text: import numpy as np
  import scipy.linalg
  
  
  class KalmanFilter:
      """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):
          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)
          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)) + innovation_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):
          """Compute gating distance between state distribution and measurements.
          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.
          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)
          cholesky_factor = np.linalg.cholesky(covariance)
          d = measurements - mean
          z = scipy.linalg.solve_triangular(cholesky_factor, d.T, lower=True, check_finite=False, overwrite_b=True)
          squared_maha = np.sum(z * z, axis=0)
          return squared_maha