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Keywords:

  • 3D image;
  • correlation analysis;
  • volume image;
  • intrinsic volumes;
  • local fibre direction;
  • microstructure

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Microcomputed tomography
  5. Segmentation
  6. Scope of quantitative analysis
  7. Use cases
  8. Outlook
  9. Acknowledgments
  10. References

In this paper, the field of quantitative microcomputed tomography arising from the combination of microcomputed tomography and quantitative 3D image analysis, is summarized with focus on materials science applications. Opportunities and limitations as well as typical application scenarios are discussed. Selected examples provide an insight into commonly used as well as recent methods from mathematical morphology and stochastic geometry.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Microcomputed tomography
  5. Segmentation
  6. Scope of quantitative analysis
  7. Use cases
  8. Outlook
  9. Acknowledgments
  10. References

Quantitative microcomputed tomography is concerned with the quantitative characterization of microstructures based on 3D image data obtained by microcomputed tomography (μCT). Three examples are shown in Figure 1. Visualization of the 3D images is of course indispensable and in many cases yields new insight already. Quantitative microcomputed tomography yields however a quantitative description and thus goes beyond pure visualization.

image

Figure 1. Reconstructed μCT images of material micro-structures. Left: Refractory concrete, dark – pores, light – alumina inclusions. The sample from TU Bergakademie Freiberg was imaged at Fraunhofer IZFP with a pixel size of 47μm resulting in sample edge length 11.7 mm. Centre: Aluminium foam in early foaming stage imaged using synchrotron radiation by L. Helfen at ESRF. Pixel size is 0.7 μm, sample edge length 4.6 mm. Visualised are the pores (red) and leftovers of the foaming agent (green). Right: Silica gel fibers with color coded local fiber directions. The sample from Fraunhofer ISC was imaged at Fraunhofer ITWM. Pixel size is 6 μm, sample size approx. 6 mm × 6 mm × 1.5 mm.

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One way of geometric characterization is metrology. That is, dimensioning of parts and comparison between actual and nominal dimensions of a part, where the former are measured in the reconstructed μCT image and the latter are, for example, given by CAD data. Opposite to metrology, quantitative μCT is focused on microscopically heterogeneous structures which are described mathematically by random sets. In this sense quantitative μCT extends classical materialography to 3D. Although classical materialography suffered from many restrictions as it is based on 2D images, quantitative μCT can exploit the full spatial information contained in 3D images. This allows, for example, detailed directional analyses, estimation of particle size distributions without shape assumptions, or judging the 3D connectivity of a structure, just to name a few.

Moreover, macroscopic materials properties like mechanical strength, permeability, or acoustic absorption, can be simulated in the 3D images or in geometric models fit to the microstructure. On the other hand, many of the image processing and analysis algorithms applied are algorithmically and computationally one magnitude more complex than in 2D. Even worse, basic image processing concepts like neighbourhoods have to be rethought when moving on to the third dimension. Nevertheless, quantitative μCT is able to answer new questions, to analyse more precisely, and to characterize the microstructures of very weak or highly porous materials for the first time.

Typical questions arising from applications are for instance: Do pores in foams first arise where the foaming agent is situated? Do the reinforcing fibres in a composite obey the desired directional distribution? Is the grain size distribution in a powder homogeneous? To answer them, the whole chain consisting of μCT imaging, tomographic reconstruction, image processing, and quantitative analysis has to be completed. Visualization of the 3D image data is indispensable when dealing with unknown materials or new analysis tasks. Most analysis methods require image segmentation – the identification of the microstructure component or object of interest – as a prerequisite. Finally the processing chain might be completed by geometric modelling and numerical simulation of macroscopic properties.

Microcomputed tomography

  1. Top of page
  2. Summary
  3. Introduction
  4. Microcomputed tomography
  5. Segmentation
  6. Scope of quantitative analysis
  7. Use cases
  8. Outlook
  9. Acknowledgments
  10. References

Computed tomography (CT) in the strict sense denotes nondestructive imaging techniques reconstructing 3D image data from projection images using tomographic reconstruction. The reconstructed CT image consists of grey values assigned to the vertices in a – usually cubic – lattice in 3D, intersected by a cuboidal observation window. The vertices of the lattice are the pixels (often also called voxels). Alternatively, pixel names the cubic Voronoi cell generated by a vertex. The lattice spacing is therefore often called pixel edge length or pixel size for short.

CT in the wider sense comprises methods using information beyond projection images like 3D X-ray diffraction and even destructive methods like 3D atom probe, too, see Banhart (2008). Here, we stick to the strict definition. The term microcomputed tomography (μCT) stands for high resolution CT with pixel sizes in the micrometre range.

The rapid development of CT devices over the last 10 years resulted in powerful systems enabling 3D imaging of samples from complete cars to living flies. Laboratory μCT broke the long-term barrier of 1 μm pixel edge length. So-called nano-CT laboratory devices allow pixel sizes down to 250 nm. All laboratory devices use absorption contrast. That is, radiographic projection images of the linear X-ray attenuation coefficient from a set of different projection angles are used to reconstruct the mass distribution within the sample.

μCT with synchrotron radiation can take advantage of the electron flux being by an order of magnitude higher than in lab devices. The flux enables to mono-chromatize the radiation thus not only improving the signal-to-noise ratio considerably but rendering alternative contrast regimes possible, too. In particular, phase-contrast and holotomography exploiting the local electron density yield emphasized interfaces and grey value differences between components not distinguishable by just the absorption contrast. See Banhart (2008) for details and Fig. 2 for an example.

image

Figure 2. Rack et al. (2008a): Corresponding slices through μCT images of an magnesium–aluminium alloy. From left to right: absorption contrast, phase-contrast tomography a holotomography. Reprinted with permission by Elsevier. See also Rack et al. (2008b).

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Consequently, the major restriction for microstructure analysis based on μCT is neither insufficient resolution nor insufficient contrast but rather the size of the sample that can be imaged. A pixel size of 1 μm results in a cylindrical sample of 2-mm diameter and 2-mm height. This hampers, for example, measuring the length distribution of glass fibres (typically 10-μm thick, several millimetres long) or the size distribution of particles from different scales. Remedies like extending the measurement range have been developed but are not yet widely available.

Segmentation

  1. Top of page
  2. Summary
  3. Introduction
  4. Microcomputed tomography
  5. Segmentation
  6. Scope of quantitative analysis
  7. Use cases
  8. Outlook
  9. Acknowledgments
  10. References

Segmenting a grey value image means either to find the component or image segment of interest or to identify connected objects or regions. The first type of segmentation is often called binarization as it results in a binary (black-and-white) image having the segment of interest as foreground. Segmentation is a crucial step in quantitative μCT since most analysis methods work on segmented images. However, segmentation is a difficult and in the mathematical sense ill-posed problem. A perfect solution for all applications is not in sight. In medical applications, expert interaction is desired and in many cases indispensable. Tailor-suited solutions for one organ like skull or liver prevail. For materials science automatic segmentation is preferred for its objectivity and the reproducibility of results.

Specific problems with image data of dimension higher than two are due to the difficult and sometimes even misleading verification of results which is often just done visually in 2D. For the same reason, interactive segmentation is much more demanding in 3D. Finally, 3D segmentation algorithms have to be much more efficient to ensure immediate response. Interactive segmentation in 3D combines low level segmentation, for example, by the watershed transform with graph cuts or statistical learning. Automatic segmentation typically uses thresholding and region growing schemes.

Specific problems with CT image data are the imaging artefacts inherent to this set of imaging techniques, most prominent ring artefacts, metal artefacts and beam hardening artefacts. Ideally, these types of artefacts are avoided by optimizing the imaging conditions or corrected for during reconstruction, see Banhart (2008). However, preprocessing can help to suppress them, too. The loss of grey value intensity towards the sample centre due to beam hardening can be corrected by shading correction methods. Ring artefacts can be weakened by a morphological opening with a structuring element parallel to the axis of sample rotation, because they are due to pixel defects in the detector and thus perpendicular to this axis.

Scope of quantitative analysis

  1. Top of page
  2. Summary
  3. Introduction
  4. Microcomputed tomography
  5. Segmentation
  6. Scope of quantitative analysis
  7. Use cases
  8. Outlook
  9. Acknowledgments
  10. References

Geometric characterization

For the actual quantitative analysis of 3D image data a wide variety of methods is available, whose choice depends on both the problem and the type of microstructure – its invariance properties, the number of components it consists of and on whether it has a natural object structure or not. The easiest case is a macroscopically homogeneous and isotropic system of particles. Macroscopically homogeneous roughly means that the random structure observed in the sample does on average not depend on the position of the sample within the specimen. Isotropic are structures which are – again on average – invariant with respect to rotations of the sample, too.

For a system of separate objects like the pores in the refractory concrete sample from Fig. 3, the typical procedure starts with a binarization yielding the object system as foreground, followed by a labelling which assigns to each pixel the number of the object it belongs to. See Fig. 3 for the labelled pores. Subsequently, size and shape of each object can be characterized by measuring geometric features like volume or diameter or by fitting regular bodies like cuboids or ellipsoids.

image

Figure 3. Slices through the reconstructed image of the refractory concrete from Fig. 1. Left: Original gray value image. Centre: Binarized, black – pores, white – matrix and alumina inclusions. Right: Labeled pores, colors just code object numbers.

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Correlations between microstructure components can be quantified by distance methods, the set covariance, n-point correlation functions (Torquato 2002). The spatial arrangement of systems of objects can be described by spatial statistics applied, for example, to the object centres.

Larger samples imaged at higher resolutions result in more and more inhomogeneous samples demanding additional local analysis. This can be achieved by tessellating the sample by subvolumes, sliding a subvolume through the sample or assigning local measurement values to each pixel. Hilfer (2000) defines local porosity in each pixel as the porosity in a cubic subvolume centred at this pixel. The spherical granulometry assigns to each pixel the size of the largest ball inscribed in the structure and covering it. This morphological size measurement can be interpreted as local pore size or wall thickness. Local fibre directions can be calculated based on grey-value gradients, their derivatives or anisotropic filters (Wirjadi et al. 2009; Krause et al. 2010).

Nonadditive functionals like the diameter of percolating pores or the geometric tortuosity can be estimated based on 3D image data, too.

Materials properties

The microstructure of a material accounts strongly for its properties like thermal conductivity, elasticity, or filtration efficiency. Image analysis can provide microstructure information needed as input for dimensioning at the macroscale. The most prominent example are fibre orientation tensors obtained by averaging local fibre directions over subvolumes of the 3D image (Wirjadi et al. 2009). Usually however, numerical simulations at the microscale yield the properties which in turn serve as input for macroscale simulations. This approach is called homogenization since the partial differential equations describing the respective property have to be solved. Simulations can be based directly on segmented 3D image data like the lattice Boltzmann methods for flow phenomena. Finite element methods rely on a meshing of either the surface by triangles or the volume by tetrahedra. Given a proper binarization, a surface mesh is easily obtained, for instance by a marching cube algorithm. These meshes however, consist of millions of triangles. The challenge is therefore to simplify the mesh considerably without removing essential geometric features. Another crucial point is to ensure that the sample size suffices to properly represent the microstructure. This size of the representative volume element can be obtained from the covariance estimated from the 3D image. See (Jeulin 2005) for a comprehensive overview.

The ultimate goal would be of course to predict physical properties directly from geometric features. Due to the complex interplay of chemical composition, molecular structure, microstructure on the one hand, and the multiscale nature of many physical phenomena, this can however be achieved for special cases of materials, microstructures and properties, only.

Microstructure optimization

In order to get a deeper understanding of the relation between microstructure on the one hand and materials properties on the other hand, geometric modelling or reconstruction allows to go one step further. A stochastic geometric model is fitted to the microstructure based on the characteristics measured in the 3D image. Subsequently, changing the model parameters allows selective alternation of the microstructure and to study the effect on the physical property by numerical simulations in realizations of this new microstructure. Iterating this procedure, the microstructure can be optimized with respect to the desired materials property. This approach is sometimes also called virtual materials design. Frequently used models are random systems of cylinders for fibres, random dense packings for granular media, and random Laguerre tessellations for foams (Redenbach 2009).

Use cases

  1. Top of page
  2. Summary
  3. Introduction
  4. Microcomputed tomography
  5. Segmentation
  6. Scope of quantitative analysis
  7. Use cases
  8. Outlook
  9. Acknowledgments
  10. References

The intrinsic volumes and their densities

A very attractive set of basic characteristics are the intrinsic volumes (also Minkowski functionals or quermass integrals) – volume V, surface area S, the integral of mean curvature M and the Euler number χ. For a convex body, M is up to a constant the mean width. The Euler number describes the topology by alternate counting of the connected components, the tunnels, and the holes of the particle. Thus for a convex body χ= 1, for a torus χ= 1 − 1 = 0, and for a sphere χ= 1 + 1 = 2. The intrinsic volumes can be measured very efficiently based on 3D image data (Ohser & Schladitz 2009). For the Euler number estimation procedure see (Toriwaki & Yoshida 2009), too. Useful further characteristics like the isoperimetric shape factors inline image can be derived from them. These shape factors are normalized such that f1=f2=f3= 1 for a ball. Deviations from 1 thus describe various aspects of deviations from ball shape. Other shape characteristics are, for example, the ratio of the volumes of an object and its convex hull characterizing the convexity of the object or the ratios of the edge lengths of the minimal-volume bounding cuboid.

Typically, particles are not separated in the 3D image. For example, the particles in a powder touch and appear connected in the binarized image. Nevertheless, they can be separated following exactly the same image processing route as in 2D – a combination of the Euclidean distance transform with the watershed transform. See Fig. 4 for the results of a straightforward labelling of the multiply connected alumina inclusions and the separation result. Fig. 5 shows analysis results. The very same strategy can be applied successfully to the pore space of open foams in order to reconstruct the foam cells. Even if the sizes of the objects to be separated differ strongly, this procedure still works out given the crucial smoothing step for avoiding oversegmentation is made size-dependent. Grey value morphology lends the necessary tools for that. This method for particle separation fails however for particles like fibres whose shape deviates strongly from ball shape.

image

Figure 4. Slices through the reconstructed image of the refractory concrete from Fig. 1. Left: Binarized, black – pores and matrix, white – alumina inclusions. Centre: Labeled alumina inclusions. Right: Separated alumina inclusions.

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image

Figure 5. Exemplary simple analysis results for the refractory concrete from Fig. 1. Left and centre: Size and shape distributions of alumina inclusions, respectively. Right: Size distribution of pores.

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For components of macroscopically homogeneous microstructures without a natural object structure, the densities of the intrinsic volumes are the basic geometric characteristics. Instead of the absolute values of the four functionals, now their ratio to the sample volume is considered: VV the volume fraction, SV the specific surface area, MV and χV the densities of the integral of mean curvature and the Euler number, respectively. For porous media, the volume fraction is 1-porosity. A shape factor – the structure model index – can be derived, too: fSMI= 4πVVMV/S2V. It assumes values 4, 3 and 0 for ideal systems of nonoverlapping balls, cylinders, and planes, respectively. For the silica fibres it yields 2.57, for the separated alumina inclusions in the refractory concrete 3.89.

Local directional analysis

Fibre systems like the glass fibre component in fibre reinforced composites, nonwoven, or the silica gel fibres from Fig. 1 can be mathematically modelled as an anisotropic randomly oriented fibre system Ξ in 3D Euclidean space inline image. The directional distribution of the fibres is crucial for materials properties like the mechanical strength of fibre reinforced composites or the filtration efficiency of a nonwoven. Thus, based on 3D image data, we want to estimate the direction distribution inline image, where inline image is a measurable set of nonoriented directions, W denotes the observation window and ν(x) is the direction of the fibre in point x. The expectation inline image is taken with respect to the random fibre system Ξ. inline image is the indicator function of the set A, hence inline image if ν∈A and inline image otherwise.

Let gσ be an isotropic Gaussian smoothing kernel in inline image with parameter σ > 0 adjusted to the fibre radius, σ≈r. Now consider the Hessian matrix H(x) of second derivatives of the image f smoothed by gσ:

  • image

The eigenvectors of H carry information about directions of the random field Ξ at x. For a fibrous structure, the least grey value variation is expected along the fibre. Thus the eigenvector to the smallest eigenvalue of the Hessian matrix H(x) at x is interpreted as the local direction ν(x). In Fig. 1 (right), the longitudinal component of the local fibre direction is visualized by assigning a colour map.

Correlation analysis

In this section, an exemplary example for judging spatial correlation between constituents of microstructures is considered. Samples of aluminium foams, produced via the powder foaming method and stopped during early pore formation, are to be analysed with the goal to get better insight into the pore formation. The μCT images clearly show three components – metal matrix, foaming agent TiH2, and pores, see Fig. 6. Do the pores develop where the foaming agent is or is there some other pore generating effect?

image

Figure 6. Volume renderings of aluminium foam samples AlSi7 (left) and AW-6061 (centre). Right: The corresponding functions t indicating strong correlation between foaming agent and pores for AW-6061.

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To answer this type of question, second-order characteristics like the covariance function covV are used. Let Ξ be a macroscopically homogeneous random closed set. The covariance function covV of Ξ depends on the difference xy of the positions x and y only, inline image for inline image and with inline image not depending on x. Torquato (2002) calls the covariance function two-point correlation function.

In the same spirit, the cross-covariance function of two macroscopically homogeneous random closed sets Ξ, Ψ is defined by

  • image

Let now Ξ be the pore system of the aluminium foam. Then the closure of its complement inline image is the solid component – aluminium and TiH2 leftovers. Denote by Ψ the foaming agent TiH2. We are interested in the cross-covariance of Ξ and Ψ. However, Ψ cannot be observed. Instead, we observe the TiH2 particles within the solid component inline image. The sets Ξ and Ψ′ are correlated even if Ξ and Ψ are independent. In the latter case we have however inline image, which is equivalent to

  • image

Thus, the relationship t≡ 1 can serve as a necessary (but not sufficient) criterion for the sets Ξ and Ψ to be independent.

Fig. 6 shows visualizations of foam samples produced using the two different precursor materials AW-6061 and AlSi7. The shapes of t(r) indicate that for the AW-6061 foam sample there is a strong short range interaction between the pore space and the blowing agent particles while we observe a weak dependence for the AlSi7 sample.

Outlook

  1. Top of page
  2. Summary
  3. Introduction
  4. Microcomputed tomography
  5. Segmentation
  6. Scope of quantitative analysis
  7. Use cases
  8. Outlook
  9. Acknowledgments
  10. References

In the future, even larger images will be produced. Detectors with 40962 pixels are already on the market, resulting in an approximate image size of 140 GB at the usual 16bit grey value range for each pixel. Time series of μCT images yield nonscalar pixel values. Moreover, the rapidly increasing number of CT devices in use as well as the rise of high-throughput devices not only strongly increase the number of 3D image data but also demand automatic analysis.

Multicomponent materials, tailored to special applications exhibit more and more intricate microstructures whose correct realization is crucial for functionality and durability of the products made from them. Consequently, more difficult analysis tasks are to be expected. Fusion of image data from different sources might be needed to answer them.

Finally, tomographic techniques different from μCT based on X-ray absorption gain importance. This comprises both tomographic techniques in the strict sense like electron tomography as well as destructive methods like FIB-tomography, serial block-face scanning electron microscopy (SBF-SEM) or optical coherence tomography based on interferometry. Here FIB-tomography stands for all 3D imaging methods combining sectioning using a focused ion beam and microscopy while SBF-SEM uses an ultra-microtome within the electron microscope for sectioning.

Acknowledgments

  1. Top of page
  2. Summary
  3. Introduction
  4. Microcomputed tomography
  5. Segmentation
  6. Scope of quantitative analysis
  7. Use cases
  8. Outlook
  9. Acknowledgments
  10. References

The author acknowledges support by (CM)2 Center for Mathematical + Computational Modelling and by the German Federal Ministry of Education and Research through project 03MS603D. The author thanks Alexander Rack for providing Fig. 2 and Oliver Wirjadi for helpful comments and Fig. 1(right).

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Microcomputed tomography
  5. Segmentation
  6. Scope of quantitative analysis
  7. Use cases
  8. Outlook
  9. Acknowledgments
  10. References
  • Banhart, J. (Ed.) (2008) Advanced Tomographic Methods in Materials Research and Engineering. Oxford University Press, Oxford .
  • Hilfer, R. (2000) Local porosity theory and stochastic reconstruction. Statistical Physics and Spatial Statistics (ed. by K.R.Mecke & D.Stoyan), vol. 554 of LNP, pp. 203241. Springer, Heidelberg .
  • Jeulin, D. (2005) Random structures in physics. Space, Structure and Randomness (ed. by M.Bilodeau, F.Meyer, & M.Schmitt), vol. 183 of Lecture Notes in Statistics, pp. 183222. Springer, New York . Contributions in Honor of Georges Matheron in the Fields of Geostatistics, Random Sets, and Mathematical Morphology.
  • Krause, M., Hausherr, J., Burgeth, B., Herrmann, C. & Krenkel, W. (2010) Determination of the fibre orientation in composites using the structure tensor and local x-ray transform. J. Mater. Sci. 45(4), 888896.
  • Ohser, J. & Schladitz, K. (2009) 3D Images of Materials Structures – Processing and Analysis. Wiley VCH, Weinheim .
  • Rack, A., Zabler, S., Müller, B. et al. (2008a) High resolution synchrotron-based radiography and tomography using hard X-rays at the bamline (Bessy ii). Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 586(2), 327344.
  • Rack, A., Halfen, L., Baumbach, T., Kirste, S., Banhart, J., Schladitz, K. & Ohser, J. (2008b) Analysis of spatial cross-correlations in multi-constituent volume data. J. Microsc. 232, 282292.
  • Redenbach, C. (2009) Modelling foam structures using random tessellations. Stereology and Image Analysis. Ecs10: Proceeding of the 10th European Conference of ISS (ed. by V.C.  et al.), vol. 4 of The MIRIAM Project Series. ESCULAPIO Pub. Co., Bologna .
  • Toriwaki, J. & Yoshida, H. (2009) Fundamentals of Three-dimensional Digital Image Processing. Springer, Berlin .
  • Torquato, S. (2002) Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New York .
  • Wirjadi, O., Schladitz, K., Rack, A. & Breuel, T. (2009) Applications of anisotropic image filters for computing 2d and 3d-fibre orientations. Stereology and Image Analysis – 10th European Congress of ISS (ed. by V.Capasso, G.Aletti, & A.Micheletti), pp. 107112. Milan, Italy .