  We show that crowd counting can be viewed as a decomposable point querying
process. This formulation enables arbitrary points as input and jointly reasons
whether the points are crowd and where they locate. The querying processing,
however, raises an underlying problem on the number of necessary querying
points. Too few imply underestimation; too many increase computational
overhead. To address this dilemma, we introduce a decomposable structure, i.e.,
the point-query quadtree, and propose a new counting model, termed Point quEry
Transformer (PET). PET implements decomposable point querying via
data-dependent quadtree splitting, where each querying point could split into
four new points when necessary, thus enabling dynamic processing of sparse and
dense regions. Such a querying process yields an intuitive, universal modeling
of crowd as both the input and output are interpretable and steerable. We
demonstrate the applications of PET on a number of crowd-related tasks,
including fully-supervised crowd counting and localization, partial annotation
learning, and point annotation refinement, and also report state-of-the-art
performance. For the first time, we show that a single counting model can
address multiple crowd-related tasks across different learning paradigms. Code
is available at https://github.com/cxliu0/PET.
