The detection and analysis of features in fluid flow are important tasks in fluid mechanics and flow visu alization. One recent class of methods to approach this problem is to first compute objective optimal reference frames, relative to which the input vector field becomes as steady as possible. However, existing methods either optimize locally over a fixed neighborhood, which might not match the extent of interesting features well, or perform global optimization, which is costly. We propose a novel objective method for the computation of optimal reference frames that automatically adapts to the flow field locally, without having to choose neighborhoods a priori. We enable adaptivity by formulating this problem as a moving least squares approximation, through which we determine a continuous field of reference frames. To incorporate fluid features into the computation of the reference frame field, we introduce the use of a scalar guidance field into the moving least squares approximation. The guidance field determines a curved manifold on which a regularly sampled input vector field becomes a set of irregularly spaced samples, which then forms the input to the moving least squares approximation. Although the guidance field can be any scalar field, by using a field that corresponds to flow features the resulting reference frame field will adapt accordingly. We show that using an FTLE field as the guidance field results in a reference frame field that adapts better to local features in the flow than prior wo rk. However, our moving least squares framework is formulated in a very general way, and therefore other types of guidance fields could be used in the future to adapt to local fluid features.

Visualizing and analyzing 3D unsteady flow fields is a very challenging task. We approach this problem by leveraging the mathematical foundations of 3D observer fields to explore and analyze 3D flows in reference frames that are more suitable to visual analysis than the input reference frame. We design novel interactive tools for determining, filtering, and combining reference frames for observer-aware 3D unsteady flow visualization. We represent the space of reference frame motions in a 3D spatial domain via a 6D parameter space, in which every observer is a time-dependent curve. Our framework supports operations in this 6D observer space by separately focusing on two 3D subspaces, for 3D translations, and 3D rotations, respectively. We show that this approach facilitates a variety of interactions with 3D flow fields. Building on the interactive selection of observers, we furthermore introduce novel techniques such as observer-aware streamline- and pathline-filtering as well as observer-aware isosurface animations of scalar fluid properties for the enhanced visualization and analysis of 3D unsteady flows. We discuss the theoretical underpinnings as well as practical implementation considerations of our approach, and demonstrate the benefits of its 6+1D observer-based methodology on several 3D unsteady flow datasets.

Abstract Vortex structures play a pivotal role in understanding complex fluid dynamics, yet defining them rigorously remains challenging. One hard criterion is that a vortex detector must be objective, i.e., it needs to be indifferent to reference frame transformations. We propose VortexTransformer, a novel deep learning approach using point transformer architectures to directly extract vortex structures from pathlines. Unlike traditional methods that rely on grid‐based velocity fields in the Eulerian frame, our approach operates entirely on a Lagrangian representation of the flow field (i.e., pathlines), enabling objective identification of both strong and weak vortex structures. To train VortexTransformer, we generate a large synthetic dataset using parametric flow models to simulate diverse vortex configurations, ensuring a robust ground truth. We compare our method against CNN and U‐Net architectures, applying the trained models to real‐world flow datasets. VortexTransformer is an end‐to‐end detector, which means that reference frame transformations as well as vortex detection are handled implicitly by the network, demonstrating the ability to extract vortex boundaries without the need for parameters such as arbitrary thresholds, or an explicit definition of a vortex. Our method offers a new approach to determining objective vortex labels by using the objective pairwise distances of material points for vortex detection and is adaptable to various flow conditions.

Abstract We present a novel technique for the extraction, visualization, and analysis of material boundaries and Lagrangian coherent structures (LCS) in 2D unsteady flow fields relative to local reference frame transformations. In addition to the input flow field, we leverage existing methods for computing reference frames adapted to local fluid features, in particular those that minimize the observed time derivative. Although, by definition, transforming objective tensor fields between reference frames does not change the tensor field, we show that transforming objective tensors, such as the finite‐time Lyapunov exponent (FTLE) or Lagrangian‐averaged vorticity deviation (LAVD), or the second‐order rate‐of‐strain tensor, into local reference frames that are naturally adapted to coherent fluid structures has several advantages: (1) The transformed fields enable analyzing LCS in space‐time visualizations that are adapted to each structure; (2) They facilitate extracting geometric features, such as iso‐surfaces and ridge lines, in a straightforward manner with high accuracy. The resulting visualizations are characterized by lower geometric complexity and enhanced topological fidelity. To demonstrate the effectiveness of our technique, we measure geometric complexity and compare it with iso‐surfaces extracted in the conventional reference frame. We show that the decreased geometric complexity of the iso‐surfaces in the local reference frame, not only leads to improved geometric and topological results, but also to a decrease in computation time.

We present a novel framework for 3D tomographic reconstruction and visualization of tomograms from noisy electron microscopy tilt-series. Our technique takes as an input aligned tilt-series from cryogenic electron microscopy and creates denoised 3D tomograms using a proximal jointly-optimized approach that iteratively performs reconstruction and denoising, relieving the users of the need to select appropriate denoising algorithms in the pre-reconstruction or post-reconstruction steps. The whole process is accelerated by exploiting parallelism on modern GPUs, and the results can be visualized immediately after the reconstruction using volume rendering tools incorporated in the framework. We show that our technique can be used with multiple combinations of reconstruction algorithms and regularizers, thanks to the flexibility provided by proximal algorithms. Additionally, the reconstruction framework is open-source and can be easily extended with additional reconstruction and denoising methods. Furthermore, our approach enables visualization of reconstruction error throughout the iterative process within the reconstructed tomogram and on projection planes of the input tilt-series. We evaluate our approach in comparison with state-of-the-art approaches and additionally show how our error visualization can be used for reconstruction evaluation.

This paper describes a novel method for detecting and visualizing vortex structures in unsteady 2D fluid flows. The method is based on an interactive local reference frame estimation that minimizes the observed time derivative of the input flow field v(x,t). A locally optimal reference frame w($x,t$) assists the user in the identification of physically observable vortex structures inObserved Line Integral Convolution(LIC) visualizations. The observed LIC visualizations are interactively computed and displayed in a user-steered vortex lens region, embedded in the context of a conventional LIC visualization outside the lens. The locally optimal reference frame is then used to detect observed critical points, where v = w, which are used to seed vortex core lines. Each vortex core line is computed as a solution of the ordinary differential equation (ODE) ˙$w(t)=w(w(t),t)$, with an observed critical point as initial condition$(w(t_0),t_0)$. During integration, we enforce a strict error bound on the difference between the extracted core line and the integration of a path line of the input vector field, i.e., a solution to the ODE ˙$v(t)=v(v(t),t)$. We experimentally verify that this error depends on the step size of the core line integration. This ensures that our method extracts Lagrangian vortex core lines that are the simultaneous solution of both ODEs with a numerical error that is controllable by the integration step size. We show the usability of our method in the context of an interactive system using a lens metaphor, and evaluate the results in comparison to state-of-the-art vortex core line extraction methods

Large-scale scientific data, such as weather and climate simulations, often comprise a large number of attributes for each data sample, like temperature, pressure, humidity, and many more. Interactive visualization and analysis require filtering according to any desired combination of attributes, in particular logical AND operations, which is challenging for large data and many attributes. Many general data structures for this problem are built for and scale with a fixed number of attributes, and scalability of joint queries with arbitrary attribute subsets remains a significant problem. We propose a flexible probabilistic framework for multivariate range queries that decouples all attribute dimensions via projection, allowing any subset of attributes to be queried with full efficiency. Moreover, our approach is output-sensitive, mainly scaling with the cardinality of the query result rather than with the input data size. This is particularly important for joint attribute queries, where the query output is usually much smaller than the whole data set. Additionally, our approach can split query evaluation between user interaction and rendering, achieving much better scalability for interactive visualization than the previous state of the art. Furthermore, even when a multi-resolution strategy is used for visualization, queries are jointly evaluated at the finest data granularity, because our framework does not limit query accuracy to a fixed spatial subdivision.

The rapidly growing size and complexity of 3D geological models has increased the need for level-of-detail techniques and compact encodings to facilitate interactive visualization. For large-scale hexahedral meshes, state-of-the-art approaches often employ wavelet schemes for level of detail as well as for data compression. Here, wavelet transforms serve two purposes: (1) they achieve substantial compression for data reduction; and (2) the multiresolution encoding provides levels of detail for visualization. However, in coarser detail levels, important geometric features, such as geological faults, often get too smoothed out or lost, due to linear translation-invariant filtering. The same is true for attribute features, such as discontinuities in porosity or permeability. We present a novel, integrated approach addressing both purposes above, while preserving critical data features of both model geometry and its attributes. Our first major contribution is that we completely decouple the computation of levels of detail from data compression, and perform nonlinear filtering in a high-dimensional data space jointly representing the geological model geometry with its attributes. Computing detail levels in this space enables us to jointly preserve features in both geometry and attributes. While designed in a general way, our framework specifically employs joint bilateral filters, computed efficiently on a high-dimensional permutohedral grid. For data compression, after the computation of all detail levels, each level is separately encoded with a standard wavelet transform. Our second major contribution is a compact GPU data structure for the encoded mesh and attributes that enables direct real-time GPU visualization without prior decoding.

Achieving high rendering quality in the visualization of large particle data, for example from large-scale molecular dynamics simulations, requires a significant amount of sub-pixel super-sampling, due to very high numbers of particles per pixel. Although it is impossible to super-sample all particles of large-scale data at interactive rates, efficient occlusion culling can decouple the overall data size from a high effective sampling rate of visible particles. However, while the latter is essential for domain scientists to be able to see important data features, performing occlusion culling by sampling or sorting the data is usually slow or error-prone due to visibility estimates of insufficient quality. We present a novel probabilistic culling architecture for super-sampled high-quality rendering of large particle data. Occlusion is dynamically determined at the sub-pixel level, without explicit visibility sorting or data simplification. We introduce confidence maps to probabilistically estimate confidence in the visibility data gathered so far. This enables progressive, confidence-based culling, helping to avoid wrong visibility decisions. In this way, we determine particle visibility with high accuracy, although only a small part of the data set is sampled. This enables extensive super-sampling of (partially) visible particles for high rendering quality, at a fraction of the cost of sampling all particles. For real-time performance with millions of particles, we exploit novel features of recent GPU architectures to group particles into two hierarchy levels, combining fine-grained culling with high frame rates.

State-of-the-art computation and visualization of vortices in unsteady fluid flow employ objective vortex criteria, which makes them independent of reference frames or observers. However, objectivity by itself, although crucial, is not sufficient to guarantee that one can identify physically-realizable observers that would perceive or detect the same vortices. Moreover, a significant challenge is that a single reference frame is often not sufficient to accurately observe multiple vortices that follow different motions. This paper presents a novel framework for the exploration and use of an interactively-chosen set of observers, of the resulting relative velocity fields, and of objective vortex structures. We show that our approach facilitates the objective detection and visualization of vortices relative to well-adapted reference frame motions, while at the same time guaranteeing that these observers are in fact physically realizable. In order to represent and manipulate observers efficiently, we make use of the low-dimensional vector space structure of the Lie algebra of physically-realizable observer motions. We illustrate that our framework facilitates the efficient choice and guided exploration of objective vortices in unsteady 2D flow, on planar as well as on spherical domains, using well-adapted reference frames.

Computing and visualizing features in fluid flow often depends on the observer, or reference frame, relative to which the input velocity field is given. A desired property of feature detectors is therefore that they are objective, meaning independent of the input reference frame. However, the standard definition of objectivity is only given for Euclidean domains and cannot be applied in curved spaces. We build on methods from mathematical physics and Riemannian geometry to generalize objectivity to curved spaces, using the powerful notion of symmetry groups as the basis for definition. From this, we develop a general mathematical framework for the objective computation of observer fields for curved spaces, relative to which other computed measures become objective. An important property of our framework is that it works intrinsically in 2D, instead of in the 3D ambient space. This enables a direct generalization of the 2D computation via optimization of observer fields in flat space to curved domains, without having to perform optimization in 3D. We specifically develop the case of unsteady 2D geophysical flows given on spheres, such as the Earth. Our observer fields in curved spaces then enable objective feature computation as well as the visualization of the time evolution of scalar and vector fields, such that the automatically computed reference frames follow moving structures like vortices in a way that makes them appear to be steady.

Flow fields are usually visualized relative to a global observer, i.e., a single frame of reference. However, often no global frame can depict all flow features equally well. Likewise, objective criteria for detecting features such as vortices often use either a global reference frame, or compute a separate frame for each point in space and time. We propose the first general framework that enables choosing a smooth trade-off between these two extremes. Using global optimization to minimize specific differential geometric properties, we compute a time-dependent observer velocity field that describes the motion of a continuous field of observers adapted to the input flow. This requires developing the novel notion of an observed time derivative. While individual observers are restricted to rigid motions, overall we compute an approximate Killing field, corresponding to almost-rigid motion. This enables continuous transitions between different observers. Instead of focusing only on flow features, we furthermore develop a novel general notion of visualizing how all observers jointly perceive the input field. This in fact requires introducing the concept of an observation time, with respect to which a visualization is computed. We develop the corresponding notions of observed stream, path, streak, and time lines. For efficiency, these characteristic curves can be computed using standard approaches, by first transforming the input field accordingly. Finally, we prove that the input flow perceived by the observer field is objective. This makes derived flow features, such as vortices, objective as well.

Molecular dynamics (MD) simulations are crucial to investigating important processes in physics and thermodynamics. The simulated atoms are usually visualized as hard spheres with Phong shading, where individual particles and their local density can be perceived well in close-up views. However, for large-scale simulations with 10 million particles or more, the visualization of large fields-of-view usually suffers from strong aliasing artifacts, because the mismatch between data size and output resolution leads to severe under-sampling of the geometry. Excessive super-sampling can alleviate this problem, but is prohibitively expensive. This paper presents a novel visualization method for large-scale particle data that addresses aliasing while enabling interactive high-quality rendering. We introduce the novel concept of screen-space normal distribution functions (S-NDFs) for particle data. S-NDFs represent the distribution of surface normals that map to a given pixel in screen space, which enables high-quality re-lighting without re-rendering particles. In order to facilitate interactive zooming, we cache S-NDFs in a screen-space mipmap (S-MIP). Together, these two concepts enable interactive, scale-consistent re-lighting and shading changes, as well as zooming, without having to re-sample the particle data. We show how our method facilitates the interactive exploration of real-world large-scale MD simulation data in different scenarios.

In this paper we present a novel GPU-based data structure for spatial indexing. Based on Fenwick trees-a special type of binary indexed trees-our data structure allows construction in linear time. Updates and prefixes can be computed in logarithmic time, whereas point queries require only constant time on average. Unlike competing data structures such as summed-area tables and spatial hashing, our data structure requires a constant amount of bits for each data element, and it offers unconstrained point queries. This property makes our data structure ideally suited for applications requiring unconstrained indexing of large data, such as block-storage of large and block-sparse volumes. Finally, we provide asymptotic bounds on both run-time and memory requirements, and we show applications for which our new data structure is useful.

Sparse volume data structures enable the efficient representation of large but sparse volumes in GPU memory for computation and visualization. However, the choice of a specific data structure for a given data set depends on several factors, such as the memory budget, the sparsity of the data, and data access patterns. In general, there is no single optimal sparse data structure, but a set of several candidates with individual strengths and drawbacks. One solution to this problem are hybrid data structures which locally adapt themselves to the sparsity. However, they typically suffer from increased traversal overhead which limits their utility in many applications. This paper presents JiTTree, a novel sparse hybrid volume data structure that uses just-in-time compilation to overcome these problems. By combining multiple sparse data structures and reducing traversal overhead we leverage their individual advantages. We demonstrate that hybrid data structures adapt well to a large range of data sets. They are especially superior to other sparse data structures for data sets that locally vary in sparsity. Possible optimization criteria are memory, performance and a combination thereof. Through just-in-time (JIT) compilation, JiTTree reduces the traversal overhead of the resulting optimal data structure. As a result, our hybrid volume data structure enables efficient computations on the GPU, while being superior in terms of memory usage when compared to non-hybrid data structures.

Abstract In the literature on optimal regular volume sampling, the Body‐Centered Cubic (BCC) lattice has been proven to be optimal for sampling spherically band‐limited signals above the Nyquist limit. On the other hand, if the sampling frequency is below the Nyquist limit, the Face‐Centered Cubic (FCC) lattice was demonstrated to be optimal in reducing the prealiasing effect. In this paper, we confirm that the FCC lattice is indeed optimal in this sense in a certain interval of the sampling frequency. By theoretically estimating the prealiasing error in a realistic range of the sampling frequency, we show that in other frequency intervals, the BCC lattice and even the traditional Cartesian Cubic (CC) lattice are expected to minimize the prealiasing. The BCC lattice is superior over the FCC lattice if the sampling frequency is not significantly below the Nyquist limit. Interestingly, if the original signal is drastically undersampled, the CC lattice is expected to provide the lowest prealiasing error. Additionally, we give a comprehensible clarification that the sampling efficiency of the FCC lattice is lower than that of the BCC lattice. Although this is a well‐known fact, the exact percentage has been erroneously reported in the literature. Furthermore, for the sake of an unbiased comparison, we propose to rotate the Marschner‐Lobb test signal such that an undue advantage is not given to either lattice.

Researchers from many domains use scientific visualization in their daily practice. Existing implementations of algorithms usually come with a graphical user interface (high-level interface), or as software library or source code (low-level interface). In this paper we present a system that integrates domain-specific languages (DSLs) and facilitates the creation of new DSLs. DSLs provide an effective interface for domain scientists avoiding the difficulties involved with low-level interfaces and at the same time offering more flexibility than high-level interfaces. We describe the design and implementation of ViSlang, an interpreted language specifically tailored for scientific visualization. A major contribution of our design is the extensibility of the ViSlang language. Novel DSLs that are tailored to the problems of the domain can be created and integrated into ViSlang. We show that our approach can be added to existing user interfaces to increase the flexibility for expert users on demand, but at the same time does not interfere with the user experience of novice users. To demonstrate the flexibility of our approach we present new DSLs for volume processing, querying and visualization. We report the implementation effort for new DSLs and compare our approach with Matlab and Python implementations in terms of run-time performance.

We present an interactive graphical approach for the explicit specification of semantics for volume visualization. This explicit and graphical specification of semantics for volumetric features allows us to visually assign meaning to both input and output parameters of the visualization mapping. This is in contrast to the implicit way of specifying semantics using transfer functions. In particular, we demonstrate how to realize a dynamic specification of semantics which allows to flexibly explore a wide range of mappings. Our approach is based on three concepts. First, we use semantic shader augmentation to automatically add rule-based rendering functionality to static visualization mappings in a shader program, while preserving the visual abstraction that the initial shader encodes. With this technique we extend recent developments that define a mapping between data attributes and visual attributes with rules, which are evaluated using fuzzy logic. Second, we let users define the semantics by analogy through brushing on renderings of the data attributes of interest. Third, the rules are specified graphically in an interface that provides visual clues for potential modifications. Together, the presented methods offer a high degree of freedom in the specification and exploration of rule-based mappings and avoid the limitations of a linguistic rule formulation.

Visualization systems often require large monitors or projection screens to display complex information. Even very sophisticated systems that exhibit complex user interfaces do usually not exploit advanced input and output devices. One of the reasons for that is the high cost of special hardware. This paper introduces Augmented Visualization, an interaction method for projection walls as well as monitors using affordable and widely available hardware such as mobile phones or tablets. The main technical challenge is the tracking of the users' devices without any special equipment or fiducial markers in the working area. We propose to track natural features of the display content with the built-in camera of mobile devices. Tracking the visualized scene allows pose estimation of the mobile devices with six degrees of freedom. The position and orientation information is then used for advanced interaction metaphors like magic lenses. For a group of experts who are analyzing the data in front of the same screen, a personal augmented view of the visualized scene is presented, for each user on his/her personal device. The prototype Augmented Visualization System achieves interactive frame rates and may lead to a greatly enhanced user experience. The paper discusses the design and implementation questions and illustrates potential application scenarios.

In this paper, we introduce a novel framework for the compositing of interactively rendered 3D layers tailored to the needs of scientific illustration. Currently, traditional scientific illustrations are produced in a series of composition stages, combining different pictorial elements using 2D digital layering. Our approach extends the layer metaphor into 3D without giving up the advantages of 2D methods. The new compositing approach allows for effects such as selective transparency, occlusion overrides, and soft depth buffering. Furthermore, we show how common manipulation techniques such as masking can be integrated into this concept. These tools behave just like in 2D, but their influence extends beyond a single viewpoint. Since the presented approach makes no assumptions about the underlying rendering algorithms, layers can be generated based on polygonal geometry, volumetric data, point-based representations, or others. Our implementation exploits current graphics hardware and permits real-time interaction and rendering.

Abstract In traditional illustration the choice of appropriate styles and rendering techniques is guided by the intention of the artist. For illustrative volume visualizations it is difficult to specify the mapping between the 3D data and the visual representation that preserves the intention of the user. The semantic layers concept establishes this mapping with a linguistic formulation of rules that directly map data features to rendering styles. With semantic layers fuzzy logic is used to evaluate the user defined illustration rules in a preprocessing step. In this paper we introduce interaction‐dependent rules that are evaluated for each frame and are therefore computationally more expensive. Enabling interaction‐dependent rules, however, allows the use of a new class of semantics, resulting in more expressive interactive illustrations. We show that the evaluation of the fuzzy logic can be done on the graphics hardware enabling the efficient use of interaction‐dependent semantics. Further we introduce the flat rendering mode and discuss how different rendering parameters are influenced by the rule base. Our approach provides high quality illustrative volume renderings at interactive frame rates, guided by the specification of illustration rules.

Direct volume rendering techniques map volumetric attributes (e.g., density, gradient magnitude, etc.) to visual styles. Commonly this mapping is specified by a transfer function. The specification of transfer functions is a complex task and requires expert knowledge about the underlying rendering technique. In the case of multiple volumetric attributes and multiple visual styles the specification of the multi-dimensional transfer function becomes more challenging and non-intuitive. We present a novel methodology for the specification of a mapping from several volumetric attributes to multiple illustrative visual styles. We introduce semantic layers that allow a domain expert to specify the mapping in the natural language of the domain. A semantic layer defines the mapping of volumetric attributes to one visual style. Volumetric attributes and visual styles are represented as fuzzy sets. The mapping is specified by rules that are evaluated with fuzzy logic arithmetics. The user specifies the fuzzy sets and the rules without special knowledge about the underlying rendering technique. Semantic layers allow for a linguistic specification of the mapping from attributes to visual styles replacing the traditional transfer function specification.

Volume rendering techniques are conventionally classified as either direct or indirect methods. Indirect methods require to transform the initial volumetric model into an intermediate geometrical model in order to efficiently visualize it. In contrast, direct volume rendering (DVR) methods can directly process the volumetric data. Modern CT scanners usually provide data as a set of samples on a rectilinear grid, which is computed from the measured projections by discrete tomographic reconstruction. Therefore the rectilinear grid can already be considered as an intermediate volume representation. In this paper we introduce direct direct volume rendering (D2VR). D2VR does not require a rectilinear grid, since it is based on an immediate processing of the measured projections. Arbitrary samples for ray casting are reconstructed from the projections by using the Filtered Back-Projection algorithm. Our method removes a lossy resampling step from the classical volume rendering pipeline. It provides much higher accuracy than traditional grid-based resampling techniques do. Furthermore we also present a novel high-quality gradient estimation scheme, which is also based on the Filtered Back-Projection algorithm.

Exploded views are an illustration technique where an object is partitioned into several segments. These segments are displaced to reveal otherwise hidden detail. In this paper we apply the concept of exploded views to volumetric data in order to solve the general problem of occlusion. In many cases an object of interest is occluded by other structures. While transparency or cutaways can be used to reveal a focus object, these techniques remove parts of the context information. Exploded views, on the other hand, do not suffer from this drawback. Our approach employs a force-based model: the volume is divided into a part configuration controlled by a number of forces and constraints. The focus object exerts an explosion force causing the parts to arrange according to the given constraints. We show that this novel and flexible approach allows for a wide variety of explosion-based visualizations including view-dependent explosions. Furthermore, we present a high-quality GPU-based volume ray casting algorithm for exploded views which allows rendering and interaction at several frames per second.