One in twenty-five patients admitted to a hospital will suffer from a hospital acquired infection. If we can intelligently track healthcare staff, patients, and visitors, we can better understand the sources of such infections. We envision a smart hospital capable of increasing operational efficiency and improving patient care with less spending. In this paper, we propose a non-intrusive vision-based system for tracking people's activity in hospitals. We evaluate our method for the problem of measuring hand hygiene compliance. Empirically, our method outperforms existing solutions such as proximity-based techniques and covert in-person observational studies. We present intuitive, qualitative results that analyze human movement patterns and conduct spatial analytics which convey our method's interpretability. This work is a first step towards a computer-vision based smart hospital and demonstrates promising results for reducing hospital acquired infections.