Uniform Velocity

Let $P$ be a 3D world point, moving with uniform linear velocity. Let ${\bf I} = \left [ \begin{array}{ccc} I_x & I_y & I_z \end{array} \right]^{\sc T}$ be its initial position and be its world velocity. These vectors are represented as homogeneous vectors and . Its position at any time instant $t$ is given by

$$P = \tilde{\bf I} + \tilde{\bf U}\; t$$ (1)

Let an affine camera observe the motion of the point. Let $p = \left [ \begin{array}{ccc} x & y & 1 \end{array} \right]^{\sc T}$ be the projection of $P$ due to the affine camera matrix ${\bf M} = \left [ \begin{array}{cc} {\bf m_{1}} & m_{14} \\ {\bf m_{2}} & m_{24} \\ {\bf0} & 1 \end{array} \right ]$. where ${\bf m_{i}}$ represents a vector of the first three elements in the $i$th row of ${\bf M}$. Then,

$$p = {\bf M}\tilde{\bf I} + {\bf M} \tilde{\bf U}\; t.$$ (2)

Differentiating with respect to $t$, we get where $\tilde{\bf v} = [v_x, v_y, 0]^{\sc T}$ is the velocity vector in the image. This implies that the velocity of a point in the image is a projection of the world velocity. The above can be expanded as

$$v_x = {\bf m_{1}} .\left [ \begin{array}{ccc} U_x & U_y & U_z \end{array}\right ]$$ (3)

$$v_y = {\bf m_{2}} . \left [ \begin{array}{ccc} U_x & U_y & U_z \end{array} \right ]$$ (4)

Equations 3 and 4 can be written as

$$\left [ \begin{array}{cccc} U_x & U_y & U_z & v_x \end{array... ... \begin{array}{cc} {\bf m_{1}} & -1 \end{array} \right ]^T = 0$$

and

$$\left [ \begin{array}{cccc} U_x & U_y & U_z & v_y \end{array... ...\begin{array}{cc} {\bf m_{2}} & -1 \end{array} \right ]^T = 0$$

If we have 4 points in the scene, $P_i,\; 1 \le i \le 4$, with world velocities and image velocities , we can define a matrix ${\bf X_v}$ as

$${\bf X_v} = \left [ \begin{array}{cccc} U_{1x} & U_{1y} & U_{1z} ... ...3y} & U_{3z} & v_{3x} \\ U_{4x} & U_{4y} & U_{4z} & v_{4x} \end{array} \right]$$ (5)

Similarly, we can define a matrix ${\bf Y_v}$ with the last column having the velocity vectors in $y$-direction. We can see that

$${\bf X_v} \left [ \begin{array}{cc} {\bf m_{1}} & -1 \end{array}\right ]^T = {\bf0} \;\;\mbox{ and }\;\;$$

$${\bf Y_v} \left [ \begin{array}{cc} {\bf m_{2}} & -1 \end{array}\right ]^T = {\bf0}.$$

For these to have a constraint, ${\bf X_v}$ and ${\bf Y_v}$ must be rank deficient, i.e., $\vert{\bf X_v}\vert = \vert{\bf Y_v}\vert = 0$. We get the following functions of the velocities of the 3D points and the image velocities by expanding the expressions for the determinants.

$$\alpha_0 v_{1x} + \alpha_1 v_{2x} + \alpha_2 v_{3x} + \alpha_3 v_{4x} = 0$$

$$\alpha_0 v_{1y} + \alpha_1 v_{2y} + \alpha_2 v_{3y} + \alpha_3 v_{4y} = 0$$ (6)

where $\alpha$'s depend only on the world velocity parameters of the 4 points. The $\alpha$'s are view-independent. That is, the above constraints hold for the image velocities of four points for the same $\alpha$'s irrespective of the pose and intrinsic parameters of the camera. They are also time-independent as time term has been eliminated. Equation 6 has four unknowns, with each view providing two equalities. Therefore, we need two views of the four points to determine all $\alpha$'s up to scale.

These results are similar to the Recognition Polynomials and Shape Tensors presented or discovered earlier. It was shown that polynomials to recognize a configuration of stationary points could be constructed from 2 views of 4 points under orthographic projections [1]. This was extended to recognize human gait using 2 views of 5 points under scaled-orthographic projections [2]. Time-dependent constraints involving a single view of 5 points with uniform velocity is presented in [11] for affine projection.. Our results yield view and time independent constraints involving 4 points in 2 views under the general affine projection.