Claudia Redenbach

Beyond imaging: on the quantitative analysis of tomographic volume data

Abstract Tomographic techniques are a valuable analytical tool as they deliver 3D spatial information on a given specimen. Both computed tomography with high spatial resolution and quantitative volume image analysis have made enormous progress during the last decade. In particular for materials and natural science applications the combination of high-resolution three-dimensional imaging and the subsequent image analysis exploiting the fully preserved spatial structural information yield new and exciting insights. In this paper, field-tested and up-to-date methods for tomographic imaging of microstructures, for processing and for quantitatively analysing three-dimensional images are reviewed. By selected applications from materials research, we shall underline the importance of volume image analysis as a crucial step in order to go beyond the images: it allows determination of spatial cross-correlations between different constituents of a specimen, investigation of orientations or derivation of statistically relevant information such as object size distributions. The core part of this work consists, besides the exemple application scenarios, in the processing chain, the tools and methods used.

3D image analysis and stochastic modelling of open foams

Abstract We present methods for the geometric characterisation and stochastic modelling of open foams based on tomographic 3D image data. In the first step, geometric characteristics of the foam microstructure are estimated from the image. Using these characteristics, a random tessellation model is fitted to the cell system of the foam. The edges of this tessellation then serve as a skeleton for the foams strut system. The focus of the paper is on the correct simulation of the foams locally varying strut thickness in the model. For this purpose, the local strut thickness and the strut profiles are estimated from the image and reproduced in the model using locally adaptable morphology.

Three‐dimensional image analytical detection of intussusceptive pillars in murine lung

A variety of diseases can lead to loss of lung tissue. Currently, this can be treated only symptomatically. In mice, a complete compensatory lung growth within 21 days after resection of the left lung can be observed. Understanding and transferring this concept of compensatory lung growth to humans would greatly improve therapeutic options. Lung growth is always accompanied by a process called angiogenesis forming new capillary blood vessels from preexisting ones. Among the processes during lung growth, the formation of transluminal tissue pillars within the capillary vessels (intussusceptive pillars) is observed. Therefore, pillars can be understood as an indicator for active angiogenesis and microvascular remodelling. Thus, their detection is very valuable when aiming at characterization of compensatory lung growth. In a vascular corrosion cast, these pillars appear as small holes that pierce the vessels. So far, pillars were detected visually only based on 2D images. Our approach relies on high‐resolution synchrotron microcomputed tomographic images. With a voxel size of 370 nm we exploit the spatial information provided by this imaging technique and present the first algorithm to semiautomatically detect intussusceptive pillars. An at least semiautomatic detection is essential in lung research, as manual pillar detection is not feasible due to the complexity and size of the 3D structure. Using our algorithm, several thousands of pillars can be detected and subsequently analysed, e.g. regarding their spatial arrangement, size and shape with an acceptable amount of human interaction. In this paper, we apply our novel pillar detection algorithm to compute pillar densities of different specimens. These are prepared such that they show different growing states. Comparing the corresponding pillar densities allows to investigate lung growth over time.

Second-order comparison of three fundamental tessellation models

We consider three random tessellation models which, due to their analytical tractability, play an important role as reference models for random tessellations: Poisson Voronoi tessellations, Poisson hyperplane tessellations, and STIT tessellations. We present a systematic comparison of second-order properties of these three tessellation models in ℝ2 and ℝ3. If possible, we make use of analytical formulae for the considered characteristics. In cases where explicit formulae are still unknown, the characteristics are studied by simulation.

On the arrangement of cells in planar STIT and Poisson line tessellations

It is well known that the distributions of the interiors of the typical cells of a Poisson line tessellation and a STIT tessellation with the same parameters coincide. In this paper, differences in the arrangement of the cells in these two tessellation models are investigated. In particular, characteristics of the set of cells neighbouring the typical cell are studied, mainly by simulation. Furthermore, the pair-correlation function and several mark correlation functions of the point processes of cell centres are estimated and compared.

Geometric and Mechanical Modeling of Fiber-Reinforced Composites

Micro-computed tomography (µCT) yields three dimensional reconstructions of the microstructures of materials down to a spatial resolution of about 1 µm. Based on the resulting image data, many mechanically relevant geometric parameters can be computed using three dimensional image analysis. These parameters include fiber density, orientation, homogeneity and thickness. We show how to fit stochastic fiber models to this image data. Such models take into account fiber densities, orientations, radii and inhomogeneities. These geometries can be realized, thus enabling numerical homogenization methods based on the Lippmann-Schwinger equations in elasticity. These yield the full elastic tensor and even nonlinear elastic behavior. With appropriate damage models, the material strength can be characterized. Such an approach has various advantages over mechanical testing. For example, it characterizes a material in every direction, instead of only the direction in which a tensile test was performed. Furthermore, material models open the path to virtual material design, where one can use computer experiments to identify the microstructural geometry which best fulfills the requirements in some given application. In this contribution, we demonstrate the entire chain consisting of image analysis, geometric and mechanical modeling for glass fiber-reinforced thermoplastics.

The maximum volume hard subset model for Poisson processes: simulation aspects

We introduce a process of non-intersecting convex particles by thinning a primary particle process such that the remaining particles are mutually non-intersecting and have maximum total volume among all such subsystems. This approach is based on the idea to construct hardcore processes by suitable dependent thinnings proposed by Matérn but generates packings with higher volume fractions than the known thinning models. Due to the enormous complexity of the computations involved, we develop a two-phase heuristic algorithm whose first phase turns out to yield a structure of Matérn III type. We focus mainly on the generation of packings with high volume fractions and present some simulation results for Poisson primary particle processes of equally sized balls in ℝ2 and ℝ3. The results are compared with the well-known random sequential adsorption model and Matérn type models.

Random Tessellations and their Application to the Modelling of Cellular Materials

This chapter introduces various tessellation models and discusses their application as models for cellular materials. First, the notion of a random tessellation, the most well-known model types (Voronoi and Laguerre tessellations, hyperplane tessellations, STIT tessellations), and their basic geometric characteristics are introduced. Assuming that a cellular material is a realisation of a suitable random tessellation model, these characteristics can be estimated from 3D images of the material. It is explained how estimates are obtained and how these characteristics can be used to fit tessellation models to the observed structure. All analysis and modelling steps are illustrated using the example of an open cell aluminium foam.

Parameter estimation for growth interaction processes using spatio-temporal information

Methods for the parameter estimation for a spatio-temporal marked point process model, the so-called growth-interaction model, are investigated. Least squares estimation methods for this model found in the literature are only concerned with fitting the mark distribution observed in the data. These methods are unable to distinguish between models which have the same birth, death, interaction and growth functions and parameters but different arrival strategies for the points. Hence, they are extended such that the spatial structure of a point pattern is also taken into account. The suggested methods are evaluated in a simulation study and applied to a small data set from forestry.

Cell shape analysis of random tessellations based on Minkowski tensors

To which degree are shape indices of individual cells of a tessellation characteristic for the stochastic process that generates them? Within the context of stochastic geometry and the physics of disordered materials, this corresponds to the question of relationships between different stochastic models. In the context of image analysis of synthetic and biological materials, this question is central to the problem of inferring information about formation processes from spatial measurements of resulting random structures. We address this question by a theory-based simulation study of shape indices derived from Minkowski tensors for a variety of tessellation models. We focus on the relationship between two indices: an isoperimetric ratio of the empirical averages of cell volume and area and the cell elongation quantified by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for these quantities, as well as for distributions thereof and for correlations of cell shape and volume, are presented for Voronoi mosaics of the Poisson point process, determinantal and permanental point processes, and Gibbs hard-core and random sequential absorption processes as well as for Laguerre tessellations of polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data are complemented by mechanically stable crystalline sphere and disordered ellipsoid packings and area-minimising foam models. We find that shape indices of individual cells are not sufficient to unambiguously identify the generating process even amongst this limited set of processes. However, we identify significant differences of the shape indices between many of these tessellation models. Given a realization of a tessellation, these shape indices can narrow the choice of possible generating processes, providing a powerful tool which can be further strengthened by density-resolved volume-shape correlations.