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Statistical depth functions have become increasingly used in nonparametric inference for multivariate data. Here the contours of such functions are studied. Structural properties of the regions enclosed by contours, such as affine equivariance, nestedness, connectedness and compactness, and almost sure convergence results for sample depth contours, are established. Also, specialized results are established for some popular depth functions, including halfspace depth, and for the case of elliptical distributions. Finally, some needed foundational results on almost sure convergence of sample depth functions are provided.

Given data in , a Tukey κ-trimmed region is the set of all points that have at least Tukey depth κ w.r.t. the data. As they are visual, affine equivariant and robust, Tukey regions are useful tools in nonparametric multivariate analysis. While these regions are easily defined and interpreted, their practical use in applications has been impeded so far by the lack of efficient computational procedures in dimension p > 2. We construct two novel algorithms to compute a Tukey κ-trimmed region, a naïve one and a more sophisticated one that is much faster than known algorithms. Further, a strict bound on the number of facets of a Tukey region is derived. In a large simulation study the novel fast algorithm is compared with the naïve one, which is slower and by construction exact, yielding in every case the same correct results. Finally, the approach is extended to an algorithm that calculates the innermost Tukey region and its barycenter, the Tukey median. Supplementary materials for this article are available online.

The aim of the research is to identify the optimal method for smoothing the surface of a hybrid digital bathymetric model (HDBM). The initiation of this research is justified by the fact that a model created from diverse types of data may have different surface textures and outliers. This diversity may cause problems in subsequent data processing stages, such as generating depth contours. As part of the adopted research methodology, fifteen filters were analysed. Filtering techniques were examined for filter type, the number of iterations, weights, and window size. The result is the adopted research methodology, which enabled the selection of the optimal filtering method. The research undertaken in this work is an extension of the methodology for developing an HDBM. An important aspect of the research is the approach to elaborating on such kinds of models in shallow and ultra-shallow waters adjacent to the land, as well as the use of data obtained by modern measurement platforms, such as unmanned surface vehicles (USV) and unmanned aerial vehicles (UAV). The studies fit into the general context of works related to the development of this type of model and undoubtedly provide a solid reference for further development or improvement of similar methods.

Generalization of nautical charts and electronic nautical charts (ENCs) is a critical process which aims at the safety of navigation and clear cartographic presentation. This paper elaborates on the problem of depth contours and coastline generalization—natural and artificial—for medium-scale charts (harbour and approach) taking into account International Hydrographic Organization (IHO) standards, hydrographic offices’ (HOs) best practices and cartographic literature. Additional factors considered are scale, depth, and seafloor characteristics. The proposed method for depth contour generalization utilizes contours created from high-resolution digital elevation models (DEMs) or those already portrayed on nautical charts. Moreover, it ensures consistency with generalized soundings. Regarding natural coastline generalization, the focus was on managing the resolution, while maintaining the shape, and on the islands. For the provision of a suitable generalization solution for the artificial shoreline, it was preprocessed in order to automatically recognize the shape of each structure as perceived by humans (e.g., a pier that looks like a T). The proposed generalization methodology is implemented with custom-developed routines utilizing standard geo-processing functions available in a geographic information system (GIS) environment and thus can be adopted by hydrographic agencies to support their ENC and nautical chart production. The methodology has been tested in the New York Lower Bay area in the U.S.A. Results have successfully delineated depth contours and coastline at scales 1:10 K, 1:20 K, 1:40 K and 1:80 K.

Electronic Navigational Chart (ENC) data are essential for safe maritime navigation and have multiple other uses in a wide range of enterprises. Charts are relied upon to be as accurate and as up-to-date as possible by the vessels moving vast amounts of products to global ports each year. However, cartographic generalization processes for updating and creating ENCs are complex and time-consuming. Increasing the efficiency of the chart production workflow has been long sought by the nautical charting community. Toward this effort, approaches must consider intended scale, data quality, various chart features, and perform consistently in different scenarios. Additionally, supporting open-science initiatives through standardized open-source workflows will increase marine data accessibility for other disciplines. Therefore, this paper reviews, improves, and integrates available open-source software, and develops new custom generalization tools, for the semi-automated processing of land and hydrographic features per nautical charting specifications. The robustness of this approach is demonstrated in two areas of very different geographic configurations and the effectiveness for use in nautical charting was confirmed by winning the first prize in an international competition. The presented rapid data processing combined with the ENC portrayal of results as a web-service provides new opportunities for applications such as the development of base-maps for marine spatial data infrastructures.

In subsurface geological mapping, it is more than advisable to compare different solutions obtained with neural and other algorithms. Here, for such comparison, we used the previously published and well-prepared dataset of subsurface data collected from the Bjelovar Subdepression, a 2900 km2 large regional macrounit in the Croatian part of the Pannonian Basin System. Data on depth were obtained for the youngest (the shallowest) Lonja Formation (Pliocene, Quaternary) and mapped using neural network (NN), inverse distance weighting (IDW), and ordinary kriging (OK) algorithms. The obtained maps were compared based on square error (using k-fold cross-validation) and the visual interpretation of isopaches. Two other algorithms were also tested, namely, random forest (RF) and extreme gradient boosting (XGB) algorithms, but they were rejected as inappropriate for this purpose solely based on the visuals of the obtained maps, which did not follow any interpretable geological structures. The results showed that NN is a highly adjustable method for interpolation, with adjustment for numerous hyperparameters. IDW showed its strength as one of the classical interpolators, and its results are always located close to the top if several methods are compared. OK is the relative winner, showing the flexibility of variogram analysis regarding the number of data points and possible clustering. The presented variogram model, even with a relatively high sill and occasional nugget effect, can be well fitted into OK, giving better results than other methods when applied to the presented area and datasets. This was not surprising because kriging is a well-established method used exclusively for interpolation. In contrast, NN and machine learning algorithms are used in many fields, and these algorithms, particularly the fitting of hyperparameters in NN, simply cannot be the best solution for all.

Mixed reality applications often require virtual objects that are partly occluded by real objects. However, previous research and commercial products have limitations in terms of performance and efficiency. To address these challenges, we propose a novel depth contour occlusion (DCO) algorithm. The proposed method is based on the sensitivity of contour occlusion and a binocular stereoscopic vision device. In this method, a depth contour map is combined with a sparse depth map obtained from a two-stage adaptive filter area stereo matching algorithm and the depth contour map of the objects extracted by a digital image stabilisation optical flow method. We also propose a quadratic optimisation model with three constraints to generate an accurate dense map of the depth contour for high-quality real-virtual occlusion. The whole process is accelerated by GPU. To evaluate the effectiveness of the algorithm, we demonstrate a time consumption statistical analysis for each stage of the DCO algorithm execution. To verify the reliability of the real-virtual occlusion effect, we conduct an experimental analysis on single-sided, enclosed, and complex occlusions. Subsequently, we compare it with the occlusion method without quadratic optimisation. With our GPU implementation for real-time DCO, the evaluation indicates that applying the presented DCO algorithm enhances the real-time performance and the visual quality of real-virtual occlusion.

For multivariate data, Tukey's half-space depth is one of the most popular depth functions available in the literature. It is conceptually simple and satisfies several desirable properties of depth functions. The Tukey median, the multivariate median associated with the half-space depth, is also a well-known measure of center for multivariate data with several interesting properties. In this article, we derive and investigate some interesting properties of half-space depth and its associated multivariate median. These properties, some of which are counterintuitive, have important statistical consequences in multivariate analysis. We also investigate a natural extension of Tukey's half-space depth and the related median for probability distributions on any Banach space (which may be finite-or infinite-dimensional) and prove some results that demonstrate anomalous behavior of half-space depth in infinite-dimensional spaces.

A class of projection-based depth functions is introduced and studied. These projection-based depth functions possess desirable properties of statistical depth functions and their sample versions possess strong and order √n uniform consistency. Depth regions and contours induced from projection-based depth functions are investigated. Structural properties of depth regions and contours and general continuity and convergence results of sample depth regions are obtained. Affine equivariant multivariate medians induced from projection-based depth functions are probed. The limiting distributions as well as the strong and order √n consistency of the sample projection medians are established. The finite sample performance of projection medians is compared with that of a leading depth-induced median, the Tukey halfspace median (induced from the Tukey halfspace depth function). It turns out that, with appropriate choices of univariate location and scale estimators, the projection medians have a very high finite sample breakdown point and relative efficiency, much higher than those of the halfspace median. Based on the results obtained, it is found that projection depth functions and projection medians behave very well overall compared with their competitors and consequently are good alternatives to statistical depth functions and affine equivariant multivariate location estimators, respectively.

A virtual particle random walking theory under variable velocities is presented in this paper. Under the proposed theory, the solutions of some two-dimensional Poisson equations, which are discretized by nine-point finite difference method and defined on the so-called spatial-temporal motion accumulative image stemming from human silhouettes, provide us the depth contour image for actions description. Although merely two-dimensional definition domain and concepts are related to the Poisson equations, both spatial and temporal evolution information of human actions are successfully included in the depth contour image owing to designating the travelling velocities of virtual particles according to the spatial-temporal motion accumulative image. In addition, it is worth noting that projecting three-dimensional human actions to the two-dimensional image descriptors contributes to much lower computation cost in the corresponding recognition algorithms, compared to those when using the three-dimensional spatial-temporal descriptors directly. In order to enhance the recognition accuracy, a hierarchical cascaded classifier is configured with cascading nearest neighbor classifiers, in each layer of which different kinds of shape and kinematic features of human actions are dealt with. Numerical experimental results on several public human action databases are illustrated to verify recognition performance improvements by means of the proposed algorithm.