Simplifying Jacobi Sets Topology and Geometry by Selective Smoothing of Bivariate 2D Scalar Fields

Local, smooth, and consistent Jacobi set simplification

Isosurface extraction and rendering of 3D scalar field is an important method in visual analysis of scalar field data. Traditional isosurface extraction algorithm does not take into account the characteristics of the scalar field itself. In this paper, a 3D scalar field multi-resolution isosurface extraction method is proposed to improve the efficiency of isosurface extraction and rendering. This method not only deals the multi-resolution of the 3D scalar field before the isosurface extraction, but also ensures the integrity of the isosurface in the subsequent isosurface extraction process. The experimental results show that the method can greatly improve the efficiency of isosurface extraction and rendering on the basis of less effect on isosurface.

A three-dimensional, time-resolved, laser-induced fluorescence (3D-LIF) technique was developed to measure the turbulent (liquid-liquid) mixing of a conserved passive scalar in the wake of an injector inserted perpendicularly into a tubular reactor with Re=4,000. In this technique, a horizontal laser sheet was traversed in its normal direction through the measurement section. Three-dimensional scalar fields were reconstructed from the 2D images captured at consecutive, closely spaced levels by means of a high-speed CCD camera. The ultimate goal of the measurements was to assess the downstream development of the 3D scalar fields (in terms of the full scalar gradient vector field and its associated scalar energy dissipation rate) in an industrial flow with significant advection velocity. As a result of this advection velocity, the measured 3D scalar field is artificially “skewed” during a scan period. A method to correct for this skewing was developed, tested and applied. Analysis of the results show consistent physical behaviour.

In this paper, we propose a technique for generating particles from user-specified transfer function for an effective point-based volume rendering. In general, a volume rendering technique utilizes an illumination model in which the 3D scalar field is characterized as a varying density emitter with a single level of scattering. This model is related to a particle system in which the particles are sufficiently small and of low albedo. A conventional volume rendering technique models the density of particles, not the particles themselves [1]. The density is defined by specifying a transfer function from a scalar data value to an opacity data value. Thus, a given scalar field is described as a continuous semitransparent gel and the accumulating order is important. This results in a considerable computational overhead. On the other hand, our rendering technique represents the 3D scalar fields as a set of particles. The particle density is derived from a userspecified transfer function, and describes the probability that a particle is present at the point. Since the particles can be considered as fully opaque, no alpha blending but only depth comparison is required during the rendering calculation, which is advantageous in the distributed processing.

We consider a problem of estimation of relationship between scalar 2D fields using gradient measure and Jacobi sets. The gradient measure method is based on estimation of alignment of gradients in every point of fields. The Jacobi set is the set of critical points of the restrictions of one function to the intersection of level sets of the other functions. We present the results of a numerical experiment for the case of multifield containing two fields: geopotential height on isobaric level 300 hPa and total ozone column, under influence of disturbing factor — intensive solar proton events in January 2005. Estimation of interrelationship by gradient measure indicates strengthening of interaction between fields for period 16th – 22th January 2005. The estimation made by Jacobi sets computation also shown strengthening of relation between analysed fields during solar proton events.

Adaptive Navigation (AN) control strategies allow an agent to autonomously alter its trajectory based on realtime measurements of the environment. Compared to conventional navigation methods, these techniques can reduce required time and energy to explore scalar characteristics of unknown and dynamic regions of interest (e.g., temperature, concentration level). Multiple Uncrewed Aerial Vehicle (UAV) approaches to AN can improve performance by exploiting synchronized spatially-dispersed measurements to generate realtime information regarding the structure of the local scalar field, which is then used to inform navigation decisions. This article presents initial results of a comprehensive program to develop, verify, and experimentally implement mission-level AN capabilities in three-dimensional (3D) space using our unique multilayer control architecture for groups of vehicles. Using our flexible formation control system, we build upon our prior 2D AN work and provide new contributions to 3D scalar field AN by a) demonstrating a wide range of 3D AN capabilities using a unified, multilayer control architecture, b) extending multivehicle 2D AN control primitives to navigation in 3D scalar fields, and c) introducing state-based sequencing of these primitive AN functions to execute 3D mission-level capabilities such as isosurface mapping and plume following. We verify functionality using high-fidelity simulations of multicopter drone clusters, accounting for vehicle dynamics, outdoor wind gust disturbances, position sensor inaccuracy, and scalar field sensor noise. This paper presents the multilayer architecture for multivehicle formation control, the 3D AN control primitives, the sequencing approaches for specific mission-level capabilities, and simulation results that demonstrate these functions.

We present a technique to generalize the 2D painting metaphor to 3D that allows the artist to treat the full 3D space as a canvas. Strokes painted in the 2D viewport window must be embedded in 3D space in a way that gives creative freedom to the artist while maintaining an acceptable level of controllability. We address this challenge by proposing a canvas concept defined implicitly by a 3D scalar field. The artist shapes the implicit canvas by creating approximate 3D proxy geometry. An optimization procedure is then used to embed painted strokes in space by satisfying different objective criteria defined on the scalar field. This functionality allows us to implement tools for painting along level set surfaces or across different level sets. Our method gives the power of fine-tuning the implicit canvas to the artist using a unified painting/sculpting metaphor. A sculpting tool can be used to paint into the implicit canvas. Rather than adding color, this tool creates a local change in the scalar field that results in outward or inward protrusions along the field's gradient direction. We address a visibility ambiguity inherent in 3D stroke rendering with a depth offsetting method that is well suited for hardware acceleration. We demonstrate results with a number of 3D paintings that exhibit effects difficult to realize with existing systems.

Spatially persisting patterns form during the downstream evolution of passive scalars in three-dimensional (3D) spatially periodic flows due to the coupled effect of stretching and folding mechanisms of the flow field. This has been investigated in many computational and theoretical studies of 2D time-periodic and 3D spatially periodic flow fields. However, experimental studies, to date, have mainly focused on flow visualization with streaks of dye rather than fully 3D scalar field measurements. Our study employs 3D particle tracking velocimetry and 3D laser-induced fluorescence to analyze the evolution of 3D flow and scalar fields and the correlation between the coherent flow/scalar field structures in a representative inline mixer, the Quatro static mixer. For this purpose an experimental setup that consists of an optically accessible test section with transparent internal elements accommodating a pressure-driven pipe flow has been built. The flow and scalar fields clearly underline the complementarity of the experimental results with numerical simulations and provide validation of the periodicity assumption needed in numerical studies. The experimental procedure employed in this investigation, which allows studying the scalar transport in the advective limit, demonstrates the suitability of the present method for exploratory mixing studies of a variety of mixing devices, beyond the Quatro static mixer.

In this paper, we segment the volume into geometrically disjoint regions that can be rendered to provide a more effective and interactive volume rendering of structured and unstructured grids. Our segmentation is based upon intervals within the scalar field, producing a set of geometrically defined interval volumes. We present many advantageous properties in using interval volumes, and provide several new rendering operations or shaders to provide effective visualizations of the 3D scalar field. In particular, we demonstrate new technologies that allow interval volumes to be rendered interactively and/or used to reduce the amount of rasterization or rendering primitives in a volume renderer. We illustrate the use of interval volumes to highlight contour boundaries or material interfaces. Several surface shaders that can easily be integrated in the volume renderer are presented. To construct the interval volumes, we cast the problem one dimension higher, using a higher-dimensional isosurface construction for interactive computation or segmentation. The algorithm is independent of the dimension and topology of the polyhedral cells comprising the grid, and thus offers an excellent enhancement to the volume rendering of unstructured grids. We present examples using hexahedral and tetrahedral cells from time-varying and multi-attribute datasets.

We present a technique to generalize the 2D painting metaphor to 3D that allows the artist to treat the full 3D space as a canvas. Strokes painted in the 2D viewport window must be embedded in 3D space in a way that gives creative freedom to the artist while maintaining an acceptable level of controllability. We address this challenge by proposing a canvas concept defined implicitly by a 3D scalar field. The artist shapes the implicit canvas by creating approximate 3D proxy geometry. An optimization procedure is then used to embed painted strokes in space by satisfying different objective criteria defined on the scalar field. This functionality allows us to implement tools for painting along level set surfaces or across different level sets. Our method gives the power of fine-tuning the implicit canvas to the artist using a unified painting/sculpting metaphor. A sculpting tool can be used to paint into the implicit canvas. Rather than adding color, this tool creates a local change in the scalar field that results in outward or inward protrusions along the field's gradient direction. We address a visibility ambiguity inherent in 3D stroke rendering with a depth offsetting method that is well suited for hardware acceleration. We demonstrate results with a number of 3D paintings that exhibit effects difficult to realize with existing systems.

In an era of quickly growing data set sizes, information reduction methods such as extracting or highlighting characteristic features become more and more important for data analysis. For single scalar fields, topological methods can fill this role by extracting and relating critical points. While such methods are regularly employed to study single scalar fields, it is less well studied how they can be extended to uncertain data, as produced, e.g., by ensemble simulations. Motivated by our previous work on visualization in climate research, we study new methods to characterize critical points in ensembles of 2D scalar fields. Previous work on this topic either assumed or required specific distributions, did not account for uncertainty introduced by approximating the underlying latent distributions by a finite number of fields, or did not allow to answer all our domain experts' questions. In this work, we use Bayesian inference to estimate the probability of critical points, either of the original ensemble or its bootstrapped mean. This does not make any assumptions on the underlying distribution and allows to estimate the sensitivity of the results to finite-sample approximations of the underlying distribution. We use color mapping to depict these probabilities and the stability of their estimation. The resulting images can, e.g., be used to estimate how precise the critical points of the mean-field are. We apply our method to synthetic data to validate its theoretical properties and compare it with other methods in this regard. We also apply our method to the data from our previous work, where it provides a more accurate answer to the domain experts' research questions.

Direct detection of dark energy or modified gravity may finally be within reach due to ultrasensitive instrumentation such as atom interferometry capable of detecting incredibly small scale accelerations. Forecasts, constraints and measurement bounds can now too perhaps be estimated from accurate numerical simulations of the fifth force and its Laplacian field at solar system scales. The cubic Galileon gravity scalar field model (CGG), which derives from the DGP braneworld model, describes modified gravity incorporating a Vainshtein screening mechanism. The nonlinear derivative interactions in the CGG equation suppress the field near regions of high density, thereby restoring general relativity (GR) while far from such regions, field enhancement is comparable to GR and the equation is dominated by a linear term. This feature of the governing PDE poses some numerical challenges for computation of the scalar potential, force and Laplacian fields even under stationary conditions. Here we present a numerical method based on finite differences for solution of the static CGG scalar field for a 2D axisymmetric Sun-Earth system and a 3D Cartesian Sun-Earth-Moon system. The method relies on gradient descent of an integrated residual based on the normal attractive branch of the CGG equation. The algorithm is shown to be stable, accurate and rapidly convergent toward the global minimum state. We hope this numerical study, which can easily be extended to include smaller bodies such as detection satellites, will prove useful to future measurement of modified gravity force fields at solar system scales.

This paper reports on a 3D temporal numerical simulation of the interaction between a jet flow and a trailing vortex. The simulation was performed from an experimental database of the vortex wake generated by a rectangular wing NACA0012 equipped with two jets beneath the airfoil. In this work, we focus on the 3D vorticity and scalar (initially included in the jet) fields in the vortex wake. The turbulence induced by the jet flow held interacts with the vortex, resulting in the large-scale structure generation outside the vortex core whereas the region very close to the core keeps its laminar state. The scalar field is subject to distortion and stretching inside the azimuthal vorticity tubes. The scalar concentration provides a reliable signature of where the helical instability occurs outside the vortex core. Later, the counter rotating vortices undergo a connection and consequently lose coherence in a very short time. The resulting scalar field concentrates then in small-scale eddies.

We propose a novel method for the computation of Jacobi sets in 2D domains. The Jacobi set is a topological descriptor based on Morse theory that captures gradient alignments among multiple scalar fields, which is useful for multi-field visualization. Previous Jacobi set computations use piecewise linear approximations on triangulations that result in discretization artifacts like zig-zag patterns. In this paper, we utilize a local bilinear method to obtain a more precise approximation of Jacobi sets by preserving the topology and improving the geometry. Consequently, zig-zag patterns on edges are avoided, resulting in a smoother Jacobi set representation. Our experiments show a better convergence with increasing resolution compared to the piecewise linear method. We utilize this advantage with an efficient local subdivision scheme. Finally, our approach is evaluated qualitatively and quantitatively in comparison with previous methods for different mesh resolutions and across a number of synthetic and real-world examples.

Building Morphological Representations for 2D and 3D Scalar Fields