We briefly discuss the dynamics of the Structure from Motion problem.
As shown earlier, it is often the case (i.e. in cinematographic
post-production, robotics, etc.) that cameras do not teleport around
the scene and objects do not move about too suddenly. These bodies
are governed by physical dynamics and it thus makes sense to constrain
the possible configurations of the camera to have some smooth temporal
changes over a causal time sequence. For instance, we consider the
typical dynamic system: ^{4}

= | (9) | ||

= | (10) |

Here, the observations are the 2D features (in ** u**,

In addition, the dynamics of the internal state are constrained. The
3D structure, 3D motion and camera geometry do not vary wildly but are
linearly dependent (via )
on their previous values at the past
time interval plus Gaussian noise. The noise process is additive with
zero-mean and covariance ** Q**. For generality, we assume that the
motion of the camera through the scene is not known a priori and thus,
is set to identity. Therefore, the internal state varies only
through some Gaussian random noise process. This can be seen as a
'random walk' type of internal state space. In other words, the vector
varies randomly but smoothly with small deltas from its past
values.

This dynamic system encodes the causal and dynamic nature of the SfM problem and allows an elegant integration of multiple frames from image sequences. It is also a probabilistic framework for representing uncertainty. These dynamical systems have been extensively studied are routinely solved via reliable Kalman Filtering (KF) techniques. In our nonlinear case, an Extended Kalman Filter (EKF) is utilized which linearizes at each time step.

The representation of the measurement vector is simply the concatenation of the 2D feature point measurements. We now turn our attention to the representation of the internal state of the unknowns of the system: the 3D structure, 3D motion and internal camera geometry. This step is critical since the effectiveness of the Kalman filtering framework depends strongly on the representation.