This system paper presents the Topology ToolKit (TTK), a software platform designed for topological data analysis in scientific visualization. TTK provides unified, generic, efficient, and robust implementation of key algorithms scalar data, including: critical points, integral lines, persistence diagrams, curves, merge trees, contour Morse-Smale complexes, fiber surfaces, continuous scatterplots, Jacobi sets, Reeb spaces, more. is easily accessible to end users due tight integration with...

Large-scale simulations are increasingly being used to study complex scientific and engineering phenomena. As a result, advanced visualization data analysis also becoming an integral part of the process. Often, key step in extracting insight from these large involves definition, extraction, evaluation features space time coordinates solution. However, many applications, involve range parameters decisions that will affect quality direction analysis. Examples include particular level sets...

An interpretation of the reduced density gradient in terms bosonic kinetic energy is presented. Contrary to other bonding indicators based on density, such as localised orbital locator (LOL) or electron localisation function, not only able identify covalent bondings, but also ionic and non-covalent interactions. This study reveals that critical points are closely connected with those LOL roots one-electron potential.

Abstract This paper presents a novel approach for fast and efficient partial shape retrieval on collection of 3D shapes. Each is represented by Reeb graph associated with geometrical signatures. Partial similarity between two shapes evaluated computing variant their maximum common sub‐graph. By investigating theory, we take advantage its intrinsic properties at levels. First, show that the segmentation provides charts disk or annulus topology only. control enables computation concise...

This paper introduces an efficient algorithm for computing the Reeb graph of a scalar function f defined on volumetric mesh M in R3. We introduce procedure called "loop surgery" that transforms into M' by sequence cuts and guarantees f(M') to be loop free. Therefore, surgery reduces computation simpler problem contour tree, which well-known algorithms exist are theoretically (O(n log n)) fast practice. Inverse reconstruct loops removed at beginning. The time complexity our is tree plus...

Interactions between atoms have a major influence on the chemical properties of molecular systems. While covalent interactions impose structural integrity molecules, noncovalent govern more subtle phenomena such as protein folding, bonding or self assembly. The understanding these types is necessary for interpretation many biological processes and design tasks. traditionally electron density analyzed to interpret quantum chemistry system, are characterized by low densities only slight...

Abstract Scientific visualization has many effective methods for examining and exploring scalar vector fields, but rather fewer bivariate fields. We report the first general purpose approach interactive extraction of geometric separating surfaces in This method is based on fiber surfaces: constructed from sets fibers, multivariate analogues isolines. show simple surface definition extraction. In particular, we a efficient algorithm Marching Cubes. also how to construct interactively with...

This paper serves as a user guide to sampling strategies for sliced optimal transport. We provide reminders and additional regularity results on the Sliced Wasserstein distance. detail construction methods, generation time complexity, theoretical guarantees, conditions each strategy. Additionally, we insights into their suitability transport in theory. Extensive experiments both simulated real-world data offer representative comparison of strategies, culminating practical recommendations best usage.

Morse-Smale (MS) complexes have been gaining popularity as a tool for feature-driven data analysis and visualization. However, the quality of their geometric embedding sole dependence on input scalar field can limit applicability when expressing application-dependent features. In this paper we introduce new combinatorial technique to compute an MS complex that conforms both additional, prior segmentation domain. The constrains computation guaranteeing boundaries in are captured separatrices...

This paper presents an efficient algorithm for the computation of Reeb space input bivariate piecewise linear scalar function f defined on a tetrahedral mesh. By extending and generalizing algorithmic concepts from univariate case to one, we report first practical, output-sensitive exact such space. The starts by identifying Jacobi set f, analogs critical points in case. Next, is computed segmenting mesh along new notion Fiber Surfaces, analog contours We additionally present simplification...

This paper presents a new algorithm for the fast, shared memory multi-core computation of augmented merge trees on triangulations. In contrast to most existing parallel algorithms, our technique computes trees. augmentation is required enable full extent tree based applications, including data segmentation. Our approach completely revisits traditional, sequential re-formulate as set independent local tasks Fibonacci heaps. results in superior time performance practice, well thanks OpenMP...

This paper presents a new algorithm for the fast, shared memory multi-threaded computation of contour trees on tetrahedral meshes. In contrast to previous algorithms, our technique computes augmented tree. Such an augmentation is required enable full extent tree based applications, including instance data segmentation. Our approach relies range-driven domain partitioning. We show how exploit such partitioning rapidly compute forests. also forests can be efficiently turned into output report...

This paper presents an efficient algorithm for the progressive approximation of Wasserstein barycenters persistence diagrams, with applications to visual analysis ensemble data. Given a set scalar fields, our approach enables computation diagram which is representative set, and visually conveys number, data ranges saliences main features interest found in set. Such diagrams are obtained by computing explicitly discrete barycenter notoriously computationally intensive task. In particular, we...

This paper presents a new algorithm for the fast, shared memory, multi-core computation of augmented contour trees on triangulations. In contrast to most existing parallel algorithms our technique computes trees, enabling full extent tree based applications including data segmentation. Our approach completely revisits traditional, sequential re-formulate all steps as set independent local tasks. includes procedure Fibonacci heaps join and split two intermediate structures used compute tree,...

In this paper, we propose to address the semantic- oriented 3D mesh hierarchical segmentation problem, using enhanced topological skeletons. This high level information drives both feature boundary computation as well hierarchy definition. Proposed scheme is based on key idea that topology of a more important decomposition criterion than its geometry. First, skeleton input triangulated surface constructed. Then it used delimit core object and identify junction areas. second step results in...

We present a combinatorial algorithm for the general topological simplification of scalar fields on surfaces. Given field f, our generates simplified g that provably admits only critical points from constrained subset singularities while guaranteeing small distance ||f - g||∞ data-fitting purpose. In contrast to previous algorithms, approach is oblivious strategy used selecting features interest and allows be removed arbitrarily. When persistence select interest, produces standard...

Abstract This paper introduces a novel, non‐local characterization of critical points and their global relation in 2D uncertain scalar fields. The is based on the analysis support probability density functions (PDF) input data. Given two fields representing reliable estimations bounds this support, our strategy identifies mandatory : spatial regions function ranges where have to occur any realization input. algorithm provides pairing scheme for which used construct join split trees . These...

We present a methodology to analyze and visualize an ensemble of finite pointset method (FPM) simulations that model the viscous fingering process salt solutions inside water. In course form structures with increased concentration value, called fingers. These are primary interest domain scientists since it is not deterministic when where fingers appear how they evolve. To explore aleatoric uncertainty embedded in we simulation runs which differ due stochastic effects. detect track derive...

This paper presents a new approach for the visualization and analysis of spatial variability features interest represented by critical points in ensemble data. Our framework, called Persistence Atlas, enables dominant patterns points, along with statistics regarding their occurrence ensemble. The persistence atlas represents geometrical domain each pattern form confidence map appearance points. As by-product, our method also provides 2-dimensional layouts entire ensemble, highlighting main...

This paper presents a robust and efficient method for tracking topological features in time-varying scalar data. Structures are tracked based on the optimal matching between persistence diagrams with respect to Wasserstein metric. fundamentally relies solving assignment problem, special case of transport, all consecutive timesteps. Our approach two main contributions. First, we revisit seminal algorithm by Kuhn Munkres which specifically adapt problem an way. Second, propose extension metric...

This paper introduces an efficient algorithm for persistence diagram computation, given input piecewise linear scalar field <inline-formula><tex-math notation="LaTeX">$f$</tex-math></inline-formula> defined on a notation="LaTeX">$d$</tex-math></inline-formula> -dimensional simplicial complex notation="LaTeX">$\mathcal {K}$</tex-math></inline-formula> , with notation="LaTeX">$d \leq 3$</tex-math></inline-formula> . Our work revisits the seminal <i>"PairSimplices"</i> (Edelsbrunner et al....

A fundamental step in stitching several pictures to form a larger mosaic is the computation of boundary seams that minimize visual artifacts transition between images. Current seam algorithms use optimization methods may be slow, sequential, memory intensive, and prone finding suboptimal solutions related local minima chosen energy function. Moreover, even when these techniques perform well, their solution not perceptually ideal (or good). Such an inflexible approach does allow possibility...