Globally Optimal Vertical Direction Estimation in Atlanta World

In man-made environments, most of the objects and structures are organized in the form of orthogonal and parallel planes. These planes can be approximated by an Atlanta world assumption, in which the normals of planes can be represented by Atlanta frames. The Atlanta 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 by a branch-and-bound (BnB) algorithm. 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, thereby guaranteeing global optimality without requiring prior knowledge of the number of Atlanta frames. In order to guarantee convergence, four novel bounds are investigated, by mapping a 3D hemisphere to a 2D region. We verify the feasibility of the proposed method using various challenging synthetic and real-world data.