We define several new models for how to define anomalous regions among enormous sets of trajectories. These are based on spatial scan statistics, and identify a geometric region which captures a subset of trajectories which are significantly different in a measured characteristic from the background population. The model definition depends on how much a geometric region is contributed to by some overlapping trajectory. This contribution can be the full trajectory, proportional to the time spent in the spatial region, or dependent on the flux across the boundary of that spatial region. Our methods are based on and significantly extend a recent two-level sampling approach which provides high accuracy at enormous scales of data. We support these new models and algorithms with extensive experiments on millions of trajectories and also theoretical guarantees.
Our algorithms build upon the recent two-level sampling framework for spatial scan statistics by Matheny al @cite_18 @cite_3 . This focused on making scalable @math -approximate spatial scan statistics over point sets. This line of work provides improvements over non-approximate variants @cite_2 @cite_13 , in terms of the number samples, and the runtime of the scanning algorithms for the same shapes we study: halfspaces, disks, and rectangles. The introduction of two-level sample makes these approaches tractable, and better coresets or faster scanning makes them extremely efficient. However, these fast SSS methods only apply for point sets.
@cite_18 @cite_13 @cite_3 @cite_2
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Recent advances in scanning technologies, in particular devices that extract depth through active sensing, allow fast scanning of urban scenes. Such rapid acquisition incurs imperfections: large regions remain missing, significant variation in sampling density is common, and the data is often corrupted with noise and outliers. However, buildings often exhibit large scale repetitions and self-similarities. Detecting, extracting, and utilizing such large scale repetitions provide powerful means to consolidate the imperfect data. Our key observation is that the same geometry, when scanned multiple times over reoccurrences of instances, allow application of a simple yet effective non-local filtering. The multiplicity of the geometry is fused together and projected to a base-geometry defined by clustering corresponding surfaces. Denoising is applied by separating the process into off-plane and in-plane phases. We show that the consolidation of the reoccurrences provides robust denoising and allow reliable completion of missing parts. We present evaluation results of the algorithm on several LiDAR scans of buildings of varying complexity and styles. In high-dimensional magnetic resonance imaging applications, time-consuming, sequential acquisition of data samples in the spatial frequency domain ( @math -space) can often be accelerated by accounting for dependencies in linear reconstruction, at the cost of noise amplification that depends on the sampling pattern. Common examples are support-constrained, parallel, and dynamic MRI, and @math -space sampling strategies are primarily driven by image-domain metrics that are expensive to compute for arbitrary sampling patterns. It remains challenging to provide systematic and computationally efficient automatic designs of arbitrary multidimensional Cartesian sampling patterns that mitigate noise amplification, given the subspace to which the object is confined. To address this problem, this paper introduces a theoretical framework that describes local geometric properties of the sampling pattern and relates them to the spread in the eigenvalues of the information matrix described by its first two spectral moments. This new criterion is then used for very efficient optimization of complex multidimensional sampling patterns that does not require reconstructing images or explicitly mapping noise amplification. Experiments with in vivo data show strong agreement between this criterion and traditional, comprehensive image-domain- and @math -space-based metrics, indicating the potential of the approach for computationally efficient (on-the-fly), automatic, and adaptive design of sampling patterns. We present a method for fusing two acquisition modes, 2D photographs and 3D LiDAR scans, for depth-layer decomposition of urban facades. The two modes have complementary characteristics: point cloud scans are coherent and inherently 3D, but are often sparse, noisy, and incomplete; photographs, on the other hand, are of high resolution, easy to acquire, and dense, but view-dependent and inherently 2D, lacking critical depth information. In this paper we use photographs to enhance the acquired LiDAR data. Our key observation is that with an initial registration of the 2D and 3D datasets we can decompose the input photographs into rectified depth layers. We decompose the input photographs into rectangular planar fragments and diffuse depth information from the corresponding 3D scan onto the fragments by solving a multi-label assignment problem. Our layer decomposition enables accurate repetition detection in each planar layer, using which we propagate geometry, remove outliers and enhance the 3D scan. Finally, the algorithm produces an enhanced, layered, textured model. We evaluate our algorithm on complex multi-planar building facades, where direct autocorrelation methods for repetition detection fail. We demonstrate how 2D photographs help improve the 3D scans by exploiting data redundancy, and transferring high level structural information to (plausibly) complete large missing regions. This paper reports a novel mutual information (MI) based algorithm for automatic registration of unstructured 3D point clouds comprised of co-registered 3D lidar and camera imagery. The proposed method provides a robust and principled framework for fusing the complementary information obtained from these two different sensing modalities. High-dimensional features are extracted from a training set of textured point clouds (scans) and hierarchical k-means clustering is used to quantize these features into a set of codewords. Using this codebook, any new scan can be represented as a collection of codewords. Under the correct rigid-body transformation aligning two overlapping scans, the MI between the codewords present in the scans is maximized. We apply a James-Stein-type shrinkage estimator to estimate the true MI from the marginal and joint histograms of the codewords extracted from the scans. Experimental results using scans obtained by a vehicle equipped with a 3D laser scanner and an omnidirectional camera are used to validate the robustness of the proposed algorithm over a wide range of initial conditions. We also show that the proposed method works well with 3D data alone.