Anisotropic noise samples

Visualization of 3D Stress Tensor Fields Using Superquadric Glyphs on Displacement Streamlines

Stress tensor fields play a central role in solid mechanics studies, but their visualization in 3D space remains challenging as the information-dense multi-variate tensor needs to be sampled in 3D space while avoiding clutter. Taking cues from current tensor visualizations, we adapted glyph-based visualization for stress tensors in 3D space. We also developed a testing framework and performed user studies to evaluate the various glyph-based tensor visualizations for objective accuracy measures, and subjective user feedback for each visualization method. To represent the stress tensor, we color encoded the original superquadric glyph, and in the user study, we compared it to superquadric glyphs developed for second-order symmetric tensors. We found that color encoding improved the user accuracy measures, while the users also rated our method the highest. We compared our method of placing stress tensor glyphs on displacement streamlines to the glyph placement on a 3D grid. In the visualization, we modified the glyph to show both the stress tensor and the displacement vector at each sample point. The participants preferred our method of glyph placement on displacement streamlines as it highlighted the underlying continuous structure in the tensor field.

Fiber visualization for preoperative glioma assessment: Tractography versus local connectivity mapping

In diffusion MRI, the advent of high angular resolution diffusion imaging (HARDI) and HARDI with compressed sensing (HARDI+CS) has led to clinically practical signal acquisition techniques which allow for the assessment of white matter architecture in routine patient studies. However, the reconstruction and visualization of fiber pathways by tractography has not yet been established as a standard methodology which can easily be applied. This is due to various algorithmic problems, such as a lack of robustness, error propagation and the necessity of fine-tuning parameters depending on the clinical question. In the framework of a clinical study of glioma patients, we compare two different whole-brain tracking methods to a local connectivity mapping approach which has recently shown promising results in an adaptation to diffusion MRI. The ability of the three methods to correctly depict fiber affection is analyzed by comparing visualization results to representations of local diffusion profiles provided by orientation distribution functions (ODFs). Our results suggest that methods beyond fiber tractography, which visualize local connectedness rather than global connectivity, should be evaluated further for pre-surgical assessment of fiber affection.

Fiber Visualization with LIC Maps Using Multidirectional Anisotropic Glyph Samples

Line integral convolution (LIC) is used as a texture-based technique in computer graphics for flow field visualization. In diffusion tensor imaging (DTI), LIC bridges the gap between local approaches, for example directionally encoded fractional anisotropy mapping and techniques analyzing global relationships between brain regions, such as streamline tracking. In this paper an advancement of a previously published multikernel LIC approach for high angular resolution diffusion imaging visualization is proposed: a novel sampling scheme is developed to generate anisotropic glyph samples that can be used as an input pattern to the LIC algorithm. Multicylindrical glyph samples, derived from fiber orientation distribution (FOD) functions, are used, which provide a method for anisotropic packing along integrated fiber lines controlled by a uniform random algorithm. This allows two- and three-dimensional LIC maps to be generated, depicting fiber structures with excellent contrast, even in regions of crossing and branching fibers. Furthermore, a color-coding model for the fused visualization of slices from T1 datasets together with directionally encoded LIC maps is proposed. The methodology is evaluated by a simulation study with a synthetic dataset, representing crossing and bending fibers. In addition, results from in vivo studies with a healthy volunteer and a brain tumor patient are presented to demonstrate the method's practicality.

A NEW ALGORITHM FOR COMPUTING RIEMANNIAN GEODESIC DISTANCE IN RECTANGULAR 2-D AND 3-D GRIDS

We present a novel way to efficiently compute Riemannian geodesic distance over a two- or three-dimensional domain. It is based on a previously presented method for computation of geodesic distances on surface meshes. Our method is adapted for rectangular grids, equipped with a variable anisotropic metric tensor. Processing and visualization of such tensor fields is common in certain applications, for instance structure tensor fields in image analysis and diffusion tensor fields in medical imaging.

The included benchmark study shows that our method provides significantly better results in anisotropic regions in 2-D and 3-D and is faster than current stat-of-the-art solvers in 2-D grids. Additionally, our method is straightforward to code; the test implementation is less than 150 lines of C++ code. The paper is an extension of a previously presented conference paper and includes new sections on 3-D grids in particular.

Anisotropic sampling of planar and two-manifold domains for texture generation and glyph distribution

We present a new method for the generation of anisotropic sample distributions on planar and two-manifold domains. Most previous work that is concerned with aperiodic point distributions is designed for isotropically shaped samples. Methods focusing on anisotropic sample distributions are rare, and either they are restricted to planar domains, are highly sensitive to the choice of parameters, or they are computationally expensive. In this paper, we present a time-efficient approach for the generation of anisotropic sample distributions that only depends on intuitive design parameters for planar and two-manifold domains. We employ an anisotropic triangulation that serves as basis for the creation of an initial sample distribution as well as for a gravitational-centered relaxation. Furthermore, we present an approach for interactive rendering of anisotropic Voronoi cells as base element for texture generation. It represents a novel and flexible visualization approach to depict metric tensor fields that can be derived from general tensor fields as well as scalar or vector fields.

An intrinsic algorithm for parallel Poisson disk sampling on arbitrary surfaces

Poisson disk sampling has excellent spatial and spectral properties, and plays an important role in a variety of visual computing. Although many promising algorithms have been proposed for multidimensional sampling in euclidean space, very few studies have been reported with regard to the problem of generating Poisson disks on surfaces due to the complicated nature of the surface. This paper presents an intrinsic algorithm for parallel Poisson disk sampling on arbitrary surfaces. In sharp contrast to the conventional parallel approaches, our method neither partitions the given surface into small patches nor uses any spatial data structure to maintain the voids in the sampling domain. Instead, our approach assigns each sample candidate a random and unique priority that is unbiased with regard to the distribution. Hence, multiple threads can process the candidates simultaneously and resolve conflicts by checking the given priority values. Our algorithm guarantees that the generated Poisson disks are uniformly and randomly distributed without bias. It is worth noting that our method is intrinsic and independent of the embedding space. This intrinsic feature allows us to generate Poisson disk patterns on arbitrary surfaces in IR(n). To our knowledge, this is the first intrinsic, parallel, and accurate algorithm for surface Poisson disk sampling. Furthermore, by manipulating the spatially varying density function, we can obtain adaptive sampling easily.

Generalized anisotropic stratified surface sampling

We introduce a novel stratified sampling technique for mesh surfaces that gives the user control over sampling density and anisotropy via a tensor field. Our approach is based on sampling space-filling curves mapped onto mesh segments via parametrizations aligned with the tensor field. After a short preprocessing step, samples can be generated in real time. Along with visual examples, we provide rigorous spectral analysis and differential domain analysis of our sampling. The sample distributions are of high quality: they fulfil the blue noise criterion, so have minimal artifacts due to regularity of sampling patterns, and they accurately represent isotropic and anisotropic densities on the plane and on mesh surfaces. They also have low discrepancy, ensuring that the surface is evenly covered.

Derived Metric Tensors for Flow Surface Visualization

Integral flow surfaces constitute a widely used flow visualization tool due to their capability to convey important flow information such as fluid transport, mixing, and domain segmentation. Current flow surface rendering techniques limit their expressiveness, however, by focusing virtually exclusively on displacement visualization, visually neglecting the more complex notion of deformation such as shearing and stretching that is central to the field of continuum mechanics. To incorporate this information into the flow surface visualization and analysis process, we derive a metric tensor field that encodes local surface deformations as induced by the velocity gradient of the underlying flow field. We demonstrate how properties of the resulting metric tensor field are capable of enhancing present surface visualization and generation methods and develop novel surface querying, sampling, and visualization techniques. The provided results show how this step towards unifying classic flow visualization and more advanced concepts from continuum mechanics enables more detailed and improved flow analysis.

Asymmetric tensor field visualization for surfaces

Asymmetric tensor field visualization can provide important insight into fluid flows and solid deformations. Existing techniques for asymmetric tensor fields focus on the analysis, and simply use evenly-spaced hyperstreamlines on surfaces following eigenvectors and dual-eigenvectors in the tensor field. In this paper, we describe a hybrid visualization technique in which hyperstreamlines and elliptical glyphs are used in real and complex domains, respectively. This enables a more faithful representation of flow behaviors inside complex domains. In addition, we encode tensor magnitude, an important quantity in tensor field analysis, using the density of hyperstreamlines and sizes of glyphs. This allows colors to be used to encode other important tensor quantities. To facilitate quick visual exploration of the data from different viewpoints and at different resolutions, we employ an efficient image-space approach in which hyperstreamlines and glyphs are generated quickly in the image plane. The combination of these techniques leads to an efficient tensor field visualization system for domain scientists. We demonstrate the effectiveness of our visualization technique through applications to complex simulated engine fluid flow and earthquake deformation data. Feedback from domain expert scientists, who are also co-authors, is provided.

Superquadric glyphs for symmetric second-order tensors

Symmetric second-order tensor fields play a central role in scientific and biomedical studies as well as in image analysis and feature-extraction methods. The utility of displaying tensor field samples has driven the development of visualization techniques that encode the tensor shape and orientation into the geometry of a tensor glyph. With some exceptions, these methods work only for positive-definite tensors (i.e. having positive eigenvalues, such as diffusion tensors). We expand the scope of tensor glyphs to all symmetric second-order tensors in two and three dimensions, gracefully and unambiguously depicting any combination of positive and negative eigenvalues. We generalize a previous method of superquadric glyphs for positive-definite tensors by drawing upon a larger portion of the superquadric shape space, supplemented with a coloring that indicates the quadratic form (including eigenvalue sign). We show that encoding arbitrary eigenvalue magnitudes requires design choices that differ fundamentally from those in previous work on traceless tensors that arise in the study of liquid crystals. Our method starts with a design of 2-D tensor glyphs guided by principles of scale-preservation and symmetry, and creates 3-D glyphs that include the 2-D glyphs in their axis-aligned cross-sections. A key ingredient of our method is a novel way of mapping from the shape space of three-dimensional symmetric second-order tensors to the unit square. We apply our new glyphs to stress tensors from mechanics, geometry tensors and Hessians from image analysis, and rate-of-deformation tensors in computational fluid dynamics.