Multidimensional Measure Matching for Crowd Counting

This article addresses the challenge of scale variations in crowd-counting problems from a multidimensional measure-theoretic perspective. We start by formulating crowd counting as a measure-matching problem, based on the assumption that discrete measures can express the scattered ground truth and the predicted density map. In this context, we introduce the Sinkhorn counting loss and extend it to the semi-balanced form, which alleviates the problems including entropic bias, distance destruction, and amount constraints. We then model the measure matching under the multidimensional space, in order to learn the counting from both location and scale. To achieve this, we extend the traditional 2-D coordinate support to 3-D, incorporating an additional axis to represent scale information, where a pyramid-based structure will be leveraged to learn the scale value for the predicted density. Extensive experiments on four challenging crowd-counting datasets, namely, ShanghaiTech A, UCF-QNRF, JHU++, and NWPU have validated the proposed method.