only image-level category information. In contrast, we are more interested in the incomplete supervised setting, where only a few exemplar bounding boxes are annotated and all others are unlabeled. This setting is practically more useful when the number of instances is large, for example to count the number of products on a retail store shelf, or the number of cars in a large parking lot.

Approach
========

Problem Definition
------------------

Although detection-based object counting resembles the general object detection problem, it still presents unique challenges when the total number of object instances in an image is large, e.g. from tens to hundreds, and the size of each instance is relatively small comparing with the image size. To save the tedious and costly labeling effort, we assume each image $I_n$ in the training set $\{I_n|n=1,\ldots,N\}$ is only labeled with a few instances (e.g. less than ten) though there might be tens or hundreds of instances in one image. For image $I_n$, denote by $\mathbf{x}_i^{n}$ its $i$-th annotated bounding box ($i = 1, \ldots, M_1(n)$). Since not all instances are annotated, the rest region could also contain the object instances. Without loss of generality, we assume the number of object categories is 