  Reliable spatial uncertainty evaluation of object detection models is of
special interest and has been subject of recent work. In this work, we review
the existing definitions for uncertainty calibration of probabilistic
regression tasks. We inspect the calibration properties of common detection
networks and extend state-of-the-art recalibration methods. Our methods use a
Gaussian process (GP) recalibration scheme that yields parametric distributions
as output (e.g. Gaussian or Cauchy). The usage of GP recalibration allows for a
local (conditional) uncertainty calibration by capturing dependencies between
neighboring samples. The use of parametric distributions such as as Gaussian
allows for a simplified adaption of calibration in subsequent processes, e.g.,
for Kalman filtering in the scope of object tracking.
  In addition, we use the GP recalibration scheme to perform covariance
estimation which allows for post-hoc introduction of local correlations between
the output quantities, e.g., position, width, or height in object detection. To
measure the joint calibration of multivariate and possibly correlated data, we
introduce the quantile calibration error which is based on the Mahalanobis
distance between the predicted distribution and the ground truth to determine
whether the ground truth is within a predicted quantile.
  Our experiments show that common detection models overestimate the spatial
uncertainty in comparison to the observed error. We show that the simple
Isotonic Regression recalibration method is sufficient to achieve a good
uncertainty quantification in terms of calibrated quantiles. In contrast, if
normal distributions are required for subsequent processes, our GP-Normal
recalibration method yields the best results. Finally, we show that our
covariance estimation method is able to achieve best calibration results for
joint multivariate calibration.
