Andrea Kratz

C. elegans chromosomes connect to centrosomes by anchoring into the spindle network.

The mitotic spindle ensures the faithful segregation of chromosomes. Here we combine the first large-scale serial electron tomography of whole mitotic spindles in early C. elegans embryos with live-cell imaging to reconstruct all microtubules in 3D and identify their plus- and minus-ends. We classify them as kinetochore (KMTs), spindle (SMTs) or astral microtubules (AMTs) according to their positions, and quantify distinct properties of each class. While our light microscopy and mutant studies show that microtubules are nucleated from the centrosomes, we find only a few KMTs directly connected to the centrosomes. Indeed, by quantitatively analysing several models of microtubule growth, we conclude that minus-ends of KMTs have selectively detached and depolymerized from the centrosome. In toto, our results show that the connection between centrosomes and chromosomes is mediated by an anchoring into the entire spindle network and that any direct connections through KMTs are few and likely very transient.

Visualization and Analysis of Second-Order Tensors: Moving Beyond the Symmetric Positive-Definite Case

Tensors provide a powerful language to describe physical phenomena. Consequently, they have a long tradition in physics and appear in various application areas, either as the final result of simulations or as intermediate product. Due to their complexity, tensors are hard to interpret. This motivates the development of well‐conceived visualization methods. As a sub‐branch of scientific visualization, tensor field visualization has been especially pushed forward by diffusion tensor imaging. In this review, we focus on second‐order tensors that are not diffusion tensors. Until now, these tensors, which might be neither positive‐definite nor symmetric, are under‐represented in visualization and existing visualization tools are often not appropriate for these tensors. Hence, we discuss the strengths and limitations of existing methods when dealing with such tensors as well as challenges introduced by them. The goal of this paper is to reveal the importance of the field and to encourage the development of new visualization methods for tensors from various application fields.

A Visual Approach to Analysis of Stress Tensor Fields

We present a visual approach for the exploration of stress tensor fields. In contrast to common tensor visualization methods that only provide a single view to the tensor field, we pursue the idea of providing various perspectives onto the data in attribute and object space. Especially in the context of stress tensors, advanced tensor visualization methods have a young tradition. Thus, we propose a combination of visualization techniques domain experts are used to with statistical views of tensor attributes. The application of this concept to tensor fields was achieved by extending the notion of shape space. It provides an intuitive way of finding tensor invariants that represent relevant physical properties. Using brushing techniques, the user can select features in attribute space, which are mapped to displayable entities in a three-dimensional hybrid visualization in object space. Volume rendering serves as context, while glyphs encode the whole tensor information in focus regions. Tensorlines can be included to emphasize directionally coherent features in the tensor field. We show that the benefit of such a multi-perspective approach is manifold. Foremost, it provides easy access to the complexity of tensor data. Moreover, including well-known analysis tools, such as Mohr diagrams, users can familiarize themselves gradually with novel visualization methods. Finally, by employing a focus-driven hybrid rendering, we significantly reduce clutter, which was a major problem of other three-dimensional tensor visualization methods.

Tensor Visualization Driven Mechanical Component Design

This paper is the result of a close collaboration of mechanical engineers and visualization researchers. It showcases how interdisciplinary work can lead to new insight and progress in both fields. Our case is concerned with one step in the product development process. Its goal is the design of mechanical parts that are functional, meet required quality measures and can be manufactured with standard production methods. The collaboration started with unspecific goals and first experiments with the available data and visualization methods. During the course of the collaboration many concrete questions arose and in the end a hypothesis was developed which will be discussed and evaluated in this paper. We facilitate a case study to validate our hypothesis. For the case study we consider the design of a reinforcement structure of a brake lever, a plastic ribbing. Three new lever geometries are developed on basis of our hypothesis and are compared against each other and against a reference model. The validation comprises standard numerical and experimental tests. In our case, all new structures outperform the reference geometry. The results are very promising and suggest potential to impact the product development process also for more complex scenarios.

Kinetochore Microtubules indirectly link Chromosomes and Centrosomes in C. elegans Mitosis

The mitotic spindle is a dynamic microtubule-based apparatus that ensures the faithful segregation of chromosomes by connecting chromosomes to spindle poles. How this pivotal connection is established and maintained during mitosis is currently debated. Here we combined large-scale serial electron tomography with live-cell imaging to uncover the spatial and dynamic organization of microtubules in the mitotic spindles in C. elegans. With this we quantified the position of microtubule minus and plus-ends as well as distinguished the different classes of microtubules, such as kinetochore, astral and spindle microtubules with their distinct properties. Although microtubules are nucleated from the centrosomes, we find only a few, if any, kinetochore microtubules directly connected to the spindle poles, suggesting an indirect pole to chromosome connection. We propose a model of kinetochore microtubule assembly and disassembly, in which microtubules undergo minus-end depolymerisation, resulting in a detachment from the centrosome. Our reconstructions and analyses of complete spindles expand our understanding of spindle architecture beyond the light microscopic limit.The mitotic spindle ensures the faithful segregation of chromosomes. To discover the nature of the crucial centrosome-to-chromosome connection during mitosis, we combined the first large-scale serial electron tomography of whole mitotic spindles in early C. elegans embryos with live-cell imaging. Using tomography, we reconstructed the positions of all microtubules in 3D, and identified their plus- and minus-ends. We classified them as kinetochore (KMTs), spindle (SMTs), or astral microtubules (AMTs) according to their positions, and quantified distinct properties of each class. While our light microscopy and mutant studies show that microtubules are nucleated from the centrosomes, we find only a few KMTs are directly connected to the centrosomes. Indeed, by quantitatively analysing several models of microtubule growth, we conclude that minus-ends of KMTs have selectively detached and depolymerized from the centrosome. In toto, our results show that the connection between centrosomes and chromosomes is mediated by an anchoring into the entire spindle network and that any direct connections through KMTs are few and likely very transient.

Automatic, tensor-guided illustrative vector field visualization

This paper proposes a vector field visualization, which mimics a sketch-like representation. The visualization combines two major perspectives: Large scale trends based on a strongly simplified field as background visualization and a local visualization highlighting strongly expressed features at their exact position. Each component considers the vector field itself and its spatial derivatives. The derivate is an asymmetric tensor field, which allows the deduction of scalar quantities reflecting distinctive field properties like strength of rotation or shear. The basis of the background visualization is a vector and scalar clustering approach. The local features are defined as the extrema of the respective scalar fields. Applying scalar field topology provides a profound mathematical basis for the feature extraction. All design decisions are guided by the goal of generating a simple to read visualization. To demonstrate the effectiveness of our approach, we show results for three different data sets with different complexity and characteristics.

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.

Top Challenges in the Visualization of Engineering Tensor Fields

In this chapter we summarize the top research challenges in creating successful visualization tools for tensor fields in engineering. The analysis is based on our collective experiences and on discussions with both domain experts and visualization practitioners. We find that creating visualization tools for engineering tensors often involves solving multiple different technical problems at the same time—including visual intuitiveness, scalability, interactivity, providing both detail and context, integration with modeling and simulation, representing uncertainty and managing multi-fields; as well as overcoming terminology barriers and advancing research in the mathematical aspects of tensor field processing. We further note the need for tools and data repositories to encourage faster advances in the field. Our interest in creating and proposing this list is to initiate a discussion about important research issues within the visualization of engineering tensor fields.

Tensor Lines in Engineering: Success, Failure, and Open Questions

Today, product development processes in mechanical engineering are almost entirely carried out via computer-aided simulations. One essential output of these simulations are stress tensors, which are the basis for the dimensioning of the technical parts. The tensors contain information about the strength of internal stresses as well as their principal directions. However, for the analysis they are mostly reduced to scalar key metrics. The motivation of this work is to put the tensorial data more into focus of the analysis and demonstrate its potential for the product development process. In this context we resume a visualization method that has been introduced many years ago, tensor lines. Since tensor lines have been rarely used in visualization applications, they are mostly considered as physically not relevant in the visualization community. In this paper we challenge this point of view by reporting two case studies where tensor lines have been applied in the process of the design of a technical part. While the first case was a real success, we could not reach similar results for the second case. It became clear that the first case cannot be fully generalized to arbitrary settings and there are many more questions to be answered before the full potential of tensor lines can be realized. In this chapter, we review our success story and our failure case and discuss some directions of further research.

Tensor Invariants and Glyph Design

Tensors provide a mathematical language for the description of many physical phenomena. They appear everywhere where the dependence of multiple vector fields is approximated as linear. Due to this generality they occur in various application areas, either as result or intermediate product of simulations. As different as these applications, is the physical meaning and relevance of particular mathematical properties. In this context, domain specific tensor invariants that describe the entities of interest play a crucial role. Due to their importance, we propose to build any tensor visualization upon a set of carefully chosen tensor invariants. In this chapter we focus on glyph-based representations, which still belong to the most frequently used tensor visualization methods. For the effectiveness of such visualizations the right choice of glyphs is essential. This chapter summarizes some common glyphs, mostly with origin in mechanical engineering, and link their interpretation to specific tensor invariants.