Crowd-sensing enables to collect a vast amount of data from the crowd by allowing a wide variety of sources to contribute data. However, the openness of crowd-sensing exposes the system to malicious and erroneous participations, inevitably resulting in poor data quality. This brings forth an important issue of false data detection and correction in crowd-sensing. Furthermore, data collected by participants normally include considerable missing values, which poses challenges for accurate false data detection. In this work, we propose DECO, a general framework to detect false values for crowd-sensing in the presence of missing data. By applying a tailored spatio-temporal compressive sensing technique, DECO is able to accurately detect the false data and estimate both false and missing values for data correction. Through comprehensive performance evaluations, we demonstrate the efficacy of DECO in achieving false data detection and correction for crowd-sensing applications with incomplete sensory data.