Gaussian random field-based log odds occupancy mapping Hongjun Li ∗ , Miguel Bar˜ ao ∗, † and Lu´ ıs Rato ∗ ∗ Department of Informatics, University of Evora, Evora, Portugal † Control of Dynamical Systems Group, INESC-ID, Lisboa, Portugal li.hongjun@foxmail.com, mjsb@uevora.pt, lmr@uevora.pt Abstract—This paper focuses on mapping problem with known robot pose in static environments and proposes a Gaussian random field-based log odds occupancy mapping (GRF-LOOM). In this method, occupancy probability is regarded as an unknown parameter and the dependence between parameters are consid- ered. Given measurements and the dependence, the parameters of not only observed space but also unobserved space can be predicted. The occupancy probabilities in log odds form are regarded as a GRF. This mapping task can be solved by the well- known prediction equation in Gaussian processes, which involves an inverse problem. Instead of the prediction equation, a new recursive algorithm is also proposed to avoid the inverse problem. Finally, the proposed method is evaluated in simulations. Index Terms—Binary Bayes filter, Gaussian random field, Log odds occupancy mapping. I. I NTRODUCTION When robots explore unknown environments, mapping is the fundamental problem. Without an accurate map, they cannot do further work, such as navigation. Occupancy grid maps [1] have been widely applied to represent environments by many researchers. In occupancy grid maps, environments are divided into many grid cells and it is convenient to do path planning. Normally, the grid cells are assumed to be independent of each other, which leads to inconsistent mapping. Gaussian random fields have been applied to consider the dependence between grid cells. Gaussian process occupancy map (GPOM) [2] is a continuous occupancy representation of the environment, which overcomes some of the limitations with occupancy grids. It considers the occupancy mapping as a binary classification problem and can predict the classification of unobserved space based on observed space. With increasing the number of training data, Gaussian processes will take more time to deal with the problem. For large-scale environments, training data is divided into small subsets and a mixture of Gaussian processes is presented in [3]. Similarly, local Gaussian processes are applied to the subsets and overlap- ping clusters is proposed to ensure continuity [4]. Reference [5] proposes a recursive method to update occupancy maps and surface meshes using Gaussian processes and Bayesian Committee Machines. A multi-support kernel, which enables This work was supported by EACEA under the Erasmus Mundus Action 2, Strand 1 project LEADER - Links in Europe and Asia for engineering, eDucation, Enterprise and Research exchanges. traditional covariance functions to accept two-dimensional regions, is introduced to reduce the size of covariance matrices and accelerate Gaussian process inference and learning [6]. Reference [7] proposes a nested Bayesian committee machine to online learning 3D occupancy maps using Gaussian pro- cesses. In GPOMs, observations have two possible values: occupied and free. These values are in discrete space. In this paper, the values are extended into continuous space based on log odds form [8]. By log odds form, probability with range (0, 1) is transformed into into (−∞, +∞). In this paper, a GRF- LOOM is proposed taking the advantages of the log odds form. In Section II, occupancy grid mapping, binary Bayes filter, and log odds form are introduced. The new method is proposed in Section III. The map is regarded as a GRF where the random variables are occupancy probabilities in log odds form. The training data for the GRF is obtained from the result of binary Bayes filter. A GRF model is built based on Bayes rule. Based on Sherman-Morrison equation, a novel algorithm is also proposed to solve the mapping problem, which can avoid the inverse problem. Simulations are done in IV. II. BACKGROUND In occupancy grid mapping, maps are divided into many grid cells as Figure 1. Each grid cell has two possible states: free and occupied. The binary occupancy value of each grid cell specifies whether or not a location is occupied with an object. The darkness of each grid cell corresponds to the likelihood of occupancy. If one grid cell is free surely, the darkness is 0. If it is occupied surely, the darkness is 1. Fig. 1: A grid map In classical occupancy grid mapping, the states between different grid cells are assumed to be independent of each