Low-cost robots such as floor cleaners generally rely on limited perception and simple algorithms, but some new models now have enough sensing capability and computation power to enable Simultaneous Localisation And Mapping (SLAM) and intelligent guided navigation. In particular, computer vision is now a serious option in low cost robotics, though its use to date has been limited to feature-based mapping for localisation. Dense environment perception such as free space finding has required additional specialised sensors, adding expense and complexity. Here we show that a robot with a single passive omnidirectional camera can perform rapid global free-space reasoning within typical rooms. Upon entering a new room, the robot makes a circular movement to capture a closely-spaced omni image sequence with disparity in all horizontal directions. feature-based visual SLAM procedure obtains accurate poses for these frames before passing them to a dense matching step, 3D semi-dense reconstruction and visibility reasoning. The result is turned into a 2D occupancy map, which can be improved and extended if necessary through further movement. This rapid, passive technique can capture high quality free space information which gives a robot a global understanding of the space around it. We present results in several scenes, including quantitative comparison with laser-based mapping.