Uniform Acceleration

We now derive relationships between points when they move with constant acceleration. Let $P$ be a 3D world point, moving with uniform linear acceleration. Let its position at any time instant $t$ be given by

$$P = \tilde{\bf I} + \tilde{\bf U} t + \frac{1}{2} \tilde{\bf A} t^2$$ (7)

where $\tilde{\bf I}$ is the initial position of the point, $\tilde{\bf U}$ is its initial velocity and is its constant acceleration.

Proceeding the same way as in the previous subsection, we get a singular matrix ${\bf X_a}$

$${\bf X_a} = \left [ \begin{array}{cccc} (U_{1x}+A_{1x} t) & (... ...4y}+A_{4y} t) & (U_{4z}+A_{4z} t) & v_{4x} \end{array} \right]$$

A similar singular matrix ${\bf Y_a}$ with motion parameters in $y$-direction also exists.

Expanding $\vert{\bf X_a}\vert$ and $\vert{\bf Y_a}\vert$, we get

$$(\beta_0 + \beta_1 t + \beta_2 t^2 + \beta_3 t^3) v_{1x} +$$

$$(\beta_4 + \beta_5 t + \beta_6 t^2 + \beta_7 t^3) v_{2x} +$$

$$(\beta_8 + \beta_9 t + \beta_{10} t^2 + \beta_{11} t^3) v_{3x} +$$

$$(\beta_{12} + \beta_{13} t + \beta_{14} t^2 + \beta_{15} t^3) v_{4x} = 0$$ (8)

$$(\beta_0 + \beta_1 t + \beta_2 t^2 + \beta_3 t^3) v_{1y} +$$

$$(\beta_4 + \beta_5 t + \beta_6 t^2 + \beta_7 t^3) v_{2y} +$$

$$(\beta_8 + \beta_9 t + \beta_{10} t^2 + \beta_{11} t^3) v_{3y} +$$

$$(\beta_{12} + \beta_{13} t + \beta_{14} t^2 + \beta_{15} t^3) v_{4y} = 0$$ (9)

where $\beta$'s depend only on the parameters of motion of the 3D points in the world. The above relations are time-dependent and view-independent. That is, the same $\beta$'s hold no matter what the pose and intrinsic parameters of the affine camera used to view them. There are 16 unknowns ( $\beta_0 \ldots \beta_{15}$) in the above relation, with each time instant providing 2 equations. We, therefore, need the velocities of 4 points for 8 time instants (9 frames) for computing the $\beta$'s. Note that these $\beta$'s can be computed from a single view, as opposed to the $\alpha$'s for the case of constant velocity, which needed two views. This is the direct result of time-dependence.

We now proceed to derive time-independent constraints for the case of constant linear acceleration in the world. For this, we differentiate the constant acceleration motion equation (Equation. 7) twice to get

$$a_x = {\bf m_{1}}. \left [ \begin{array}{ccc} A_x & A_y & A_z \end{array} \right]$$ (10)

$$a_y = {\bf m_{2}}. \left [ \begin{array}{ccc} A_x & A_y & A_z \end{array} \right]$$ (11)

where $a_x$ and $a_y$ are the image-accelerations of the projection of the point. We can similarly define the singular matrices ${\bf X_a^\prime}$ and ${\bf Y_a^\prime}$ for the four points $P_i$, $1 \le i \le 4$. ${\bf X_a^\prime}$ is given below.

$${\bf X_a^\prime} = \left [ \begin{array}{cccc} A_{1x} & A_{1y} & ... ...3y} & A_{3z} & a_{3x} \\ A_{4x} & A_{4y} & A_{4z} & a_{4x} \end{array} \right]$$ (12)

Expanding the determinants of ${\bf X_a^\prime}$ and ${\bf Y_a^\prime}$, we get

$$\gamma_0 a_{1x} + \gamma_1 a_{2x} + \gamma_2 a_{3x} + \gamma_3 a_{4x} = 0$$

$$\gamma_0 a_{1y} + \gamma_1 a_{2y} + \gamma_2 a_{3y} + \gamma_3 a_{4y} = 0,$$ (13)

where $\gamma$'s are functions of world accelerations parameters of the 4 points only. The $\gamma$'s are view and time independent. The system of Equations 13 has four unknowns and we need 2 views of the 4 points to determine all the $\gamma$'s.

Levin et al. [11] derive constraints for motion constrained to elliptic paths. It is the first time that view or time independent constraints for points moving with constant linear acceleration have been derived.