation is only up to a scale factor can be written as a 3x4 projection matrix that represents a map from 3D to 2D.
\begin{equation}
\begin{bmatrix}
x \\
y \\
w
\end{bmatrix}=
\begin{bmatrix}
f & 0 & 0 & 0 \\
0 & f & 0 & 0 \\
0 & 0 & 1 & 0
\end{bmatrix}
\begin{bmatrix}
X_c \\
Y_c \\
Z_c \\
1
\end{bmatrix}
\end{equation}
However, I am unclear as to how an affine camera differs and how to fit a linear mapping based on the above given governing equation.

 

[jambaugh]

Remember you want your original vector plus the displacement. Start with the identity matrix components in the first three columns and then ask what effect a non-zero value in the fourth column will have.