Cartesian Velocity Moments

The Cartesian velocity moments [4] are computed from a sequence of images as:

$$vm_{pq\mu\gamma} = \sum_{i=2}^{Images} \sum_{x=1}^{M} \sum_{y=1}^N U(i,\mu,\gamma)~C(i,p,q)~P_{i_{xy}}$$ (13)

$C(i,p,q)$ arises from the centralised moments:

$$C(i,p,q)=(x-\overline{x_{i}})^{p}(y-\overline{y_{i}})^q$$ (14)

$U(i,\mu,\gamma)$ introduces velocity as:

$$U(i,\mu,\gamma)=(\overline{x_{i}} - \overline{x_{i-1}} )^\mu (\overline{y_{i}} - \overline{y_{i-1}} )^\gamma$$ (15)

$\overline{x_{i}}$ is the current COM (Centre of Mass - as defined by the first order Cartesian moment) in the $x$ direction, while $\overline{x_{i-1}}$ is the previous COM in the $x$ direction, $\overline{y_{i}}$ and $\overline{y_{i-1}}$ are the equivalent values for the $y$ direction. It can be seen that the equation can easily be decomposed into averaged centralised moments ($vm_{1100}$), and then further into an averaged Cartesian moment ($vm_{1100}$ with $\overline{x_{i}} = \overline{y_{i}} = 0$). The zero order velocity moments for which $\mu=0$ and $\gamma=0$ are then:

$$vm_{pq00} = \sum_{i=2}^{Images} \sum_{x=1}^{M} \sum_{y=1}^N (x-\overline{x_{i}})^{p} (y-\overline{y_{i}})^q P_{i_{xy}}$$ (16)

which are the averaged centralised moments. The zero order components for which $p=0$ and $q=0$ are:

$$vm_{00\mu\gamma} = \sum_{i=2}^{Images} \sum_{x=1}^{M} \sum_... ...mu (\overline{y_{i}} - \overline{y_{i-1}} )^\gamma P_{i_{xy}}$$ (17)

which is a summation of the difference between COMs of successive images (ie the velocity). The structure of Equation 13 allows the image structure to be described together with velocity information from both the $x$ and $y$ directions. These results are averaged by normalising with respect to the number of images and the average area of the object. This results in pixel values for the velocity terms, where the velocity is measured in pixels per image. The normalisation is expressed as:

$$\overline{vm_{pq\mu\gamma}} = \frac{vm_{pq\mu\gamma}}{A\cdot I}$$ (18)

where $A$ is the average area (in numbers of pixels) of the moving object, $I$ is the number of images and $\overline{vm_{pq\mu\gamma}}$ is the normalised Cartesian velocity moment.