In this paper, we investigate the following problem: given the image of a scene, what is the trajectory that a robot-mounted camera should follow to allow optimal dense depth estimation? The solution we propose is based on maximizing the information gain over a set of candidate trajectories. In order to estimate the information that we expect from a camera pose, we introduce a novel formulation of the measurement uncertainty that accounts for the scene appearance (i.e., texture in the reference view), the scene depth and the vehicle pose. We successfully demonstrate our approach in the case of real-time, monocular reconstruction from a micro aerial vehicle and validate the effectiveness of our solution in both synthetic and real experiments. To the best of our knowledge, this is the first work on active, monocular dense reconstruction, which chooses motion trajectories that minimize perceptual ambiguities inferred by the texture in the scene.