General Planar Motion

Figure 1: Two image sequences of an exploding pot

If points undergoing general planar motion can be tracked across time, the trajectory of each forms a contour which is viewed by both video cameras. Both videos see the same contour, but the starting points are different. The problem reduces to contour matching under affine homography and unknown shift. Solutions for this situation using a measure similar to $\kappa$ has been presented in [10]. The new measure is

$$\kappa^{\prime}_p(l)[k] = ({\bf X}^l[k])^{*{\sc T}} \left[ \begin{array}{cl} 0 & 1 -1\;\; & 0 \end{array}\right] {\bf X}^l[p]$$

$$= \vert{\bf A}_l\vert\; \kappa^{\prime}_p(0)[k]\; e^{ - j2\pi\lambda_l (k - p) / N}$$ (23)

where p is a constant (typically 1 or 2) and $\lambda_l$ is the unknown shift in view $l$ compared to the reference view $0$. The ratio $\frac{\kappa^{\prime}_p(l)[k]}{\kappa^{\prime}_p(0)[k]}$ will be a complex sinusoid. The shift $\lambda_l$ can be recovered from the inverse Fourier transform of this ratio. For more details, see [10].