Joachim Ohser

3D images of materials structures : processing and analysis

PERFACE INTRODUCTION PRELIMINARIES General Notation Characteristics of Sets Random Sets Fourier Analysis LATTICES, ADJACENCY OF LATTICE POINTS, AND IMAGES Introduction Point Lattices, Digitizations and Pixel Configurations Adjacency and Euler Number The Euler Number of Microstructure Constituents Image Data Rendering IMAGE PROCESSING Fourier Transform of an Image Filtering Segmentation MEASUREMENT OF INTRINSIC VOLUMES AND RELATED QUANTITIES Introduction Intrinsic Volumes Intrinsic Volume Densities Directional Analysis Distances Between Random Sets and Distance Distributions SPECTRAL ANALYSIS Introduction Second-Order Characteristics of a Random Volume Measure Correlations Between Random Structures Second-Order Characteristics of Random Surfaces Second-Order Characteristics of Random Point Fields MODEL-BASED IMAGE ALANYSIS Introduction,Motivation Point Field Models MacroscopicallyHomogeneous Systems of Non-overlapping Particles Macroscopically Homogeneous Systems of Overlapping Particles Macroscopically Homogeneous Fibre Systems Tessellations SIMULATION OF MATERIAL PROPERTIES Introduction Effective Conductivity of Polycrystals by StochasticHomogenization Computation of Effective Elastic Moduli of Porous Media by FEM Simulation REFERENCES INDEX

On the analysis of spatial binary images

This paper deals with the analysis of spatial images taken from microscopically heterogeneous but macroscopically homogeneous microstructures. A new method is presented, which is strictly based on integral‐geometric formulae such as Croftons intersection formulae and Hadwigers recursive definition of the Euler number. By means of this approach the quermassdensities can be expressed as the inner products of two vectors where the first vector carries the ‘integrated local knowledge’ about the microstructure and the second vector depends on the lateral resolution of the image as well as the quadrature rules used in the discretization of the integral‐geometric formulae. As an example of application we consider the analysis of spatial microtomographic images obtained from natural sandstones.

The “sun-effect”: microclimatic alterations predispose forest edges to bark beetle infestations

Bark beetle dispersal and host selection behaviour are a complex and poorly understood process, resulting in specific spatio-temporal infestation patterns in forests. Aerial images from the Bavarian Forest National Park (Germany) provide a high-resolution, that is, tree-scale data set for the period 2001–2010, including information about Ips typographus (Col., Curculio., Scolytinae) infestation, the application of sanitary logging, natural forest edges and the area of living spruce susceptible to bark beetle infestation. We combined methods of GIS and image analysis to investigate the infestation probabilities at three types of forest edges under spatial and temporal aspects and compared them to the corresponding probabilities at the stand interior. Our results showed a pronounced infestation predisposition of such edge trees delimiting infestation patches cleared by sanitary logging measures, in particular at the south-facing edge sector. In contrast, edges adjacent to non-cleared infestation were revealed as less attractive for subsequent infestations, but nonetheless more attractive than permanent forest edges or the stand interior. Additionally, we measured near-bark surface air temperature to determine microclimatic differences at those edge- or non-edge sites and related them to predisposition results. Finally, our study emphasized favourable microclimatic conditions—summarized as the “sun-effect”—as a decisive factor enhancing the local infestation probability at recent forest edges in multiple ways. Both insect- and host tree-related reactions to suddenly altered microclimate are supposed to bias arbitrary colonization behaviour at patch and tree level, thereby mainly explaining observed infestation patterns. From the forester’s point of view, our results may contribute to precise bark beetle risk assessment and thus facilitate decision making in forest management.

Efficient texture analysis of binary images

A new method of determining some characteristics of binary images is proposed based on a special linear filtering. This technique enables estimation of the area fraction, the specific line length and the specific integral of curvature. Furthermore, the specific length of the total projection is obtained, which gives detailed information about the texture of the image. The influence of lateral and directional resolution depending on the size of the applied filter mask is discussed in detail. The technique includes a method of increasing directional resolution for texture analysis while keeping lateral resolution as high as possible.

THE ESTIMATION OF THE EULER-POINCARE CHARACTERISTIC FROM OBSERVATIONS ON PARALLEL SECTIONS

An unbiased estimator of the Euler‐Poincaré characteristic (Euler number) of an arbitrary object or a random structure, respectively, is given. The estimator is based on joint observations of pairs of parallel section profiles. Thus the present paper extends the use of Sterios disector from counting particles to determining the Euler number for a wide class of probes. The correctness of the given formulae is proved with mathematical strictness. Furthermore, the feasibility of the method is illustrated by an example from materials sciences.

Spectral theory for random closed sets and estimating the covariance via frequency space

A spectral theory for stationary random closed sets is developed and provided with a sound mathematical basis. The definition and a proof of the existence of the Bartlett spectrum of a stationary random closed set as well as the proof of a Wiener-Khinchin theorem for the power spectrum are used to two ends. First, well-known second-order characteristics like the covariance can be estimated faster than usual via frequency space. Second, the Bartlett spectrum and the power spectrum can be used as second-order characteristics in frequency space. Examples show that in some cases information about the random closed set is easier to obtain from these characteristics in frequency space than from their real-world counterparts.

The Euler Number Of Discretized Sets - On The Choice Of Adjacency In Homogeneous Lattices

Two approaches for determining the Euler-Poincare characteristic of a set observed on lattice points are considered in the context of image analysis { the integral geometric and the polyhedral approach. Information about the set is assumed to be available on lattice points only. In order to retain properties of the Euler number and to provide a good approximation of the true Euler number of the original set in the Euclidean space, the appropriate choice of adjacency in the lattice for the set and its background is crucial. Adjacencies are defined using tessellations of the whole space into polyhedrons. In R 3 , two new 14 adjacencies are introduced additionally to the well known 6 and 26 adjacencies. For the Euler number of a set and its complement, a consistency relation holds. Each of the pairs of adjacencies (14:1; 14:1), (14:2; 14:2), (6; 26), and (26; 6) is shown to be a pair of complementary adjacencies with respect to this relation. That is, the approximations of the Euler numbers are consistent if the set and its background (complement) are equipped with this pair of adjacencies. Furthermore, sufficient conditions for the correctness of the approximations of the Euler number are given. The analysis of selected microstructures and a simulation study illustrate how the estimated Euler number depends on the chosen adjacency. It also shows that there is not a uniquely best pair of adjacencies with respect to the estimation of the Euler number of a set in Euclidean space.

Analysis of spatial cross-correlations in multi-constituent volume data

We investigate spatial cross‐correlations between two constituents, both belonging to the same microstructure. These investigations are based on two approaches: one via the measurement of the cross‐correlation function and the other uses the spatial distances between the constituents. The cross‐correlation function can be measured using the fast Fourier transform, whereas the distances are determined via the Euclidean distance transform. The characteristics are derived from volume images obtained by synchrotron microtomography. As an example we consider pore formation in metallic foams, knowledge of which is important to control the foam production process. For this example, we discuss the spatial cross‐correlation between the pore space and the blowing agent particles in detail.

Measuring intrinsic volumes in digital 3d images

The intrinsic volumes – in 3d up to constants volume, surface area, integral of mean curvature, and Euler number – are a very useful set of geometric characteristics Combining integral and digital geometry we develop a method for efficient simultanous calculation of the intrinsic volumes of sets observed in binary images In order to achieve consistency in the derived intrinsic volumes for both foreground and background, suitable pairs of discrete connectivities have to be used To make this rigorous, the concepts discretization w.r.t an adjacency system and complementarity of adjacency systems are introduced.

The chord length transform and the segmentation of crossing fibres

Segmentation of crossing fibres is a complex problem of image processing. In the present paper, various solutions are presented basing on tools of morphological image processing. Two new image transforms are introduced – the lineal distance transform and the chord length transform. Both transforms are applied to two‐dimensional images and their results are three‐dimensional images. Thus, the segmentation problem originally formulated for crossing fibres observed in a two‐dimensional image can be reformulated as a segmentation problem in a three‐dimensional image. This can be solved by a segmentation in the three‐dimensional image. Algorithms for the lineal distance transform and the chord length transform are given and their use in image analysis is demonstrated. Furthermore, the chord length distribution function of the foreground of a binary image can efficiently be estimated via the chord length transform.