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Objectives: Obtaining valley (ridge) lines from contour data has important practical significance and high production value for terrain analysis, slope aspect calculation and reservoir area designing. In view of the fact that current contour-oriented geographic lines extraction mostly adopts feature point direct-connecting method, and the extracted geographic lines are prone to spatial logical conflicts with contour lines. Meantime, in areas where the contour lines are more curved and more complex, the extracted valley (ridge) line does not conform to the bending characteristics of contour lines, and even deviates dramatically in certain local areas. Methods: Therefore, after overall shape and local shape characteristics of the contour line are analyzed sufficiently based on Tobler ' s first law, a novel methodology of extracting geographic lines from contour data based on Morphing transformation techniques is proposed, which takes the contour line on both sides of a valley (ridge) into consideration, as well as constraint of feature points’ location. First, a contour tree is constructed based on the elevation and hierarchical relationship of contours, and the con‐ tour curve segmentation is carried out according to the concave and convex attributes of nodes on the contour line. Then, the terrain feature points are extracted efficiently and accurately. Second, each contour line is split into two parts with the terrain feature points, where after, an optimal correspondence of the nodes on both parts of a contour line are computed. Third, with respect to the constraints of terrain feature points, the geographic lines are extracted by performing Morphing transformation with a gradient shifting distance value. Results: The experimental results show that the methodology extracts complete geographic lines accurately, which are consistent with the actual terrain and take into account the overall shape and trend of contour lines. Conclusions: The geographic lines extraction based on Morphing transformation techniques method can take into account the overall shape and trend of contour lines and unique shape at feature points to ex‐ tract complete geographic lines consistent with the actual terrain, and it is robust. When the local terrain changes are complex, the contour shape is also complex. At this time, the allocation of Morphing transformation value for parameter t considering more factors will be involved, which is also the content to be further studied in the follow-up work. [ABSTRACT FROM AUTHOR]

Contour trees are used for topological data analysis in scientific visualization. While originally computed with serial algorithms, recent work has introduced a vector-parallel algorithm. However, this algorithm is relatively slow for fully augmented contour trees which are needed for many practical data analysis tasks. We therefore introduce a representation called the hyperstructure that enables efficient searches through the contour tree and use it to construct a fully augmented contour tree in data parallel, with performance on average 6 times faster than the state-of-the-art parallel algorithm in the TTK topological toolkit.

As data sets grow to exascale, automated data analysis and visualization are increasingly important, to intermediate human understanding and to reduce demands on disk storage via in situ analysis. Trends in architecture of high performance computing systems necessitate analysis algorithms to make effective use of combinations of massively multicore and distributed systems. One of the principal analytic tools is the contour tree, which analyses relationships between contours to identify features of more than local importance. Unfortunately, the predominant algorithms for computing the contour tree are explicitly serial, and founded on serial metaphors, which has limited the scalability of this form of analysis. While there is some work on distributed contour tree computation, and separately on hybrid GPU-CPU computation, there is no efficient algorithm with strong formal guarantees on performance allied with fast practical performance. We report the first shared SMP algorithm for fully parallel contour tree computation, with formal guarantees of O(lg V lg t) parallel steps and O(V lg V) work for data with V samples and t contour tree supernodes, and implementations with more than 30× parallel speed up on both CPU using TBB and GPU using Thrust and up 70× speed up compared to the serial sweep and merge algorithm.

As data sets grow to exascale, automated data analysis and visualisation are increasingly important, to intermediate human understanding and to reduce demands on disk storage via in situ analysis. Trends in architecture of high performance computing systems necessitate analysis algorithms to make effective use of combinations of massively multicore and distributed systems. One of the principal analytic tools is the contour tree, which analyses relationships between contours to identify features of more than local importance. Unfortunately, the predominant algorithms for computing the contour tree are explicitly serial, and founded on serial metaphors, which has limited the scalability of this form of analysis. While there is some work on distributed contour tree computation, and separately on hybrid GPU-CPU computation, there is no efficient algorithm with strong formal guarantees on performance allied with fast practical performance. We report the first shared SMP algorithm for fully parallel contour tree computation, withfor-mal guarantees of O(lgnlgt) parallel steps and O(n lgn) work, and implementations with up to 10x parallel speed up in OpenMP and up to 50x speed up in NVIDIA Thrust.

As data sets grow to exascale, automated data analysis and visualisation are increasingly important, to intermediate human understanding and to reduce demands on disk storage via in situ analysis. Trends in architecture of high performance computing systems necessitate analysis algorithms to make effective use of combinations of massively multicore and distributed systems. One of the principal analytic tools is the contour tree, which analyses relationships between contours to identify features of more than local importance. Unfortunately, the predominant algorithms for computing the contour tree are explicitly serial, and founded on serial metaphors, which has limited the scalability of this form of analysis. While there is some work on distributed contour tree computation, and separately on hybrid GPU-CPU computation, there is no efficient algorithm with strong formal guarantees on performance allied with fast practical performance. We report the first shared SMP algorithm for fully parallel contour tree computation, withfor-mal guarantees of O(lgnlgt) parallel steps and O(n lgn) work, and implementations with up to 10x parallel speed up in OpenMP and up to 50x speed up in NVIDIA Thrust.

Objectives: Quantitative expression of topological relations of contour is vital to spatial cognition, topological feature mining and map generalization. However, previous studies focus on the construction of contour tree and the generalization algorithm of contour. However, its lack of the quantitative expression of topological relations and the research of the quantitative relationship between it and map scale change, and it is still unknown whether the topological indexes of different geomorphic types vary with map scale change or not. Methods: By preprocessing of non - closed contour and constructing directed contour tree, such as topology checking, matching and non-closed contour closure, a quantitative expression model of them was proposed and constructed, and 60 groups multi-scale contour datasets from different geomorphic types in Gansu were used to explore the consistency of the relationship. Results and Conclusions: Experimental results show that the topological indexes change rules of adjacent scale is consistent with the space cognitive and reasoning, and the same power function can be used to express the quantitative relation-ship of contour from different geomorphic types, which further demonstrates that it is rationality and possibility to realization the completely automatic generalization of contour based on multi-scale spatial similarity relation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Constructing the access roads between wind turbines requires a significant cost when a wind farm is built in hills or mountains. An optimized design of road network can substantially reduce construction costs and increase investment returns. In this paper, we consider a challenging problem of the road network design for a wind farm with complex topography. An automatic contour-based model is developed for road network design, and is incorporated into the optimization of wind farm micrositing to maximize investment returns of the wind farm. The directions of access roads are first deduced from a contour tree, the route projections are then designed by a gradient-bounded algorithm, and the most cost-effective road network is finally obtained by the minimum spanning tree algorithm. The topographic speedup, wind resource grid and Park wake model are incorporated to accurately evaluate the power production of the wind farm. Both the profit of wind energy sales and the cost of access roads are combined into a cost function called net present value, which is optimized using genetic algorithms. Simulation results illustrate that the length of access roads is decreased significantly while the locations of wind turbines are optimized. The proposed method simultaneously optimizes the turbine layout and road network of the wind farm, therefore achieving a more practical and profitable wind farm micrositing.

The topology of isosurfaces changes at isovalues of critical points, making such points an important feature when building contour trees or Morse-Smale complexes. Hexahedral elements with linear interpolants can contain additional off-vertex critical points in element bodies and on element faces. Moreover, a point on the face of a hexahedron which is critical in the element-local context is not necessarily critical in the global context. Weber et al. (2002) introduce a method to determine whether critical points on faces are also critical in the global context, based on the gradient of the asymptotic decider (G. M. Nielson and B. Hamann) (1991) in each element that shares the face. However, as defined, the method of Weber et al. contains an error, and can lead to incorrect results. In this work we correct the error. [ABSTRACT FROM AUTHOR]

Topology provides a foundation for the development of mathematically sound tools for processing and exploration of scalar fields. Existing topology-based methods can be used to identify interesting features in volumetric data sets, to find seed sets for accelerated isosurface extraction, or to treat individual connected components as distinct entities for isosurfacing or interval volume rendering. We describe a framework for direct volume rendering based on segmenting a volume into regions of equivalent contour topology and applying separate transfer functions to each region. Each region corresponds to a branch of a hierarchical contour tree decomposition, and a separate transfer function can be defined for it. The novel contributions of our work are 1) a volume rendering framework and interface where a unique transfer function can be assigned to each subvolume corresponding to a branch of the contour tree, 2) a runtime method for adjusting data values to reflect contour tree simplifications, 3) an efficient way of mapping a spatial location into the contour tree to determine the applicable transfer function, and 4) an algorithm for hardware-accelerated direct volume rendering that visualizes the contour tree-based segmentation at interactive frame rates using graphics processing units (GPUs) that support loops and conditional branches in fragment programs.

As the most important manifestation of the activities of the life service industry, the reasonable layout of spatial agglomeration and dispersion of the accommodation and catering industry plays an important role in guiding the spatial structure of the urban industry and population. Applying the contour tree and location quotient index methods, based on points of interest (POI) data of the accommodation and catering industry in Beijing and on the identification of the spatial structure and cluster center of the accommodation and catering industry, we investigated the distribution and agglomeration characteristics of the urban accommodation and catering industry from the perspective of industrial spatial differentiation. The results show that: (1) the accommodation and catering industry in Beijing presents a polycentric agglomeration pattern in space, mainly distributed within a radius of 20 km from the city center and on a relatively large scale; areas beyond this distance contain isolated single cluster centers. (2) From the perspective of the industry, the cluster centers close to the core area of the city are characterized by the agglomeration of multiple advantageous industries, while those in the outer suburbs of the city are more prominent in a single industry. (3) From the perspective of the location quotient of cluster centers, the leisure catering industries are mainly located close to the urban centers. On the contrary, the cluster centers in the outer suburbs and counties are relatively small and dominated by restaurants and fast food industries. Commercial accommodation businesses are mainly distributed in the transportation hub centers and in entertainment and leisure areas.

The Reeb graph of a smooth function on a connected smooth closed orientable n-manifold is obtained by contracting the connected components of the level sets to points. The number of loops in the Reeb graph is defined as its first Betti number. We describe the set of possible values of the number of loops in the Reeb graph in terms of the co-rank of the fundamental group of the manifold and show that all such values are realized for Morse functions and, except on surfaces, even for simple Morse functions. For surfaces, we describe the set of Morse functions with the number of loops in the Reeb graph equal to the genus of the surface. [ABSTRACT FROM AUTHOR]

Drumlins are ubiquitous landforms in previously glaciated regions, formed through a series of complex subglacial processes operating underneath the paleo-ice sheets. Accurate delineation and characterization of drumlins are essential for understanding the formation mechanism of drumlins as well as the flow behaviors and basal conditions of paleo-ice sheets. Automated mapping of drumlins is particularly important for examining the distribution patterns of drumlins across large spatial scales. This paper presents an automated vector-based approach to mapping drumlins from high-resolution light detection and ranging (LiDAR) data. The rationale is to extract a set of concentric contours by building localized contour trees and establishing topological relationships. This automated method can overcome the shortcomings of previously manual and automated methods for mapping drumlins, for instance, the azimuthal biases during the generation of shaded relief images. A case study was carried out over a portion of the New York Drumlin Field. Overall 1181 drumlins were identified from the LiDAR-derived DEM across the study region, which had been underestimated in previous literature. The delineation results were visually and statistically compared to the manual digitization results. The morphology of drumlins was characterized by quantifying the length, width, elongation ratio, height, area, and volume. Statistical and spatial analyses were conducted to examine the distribution pattern and spatial variability of drumlin size and form. The drumlins and the morphologic characteristics exhibit significant spatial clustering rather than randomly distributed patterns. The form of drumlins varies from ovoid to spindle shapes towards the downstream direction of paleo ice flows, along with the decrease in width, area, and volume. This observation is in line with previous studies, which may be explained by the variations in sediment thickness and/or the velocity increases of ice flows towards ice front. [ABSTRACT FROM AUTHOR]

The recently published standard ISO 25178-2 distinguishes between field parameters and feature parameters for surface texture characterisation, whereby the main difference between these two types is due to the fact that the parameters belonging to the first group are deduced from all points of a scale-limited surface, while the parameters belonging to the second group are deduced from only a subset of predefined topological surface features. As specified in ISO 25178-2, two prerequisites are indispensable for the determination of the feature parameters, viz., an adequate data structure for surface characterisation and a suitable formal method for surface generalisation , i.e. for the successive elimination of the less important surface features. Within ISO 25178-2 change trees are proposed for describing the surface topography, while Wolf pruning is suggested for surface simplification (cf. also ISO 16610-85). Apart from the techniques specified in ISO 25178-2 and ISO 16610-85, the present paper describes a second geometrical–topological approach for the characterisation and generalisation of surfaces that has its origin within the geosciences and is based on weighted surface networks and w-contractions . In addition, it is revealed how the two approaches, both of which have their foundations in graph theory, are interrelated to each other and how, from a historical point of view, the GIScience approach forms the basis of the one applied within surface metrology. Finally, some applications within precision engineering are described. [ABSTRACT FROM AUTHOR]

3D building model reconstruction is of great importance for environmental and urban applications. Airborne light detection and ranging (LiDAR) is a very useful data source for acquiring detailed geometric and topological information of building objects. In this study, we employed a graph-based method based on hierarchical structure analysis of building contours derived from LiDAR data to reconstruct urban building models. The proposed approach first uses a graph theory-based localized contour tree method to represent the topological structure of buildings, then separates the buildings into different parts by analyzing their topological relationships, and finally reconstructs the building model by integrating all the individual models established through the bipartite graph matching process. Our approach provides a more complete topological and geometrical description of building contours than existing approaches. We evaluated the proposed method by applying it to the Lujiazui region in Shanghai, China, a complex and large urban scene with various types of buildings. The results revealed that complex buildings could be reconstructed successfully with a mean modeling error of 0.32 m. Our proposed method offers a promising solution for 3D building model reconstruction from airborne LiDAR point clouds. [ABSTRACT FROM AUTHOR]

Individual tree crown delineation is of great importance for forest inventory and management. The increasing availability of high-resolution airborne light detection and ranging (LiDAR) data makes it possible to delineate the crown structure of individual trees and deduce their geometric properties with high accuracy. In this study, we developed an automated segmentation method that is able to fully utilize high-resolution LiDAR data for detecting, extracting, and characterizing individual tree crowns with a multitude of geometric and topological properties. The proposed approach captures topological structure of forest and quantifies topological relationships of tree crowns by using a graph theory-based localized contour tree method, and finally segments individual tree crowns by analogy of recognizing hills from a topographic map. This approach consists of five key technical components: (1) derivation of canopy height model from airborne LiDAR data; (2) generation of contours based on the canopy height model; (3) extraction of hierarchical structures of tree crowns using the localized contour tree method; (4) delineation of individual tree crowns by segmenting hierarchical crown structure; and (5) calculation of geometric and topological properties of individual trees. We applied our new method to the Medicine Bow National Forest in the southwest of Laramie, Wyoming and the HJ Andrews Experimental Forest in the central portion of the Cascade Range of Oregon, U.S. The results reveal that the overall accuracy of individual tree crown delineation for the two study areas achieved 94.21% and 75.07%, respectively. Our method holds great potential for segmenting individual tree crowns under various forest conditions. Furthermore, the geometric and topological attributes derived from our method provide comprehensive and essential information for forest management. [ABSTRACT FROM AUTHOR]

Sinkhole mapping is critical for understanding hydrological processes and mitigating geological hazards in karst landscapes. Current methods for identifying sinkholes are primarily based on visual interpretation of low-resolution topographic maps and aerial photographs with subsequent field verification, which is labor-intensive and time-consuming. The increasing availability of high-resolution LiDAR-derived digital elevation data allows for an entirely new level of detailed delineation and analyses of small-scale geomorphologic features and landscape structures at fine scales. In this paper, we present a localized contour tree method for automated extraction of sinkholes in karst landscapes. One significant advantage of our automated approach for sinkhole extraction is that it may reduce inconsistencies and alleviate repeatability concerns associated with visual interpretation methods. In addition, the proposed method has contributed to improving the sinkhole inventory in several ways: (1) detection of non-inventoried sinkholes; (2) identification of previously inventoried sinkholes that have been filled; (3) delineation of sinkhole boundaries; and (4) characterization of sinkhole morphometric properties. We applied the method to Fillmore County in southeastern Minnesota, USA, and identified three times as many sinkholes as the existing database for the same area. The results suggest that previous visual interpretation method might significantly underestimate the number of potential sinkholes in the region. Our method holds great potential for creating and updating sinkhole inventory databases at a regional scale in a timely manner. [ABSTRACT FROM AUTHOR]

Surface depressions are abundant in topographically complex landscapes, and they exert significant influences on hydrological, ecological, and biogeochemical processes at local and regional scales. The increasing availability of high-resolution topographical data makes it possible to resolve small surface depressions. By analogy with the reasoning process of a human interpreter to visually recognize surface depressions from a topographic map, we developed a localized contour tree method that is able to fully exploit high-resolution topographical data for detecting, delineating, and characterizing surface depressions across scales with a multitude of geometric and topological properties. In this research, we introduce a new concept ‘pour contour’ and a graph theory-based contour tree representation for the first time to tackle the surface depression detection and delineation problem. Beyond the depression detection and filling addressed in the previous raster-based methods, our localized contour tree method derives the location, perimeter, surface area, depth, spill elevation, storage volume, shape index, and other geometric properties for all individual surface depressions, as well as the nested topological structures for complex surface depressions. The combination of various geometric properties and nested topological descriptions provides comprehensive and essential information about surface depressions across scales for various environmental applications, such as fine-scale ecohydrological modeling, limnological analyses, and wetland studies. Our application example demonstrated that our localized contour tree method is functionally effective and computationally efficient. [ABSTRACT FROM PUBLISHER]

Contour Trees and Reeb Graphs are firmly embedded in scientific visualization for analysing univariate (scalar) fields. We generalize this analysis to multivariate fields with a data structure called the Joint Contour Net that quantizes the variation of multiple variables simultaneously. We report the first algorithm for constructing the Joint Contour Net, and demonstrate some of the properties that make it practically useful for visualisation, including accelerating computation by exploiting a relationship with rasterisation in the range of the function. [ABSTRACT FROM PUBLISHER]

Abstract: We present a feature-based interactive technique for comparing multiple scalar fields. It extracts largest contour features from the topologically simplified contour trees of each field, then measures similarities between features using information theory. These similarities are stored in a weighted graph on which we apply clustering. The clustering identifies coherent features across fields. Finally, the clusters are ranked and presented in a thumbnail gallery providing a quick overview of the whole dataset. We demonstrate the utility of our approach by applying it to a number of complex fluid flow datasets. [Copyright &y& Elsevier]

We present an algorithm to compute the connected components of upper or lower level sets from level lines. The running time is O (n + \vert \tau \vert), where n is the size of the image and \vert \tau \vert is the number of level lines. [ABSTRACT FROM AUTHOR]

Transfer functions facilitate the volumetric data visualization by assigning optical properties to various data features and scalar values. Automation of transfer function specifications still remains a challenge in volume rendering. This paper presents an approach for automating transfer function generations by utilizing topological attributes derived from the contour tree of a volume. The contour tree acts as a visual index to volume segments, and captures associated topological attributes involved in volumetric data. A residue flow model based on Darcy's Law is employed to control distributions of opacity between branches of the contour tree. Topological attributes are also used to control color selection in a perceptual color space and create harmonic color transfer functions. The generated transfer functions can depict inclusion relationship between structures and maximize opacity and color differences between them. The proposed approach allows efficient automation of transfer function generations and exploration on the data to be carried out based on controlling of opacity residue flow rate instead of complex low-level transfer function parameter adjustments. Experiments on various data sets demonstrate the practical use of our approach in transfer function generations. [ABSTRACT FROM AUTHOR]

Understanding fluid flow data, especially vortices, is still a challenging task. Sophisticated visualization tools help to gain insight. In this paper, we present a novel approach for the interactive comparison of scalar fields using isosurfaces, and its application to fluid flow datasets. Features in two scalar fields are defined by largest contour segmentation after topological simplification. These features are matched using a volumetric similarity measure based on spatial overlap of individual features. The relationships defined by this similarity measure are ranked and presented in a thumbnail gallery of feature pairs and a graph representation showing all relationships between individual contours. Additionally, linked views of the contour trees are provided to ease navigation. The main render view shows the selected features overlapping each other. Thus, by displaying individual features and their relationships in a structured fashion, we enable exploratory visualization of correlations between similar structures in two scalar fields. We demonstrate the utility of our approach by applying it to a number of complex fluid flow datasets, where the emphasis is put on the comparison of vortex related scalar quantities. [ABSTRACT FROM AUTHOR]

Differential topology, and specifically Morse theory, provide a suitable setting for formalizing and solving several problems related to shape analysis. The fundamental idea behind Morse theory is that of combining the topological exploration of a shape with quantitative measurement of geometrical properties provided by a real function defined on the shape. The added value of approaches based on Morse theory is in the possibility of adopting different functions as shape descriptors according to the properties and invariants that one wishes to analyze. In this sense, Morse theory allows one to construct a general framework for shape characterization, parametrized with respect to the mapping function used, and possibly the space associated with the shape. The mapping function plays the role of a lens through which we look at the properties of the shape, and different functions provide different insights. In the last decade, an increasing number of methods that are rooted in Morse theory and make use of properties of real-valued functions for describing shapes have been proposed in the literature. The methods proposed range from approaches which use the configuration of contours for encoding topographic surfaces to more recent work on size theory and persistent homology. All these have been developed over the years with a specific target domain and it is not trivial to systematize this work and understand the links, similarities, and differences among the different methods. Moreover, different terms have been used to denote the same mathematical constructs, which often overwhelm the understanding of the underlying common framework. The aim of this survey is to provide a clear vision of what has been developed so far, focusing on methods that make use of theoretical frameworks that are developed for classes of real functions rather than for a single function, even if they are applied in a restricted manner. The term geometrical-topological used in the title is meant to underline that both levels of information content are relevant for the applications of shape descriptions: geometrical, or metrical, properties and attributes are crucial for characterizing specific instances of features, while topological properties are necessary to abstract and classify shapes according to invariant aspects of their geometry. The approaches surveyed will be discussed in detail, with respect to theory, computation, and application. Several properties of the shape descriptors will be analyzed and compared. We believe this is a crucial step to exploit fully the potential of such approaches in many applications, as well as to identify important areas of future research. [ABSTRACT FROM AUTHOR]

Scientific visualization and illustration tools are designed to help people understand the structure and complexity of scientific data with images that are as informative and intuitive as possible. In this context the use of metaphors plays an important role since they make complex information easily accessible by using commonly known concepts. In this paper we propose a new metaphor, called "Topological Landscapes," which facilitates understanding the topological structure of scalar functions. The basic idea is to construct a terrain with the same topology as a given dataset and to display the terrain as an easily understood representation of the actual input data. In this projection from an n-dimensional scalar function to a two-dimensional (20) model we preserve function values of critical points, the persistence (function span) of topological features, and one possible additional metric property (in our examples volume). By displaying this topologically equivalent landscape together with the original data we harness the natural human proficiency in understanding terrain topography and make complex topological information easily accessible. [ABSTRACT FROM AUTHOR]

We introduce an optimal topdown algorithm for computing and representing level line trees of 2-D intensity images. The running time of the algorithm is O(n + t), where n is the size of the input image and t is the total length of all level lines. The properties of level line trees are also investigated. The efficiency of the algorithm is illustrated by experiments on images of different sizes and scenes. [ABSTRACT FROM AUTHOR]

In this paper we present a method for the multiresolution comparison of biomolecular electrostatic potentials without the need for global structural alignment of the biomolecules. The underlying computational geometry algorithm uses multiresolution attributed contour trees (MACTs) to compare the topological features of volumetric scalar fields. We apply the MACTs to compute electrostatic similarity metrics for a large set of protein chains with varying degrees of sequence, structure, and function similarity. For calibration, we also compute similarity metrics for these chains by a more traditional approach based upon 3D structural alignment and analysis of Carbo similarity indices. Moreover, because the MACT approach does not rely upon pairwise structural alignment, its accuracy and efficiency promise to perform well on future large-scale classification efforts across groups of structurally diverse proteins. The MACT method discriminates between protein chains at a level comparable to the Carbo similarity index method; i.e., it is able to accurately cluster proteins into functionally relevant groups which demonstrate strong dependence on ligand binding sites. The results of the analyses are available from the linked web databases http:// ccvweb.cres.utexas.edu/MolSignature/ and http://agave.wustl.edu/similarity/. The MACT analysis tools are available as part of the public domain library of the Topological Analysis and Quantitative Tools (TAQT) from the Center of Computational Visualization at the University of Texas at Austin (http://ccvweb.csres.utexas.edu/software). The Carbo software is available for download with the open-source APBS software package at http://apbs.sf.net/. [ABSTRACT FROM AUTHOR]

The contour tree has been used to compute the topology of isosurfaces, generate a minimal seed set for accelerated isosurface extraction, and provide a user interface to segment individual contour components in a scalar field. In this paper, we extend the benefits of the contour tree to time-varying data visualization. We define temporal correspondence of contour components and describe an algorithm to compute the correspondence information in time-dependent contour trees. A graph representing the topology changes of time-varying isosurfaces is constructed in real-time for any selected isovalue using the precomputed correspondence information. Quantitative properties, such as surface area and volume of contour components, are computed and labeled on the graph. This topology change graph helps users to detect significant topological and geometric changes in time-varying isosurfaces. The graph is also used as an interactive user interface to segment, track, and visualize the evolution of any selected contour components over time. [ABSTRACT FROM AUTHOR]

A contour tree is a good graphical tool for representing the spatial relations of contour lines and has many applications in map generalization, map annotation, terrain analysis, etc. A new method for generating contour trees by introducing a Voronoi-based interior adjacency concept is proposed in this paper. The immediate interior adjacency set is employed to identify all of the children contours of each contour without contour elevations. It has advantages over existing methods such as the geometric method and the region growing-based method. [ABSTRACT FROM AUTHOR]

A contour tree is a good graphical tool for representing the spatial relations of contour lines and has found many applications in map generalization, map annotation, terrain analysis, etc. A new approach for generating contour trees by introducing a Voronoi-based interior adjacency set concept is proposed in this paper. The immediate interior adjacency set is employed to identify all of the children contours of each contour without contour elevations. It has advantages over existing methods such as the point-in-polygon method and the region growing-based method. This new approach can be used for spatial data mining and knowledge discovering, such as the automatic extraction of terrain features and construction of multi-resolution digital elevation model. [Copyright &y& Elsevier]

As the most important manifestation of the activities of the life service industry, the reasonable layout of spatial agglomeration and dispersion of the accommodation and catering industry plays an important role in guiding the spatial structure of the urban industry and population. Applying the contour tree and location quotient index methods, based on points of interest (POI) data of the accommodation and catering industry in Beijing and on the identification of the spatial structure and cluster center of the accommodation and catering industry, we investigated the distribution and agglomeration characteristics of the urban accommodation and catering industry from the perspective of industrial spatial differentiation. The results show that: (1) the accommodation and catering industry in Beijing presents a polycentric agglomeration pattern in space, mainly distributed within a radius of 20 km from the city center and on a relatively large scale; areas beyond this distance contain isolated single cluster centers. (2) From the perspective of the industry, the cluster centers close to the core area of the city are characterized by the agglomeration of multiple advantageous industries, while those in the outer suburbs of the city are more prominent in a single industry. (3) From the perspective of the location quotient of cluster centers, the leisure catering industries are mainly located close to the urban centers. On the contrary, the cluster centers in the outer suburbs and counties are relatively small and dominated by restaurants and fast food industries. Commercial accommodation businesses are mainly distributed in the transportation hub centers and in entertainment and leisure areas. [ABSTRACT FROM AUTHOR]