Abstract: Missing data due to occlusion is a key challenge in 3D acquisition, particularly in cluttered man-made scenes. Such partial information about the scenes limits our ability to analyze and understand them. In this work we abstract such environments as collections of cuboids and hallucinate geometry in the occluded regions by globally analyzing the physical stability of the resultant arrangements of the cuboids. Our algorithm extrapolates the cuboids into the un-seen regions to infer both their corresponding geometric attributes (e.g., size, orientation) and how the cuboids topologically interact with each other (e.g., touch or fixed). The resultant arrangement provides an abstraction for the underlying structure of the scene that can then be used for a range of common geometry processing tasks. We evaluate our algorithm on a large number of test scenes with varying complexity, validate the results on existing benchmark datasets, and demonstrate the use of the recovered cuboid-based structures towards object retrieval, scene completion, etc.