## Image Analysis & Stereology

- EDITORIAL

*Katja Schladitz, Claudia Redenbach* - NUMERICAL EXPERIMENTS FOR THE ESTIMATION OF MEAN DENSITIES OF RANDOM SETS

*Federico Camerlenghi, Vincenzo Capasso, Elena Villa*

Many real phenomena may be modelled as random closed sets in ℝd, of different Hausdorff dimensions. The problem of the estimation of pointwise mean densities of absolutely continuous, and spatially inhomogeneous, random sets with Hausdorff dimension n < d, has been the subject of extended mathematical analysis by the authors. In particular, two different kinds of estimators have been recently proposed, the first one is based on the notion of Minkowski content, the second one is a kernel-type estimator generalizing the well-known kernel density estimator for random variables. The specific aim of the present paper is to validate the theoretical results on statistical properties of those estimators by numerical experiments. We provide a set of simulations which illustrates their valuable properties via typical examples of lower dimensional random sets.

- STRUCTURE TENSOR IMAGE FILTERING USING RIEMANNIAN L1 AND L∞ CENTER-OF-MASS

*Jesus Angulo*

Structure tensor images are obtained by a Gaussian smoothing of the dyadic product of gradient image. These images give at each pixel a n×n symmetric positive definite matrix SPD(n), representing the local orientation and the edge information. Processing such images requires appropriate algorithms working on the Riemannian manifold on the SPD(n) matrices. This contribution deals with structure tensor image filtering based on Lp geometric averaging. In particular, L1 center-of-mass (Riemannian median or Fermat-Weber point) and L∞ center-of-mass (Riemannian circumcenter) can be obtained for structure tensors using recently proposed algorithms. Our contribution in this paper is to study the interest of L1 and L∞ Riemannian estimators for structure tensor image processing. In particular, we compare both for two image analysis tasks: (i) structure tensor image denoising; (ii) anomaly detection in structure tensor images.

- A QUASI-LIKELIHOOD APPROACH TO PARAMETER ESTIMATION FOR SIMULATABLE STATISTICAL MODELS

*Markus Baaske, Felix Ballani, Karl Gerald van den Boogaart*

This paper introduces a parameter estimation method for a general class of statistical models. The method exclusively relies on the possibility to conduct simulations for the construction of interpolation-based metamodels of informative empirical characteristics and some subjectively chosen correlation structure of the underlying spatial random process. In the absence of likelihood functions for such statistical models, which is often the case in stochastic geometric modelling, the idea is to follow a quasi-likelihood (QL) approach to construct an optimal estimating function surrogate based on a set of interpolated summary statistics. Solving these estimating equations one can account for both the random errors due to simulations and the uncertainty about the meta-models. Thus, putting the QL approach to parameter estimation into a stochastic simulation setting the proposed method essentially consists of finding roots to a sequence of approximating quasiscore functions. As a simple demonstrating example, the proposed method is applied to a special parameter estimation problem of a planar Boolean model with discs. Here, the quasi-score function has a half-analytical, numerically tractable representation and allows for the comparison of the model parameter estimates found by the simulation-based method and obtained from solving the exact quasi-score equations.

- 3D RECONSTRUCTION OF A MULTISCALE MICROSTRUCTURE BY ANISOTROPIC TESSELLATION MODELS

*Hellen Altendorf, Felix Latourte, Dominique Jeulin, Matthieu Faessel, Lucie Saintoyant*

In the area of tessellation models, there is an intense activity to fully understand the classical models of Voronoi, Laguerre and Johnson-Mehl. Still, these models are all simulations of isotropic growth and are therefore limited to very simple and partly convex cell shapes. The here considered microstructure of martensitic steel has a much more complex and highly non convex cell shape, requiring new tessellation models. This paper presents a new approach for anisotropic tessellation models that resolve to the well-studied cases of Laguerre and Johnson-Mehl for spherical germs. Much better reconstructions can be achieved with these models and thus more realistic microstructure simulations can be produced for materials widely used in industry like martensitic and bainitic steels.

- INTERRUPTED IN-SITU COMPRESSIVE DEFORMATION EXPERIMENTS ON MMC FOAMS IN AN XCT: EXPERIMENTS AND ESTIMATION OF DISPLACEMENT FIELDS

*Katharina Losch, Katja Schladitz, Uta Ballaschk, Harra Berek, Christos G. Aneziris*

The mechanical properties of a metal-matrix composite foam are investigated by interrupted in-situ compressive deformation experiments within an X-ray computed tomography device (XCT). Each in-situ experiment generates a sequence of reconstructed 3D images of the foam microstructure. From these data, the deformation field is estimated by registring the images corresponding to three consecutive steps. To this end, the generic registration framework of the itk software suite is exploited and combined with several image preprocessing steps. Both segmented (binary) images having just two grey values for foreground (strut structure) and background (pore space) and the result of the Euclidean distance transform (EDT) on pore space and solid phase are used. The estimation quality is evaluated based on a sequence of synthetic data sets, where the foam’s microstructure is modelled by a random Laguerre tessellation. For large deformations, a combination of non-rigid registration for the EDT images and partwise-rigid registration on strongly deformed regions of the binary images, yields surprisingly small estimation errors.

- INCLUSION RATIO BASED ESTIMATOR FOR THE MEAN LENGTH OF THE BOOLEAN LINE SEGMENT MODEL WITH AN APPLICATION TO NANOCRYSTALLINE CELLULOSE

*Mikko Niilo-Rämä, Salme Kärkkäinen, Dario Gasbarra, Timo Lappalainen*

A novel estimator for estimating the mean length of fibres is proposed for censored data observed in square shaped windows. Instead of observing the fibre lengths, we observe the ratio between the intensity estimates of minus-sampling and plus-sampling. It is well-known that both intensity estimators are biased. In the current work, we derive the ratio of these biases as a function of the mean length assuming a Boolean line segment model with exponentially distributed lengths and uniformly distributed directions. Having the observed ratio of the intensity estimators, the inverse of the derived function is suggested as a new estimator for the mean length. For this estimator, an approximation of its variance is derived. The accuracies of the approximations are evaluated by means of simulation experiments. The novel method is compared to other methods and applied to real-world industrial data from nanocellulose crystalline.

- QUANTIFICATION OF THE 3D MORPHOLOGY OF THE BONE CELL NETWORK FROM SYNCHROTRON MICRO-CT IMAGES

*Pei Dong, Alexandra Pacureanu, Maria Alejandra Zuluaga, Cécile Olivier, Quentin Grimal, Françoise Peyrin*

In the context of bone diseases research, recent works have highlighted the crucial role of the osteocyte system. This system, hosted in the lacuno-canalicular network (LCN), plays a key role in the bone remodeling process. However, few data are available on the LCN due to the limitations of current microscopy techniques, and have mainly only been obtained from 2D histology sections. Here we present, for the first time, an automatic method to quantify the LCN in 3D from synchrotron radiation micro-tomography images. After segmentation of the LCN, two binary images are generated, one of lacunae (hosting the cell body) and one of canaliculi (small channels linking the lacunae). The binary image of lacunae is labeled, and for each object, lacunar descriptors are extracted after calculating the second order moments and the intrinsic volumes. Furthermore, we propose a specific method to quantify the ramification of canaliculi around each lacuna. To this aim, a signature of the numbers of canaliculi at different distances from the lacunar surface is estimated through the calculation of topological parameters. The proposed method was applied to the 3D SR micro-CT image of a human femoral mid-diaphysis bone sample. Statistical results are reported on 399 lacunae and their surrounding canaliculi.