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In order to test for non-Gaussianities with respect to scale-dependencies we use so-called surrogate maps, in which possible phase correlations of the Fourier phases of the original WMAP data and simulations, respectively, are destroyed by applying a shuffling scheme to the maps. A statistical comparison of the original maps with the surrogate maps then allows to test for the existence of higher order correlations in the original maps, also and especially on well-defined Fourier modes. Using Minkowski functionals as an image analysis technique we calculate the deviation between the original data and 500 surrogates for different hemispheres in the sky and find ecliptic hemispherical asymmetries between northern and southern ecliptic sky. We find strong deviations from Gaussianity in the WMAP data when considering the low-l range with l = [2,20]. The analysis technique of the scaling indices leads to a slightly lower deviation. Although the underlying foreground reduction methods of the maps differ from each other, we find similar results for the WMAP 7yr ILC map and the WMAP 7yr (needlet-based) NILC map in the low-l range. Our results point once more to a cosmological nature of the signal. For a higher l range with l = [120,300] the results differ between the two image analysis techniques and between the two maps which makes an intrinsic nature of the signal on this l range less likely. When we decrease the size of the analysed sky regions for the low-l study, we do not find signatures of NG in the northern sky. In the south we find individual spots which show deviations from Gaussianity. In addition, we investigate non-Gaussian CMB simulations that depend on the f_NL parameter of the local type. These simulations cannot account for the detected signatures on the low-l range., Comment: 13 pages, 10 figures, accepted for publication in MNRAS

In this study, we investigated the scaling relations between trabecular bone volume fraction (BV/TV) and parameters of the trabecular microstructure at different skeletal sites. Cylindrical bone samples with a diameter of 8mm were harvested from different skeletal sites of 154 human donors in vitro: 87 from the distal radius, 59/69 from the thoracic/lumbar spine, 51 from the femoral neck, and 83 from the greater trochanter. μCT images were obtained with an isotropic spatial resolution of 26μm. BV/TV and trabecular microstructure parameters (TbN, TbTh, TbSp, scaling indices ( and σ of α and αz), and Minkowski Functionals (Surface, Curvature, Euler)) were computed for each sample. The regression coefficient β was determined for each skeletal site as the slope of a linear fit in the double-logarithmic representations of the correlations of BV/TV versus the respective microstructure parameter. Statistically significant correlation coefficients ranging from r=0.36 to r=0.97 were observed for BV/TV versus microstructure parameters, except for Curvature and Euler. The regression coefficients β were 0.19 to 0.23 (TbN), 0.21 to 0.30 (TbTh), -0.28 to -0.24 (TbSp), 0.58 to 0.71 (Surface) and 0.12 to 0.16 ( ), 0.07 to 0.11 ( ), -0.44 to -0.30 (σ(α)), and -0.39 to -0.14 (σ(αz)) at the different skeletal sites. The 95% confidence intervals of β overlapped for almost all microstructure parameters at the different skeletal sites. The scaling relations were independent of vertebral fracture status and similar for subjects aged 60-69, 70-79, and >79years. In conclusion, the bone volume fraction-microstructure scaling relations showed a rather universal character.

According to Wolff's law bone remodels in response to the mechanical stresses it experiences so as to produce a minimal-weight structure that is adapted to its applied stresses. Consequently, the inner bone structure should show signs of adaptation to external forces acting on the bone. To test this paradigm, we investigate the relations between bone volume and structure for the trabecular bone using 3D μCT images taken from two different sites in the femur in vitro, namely from the femoral neck (88 specimens) and femoral trochanter (126 specimens). We determine the local structure of the trabecular network as well as its alignment with the direction of the external force acting on the bone by calculating isotropic (α) and anisotropic scaling indices (αz). Comparing global structure measures derived from the scaling indices (mean, variance) with the bone mass (BV/TV) we find that all correlations obey very accurately power laws with scaling exponents of 0.48 and 0.45 ( ), -1.45 and -1.59 (var(μz)), 0.50 and 0.44 ( ) and -1,47 and -1.32 (var(μz)) (neck and trochanter respectively). Thus, the relations for the isotropic scaling indices turn out to be siteindependent, albeit the mechanical stress to which the femoral neck is exposed is much larger than that for the trochanter. We find, however, differences in the degree of alignment of the trabeculae as reflected by the moments of the distribution of the anisotropic scaling indices. In summary, the mass-structure scaling relations of the bone probes taken from the two different sites of the femur show surprisingly small variations. Thus, a naive interpretation of Wolff's law may not universally valid.

We analyze μ-CT tomographic images of human trabecular bone in vitro. We consider a sample consisting of 201 bone specimens harvested from six different skeletal sites with bone fraction in the range BV/TV e [0.04, 0.075]. Using the local characterization of the bone trabecular network given by isotropic and anisotropic scaling indices, we apply classification algorithms in order to reveal structural similarities in the sample. The classification procedures based on isotropic and anisotropic scaling indices lead to different clustering solutions. This comparison helps revealing interesting site specific structural features connected to the intrinsic anisotropy of the trabecular network.

The development of high-resolution visualisation techniques, such as magnetic resonance (MR) and computer tomography (CT) opens the possibility for non-destructive studies of the inner structure of the objects of different nature and origin. The deterioration of the structure with time, deformations and loss of strength under the load and other processes lead to morphological and topological changes inside materials, which require both qualitative analysis and quantitative evaluation. Assessment and proper description of the topological and morphological characteristics of materials with porous and irregular architecture is of great importance for many scientific and engineering studies. The Scaling Index Method (SIM) is a novel numerical tool for characterising the local topology of an arbitrary structure. By evaluating the local dimensionality of each point, the SIM indicates topologically different substructures: unstructured background, onedimensional (rod-like) and two-dimensional (plate-like) elements. By changing the parameters of the SIM one can distinguish the outer surface from inner points of the structure, describe structures at different scales, and include anisotropic features of the tissue. This method can be applied to both binary and greyscale multidimensional images. To demonstrate the scientific performance of the method we apply numerical techniques based on the SIM to tree-dimensional μCT images of the bone tissue engineering scaffolds (as an example of designed porous structure) and to trabecular bone specimens taken from the human vertebrae in vitro (as an example of biological tissue with very irregular and complicated structure). A proper description of the global and local structural characteristics of the trabecular bone network, which carries and redistributes mechanical load inside the bone, helps to evaluate the deterioration of bone tissue caused by osteoporosis and to predict the most frequent complications of this disease, namely spine and hip fractures. Because of the porous and very irregular architecture of the trabecular bone tissue, a detailed assessment of such a structure requires the use of many texture measures derived from different morphological, biomechanical, topological, and statistical concepts. We show that the Scaling Index Method provides complementary information to the existing wellestablished techniques.

Osteoporosis is a metabolic bone disorder in which bones become brittle and prone to fracture. According to the World Health Organization, it is characterized by the loss of bone mineral density and the deterioration of the bone micro-architecture (Prevention and Management of Osteoporosis, 2003). One of the most important factors in determining the risk of fracture is the bone strength. In clinical practice, the risk of fracture and the effects of drug therapy are assessed using only densitometric techniques as a quantitative measure (Kanis, 2002; Kanis, 2007). Modern high-resolution imaging modalities like High-Resolution Computer Tomography (HRCT) and High-Resolution Magnetic Resonance Imaging (HRMRI) open up new possibilities to improve diagnostic techniques of osteoporosis since they are able to depict the architecture of trabecular bone. In particular, recent advances in MRI technology allow us to obtain images with in-plane spatial resolution as high as 50 μm in vitro and 150 μm in vivo, and a slice thickness of 128 μm in vitro and 280 μm in vivo (Link, et al., 1999; Carballido-Gamio et al., 2006; Wehrli, 2007; Krug et al., 2006). These figures should be compared with the actual thickness of the trabeculae, which ranges from 80 to 200 μm with an average size of approximately 120 μm. It has been shown that these imaging modalities enable the assessment of image data with respect to textural properties to detect structural differences (Boutry et al., 2003; Majumdar et al., 1996; Vieth et al., 2001). HRCT is the preferred methodology to image bones. In contrast to MRI scans, CT scans can easily be calibrated using phantoms, thus rendering images with a standardize grey level distribution. In addition, CT images of the bone are free from the susceptibility artifacts and, in case of in vivo applications, have a less stringent requirement for patients to remain absolutely motionless during the scan as compared to MRI.

According to Wolff's law bone remodels in response to the mechanical stresses it experiences so as to produce a minimal-weight structure that is adapted to its applied stresses. Here, we investigate the relations between bone volume and structure for the trabecular bone using 3D μCT images taken from different skeletal sites in vitro, namely from the distal radii (96 specimens), thoracic (73 specimens) and lumbar vertebrae (78 specimens). We determine the local structure of the trabecular network by calculating isotropic and anisotropic scaling indices (α, αz). These measures have been proven to be able to discriminate rod- from sheet-like structures and to quantify the alignment of structures with respect to a preferential direction as given by the direction of the external force. Comparing global structure measures derived from the scaling indices (mean, standard deviation) with the bone mass (BV/TV) we find that all correlations obey very accurately power laws with scaling exponents of 0.14, 0.12, 0.15 (

We present a model-independent method to test for scale-dependent non-Gaussianities in combination with scaling indices as test statistics. Therefore, surrogate data sets are generated, in which the power spectrum of the original data is preserved, while the higher order correlations are partly randomized by applying a scale-dependent shuffling procedure to the Fourier phases. We apply this method to the Wilkinson Microwave Anisotropy Probe data of the cosmic microwave background and find signatures for non-Gaussianities on large scales. Further tests are required to elucidate the origin of the detected anomalies.

The dynamical onset of lane formation is studied in experiments with binary complex plasmas under microgravity conditions. Small microparticles are driven and penetrate into a cloud of big particles, revealing a strong tendency towards lane formation. The observed time-resolved lane formation process is in good agreement with computer simulations of a binary Yukawa model with Langevin dynamics. The laning is quantified in terms of the anisotropic scaling index, leading to a universal order parameter for driven systems., Comment: 4 pages, 3 figures, movies available at http://www.mpe.mpg.de/pke/lane-formation/

Methods derived from nonlinear time series analyses are applied to three-dimensional point distributions as they are typical in the analysis of the cosmic large scale structure. Using the technique of constrained randomisation we generate for a given data set surrogate data sets which have the same linear properties (power spectrum) as well as the same density amplitude distribution but different morphological features. It is shown that the original data set can be discriminated from the surrogates by analysing the local scaling properties of the point sets as measured by weighted scaling indices.