  We address the problem of crowd localization, i.e., the prediction of dots
corresponding to people in a crowded scene. Due to various challenges, a
localization method is prone to spatial semantic errors, i.e., predicting
multiple dots within a same person or collapsing multiple dots in a cluttered
region. We propose a topological approach targeting these semantic errors. We
introduce a topological constraint that teaches the model to reason about the
spatial arrangement of dots. To enforce this constraint, we define a
persistence loss based on the theory of persistent homology. The loss compares
the topographic landscape of the likelihood map and the topology of the ground
truth. Topological reasoning improves the quality of the localization algorithm
especially near cluttered regions. On multiple public benchmarks, our method
outperforms previous localization methods. Additionally, we demonstrate the
potential of our method in improving the performance in the crowd counting
task.
