Crease surfaces: from theory to extraction and application to diffusion tensor MRI

Diffusion MRI visualization

Modern diffusion magnetic resonance imaging (dMRI) acquires intricate volume datasets and biological meaning can only be found in the relationship between its different measurements. Suitable strategies for visualizing these complicated data have been key to interpretation by physicians and neuroscientists, for drawing conclusions on brain connectivity and for quality control. This article provides an overview of visualization solutions that have been proposed to date, ranging from basic grayscale and color encodings to glyph representations and renderings of fiber tractography. A particular focus is on ongoing and possible future developments in dMRI visualization, including comparative, uncertainty, interactive and dense visualizations.

SurfCut: Surfaces of Minimal Paths from Topological Structures

We present SurfCut, an algorithm for extracting a smooth, simple surface with an unknown 3D curve boundary from a noisy 3D image and a seed point. Our method is built on the novel observation that ridge curves of the Euclidean length of minimal paths ending on a level set of the solution of the eikonal equation lie on the surface. Our method extracts these ridges and cuts them to form the surface boundary. Our surface extraction algorithm is built on the novel observation that the surface lies in a valley of the eikonal equation solution. The resulting surface is a collection of minimal paths. Using the framework of cubical complexes and Morse theory, we design algorithms to extract ridges and valleys robustly. Experiments on three 3D datasets show the robustness of our method, and that it achieves higher accuracy with lower computational cost than state-of-the-art.

DT-MRI Streamsurfaces Revisited

DT-MRI streamsurfaces, defined as surfaces that are everywhere tangential to the major and medium eigenvector fields, have been proposed as a tool for visualizing regions of predominantly planar behavior in diffusion tensor MRI. Even though it has long been known that their construction assumes that the involved eigenvector fields satisfy an integrability condition, it has never been tested systematically whether this condition is met in real-world data. We introduce a suitable and efficiently computable test to the visualization literature, demonstrate that it can be used to distinguish integrable from nonintegrable configurations in simulations, and apply it to whole-brain datasets of 15 healthy subjects. We conclude that streamsurface integrability is approximately satisfied in a substantial part of the brain, but not everywhere, including some regions of planarity. As a consequence, algorithms for streamsurface extraction should explicitly test local integrability. Finally, we propose a novel patch-based approch to streamsurface visualization that reduces visual artifacts, and is shown to more fully sample the extent of streamsurfaces.

Dancing Styles of Collective Cell Migration: Image-Based Computational Analysis of JRAB/MICAL-L2

Collective cell migration is observed during morphogenesis, angiogenesis, and wound healing, and this type of cell migration also contributes to efficient metastasis in some kinds of cancers. Because collectively migrating cells are much better organized than a random assemblage of individual cells, there seems to be a kind of order in migrating clusters. Extensive research has identified a large number of molecules involved in collective cell migration, and these factors have been analyzed using dramatic advances in imaging technology. To date, however, it remains unclear how myriad cells are integrated as a single unit. Recently, we observed unbalanced collective cell migrations that can be likened to either precision dancing or awa-odori, Japanese traditional dancing similar to the style at Rio Carnival, caused by the impairment of the conformational change of JRAB/MICAL-L2. This review begins with a brief history of image-based computational analyses on cell migration, explains why quantitative analysis of the stylization of collective cell behavior is difficult, and finally introduces our recent work on JRAB/MICAL-L2 as a successful example of the multidisciplinary approach combining cell biology, live imaging, and computational biology. In combination, these methods have enabled quantitative evaluations of the “dancing style” of collective cell migration.

Quantifying the brain's sheet structure with normalized convolution

The hypothesis that brain pathways form 2D sheet-like structures layered in 3D as "pages of a book" has been a topic of debate in the recent literature. This hypothesis was mainly supported by a qualitative evaluation of "path neighborhoods" reconstructed with diffusion MRI (dMRI) tractography. Notwithstanding the potentially important implications of the sheet structure hypothesis for our understanding of brain structure and development, it is still considered controversial by many for lack of quantitative analysis. A means to quantify sheet structure is therefore necessary to reliably investigate its occurrence in the brain. Previous work has proposed the Lie bracket as a quantitative indicator of sheet structure, which could be computed by reconstructing path neighborhoods from the peak orientations of dMRI orientation density functions. Robust estimation of the Lie bracket, however, is challenging due to high noise levels and missing peak orientations. We propose a novel method to estimate the Lie bracket that does not involve the reconstruction of path neighborhoods with tractography. This method requires the computation of derivatives of the fiber peak orientations, for which we adopt an approach called normalized convolution. With simulations and experimental data we show that the new approach is more robust with respect to missing peaks and noise. We also demonstrate that the method is able to quantify to what extent sheet structure is supported for dMRI data of different species, acquired with different scanners, diffusion weightings, dMRI sampling schemes, and spatial resolutions. The proposed method can also be used with directional data derived from other techniques than dMRI, which will facilitate further validation of the existence of sheet structure.

Sheet Probability Index (SPI): Characterizing the geometrical organization of the white matter with diffusion MRI

The question whether our brain pathways adhere to a geometric grid structure has been a popular topic of debate in the diffusion imaging and neuroscience societies. Wedeen et al. (2012a, b) proposed that the brain's white matter is organized like parallel sheets of interwoven pathways. Catani et al. (2012) concluded that this grid pattern is most likely an artifact, resulting from methodological biases that cause the tractography pathways to cross in orthogonal angles. To date, ambiguities in the mathematical conditions for a sheet structure to exist (e.g. its relation to orthogonal angles) combined with the lack of extensive quantitative evidence have prevented wide acceptance of the hypothesis. In this work, we formalize the relevant terminology and recapitulate the condition for a sheet structure to exist. Note that this condition is not related to the presence or absence of orthogonal crossing fibers, and that sheet structure is defined formally as a surface formed by two sets of interwoven pathways intersecting at arbitrary angles within the surface. To quantify the existence of sheet structure, we present a novel framework to compute the sheet probability index (SPI), which reflects the presence of sheet structure in discrete orientation data (e.g. fiber peaks derived from diffusion MRI). With simulation experiments we investigate the effect of spatial resolution, curvature of the fiber pathways, and measurement noise on the ability to detect sheet structure. In real diffusion MRI data experiments we can identify various regions where the data supports sheet structure (high SPI values), but also areas where the data does not support sheet structure (low SPI values) or where no reliable conclusion can be drawn. Several areas with high SPI values were found to be consistent across subjects, across multiple data sets obtained with different scanners, resolutions, and degrees of diffusion weighting, and across various modeling techniques. Under the strong assumption that the diffusion MRI peaks reflect true axons, our results would therefore indicate that pathways do not form sheet structures at every crossing fiber region but instead at well-defined locations in the brain. With this framework, sheet structure location, extent, and orientation could potentially serve as new structural features of brain tissue. The proposed method can be extended to quantify sheet structure in directional data obtained with techniques other than diffusion MRI, which is essential for further validation.

Feature Surfaces in Symmetric Tensor Fields Based on Eigenvalue Manifold

Three-dimensional symmetric tensor fields have a wide range of applications in solid and fluid mechanics. Recent advances in the (topological) analysis of 3D symmetric tensor fields focus on degenerate tensors which form curves. In this paper, we introduce a number of feature surfaces, such as neutral surfaces and traceless surfaces, into tensor field analysis, based on the notion of eigenvalue manifold. Neutral surfaces are the boundary between linear tensors and planar tensors, and the traceless surfaces are the boundary between tensors of positive traces and those of negative traces. Degenerate curves, neutral surfaces, and traceless surfaces together form a partition of the eigenvalue manifold, which provides a more complete tensor field analysis than degenerate curves alone. We also extract and visualize the isosurfaces of tensor modes, tensor isotropy, and tensor magnitude, which we have found useful for domain applications in fluid and solid mechanics. Extracting neutral and traceless surfaces using the Marching Tetrahedra method can cause the loss of geometric and topological details, which can lead to false physical interpretation. To robustly extract neutral surfaces and traceless surfaces, we develop a polynomial description of them which enables us to borrow techniques from algebraic surface extraction, a topic well-researched by the computer-aided design (CAD) community as well as the algebraic geometry community. In addition, we adapt the surface extraction technique, called A-patches, to improve the speed of finding degenerate curves. Finally, we apply our analysis to data from solid and fluid mechanics as well as scalar field analysis.

Development and Analysis of Patient-Based Complete Conducting Airways Models

The analysis of high-resolution computed tomography (CT) images of the lung is dependent on inter-subject differences in airway geometry. The application of computational models in understanding the significance of these differences has previously been shown to be a useful tool in biomedical research. Studies using image-based geometries alone are limited to the analysis of the central airways, down to generation 6-10, as other airways are not visible on high-resolution CT. However, airways distal to this, often termed the small airways, are known to play a crucial role in common airway diseases such as asthma and chronic obstructive pulmonary disease (COPD). Other studies have incorporated an algorithmic approach to extrapolate CT segmented airways in order to obtain a complete conducting airway tree down to the level of the acinus. These models have typically been used for mechanistic studies, but also have the potential to be used in a patient-specific setting. In the current study, an image analysis and modelling pipeline was developed and applied to a number of healthy (n = 11) and asthmatic (n = 24) CT patient scans to produce complete patient-based airway models to the acinar level (mean terminal generation 15.8 ± 0.47). The resulting models are analysed in terms of morphometric properties and seen to be consistent with previous work. A number of global clinical lung function measures are compared to resistance predictions in the models to assess their suitability for use in a patient-specific setting. We show a significant difference (p < 0.01) in airways resistance at all tested flow rates in complete airway trees built using CT data from severe asthmatics (GINA 3-5) versus healthy subjects. Further, model predictions of airways resistance at all flow rates are shown to correlate with patient forced expiratory volume in one second (FEV1) (Spearman ρ = -0.65, p < 0.001) and, at low flow rates (0.00017 L/s), FEV1 over forced vital capacity (FEV1/FVC) (ρ = -0.58, p < 0.001). We conclude that the pipeline and anatomical models can be used directly in mechanistic modelling studies and can form the basis for future patient-based modelling studies.

Surface-Based Structure Analysis and Visualization for Multifield Time-Varying Datasets

This paper introduces a new feature analysis and visualization method for multifield datasets. Our approach applies a surface-centric model to characterize salient features and form an effective, schematic representation of the data. We propose a simple, geometrically motivated, multifield feature definition. This definition relies on an iterative algorithm that applies existing theory of skeleton derivation to fuse the structures from the constitutive fields into a coherent data description, while addressing noise and spurious details. This paper also presents a new method for non-rigid surface registration between the surfaces of consecutive time steps. This matching is used in conjunction with clustering to discover the interaction patterns between the different fields and their evolution over time. We document the unified visual analysis achieved by our method in the context of several multifield problems from large-scale time-varying simulations.

On mesh-free valley surface extraction with application to low frequency sound simulation

Crease surfaces describe extremal structures of 3D scalar fields. We present a new region-growing-based approach to the meshless extraction of adaptive nonmanifold valley and ridge surfaces that overcomes limitations of previous approaches by decoupling point seeding and triangulation of the surface. Our method is capable of extracting valley surface skeletons as connected minimum structures. As our algorithm is inherently mesh-free and curvature adaptive, it is suitable for surface construction in fields with an arbitrary neighborhood structure. As an application for insightful visualization with valley surfaces, we choose a low frequency acoustics simulation. We use our valley surface construction approach to visualize the resulting complex-valued scalar pressure field for arbitrary frequencies to identify regions of sound cancellation. This provides an expressive visualization of the topology of wave node and antinode structures in simulated acoustics.

Interactive separating streak surfaces

Streak surfaces are among the most important features to support 3D unsteady flow exploration, but they are also among the computationally most demanding. Furthermore, to enable a feature driven analysis of the flow, one is mainly interested in streak surfaces that show separation profiles and thus detect unstable manifolds in the flow. The computation of such separation surfaces requires to place seeding structures at the separation locations and to let the structures move correspondingly to these locations in the unsteady flow. Since only little knowledge exists about the time evolution of separating streak surfaces, at this time, an automated exploration of 3D unsteady flows using such surfaces is not feasible. Therefore, in this paper we present an interactive approach for the visual analysis of separating streak surfaces. Our method draws upon recent work on the extraction of Lagrangian coherent structures (LCS) and the real-time visualization of streak surfaces on the GPU. We propose an interactive technique for computing ridges in the finite time Lyapunov exponent (FTLE) field at each time step, and we use these ridges as seeding structures to track streak surfaces in the time-varying flow. By showing separation surfaces in combination with particle trajectories, and by letting the user interactively change seeding parameters such as particle density and position, visually guided exploration of separation profiles in 3D is provided. To the best of our knowledge, this is the first time that the reconstruction and display of semantic separable surfaces in 3D unsteady flows can be performed interactively, giving rise to new possibilities for gaining insight into complex flow phenomena.

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.

Sampling and visualizing creases with scale-space particles

Particle systems have gained importance as a methodology for sampling implicit surfaces and segmented objects to improve mesh generation and shape analysis. We propose that particle systems have a significantly more general role in sampling structure from unsegmented data. We describe a particle system that computes samplings of crease features (i.e. ridges and valleys, as lines or surfaces) that effectively represent many anatomical structures in scanned medical data. Because structure naturally exists at a range of sizes relative to the image resolution, computer vision has developed the theory of scale-space, which considers an n-D image as an (n+1)-D stack of images at different blurring levels. Our scale-space particles move through continuous four-dimensional scale-space according to spatial constraints imposed by the crease features, a particle-image energy that draws particles towards scales of maximal feature strength, and an inter-particle energy that controls sampling density in space and scale. To make scale-space practical for large three-dimensional data, we present a spline-based interpolation across scale from a small number of pre-computed blurrings at optimally selected scales. The configuration of the particle system is visualized with tensor glyphs that display information about the local Hessian of the image, and the scale of the particle. We use scale-space particles to sample the complex three-dimensional branching structure of airways in lung CT, and the major white matter structures in brain DTI.