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Given earth imagery with spectral features on a terrain surface, this paper studies surface segmentation based on both explanatory features and surface topology. The problem is important in many spatial and spatiotemporal applications such as flood extent mapping in hydrology. The problem is uniquely challenging for several reasons: first, the size of earth imagery on a terrain surface is often much larger than the input of popular deep convolutional neural networks; second, there exists topological structure dependency between pixel classes on the surface, and such dependency can follow an unknown and non-linear distribution; third, there are often limited training labels. Existing methods for earth imagery segmentation often divide the imagery into patches and consider the elevation as an additional feature channel. These methods do not fully incorporate the spatial topological structural constraint within and across surface patches and thus often show poor results, especially when training labels are limited. Existing methods on semi-supervised and unsupervised learning for earth imagery often focus on learning representation without explicitly incorporating surface topology. In contrast, we propose a novel framework that explicitly models the topological skeleton of a terrain surface with a contour tree from computational topology, which is guided by the physical constraint (e.g., water flow direction on terrains). Our framework consists of two neural networks: a convolutional neural network (CNN) to learn spatial contextual features on a 2D image grid, and a graph neural network (GNN) to learn the statistical distribution of physics-guided spatial topological dependency on the contour tree. The two models are co-trained via variational EM. Evaluations on the real-world flood mapping datasets show that the proposed models outperform baseline methods in classification accuracy, especially when training labels are limited. 

Spatial classification with limited observations is important in geographical applications where only a subset of sensors are deployed at certain spots or partial responses are collected in field surveys. For example, in observation-based flood inundation mapping, there is a need to map the full flood extent on geographic terrains based on earth imagery that partially covers a region. Existing research mostly focuses on addressing incomplete or missing data through data cleaning and imputation or modeling missing values as hidden variables in the EM algorithm. These methods, however, assume that missing feature observations are rare and thus are ineffective in problems whereby the vast majority of feature observations are missing. To address this issue, we recently proposed a new approach that incorporates physics-aware structural constraint into the model representation. We design efficient learning and inference algorithms. This paper extends our recent approach by allowing feature values of samples in each class to follow a multi-modal distribution. Evaluations on real-world flood mapping applications show that our approach significantly outperforms baseline methods in classification accuracy, and the multi-modal extension is more robust than our early single-modal version. Computational experiments show that the proposed solution is computationally efficient on large datasets. 

Accurate maps of regional surface water features are integral for advancing ecologic, atmospheric and land development studies. The only comprehensive surface water feature map of Alaska is the National Hydrography Dataset (NHD). NHD features are often digitized representations of historic topographic map blue lines and may be outdated. Here we test deep learning methods to automatically extract surface water features from airborne interferometric synthetic aperture radar (IfSAR) data to update and validate Alaska hydrographic databases. U-net artificial neural networks (ANN) and high-performance computing (HPC) are used for supervised hydrographic feature extraction within a study area comprised of 50 contiguous watersheds in Alaska. Surface water features derived from elevation through automated flow-routing and manual editing are used as training data. Model extensibility is tested with a series of 16 U-net models trained with increasing percentages of the study area, from about 3 to 35 percent. Hydrography is predicted by each of the models for all watersheds not used in training. Input raster layers are derived from digital terrain models, digital surface models, and intensity images from the IfSAR data. Results indicate about 15 percent of the study area is required to optimally train the ANN to extract hydrography when F1-scores for tested watersheds average between 66 and 68. Little benefit is gained by training beyond 15 percent of the study area. Fully connected hydrographic networks are generated for the U-net predictions using a novel approach that constrains a D-8 flow-routing approach to follow U-net predictions. This work demonstrates the ability of deep learning to derive surface water feature maps from complex terrain over a broad area. 

Spatial classification with limited feature observations has been a challenging problem in machine learning. The problem exists in applications where only a subset of sensors are deployed at certain regions or partial responses are collected in field surveys. Existing research mostly focuses on addressing incomplete or missing data, e.g., data cleaning and imputation, classification models that allow for missing feature values, or modeling missing features as hidden variables and applying the EM algorithm. These methods, however, assume that incomplete feature observations only happen on a small subset of samples, and thus cannot solve problems where the vast majority of samples have missing feature observations. To address this issue, we propose a new approach that incorporates physics-aware structural constraints into the model representation. Our approach assumes that a spatial contextual feature is observed for all sample locations and establishes spatial structural constraint from the spatial contextual feature map. We design efficient algorithms for model parameter learning and class inference. Evaluations on real-world hydrological applications show that our approach significantly outperforms several baseline methods in classification accuracy, and the proposed solution is computationally efficient on a large data volume.