Generalized Characteristic Function Loss for Crowd Analysis in the Frequency Domain

Typical approaches that learn crowd density maps are limited to extracting the supervisory information from the loosely organized spatial information in the crowd dot/density maps. This paper tackles this challenge by performing the supervision in the frequency domain. More specifically, we devise a new loss function for crowd analysis called generalized characteristic function loss (GCFL). This loss carries out two steps: 1) transforming the spatial information in density or dot maps to the frequency domain; 2) calculating a loss value between their frequency contents. For step 1, we establish a series of theoretical fundaments by extending the definition of the characteristic function for probability distributions to density maps, as well as proving some vital properties of the extended characteristic function. After taking the characteristic function of the density map, its information in the frequency domain is well-organized and hierarchically distributed, while in the spatial domain it is loose-organized and dispersed everywhere. In step 2, we design a loss function that can fit the information organization in the frequency domain, allowing the exploitation of the well-organized frequency information for the supervision of crowd analysis tasks. The loss function can be adapted to various crowd analysis tasks through the specification of its window functions. In this paper, we demonstrate its power in three tasks: Crowd Counting, Crowd Localization and Noisy Crowd Counting. We show the advantages of our GCFL compared to other SOTA losses and its competitiveness to other SOTA methods by theoretical analysis and empirical results on benchmark datasets. Our codes are available at https://github.com/wbshu/Crowd_Counting_in_the_Frequency_Domain.