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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. [ABSTRACT FROM AUTHOR]

We propose a new linear RGB-D simultaneous localization and mapping (SLAM) formulation by utilizing planar features of the structured environments. The key idea is to understand a given structured scene and exploit its structural regularities such as the Manhattan world. This understanding allows us to decouple the camera rotation by tracking structural regularities, which makes SLAM problems free from being highly nonlinear. Additionally, it provides a simple yet effective cue for representing planar features, which leads to a linear SLAM formulation. Given an accurate camera rotation, we jointly estimate the camera translation and planar landmarks in the global planar map using a linear Kalman filter. Our linear SLAM method, called L-SLAM, can understand not only the Manhattan world but the more general scenario of the Atlanta world, which consists of a vertical direction and a set of horizontal directions orthogonal to the vertical direction. To this end, we introduce a novel tracking-by-detection scheme that infers the underlying scene structure by Atlanta representation. With efficient Atlanta representation, we formulate a unified linear SLAM framework for structured environments. We evaluate L-SLAM on a synthetic dataset and RGB-D benchmarks, demonstrating comparable performance to other state-of-the-art SLAM methods without using expensive nonlinear optimization. We assess the accuracy of L-SLAM on a practical application of augmented reality. [ABSTRACT FROM AUTHOR]