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.

This paper investigates crowd counting in the frequency domain, which is a novel direction compared to the traditional view in the spatial domain. By transforming the density map into the frequency domain and using the properties of the characteristic function, we propose a novel method that is simple, effective, and efficient. The solid theoretical analysis ends up as an implementation-friendly loss function, which requires only standard tensor operations in the training process. We prove that our loss function is an upper bound of the pseudo sup norm metric between the ground truth and the prediction density map (over all of their sub-regions), and demonstrate its efficacy and efficiency versus other loss functions. The experimental results also show its competitiveness to the state-of-the-art on five benchmark data sets: ShanghaiTech A & B, UCF-QNRF, JHU++, and NWPU. Our codes will be available at: wb-shu/Crowd_Couniing_in_the_Frequency_Domain

In the research area of time series classification, the ensemble shapelet transform algorithm is one of state-of-the-art algorithms for classification. However, its high time complexity is an issue to hinder its application since its base classifier shapelet transform includes a high time complexity of a distance calculation and shapelet selection. Therefore, in this paper we introduce a novel algorithm, i.e. short isometric shapelet transform, which contains two strategies to reduce the time complexity. The first strategy of SIST fixes the length of shapelet based on a simplified distance calculation, which largely reduces the number of shapelet candidates as well as speeds up the distance calculation in the ensemble shapelet transform algorithm. The second strategy is to train a single linear classifier in the feature space instead of an ensemble classifier. The theoretical evidences of these two strategies are presented to guarantee a near-lossless accuracy under some preconditions while reducing the time complexity. Furthermore, empirical experiments demonstrate the superior performance of the proposed algorithm.