API¶

Gaussian Component¶

class pymrt.tracking.utils.GaussianComponent(n, weight, mean, cov)¶

Gaussian Mixture Component

A Gaussian mixture component is composed of weight ($w$), mean vector ($\mu$), and covariance matrix ($\Sigma$). A multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$ evaluated at position $x$ is given by

\[f(x) = \frac{1}{\sqrt{(2\pi) ^ k \left| \Sigma \right|}} e^{-\frac{1}{2} (x - \mu) ^ T \Sigma ^ {-1} (x - \mu)} \]

A multivariate Gaussian Mixture Component, denoted $\mathcal{N}(w, \mu, \Sigma)$ is a equivalent to a multivariate Gaussian distribution with a weight factor, i.e. $w \mathcal{N}(\mu, \Sigma)$. As a result, a GM evaluated at position $x$ is given by

\[f_{GM}(x) = w f(x) \]

where $f(x)$ is the density of a multivariate Gaussian distribution evaluated at location $x$.

n¶: int – Dimensionality of the space the Gaussian component is evaluated on.

weight¶: float – Weight of this Gaussian Component.

mean¶: numpy.ndarray – Mean vector (column) $\mu$ of shape (n, 1).

cov¶: numpy.ndarray – Covariance matrix of shape (n, n).

dmvnorm(x)¶

Density of multivariate normal distribution evaluated at location x

dmv_part1 calculates

\[\frac{1}{\sqrt{(2\pi) ^ k \left| \Sigma \right|}} e^{-\frac{1}{2} (x - \mu) ^ T \Sigma ^ {-1} (x - \mu)} \]

dmvnormcomp(x)¶: Density of multivariate GM component evaluated at location x

kalman_update(F, Q)¶

Update a GM Component based on a linear prediction model.

Assume each target follows a linear Gaussian dynamic model:

\[f_{k|k=1} (x|\eta) = N(x; F_{k-1}\eta, Q_{k-1}) \]

Tracking Models¶

Constant Velocity Model Base Class¶

class pymrt.tracking.models.CVModel(n)¶

Constant Velocity Model

Assume that the target of interest is moving at a constant velocity in a n-dimensional space. The location in the n-dimensional space is observable. Thus the dimensionality of the measurement space is $n$ and the dimensionality of the state space is $2n$.

Parameters:	n (`int`) – Dimensionality of space the target is moving in.

n¶: int – Dimensionality of the space the target is moving in. It is the same as observation space in constant velocity model.

x_dim¶: int – Dimensionality of state space.

_t¶: int – Time intervals for each sample.

_F¶: numpy.ndarray – Dynamic model linear motion multiplier.

_G¶: numpy.ndarray – Dynamic model linea error multiplier.

w_generator¶: RandomGenerator – Random generator for state update error estimation.

r_generator¶: RandomGenerator – Random generator for measurement error estimation.

generate_new_state(x_prime, noise=True, noise_array=None)¶

Generate new state based on previous state vector $x’$.

Parameters:

x_prime (numpy.ndarray) – State vector of size (self.x_dim, ).
noise (bool) – Whether the generated state includes disturbance.
noise_array (numpy.ndarray) – First order disturbance to the target state. It needs to be a column vector of size $n$. If it is set to None, the disturbance is generated by w_generator.

Returns:

Next state vector of size: (self.x_dim, ).

Return type:

x (numpy.ndarray)

generate_observation(x_prime, noise=True, noise_array=None)¶

Generate Observation based on current state vector.

Parameters:

x_prime (numpy.ndarray) – State vector of size (self.x_dim, ).
noise (bool) – Whether the generated state includes disturbance.
noise_array (numpy.ndarray) – First order disturbance to the target state. It needs to be a column vector of size $n$. If it is set to None, the disturbance is generated by d_generator.

Returns:

Measurement vector of size: (self.z_dim, ).

Return type:

z (numpy.ndarray)

Constant Velocity Model Base Class with Track ID¶

class pymrt.tracking.models.CVIDModel(n)¶

Constant Velocity Model with Track ID

Assume that the target of interest is moving at a constant velocity in a n-dimensional space. The location in the n-dimensional space is observable. However, in order to associate observation with a specific target, a track_id field needs to be added to the state vector. Thus the dimensionality of the measurement space is $n$ and the dimensionality of the state space is $2n+1$

Parameters:	n (`int`) – Dimensionality of space the target is moving in.

n¶: int – Dimensionality of the space the target is moving in. It is the same as observation space in constant velocity model.

x_dim¶: int – Dimensionality of state space.

_t¶: int – Time intervals for each sample.

_F¶: numpy.ndarray – Dynamic model linear motion multiplier.

_G¶: numpy.ndarray – Dynamic model linea error multiplier.

generate_new_state(x_prime, noise=True, noise_array=None)¶

Generate new state based on previous state vector $x’$.

Parameters:

x_prime (numpy.ndarray) – State vector of size (self.x_dim, ).
noise (bool) – Whether the generated state includes disturbance.
noise_array (numpy.ndarray) – First order disturbance to the target state. It needs to be a column vector of size $n$. If it is set to None, the disturbance is generated by w_generator.

Returns:

Next state vector of size: (self.x_dim, ).

Return type:

x (numpy.ndarray)

generate_observation(x_prime, noise=True, noise_array=None)¶

Generate Observation based on current state vector.

Parameters:

x_prime (numpy.ndarray) – State vector of size (self.x_dim, ).
noise (bool) – Whether the generated state includes disturbance.
noise_array (numpy.ndarray) – First order disturbance to the target state. It needs to be a column vector of size $n$. If it is set to None, the disturbance is generated by r_generator.

Returns:

Measurement vector of size: (self.z_dim, ).

Return type:

z (numpy.ndarray)

Base Model for GM-PHD¶

class pymrt.tracking.models.GmPhdModel¶

Defines the parameters needed for the model to be used with Gaussian-Mixture Probability Hypothesis Density Multi-target Tracking Filter.

birth¶: list – List of GMs defining the birth probability across the state space.

ps¶: float – Persistence probability (between 0 to 1)

pd¶: float – Detection probability (between 0 to 1)

lambda_c¶: float – Poisson distribution parameter for clutter process (Assume it is a Poission Point Process).

gm_T¶: float – GM truncate threshold.

gm_U¶: float – GM merge threshold.

gm_Jmax¶: int – Max number of GM to track in calculation.

Constant Velocity Model for GM-PHD¶

class pymrt.tracking.models.GmPhdCvModel(n)¶: Constant Velocity Model for Gaussian-Mixture Probability Hypothesis Density Filter