Reference for ultralytics/trackers/utils/kalman_filter.py

class KalmanFilterXYAH:
    """
    For bytetrack. A simple Kalman filter for tracking bounding boxes in image space.

    The 8-dimensional state space (x, y, a, h, vx, vy, va, vh) contains the bounding box center position (x, y),
    aspect ratio a, height h, and their respective velocities.

    Object motion follows a constant velocity model. The bounding box location (x, y, a, h) is taken as direct
    observation of the state space (linear observation model).
    """

    def __init__(self):
        """Initialize Kalman filter model matrices with motion and observation uncertainty weights."""
        ndim, dt = 4, 1.

        # Create Kalman filter model matrices.
        self._motion_mat = np.eye(2 * ndim, 2 * ndim)
        for i in range(ndim):
            self._motion_mat[i, ndim + i] = dt
        self._update_mat = np.eye(ndim, 2 * ndim)

        # Motion and observation uncertainty are chosen relative to the current state estimate. These weights control
        # the amount of uncertainty in the model. This is a bit hacky.
        self._std_weight_position = 1. / 20
        self._std_weight_velocity = 1. / 160

    def initiate(self, measurement):
        """
        Create track from unassociated measurement.

        Parameters
        ----------
        measurement : ndarray
            Bounding box coordinates (x, y, a, h) with center position (x, y),
            aspect ratio a, and height h.

        Returns
        -------
        (ndarray, ndarray)
            Returns the mean vector (8 dimensional) and covariance matrix (8x8
            dimensional) of the new track. Unobserved velocities are initialized
            to 0 mean.
        """
        mean_pos = measurement
        mean_vel = np.zeros_like(mean_pos)
        mean = np.r_[mean_pos, mean_vel]

        std = [
            2 * self._std_weight_position * measurement[3], 2 * self._std_weight_position * measurement[3], 1e-2,
            2 * self._std_weight_position * measurement[3], 10 * self._std_weight_velocity * measurement[3],
            10 * self._std_weight_velocity * measurement[3], 1e-5, 10 * self._std_weight_velocity * measurement[3]]
        covariance = np.diag(np.square(std))
        return mean, covariance

    def predict(self, mean, covariance):
        """
        Run Kalman filter prediction step.

        Parameters
        ----------
        mean : ndarray
            The 8 dimensional mean vector of the object state at the previous time step.
        covariance : ndarray
            The 8x8 dimensional covariance matrix of the object state at the previous time step.

        Returns
        -------
        (ndarray, ndarray)
            Returns the mean vector and covariance matrix of the predicted state. Unobserved velocities are
            initialized to 0 mean.
        """
        std_pos = [
            self._std_weight_position * mean[3], self._std_weight_position * mean[3], 1e-2,
            self._std_weight_position * mean[3]]
        std_vel = [
            self._std_weight_velocity * mean[3], self._std_weight_velocity * mean[3], 1e-5,
            self._std_weight_velocity * mean[3]]
        motion_cov = np.diag(np.square(np.r_[std_pos, std_vel]))

        # mean = np.dot(self._motion_mat, mean)
        mean = np.dot(mean, self._motion_mat.T)
        covariance = np.linalg.multi_dot((self._motion_mat, covariance, self._motion_mat.T)) + motion_cov

        return mean, covariance

    def project(self, mean, covariance):
        """
        Project state distribution to measurement space.

        Parameters
        ----------
        mean : ndarray
            The state's mean vector (8 dimensional array).
        covariance : ndarray
            The state's covariance matrix (8x8 dimensional).

        Returns
        -------
        (ndarray, ndarray)
            Returns the projected mean and covariance matrix of the given state estimate.
        """
        std = [
            self._std_weight_position * mean[3], self._std_weight_position * mean[3], 1e-1,
            self._std_weight_position * mean[3]]
        innovation_cov = np.diag(np.square(std))

        mean = np.dot(self._update_mat, mean)
        covariance = np.linalg.multi_dot((self._update_mat, covariance, self._update_mat.T))
        return mean, covariance + innovation_cov

    def multi_predict(self, mean, covariance):
        """
        Run Kalman filter prediction step (Vectorized version).

        Parameters
        ----------
        mean : ndarray
            The Nx8 dimensional mean matrix of the object states at the previous time step.
        covariance : ndarray
            The Nx8x8 dimensional covariance matrix of the object states at the previous time step.

        Returns
        -------
        (ndarray, ndarray)
            Returns the mean vector and covariance matrix of the predicted state. Unobserved velocities are
            initialized to 0 mean.
        """
        std_pos = [
            self._std_weight_position * mean[:, 3], self._std_weight_position * mean[:, 3],
            1e-2 * np.ones_like(mean[:, 3]), self._std_weight_position * mean[:, 3]]
        std_vel = [
            self._std_weight_velocity * mean[:, 3], self._std_weight_velocity * mean[:, 3],
            1e-5 * np.ones_like(mean[:, 3]), self._std_weight_velocity * mean[:, 3]]
        sqr = np.square(np.r_[std_pos, std_vel]).T

        motion_cov = [np.diag(sqr[i]) for i in range(len(mean))]
        motion_cov = np.asarray(motion_cov)

        mean = np.dot(mean, self._motion_mat.T)
        left = np.dot(self._motion_mat, covariance).transpose((1, 0, 2))
        covariance = np.dot(left, self._motion_mat.T) + motion_cov

        return mean, covariance

    def update(self, mean, covariance, measurement):
        """
        Run Kalman filter correction step.

        Parameters
        ----------
        mean : ndarray
            The predicted state's mean vector (8 dimensional).
        covariance : ndarray
            The state's covariance matrix (8x8 dimensional).
        measurement : ndarray
            The 4 dimensional measurement vector (x, y, a, h), where (x, y) is the center position, a the aspect
            ratio, and h the height of the bounding box.

        Returns
        -------
        (ndarray, ndarray)
            Returns the measurement-corrected state distribution.
        """
        projected_mean, projected_cov = self.project(mean, covariance)

        chol_factor, lower = scipy.linalg.cho_factor(projected_cov, lower=True, check_finite=False)
        kalman_gain = scipy.linalg.cho_solve((chol_factor, lower),
                                             np.dot(covariance, self._update_mat.T).T,
                                             check_finite=False).T
        innovation = measurement - projected_mean

        new_mean = mean + np.dot(innovation, kalman_gain.T)
        new_covariance = covariance - np.linalg.multi_dot((kalman_gain, projected_cov, kalman_gain.T))
        return new_mean, new_covariance

    def gating_distance(self, mean, covariance, measurements, only_position=False, metric='maha'):
        """
        Compute gating distance between state distribution and measurements. A suitable distance threshold can be
        obtained from `chi2inv95`. If `only_position` is False, the chi-square distribution has 4 degrees of
        freedom, otherwise 2.

        Parameters
        ----------
        mean : ndarray
            Mean vector over the state distribution (8 dimensional).
        covariance : ndarray
            Covariance of the state distribution (8x8 dimensional).
        measurements : ndarray
            An Nx4 dimensional matrix of N measurements, each in format (x, y, a, h) where (x, y) is the bounding box
            center position, a the aspect ratio, and h the height.
        only_position : Optional[bool]
            If True, distance computation is done with respect to the bounding box center position only.

        Returns
        -------
        ndarray
            Returns an array of length N, where the i-th element contains the squared Mahalanobis distance between
            (mean, covariance) and `measurements[i]`.
        """
        mean, covariance = self.project(mean, covariance)
        if only_position:
            mean, covariance = mean[:2], covariance[:2, :2]
            measurements = measurements[:, :2]

        d = measurements - mean
        if metric == 'gaussian':
            return np.sum(d * d, axis=1)
        elif metric == 'maha':
            cholesky_factor = np.linalg.cholesky(covariance)
            z = scipy.linalg.solve_triangular(cholesky_factor, d.T, lower=True, check_finite=False, overwrite_b=True)
            return np.sum(z * z, axis=0)  # square maha
        else:
            raise ValueError('invalid distance metric')

class KalmanFilterXYWH(KalmanFilterXYAH):
    """
    For BoT-SORT. A simple Kalman filter for tracking bounding boxes in image space.

    The 8-dimensional state space (x, y, w, h, vx, vy, vw, vh) contains the bounding box center position (x, y),
    width w, height h, and their respective velocities.

    Object motion follows a constant velocity model. The bounding box location (x, y, w, h) is taken as direct
    observation of the state space (linear observation model).
    """

    def initiate(self, measurement):
        """
        Create track from unassociated measurement.

        Parameters
        ----------
        measurement : ndarray
            Bounding box coordinates (x, y, w, h) with center position (x, y), width w, and height h.

        Returns
        -------
        (ndarray, ndarray)
            Returns the mean vector (8 dimensional) and covariance matrix (8x8 dimensional) of the new track.
            Unobserved velocities are initialized to 0 mean.
        """
        mean_pos = measurement
        mean_vel = np.zeros_like(mean_pos)
        mean = np.r_[mean_pos, mean_vel]

        std = [
            2 * self._std_weight_position * measurement[2], 2 * self._std_weight_position * measurement[3],
            2 * self._std_weight_position * measurement[2], 2 * self._std_weight_position * measurement[3],
            10 * self._std_weight_velocity * measurement[2], 10 * self._std_weight_velocity * measurement[3],
            10 * self._std_weight_velocity * measurement[2], 10 * self._std_weight_velocity * measurement[3]]
        covariance = np.diag(np.square(std))
        return mean, covariance

    def predict(self, mean, covariance):
        """
        Run Kalman filter prediction step.

        Parameters
        ----------
        mean : ndarray
            The 8 dimensional mean vector of the object state at the previous time step.
        covariance : ndarray
            The 8x8 dimensional covariance matrix of the object state at the previous time step.

        Returns
        -------
        (ndarray, ndarray)
            Returns the mean vector and covariance matrix of the predicted state. Unobserved velocities are
            initialized to 0 mean.
        """
        std_pos = [
            self._std_weight_position * mean[2], self._std_weight_position * mean[3],
            self._std_weight_position * mean[2], self._std_weight_position * mean[3]]
        std_vel = [
            self._std_weight_velocity * mean[2], self._std_weight_velocity * mean[3],
            self._std_weight_velocity * mean[2], self._std_weight_velocity * mean[3]]
        motion_cov = np.diag(np.square(np.r_[std_pos, std_vel]))

        mean = np.dot(mean, self._motion_mat.T)
        covariance = np.linalg.multi_dot((self._motion_mat, covariance, self._motion_mat.T)) + motion_cov

        return mean, covariance

    def project(self, mean, covariance):
        """
        Project state distribution to measurement space.

        Parameters
        ----------
        mean : ndarray
            The state's mean vector (8 dimensional array).
        covariance : ndarray
            The state's covariance matrix (8x8 dimensional).

        Returns
        -------
        (ndarray, ndarray)
            Returns the projected mean and covariance matrix of the given state estimate.
        """
        std = [
            self._std_weight_position * mean[2], self._std_weight_position * mean[3],
            self._std_weight_position * mean[2], self._std_weight_position * mean[3]]
        innovation_cov = np.diag(np.square(std))

        mean = np.dot(self._update_mat, mean)
        covariance = np.linalg.multi_dot((self._update_mat, covariance, self._update_mat.T))
        return mean, covariance + innovation_cov

    def multi_predict(self, mean, covariance):
        """
        Run Kalman filter prediction step (Vectorized version).

        Parameters
        ----------
        mean : ndarray
            The Nx8 dimensional mean matrix of the object states at the previous time step.
        covariance : ndarray
            The Nx8x8 dimensional covariance matrix of the object states at the previous time step.

        Returns
        -------
        (ndarray, ndarray)
            Returns the mean vector and covariance matrix of the predicted state. Unobserved velocities are
            initialized to 0 mean.
        """
        std_pos = [
            self._std_weight_position * mean[:, 2], self._std_weight_position * mean[:, 3],
            self._std_weight_position * mean[:, 2], self._std_weight_position * mean[:, 3]]
        std_vel = [
            self._std_weight_velocity * mean[:, 2], self._std_weight_velocity * mean[:, 3],
            self._std_weight_velocity * mean[:, 2], self._std_weight_velocity * mean[:, 3]]
        sqr = np.square(np.r_[std_pos, std_vel]).T

        motion_cov = [np.diag(sqr[i]) for i in range(len(mean))]
        motion_cov = np.asarray(motion_cov)

        mean = np.dot(mean, self._motion_mat.T)
        left = np.dot(self._motion_mat, covariance).transpose((1, 0, 2))
        covariance = np.dot(left, self._motion_mat.T) + motion_cov

        return mean, covariance

    def update(self, mean, covariance, measurement):
        """
        Run Kalman filter correction step.

        Parameters
        ----------
        mean : ndarray
            The predicted state's mean vector (8 dimensional).
        covariance : ndarray
            The state's covariance matrix (8x8 dimensional).
        measurement : ndarray
            The 4 dimensional measurement vector (x, y, w, h), where (x, y) is the center position, w the width,
            and h the height of the bounding box.

        Returns
        -------
        (ndarray, ndarray)
            Returns the measurement-corrected state distribution.
        """
        return super().update(mean, covariance, measurement)