Modeling nonstationary and nonlinear variations in spatial processes is challenging. To overcome this long-standing challenge in spatial interpolation, we introduced the spatial compound variable theory, which assumes that a spatial variable can be divided into global regular and local irregular components. We argue that nonstationary and nonlinear variations in these two components have distinct properties: the global regular component describes a nonlinear trend that is locally predictable, whereas the local irregular component represents local hotspots, cold spots, or outliers. Following the spatial compound variable theory, we developed a novel nonlinear and nonstationary spatial interpolation model by integrating Yang Chizhong filtering and an inductive graph convolution network (YangNet). Specifically, we used the Yang Chizhong filtering and interpolation method to estimate the global regular component with a nonlinear trend, built an inductive graph convolution network to model the nonlinear relationship between the estimated global regular component and the corresponding observation data, and used the nonlinear relationship to estimate the local irregular component. A case study of the Xiadian gold deposit in China demonstrates that YangNet outperforms four representative gold grade interpolation methods in terms of accuracy and smoothing effect mitigation. YangNet exhibits excellent adaptability and can be applied widely in geoscience.

In multivariate spatial interpolation, the accuracy of a variable of interest can be improved using ancillary variables. Although geostatistical methods are widely used for multivariate spatial interpolation, these methods usually require second-order stationary assumption of spatial processes, which is difficult to satisfy in practice. We developed a new multivariate spatial interpolation method based on Yang-Chizhong filtering (CoYangCZ) to overcome this limitation. CoYangCZ does not solve the multivariate spatial interpolation problem from a purely statistical point of view but integrates geometry and statistics-based strategies. First, we used a weighted moving average method based on binomial coefficients (i.e. Yang-Chizhong filtering) to fit the spatial autocorrelation structure of each spatial variable from a geometric perspective. We then quantified the spatial autocorrelation of each spatial variable and the correlations between different spatial variables by analyzing the variances of different spatial variables. Finally, we obtain the best linear unbiased estimators at the unsampled locations. Experiments on air pollution and meteorological datasets show that CoYangCZ has a higher interpolation accuracy than cokriging, regression kriging, gradient plus-inverse distance squared, sequential Gaussian co-simulation, and the kriging convolutional network. CoYangCZ can adapt to second-order non-stationary spatial processes; therefore, it has a wider scope of application than purely statistical methods.

The stationarity assumption of geostatistical methods is difficult to satisfy in practice. To overcome this limitation, this study proposed a geometric and statistical coupling strategy for modeling spatial dependence structures and developed a generalized Yang Chizhong filtering and interpolation (GYangCZ) method without the assumption of stationarity. In this work, we theoretically prove the effectiveness of Yang Chizhong filtering in fitting spatial dependence structures from a geometric perspective, and develop an orientation-constrained Yang Chizhong filtering to fit the local and discontinuous spatial dependence structures. To measure nonstationary spatial dependence structure, we define a local statistical indicator (i.e., fundamental variation function) by comparing the variance of the original data and the fitted geometric surfaces obtained under different filtering radii. The fundamental variation function is used as the kernel function to obtain the approximate best linear unbiased estimators at unobserved locations. We theoretically demonstrate that when only a linear drift exists in local areas, GYangCZ does not require the stationarity assumption. GYangCZ was used to estimate the gold grade of the Xiadian gold deposit in China. The results show that GYangCZ outperformed ordinary kriging, moving window kriging, and kriging convolution networks. GYangCZ is easy to implement with wide applications in geoscience.

The building contours, as one of the most important features for representing geometry, are widely used in various applications including urban modeling and reconstruction. Automatic extraction of high-fidelity compact contours from unstructured point clouds is rather challenging, and the existing methods are limited in generating global-aware and artifact-free building contours. Here, we approach contour extraction as a Bayesian inference problem. Two ideas are proposed to obtain building contour points and edges directly from unstructured point clouds. First, we construct a point-edge collaborative (PECo) Bayesian framework to couple the information of contour points and contour edges. The developed model fully takes local contour features, global edge structures, and global geometric priors into account. Second, given the Bayesian framework for building contouring, we leverage an expectation maximization (EM) algorithm to iteratively infer the contour edges in a maximum posteriori manner. The alternate EM iterations between point and edge domains progressively refine the local pointwise information to a global representation of contour edges. As a result, a synergistic effect between the point features and edge structures for global-aware building contouring is attained. Our approach outperforms the state-of-the-art methods in terms of geometric accuracy and structural compactness in modeling buildings with various complexities. Furthermore, the compactness of the contouring results can be more flexible and easily controlled.

In 3D urban environments, features of interest (FOIs) are often occluded by clusters of buildings, which prevent a clear overview of important spatial features. State-of-the-art disocclusion methods for urban environments fall short of preserving cityscape appearance or require time-consuming computation. These methods use only one or two operators for disocclusion and might not strike a good balance between disocclusion and distortion control. We present a novel, automatic method enabling interactive context-aware visualization of urban features of interest, which combines four effective disocclusion operators including viewpoint elevation, road shifting, building scaling, and building displacement to disocclude the features of interest. Our method provides an optimum compromise among the disocclusion operators via an efficient constrained optimization and the post-polishing phrases, which minimizes the distortions while enforcing the visibility of the FOIs. The 3D views generated at interactive frame rates ensure a resemblance in the cityscape appearance to its original ones and provide a good overview of the FOIs. The experiments with real data demonstrate that our method can greatly facilitate tasks such as navigation, wayfinding, and information overlay.