Perspective Cameras

We take a look at how the above relationships can be extended to perspective cameras. Suppose that we have a world point $P$ that moves as per Equation 1. Let this point be viewed by a perspective camera represented by the camera matrix ${\bf M}$. The projection of $P$ will trace a line in the image. Let us represent this line $l$ by Let $p$ be the projection of $P$ due to ${\bf M}$ at some time instant $t$.

$$l^T p = 0$$

$$\left [ \begin{array}{ccc} a & b & c \end{array} \right ] {\bf M}... ...rray}{cccc} (I_x+U_xt) & (I_y+U_yt) & (I_z+U_zt) & 1 \end{array} \right ]^T = 0$$

Differentiating w.r.t. $t$, we get

$$(a {\bf m_1} + b {\bf m_2} + c{\bf m_3}). \left [ \begin{array}{ccc} U_x & U_y & U_z \end{array} \right ] = 0$$

Proceeding in a similar manner as before to eliminate the projection terms $m_{ij}$, we can derive a view-independent relation for uniform velocity motion under perspective projection. However, the number of unknowns are very high and hence the number of views or time instants needed to solve for them runs into a large number. Extension to constant acceleration motion also has similar properties under perspective projection. Levin et al.  sketch the extension of their view-independent constraints to perspective cameras. Their extension required 6 points in 49 time instants under perspective projection, compared to 5 points in 8 time instants under affine projection. Similar ideas must be applied to reduce the view requirements to practical levels. We are actively pursuing this currently.