This letter considers the problem of detecting and tracking objects in a sequence of images. The problem is formulated in a filtering framework, using the output of object-detection algorithms as measurements. An extension to the filtering formulation is proposed that incorporates class information from the previous frame to robustify the classification. Further, the properties of the object-detection algorithm are exploited to quantify the uncertainty of the bounding box detection in each frame. The complete filtering method is evaluated on camera trap images of the four large Swedish carnivores, bear, lynx, wolf, and wolverine. The experiments show that the class tracking formulation leads to a more robust classification.

The uncertainty in the prediction calculated using the delta method for an over-parameterized (parametric) black-box model is shown to be larger or equal to the uncertainty in the prediction of a canonical (minimal) model. Equality holds if the additional parameters of the overparameterized model do not add flexibility to the model. As a conclusion, for an overparameterized black-box model, the calculated uncertainty in the prediction by the delta method is not underestimated. The results are shown analytically and are validated in a simulation experiment where the relationship between the normalized traction force and the wheel slip of a car is modelled using e.g., a neural network.

Neural Networks (NNS) can solve very hard classification and estimation tasks but are less well suited to solve complex sensor fusion challenges, such as end-to-end control of autonomous vehicles. Nevertheless, NN can still be a powerful tool for particular sub-problems in sensor fusion. This would require a reliable and quantifiable measure of the stochastic uncertainty in the predictions that can be compared to classical sensor measurements. However, current NN'S output some figure of merit, that is only a relative model fit and not a stochastic uncertainty. We propose to embed the NN'S in a proper stochastic system identification framework. In the training phase, the stochastic uncertainty of the parameters in the (last layers of the) NN is quantified. We show that this can be done recursively with very few extra computations. In the classification phase, Monte-Carlo (MC) samples are used to generate a set of classifier outputs. From this set, a distribution of the classifier output is obtained, which represents a proper description of the stochastic uncertainty of the predictions. We also show how to use the calculated uncertainty for outlier detection by including an artificial outlier class. In this way, the NN fits a sensor fusion framework much better. We evaluate the approach on images of handwritten digits. The proposed method is shown to be on par with MC dropout, while having lower computational complexity, and the outlier detection almost completely eliminates false classifications.

The prediction uncertainty of a neural network is considered from a classical system identification point of view. To know this uncertainty is extremely important when using a network in decision and feedback applications. The asymptotic covariance of the internal parameters in the network due to noise in the observed dependent variables (output) and model class mismatch, i.e., the true system cannot be exactly described by the model class, is first surveyed. This is then applied to the prediction step of the network to get a closed form expression for the asymptotic, in training data information, prediction variance. Another interpretation of this expression is as the non-asymptotic Cramér-Rao Lower Bound. To approximate this expression, only the gradients and residuals, already computed in the gradient descent algorithms commonly used to train neural networks, are needed. Using a toy example, it is illustrated how the uncertainty in the output of a neural network can be estimated.

Suppose data-driven black-box models, e.g., neural networks, should be used as components in safety-critical systems such as autonomous vehicles. In that case, knowing how uncertain they are in their predictions is crucial. However, this needs to be provided for standard formulations of neural networks. Hence, this thesis aims to develop a method that can, out-of-the-box, extend the standard formulations to include uncertainty in the prediction. The proposed method in the thesis is based on a local linear approximation, using a two-step linearization to quantify the uncertainty in the prediction from the neural network. First, the posterior distribution of the neural network parameters is approximated using a Gaussian distribution. The mean of the distribution is at the maximum a posteriori estimate of the parameters, and the covariance is estimated using the shape of the likelihood function in the vicinity of the estimated parameters. The second linearization is used to propagate the uncertainty in the parameters to uncertainty in the model’s output. Hence, to create a linear approximation of the nonlinear model that a neural network is. 

The first part of the thesis considers regression problems with examples of road-friction experiments using simulated and experimentally collected data. For the model-order selection problem, it is shown that the method does not under-estimate the uncertainty in the prediction of overparametrized models. 

The second part of the thesis considers classification problems. The concept of calibration of the uncertainty, i.e., how reliable the uncertainty is and how close it resembles the true uncertainty, is considered. The proposed method is shown to create calibrated estimates of the uncertainty, evaluated on classical image data sets. From a computational perspective, the thesis proposes a recursive update of the parameter covariance, enhancing the method’s viability. Furthermore, it shows how quantified uncertainty can improve the robustness of a decision process by formulating an information fusion scheme that includes both temporal correlational and correlation between classifiers. Moreover, having access to a measure of uncertainty in the prediction is essential when detecting outliers in the data, i.e., examples that the neural network has yet to see during the training. On this task, the proposed method shows promising results. Finally, the thesis proposes an extension that enables a multimodal representation of the uncertainty. 

The third part of the thesis considers the tracking of objects in image sequences, where the object is detected using standard neural network-based object detection algorithms. It formulates the problem as a filtering problem with the prediction of the class and the position of the object viewed as the measurements. The filtering formulation improves robustness towards false classifications when evaluating the method on examples from animal conservation in the Swedish forests.