Globally optimal vertical direction estimation in Atlanta World

In man-made environments, such as indoor and urban scenes, most of the objects and structures are organized in the form of orthogonal and parallel planes. These planes can be approximated by the Atlanta world assumption, in which the normals of planes can be represented by the Atlanta frames. Atlanta world assumption, which can be considered as a generalized Manhattan world assumption, has one vertical frame and multiple horizontal frames. Conventionally, given a set of inputs such as surface normals, the Atlanta frame estimation problem can be solved in one-time by branch-and-bound (BnB). However, the runtime of the BnB algorithm will increase greatly when the dimensionality (i.e., the number of horizontal frames) increases. In this paper, we estimate only the vertical direction instead of all Atlanta frames at once. Accordingly, we propose a vertical direction estimation method by considering the relationship between the vertical frame and horizontal frames. Concretely, our approach employs a BnB algorithm to search the vertical direction guaranteeing global optimality without requiring prior knowledge of the number of Atlanta frames. Four novel bounds by mapping 3D-hemisphere to a 2D region are investigated to guarantee convergence. We verify the validity of the proposed method in various challenging synthetic and real-world data.