Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize topological changes in the (sub)level sets of scalar fields. One of the biggest challenges and opportunities to advance topology-based visualization is to understand and incorporate uncertainty into such topological descriptors to effectively reason about their underlying data. In this paper, we study a structural average of a set of labeled merge trees and use it to encode uncertainty in data. Specifically, we compute a 1-center tree that minimizes its maximum distance to any other tree in the set under a well-defined metric called the interleaving distance. We provide heuristic strategies that compute structural averages of merge trees whose labels do not fully agree. We further provide an interactive visualization system that resembles a numerical calculator that takes as input a set of merge trees and outputs a tree as their structural average. We also highlight structural similarities between the input and the average and incorporate uncertainty information for visual exploration. We develop a novel measure of uncertainty, referred to as consistency, via a metric-space view of the input trees. Finally, we demonstrate an application of our framework through merge trees that arise from ensembles of scalar fields. Our work is the first to employ interleaving distances and consistency to study a global, mathematically rigorous, structural average of merge trees in the context of uncertainty visualization.
Graph-based topological descriptors. Graph-based topological descriptors include merge trees @cite_20 (also known as barrier trees @cite_37 or join trees @cite_17 ), contour trees @cite_17 , Reeb graphs @cite_16 , mapper graphs @cite_12 , and joint contour nets @cite_27 . These descriptors are graph-based representations to illustrate how the topology of level sets or sublevel sets of scalar fields changes with a scalar value parameter.
@cite_37 @cite_27 @cite_16 @cite_20 @cite_12 @cite_17
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In this paper, we propose a novel appearance-based approach for topological mapping based on a hierarchical decomposition of the environment. In our map, images with similar visual properties are grouped together in nodes, which are represented by means of an average global descriptor and an index of binary features based on a bag-of-words online approach. Each image is represented by means of a global descriptor and a set of local features, and this information is used in a two-level loop closure approach, where first global descriptors are employed to obtain the most likely nodes of the map and then binary image features are used to retrieve the most likely images inside these nodes. This hierarchical scheme enables us to reduce the search space when recognizing places, maintaining high accuracy when creating a map. Our approach is validated using several public datasets and compared against several state-of-the-art techniques. The accuracy and the sparsity of the generated maps are also discussed. This paper describes an effort to create new visualizations by exploiting hierarchical scalar topology. First, we build a hierarchical topology through synchronously constructing and simplifying Contour Tree (CT) and Morse-Smale (MS) complex of scalar fields. We then introduce three algorithms based on the hierarchical topology: (1) topology-based multi-resolution contouring — an overview provided for a scalar field by extracting iso-values from the simplified CT and tracing approximate contours across the MS complex cells; (2) topology based spaghetti plots for uncertainty — a seeding scheme based on the hierarchical topology for visualizing uncertainty among ensemble scalar data; (3) virtual ribbons — a new scheme for visualizing multivariate data invented by overlapping visual ribbons which encode the scalar variation of a region covered by uniform contours. We compare the new approaches with current alternatives. We introduce a novel RGB-D patch descriptor designed for detecting coplanar surfaces in SLAM reconstruction. The core of our method is a deep convolutional neural net that takes in RGB, depth, and normal information of a planar patch in an image and outputs a descriptor that can be used to find coplanar patches from other images.We train the network on 10 million triplets of coplanar and non-coplanar patches, and evaluate on a new coplanarity benchmark created from commodity RGB-D scans. Experiments show that our learned descriptor outperforms alternatives extended for this new task by a significant margin. In addition, we demonstrate the benefits of coplanarity matching in a robust RGBD reconstruction formulation.We find that coplanarity constraints detected with our method are sufficient to get reconstruction results comparable to state-of-the-art frameworks on most scenes, but outperform other methods on standard benchmarks when combined with a simple keypoint method. Most of the existing appearance based topological mapping algorithms produce dense topological maps in which each image stands as a node in the topological graph. Sparser maps can be built by representing groups of visually similar images as nodes of a topological graph. In this paper, we present a sparse topological mapping framework which uses Image Sequence Partitioning (ISP) techniques to group visually similar images as topological graph nodes. We present four different ISP techniques and evaluate their performance. In order to take advantage of the afore mentioned maps, we make use of Hierarchical Inverted Files (HIF) which enable efficient hierarchical loop closure. Outdoor experimental results demonstrating the sparsity, efficiency and accuracy achieved by the combination of ISP and HIF in performing loop closure are presented.