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...bilistic, continuous ground surface estimation, for sparse 3D data sets that are cluttered by non-ground objects. The use of Gaussian Process (GP) methods to model sparse terrain data was explored in =-=[18]-=-. These methods have three properties that are useful for segmentation: (1) they operate on sparse data, (2) they are probabilistic, so decision boundaries for segmentation can be specified rigorously...

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...e of the random variable represents the temperature or humidity at that point, the distribution of which is specified by some mean and covariance. Equations 2 and 3 specify the distribution of x∗ conditioned on the training data. Let f∗ and V [f∗] denote the mean and variance of the distribution, respectively. Application of the model is done by evaluating these equations at every point of interest. f∗ = k T ∗K −1 z z (2) V [f∗] = k(x∗, x∗)− kT∗K−1z k∗ (3) Here, we take k∗ to be the vector of covariances between x∗ and the training inputs and k(x∗, x∗) to be the variance of x∗. Work done in [12], [13], [14], [15] proposes the use of a KD-tree structure for a fast local approximation. For a training set of n observations, the inference process requires calculating and inverting an n× n covariance matrix, which gets computationally expensive as n gets large. For inference about x∗, the KD-tree approximation method selects the k spatially closest points to x∗, where k is some predefined number. For our work, we found this approach useful when dealing with the large, simulated data sets. It allowed us to evaluate our training methods on dense sets of simulation data in reasonable amounts...

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...ntal estimation. Stachniss et al. [14] use GP mixture models to estimate spatiotemporal gas concentration, and validate their approach with a mobile sensor in different environments. Vasudevan et al. =-=[15]-=- model a 2.5D elevation map, and Hollinger et al. [16] model the surface of detailed underwater structures. Gonzalez et al. [17] use GP regression to model several important soil properties. Martin et...

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... in Sec. V. The target data are given by Y i = [yi1, ..., yin]T , where yik 2 R, 8k. For example, in the classical 2D example of elevation maps, yk would be the elevation at a position xk in 2D space =-=[17]-=-. The GP provides a continuous representation of the output function f represented by the mean estimates f̄⇤ i (x⇤) with uncertainties Vi(f⇤(x⇤)) which can be queried at any location x⇤ 2 Rd. The pred...

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...easurements of a laser scanner or image sensors. We do not address the problem of terrain sensing in this paper, however it is a well-studied problem over the last few decades, see, e.g., [25], [26], =-=[27]-=-. Formally, a set of primitives can be modelled as a finite alphabet P . Then a k-step motion-planning algorithm is a function TQ × H → Pk, where H is a suitable function space for height maps. In a r...

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...ccupancy probability is presented in (O’Callaghan and Ramos, 2012). Another approach, (Murphy and Newman, 2010), takes advantage of GPs to build cost maps for path planning from overhead imagery. In (=-=Vasudevan et al., 2009-=-), the authors infer missing data in 3D range maps for open pit mines. And recently, GPs were used to segment point clouds in not one but multiple classes (Paul et al., 2012). In (Douillard et al., 20...