BLUES: Before-reLU-EStimates Bayesian Inference for Crowd Counting

Ensuring the trustworthiness of artificial intelligence and machine learning systems is becoming a crucial requirement given their widespread applications, including crowd counting, which we focus on in this work. This is often addressed by integrating uncertainty measures into their predictions. Most Bayesian uncertainty quantification techniques use a Gaussian approximation of the output, whose variance is interpreted as the uncertainty measure. However, in the case of neural network models for crowd counting based on density estimation, where the ReLU activation function is used for the output units, such a prior may lead to an approximated distribution with a significant mass on negative values, although they cannot be produced by the ReLU activation. Interestingly, we found that this is related to “false positive” pedestrian localisation errors in the density map. We propose to address this issue by shifting the Bayesian Inference Before the reLU EStimates (BLUES). This modification allows us to estimate a probability distribution both on the people density and the people presence in each pixel. This allows us to compute a crowd segmentation map, which we exploit for filtering out false positive localisations. Results on several benchmark data sets provide evidence that our BLUES approach allows for improving the accuracy of the estimated density map and the quality of the corresponding uncertainty measure.