Tracking & motion features

Default features

TrackId

The ID of the track an object belongs to. It is empty if the object is not part of any track.

Time

The object time in i-th frame $t_i$ (in seconds) that was used (defined in the Track Particles settings dialog) for tracking. It is typically Time but the tracking may reference different calculated feature.

Position

The object position in i-th frame $P_i=|x_i,y_i|$ for 2D and $P_i=|x_i,y_i,z_i|$ for 3D (in microns) that was used (defined in the Track Particles settings dialog) for tracking. It is typically one of the Object position features but the tracking may reference different calculated position (e.g. relative to another object).

Motion features

Motion features are calculated by Motion Features for each object (in a track) and per $i$-th frame.

All motion features are calculated from object’s Position $P_i$ and time $t_i$ in the current $i$-th position with respect to the previous $(i-1)$-th position. First $0$-th position has the features left empty.

ΔTime

$$\Delta t_i=t_i-t_{i-1}$$

ΔPosition

$$\overrightarrow{\Delta P_i}=\overrightarrow{P_i}-\overrightarrow{P_{i-1}}$$

Velocity

$$\overrightarrow{v_i}=\overrightarrow{P_i}-\overrightarrow{P_{i-1}}$$

Distance

$$d_i=\Vert \overrightarrow{\Delta P_i}\Vert =\Vert \overrightarrow{v_i}\Vert$$

Speed

$$s_i=\frac{d_i}{\Delta t_i}$$

Heading

Angle in degrees between the direction of the Velocity vector $\overrightarrow{v}$ and the positive $x$-axis. Zero degrees is in the direction of X axis (to the right) and increasing counterclockwise until 360.

$$H_i=\mathrm{mod}\left(\frac{180}{\pi }atan2(v_{i,y},v_{i,x}),360\right)$$

Elevation

Angle between the direction of the Velocity vector $\overrightarrow{v}$ and the XY plane. Positive Elevation is towards positive Z coordinates. The values range from -90 to 90.

$$E_i=\frac{180}{\pi }atan2\left(v_{i,z},\sqrt{{v_{i,x}}^2+{v_{i,y}}^2}\right)$$

Acceleration

The length of Acceleration vector $|\overrightarrow{a}|$ is a measure of total force that is acting on an object.

$$a_i=\Vert \frac{\overrightarrow{v_i}-\overrightarrow{v_{i-1}}}{\Delta t_i}\Vert$$

Resulting table:

t-index	time [s]	position	velocity	accelerations
0	$t_0$	$P_0$
1	$t_1$	$P_1$	$\overrightarrow{v_1}$
2	$t_2$	$P_2$	$\overrightarrow{v_2}$	$\overrightarrow{a_2},\overrightarrow{a_2^N},\overrightarrow{a_2^T}$

AccelerationT

Tangential acceleration $a^T$ is the part of Acceleration vector $\overrightarrow{a}$ that has the same direction as the Velocity vector $\overrightarrow{v}$. It represents the accelerating or decelerating force.

$$a_i^T=\overrightarrow{a_i}\cdot \frac{\overrightarrow{v_{i-1}}}{\Vert \overrightarrow{v_{i-1}}\Vert }$$

AccelerationN

Normal Acceleration $a^N$ is the part of Acceleration vector $\overrightarrow{a}$ that is perpendicular to the Velocity vector $\overrightarrow{v}$. It represents a deviating force.

$$a_i^N=\sqrt{a_i^2-{a_i^T}^2}$$

ΔHeading

$$\Delta H_i=H_i-H_{i-1}$$

ΔElevation

$$\Delta E_i=E_i-E_{i-1}$$

Turning Rate

In degrees per second.

$$H^{\prime }=\frac{\Delta H_i}{\Delta t_i}$$

Pitching Rate

In degrees per second.

$$E^{\prime }=\frac{\Delta E_i}{\Delta t_i}$$

Path Length

Is the sum of the segment lengths up the the $i$-th frame.

$$P{L}_i=\sum _{k=0}^i{d}_k$$

Path Speed

$$P{S}_i=\frac{{\sum }_{k=0}^i{d}_k}{t_i-t_0}$$

Line Length

Is the distance of $i$-th position from the beginning of the track.

$$L{L}_i=\Vert \overrightarrow{X_i}-\overrightarrow{X_0}\Vert$$

Line Speed

$$L{S}_i=\frac{\Vert \overrightarrow{X_i}-\overrightarrow{X_0}\Vert }{t_i-t_0}$$

Straightness

Line length over path length. Ranges from 0 to 1.

$$ST{R}_i=\frac{L{L}_i}{P{L}_i}$$

Track features

Are features measured on the whole track by the Track Features node.

Segments

Number of segments. One less teh number of objects in the track.

$$N$$

Length

Is the length of the track (sum pf all segment lengths).

$$L=\sum _{i=0}^N{d}_i$$

Duration

How long the track lasts (in seconds).

$$T=t_N-t_0=\sum _{i=0}^N\Delta t_i$$

Speed

Average track speed.

$$\bar{S}=\frac{L}{T}$$

Heading

Average track heading.

$$\bar{H}=\mathrm{mod}\left(\frac{180}{\pi }atan2(P_{N,y}-P_{0,y},P_{N,x}-P_{0,x}),360\right)$$

Elevation

Average track elevation.

$$E_i=\frac{180}{\pi }atan2\left(P_{N,z}-P_{0,z},\sqrt{(P_{N,x}-P_{0,x}{)}^2+(P_{N,y}-P_{0,y}{)}^2}\right)$$

Acceleration

Total acceleration.

$$\bar{A}=\Vert \sum _{i=0}^N\overrightarrow{a_i}\Vert$$

Line Length

Is the distance of $i$-th position from the beginning of the track.

$$LL=\Vert \overrightarrow{P_N-P_0}\Vert$$

Line Speed

$$LS=\frac{LL}{T}$$

Straightness

Line length over track length. Ranges from 0 to 1.

$$STR=\frac{LL}{L}$$