In medical imaging, image background is often defined by zero signal. Moreover, in medical images the background area - or conversely, the spatial support (the extent of the non-zero part of the image) - is often known a priori or can be easily estimated. For example, support information can be estimated from the low-resolution "scout" images typically acquired during pre-scan localization in both MRI and CT. In dynamic scans, object support in a single time-frame is often obtainable from a prior time frame, or from a composite image formed from data from multiple time frames. In this work, incorporation of cither complete or partial a priori knowledge of object spatial support into the compressive sensing (CS) framework is investigated. Following development of the augmented reconstruction model, examples of support-constrained CS reconstruction of phantom and MR images under both exact and inexact support definitions are given. For each experiment, the straightforward incorporation of the proposed spatial support constraint into the standard CS model was shown to both significantly accelerate reconstruction convergence and yield a lower terminal RMSE compared to a conventional CS reconstruction. The proposed augmented reconstruction model was also shown to be robust to inaccuracies in the estimated object support.