Minkowski Functional Research Articles

Asymptotic expansion of the expected Minkowski functional for isotropic central limit random fields

AbstractThe Minkowski functionals, including the Euler characteristic statistics, are standard tools for morphological analysis in cosmology. Motivated by cosmic research, we examine the Minkowski functional of the excursion set for an isotropic central limit random field, whose k-point correlation functions (kth-order cumulants) have the same structure as that assumed in cosmic research. Using 3- and 4-point correlation functions, we derive the asymptotic expansions of the Euler characteristic density, which is the building block of the Minkowski functional. The resulting formula reveals the types of non-Gaussianity that cannot be captured by the Minkowski functionals. As an example, we consider an isotropic chi-squared random field and confirm that the asymptotic expansion accurately approximates the true Euler characteristic density.

Minkowski functional evaluation of representative elementary volume of rock microtomography images at multiple resolutions

Pore-scale properties can be obtained by building a reliable digital twin of porous media through the digital rock physics (DRP) workflow. The two prerequisites of DRP are reliable imaging and computing power. Determining a proper image resolution that can reveal the actual pore-scale properties is challenging as there is a trade-off between the resolution and the representative elementary volume (REV). The REV is the smallest volume that reproduces the properties of the whole porous medium. The REV is a function of heterogeneities on the pore scale, the parameter of interest, and the scale range. Although the REV analysis for hydraulic properties is straightforward, it is computationally expensive. This study aims to estimate hydraulic pore-scale properties during REV evaluations by the geometric characterization of porous media using the Minkowski morphological functionals. Two sandstone and one carbonate rock samples were scanned at multiple imaging resolutions by both laboratory and synchrotron tomography. The REVs of various parameters of interest (porosity, permeability, surface area, tortuosity, Minkowski functionals) were computed, and the effect of image resolution and artificial rebinning on the final REV values was examined. After reaching the REV for porosity, the REV for the integrals of mean and total curvature agreed well with the permeability REV for large-volume image sizes. At constant porosity, the Minkowski integrals were found to be indicators for pore throat sizes. We also showed that the properties obtained from the rebinned (or coarsened) images differ entirely from that of actual scans at the same resolution.

Minkowski Functionals in Joint Galaxy Clustering & Weak Lensing Analyses

We investigate the inclusion of clustering maps in a weak lensing Minkowski functional (MF) analysis of DES-like and LSST-like simulations to constrain cosmological parameters. The standard 3x2pt approach to lensing and clustering data uses two-point correlations as its primary statistic; MFs, morphological statistics describing the shape of matter fields, provide additional information for non-Gaussian fields. Previous analyses have studied MFs of lensing convergence maps; in this project we explore their simultaneous application to clustering maps. We employ a simplified linear galaxy bias model, and using a lognormal curved sky measurement and Monte Carlo Markov Chain (MCMC) sampling process for parameter inference, we find that MFs do not yield any information in the $\Omega_{\rm m}$ -- $\sigma_8$ plane not already generated by a 3x2pt analysis. However, we expect that MFs should improve constraining power when nonlinear baryonic and other small-scale effects are taken into account. As with a 3x2pt analysis, we find a significant improvement to constraints when adding clustering data to MF-only and MF$+C_\ell$ shear measurements, and strongly recommend future higher order statistics be measured from both convergence and clustering maps.

Minkowski Functionals of SDSS-III BOSS: Hints of Possible Anisotropy in the Density Field?

Abstract We present measurements of the Minkowski functionals extracted from the SDSS-III BOSS catalogs. After defining the Minkowski functionals, we describe how an unbiased reconstruction of these statistics can be obtained from a field with masked regions and survey boundaries, validating our methodology with Gaussian random fields and mock galaxy snapshot data. From the BOSS galaxy data, we generate a set of four density fields in three dimensions corresponding to the northern and southern skies of LOWZ and CMASS catalogs, smoothing over large scales (Gaussian smoothing scale of 35 Mpc) such that the field is perturbatively non-Gaussian. We extract the Minkowski functionals from each data set separately, and measure their shapes and amplitudes by fitting a Hermite polynomial expansion. For the shape parameter of the Minkowski functional curves a 0, that is related to the bispectrum of the field, we find that the LOWZ-South data presents a systematically lower value of a 0 = −0.080 ± 0.040 than its northern sky counterpart a 0 = 0.032 ± 0.024. Although the significance of this discrepancy is not very high, it potentially indicates some systematics in the data or that the matter density field exhibits anisotropy at low redshift. By assuming a standard isotropic flat ΛCDM cosmology, the amplitudes of the Minkowski functionals from the combination of northern and southern sky data give the constraints Ωc h 2 n s = 0.110 ± 0.006 and 0.111 ± 0.008 for CMASS and LOWZ, respectively, which is in agreement with the Planck ΛCDM best-fit Ωc h 2 n s = 0.116 ± 0.001.

Morphological characterization of carbon nanowalls networks using Minkowski functionals

This paper reports the study and description of the morphology of carbon nanowalls (CNWs) based on fractal analysis and Minkowski functionals. CNWs were synthesized by chemical vapor deposition in capacitively-coupled plasma (CCP-PECVD), at various power values of the high-frequency discharge. The synthesized samples were studied using atomic force microscopy and the acquired data were analyzed using the Gwyddion 2.55 program, corresponding fractal analysis was carried out using the height–height correlation function and the power spectral density function, in addition, the Minkowski functional, which provides information on the morphology of the CNWs, was plotted.

Predicting Resistivity and Permeability of Porous Media Using Minkowski Functionals

Permeability and formation factor are important properties of a porous medium that only depend on pore space geometry, and it has been proposed that these transport properties may be predicted in terms of a set of geometric measures known as Minkowski functionals. The well-known Kozeny–Carman and Archie equations depend on porosity and surface area, which are closely related to two of these measures. The possibility of generalizations including the remaining Minkowski functionals is investigated in this paper. To this end, two-dimensional computer-generated pore spaces covering a wide range of Minkowski functional value combinations are generated. In general, due to Hadwiger’s theorem, any correlation based on any additive measurements cannot be expected to have more predictive power than those based on the Minkowski functionals. We conclude that the permeability and formation factor are not uniquely determined by the Minkowski functionals. Good correlations in terms of appropriately evaluated Minkowski functionals, where microporosity and surface roughness are ignored, can, however, be found. For a large class of random systems, these correlations predict permeability and formation factor with an accuracy of 40% and 20%, respectively.

Minkowski functional characterization and fractal analysis of surfaces of titanium nitride films

The aim of this study is to gain a deeper understanding of the micromorphology characteristics of thin titanium nitride (TiN) films sputtered on glass substrates by using ion beam sputtering (IBS). For that purpose, TiN samples were deposited onto glass substrates from gas mixtures with different contents of molecular nitrogen and argon atoms. Atomic force microscopy (AFM) was used to characterize the surface microtexture of obtained thin films at high magnification. The detailed analysis of AFM images using Minkowski functionals and fractal analysis reveals a significant effect of the preparation conditions on the surface features with non-monotonic dependences. Presented results suggest that non-classical spatial characteristics of the variability of the surface topography, including fractal dimension, corner frequency, roughness, and feature shape and size, can be tuned having control over the relative flow rates of the gas mixture during film deposition.

Fluid demixing kinetics on spherical geometry: power spectrum and Minkowski functional analysis

Dynamic density functional theory calculations of fluid–fluid demixing on spherical geometries are characterized via their angular power spectrum as well as via the Minkowski functionals (MFs) of their binarized fluid density fields. MFs form a complete set of additive, motion invariant and continuous morphological measures sensitive to nonlinear (spatial) correlations. The temporal evolution of the fluid density fields is analyzed for different sphere sizes and mixing compositions. The demixing process in the stages of early spinodal decomposition and consecutive domain growth can be characterized by both methods and a power-law domain growth is evidenced for the MF measures. The average domain size obtained by the structure factor only responds to the late stage domain growth of the demixing process. MFs provide refined insights into the demixing process: they allow the detection of distinct stages in the early spinodal decomposition, provide a precise measure of the relative species composition of the mixture and, most importantly: after a proper rescaling, they allow the detection of a universal demixing behavior for a wide range of mixture fractions and for different sphere sizes.

Minkowski Tensors in Three Dimensions: Probing the Anisotropy Generated by Redshift Space Distortion

Abstract We apply the Minkowski tensor statistics to three-dimensional Gaussian random fields. Minkowski tensors contain information regarding the orientation and shape of excursion sets that is not present in the scalar Minkowski functionals. They can be used to quantify globally preferred directions and also provide information on the mean shape of the subsets of a field. This makes them ideal statistics to measure the anisotropic signal generated by redshift space distortion in the low-redshift matter density field. We review the definition of the Minkowski tensor statistics in three dimensions, focusing on two coordinate invariant quantities, and . We calculate the ensemble average of these 3 × 3 matrices for an isotropic Gaussian random field, finding that they are proportional to products of the identity matrix and a corresponding scalar Minkowski functional. We show how to numerically reconstruct and from discretely sampled fields and apply our algorithm to isotropic Gaussian fields generated from a linear ΛCDM matter power spectrum. We then introduce anisotropy by applying a linear redshift space distortion operator to the matter density field and find that both and exhibit a distinct signal characterized by inequality between their diagonal components. We discuss the physical origin of this signal and how it can be used to constrain the redshift space distortion parameter ϒ ≡ f/b.

BOSS Great Wall: morphology, luminosity, and mass

We study the morphology, luminosity and mass of the superclusters from the BOSS Great Wall (BGW), a recently discovered very rich supercluster complex at the redshift $z = 0.47$. We have employed the Minkowski functionals to quantify supercluster morphology. We calculate supercluster luminosities and masses using two methods. Firstly, we used data about the luminosities and stellar masses of high stellar mass galaxies with $\log(M_*/h^{-1}M_\odot) \geq 11.3$. Secondly, we applied a scaling relation that combines morphological and physical parameters of superclusters to obtain supercluster luminosities, and obtained supercluster masses using the mass-to-light ratios found for local rich superclusters. We find that the BGW superclusters are very elongated systems, with shape parameter values of less than $0.2$. This value is lower than that found for the most elongated local superclusters. The values of the fourth Minkowski functional $V_3$ for the richer BGW superclusters ($V_3 = 7$ and $10$) show that they have a complicated and rich inner structure. We identify two Planck SZ clusters in the BGW superclusters, one in the richest BGW supercluster, and another in one of the poor BGW superclusters. The luminosities of the BGW superclusters are in the range of $1 - 8\times~10^{13}h^{-2}L_\odot$, and masses in the range of $0.4 - 2.1\times~10^{16}h^{-1}M_\odot$. Supercluster luminosities and masses obtained with two methods agree well. We conclude that the BGW is a complex of massive, luminous and large superclusters with very elongated shape. The search and detailed study, including the morphology analysis of the richest superclusters and their complexes from observations and simulations can help us to understand formation and evolution of the cosmic web.

Model-independent analyses of non-Gaussianity in Planck CMB maps using Minkowski functionals

Despite the wealth of Planck results, there are difficulties in disentangling the primordial non-Gaussianity of the Cosmic Microwave Background (CMB) from the secondary and the foreground non-Gaussianity (NG). For each of these forms of NG the lack of complete data introduces model-dependences. Aiming at detecting the NGs of the CMB temperature anisotropy , while paying particular attention to a model-independent quantification of NGs, our analysis is based upon statistical and morphological univariate descriptors, respectively: the probability density function , related to v0, the first Minkowski Functional (MF), and the two other MFs, v1 and v2. From their analytical Gaussian predictions we build the discrepancy functions (k = P, 0, 1, 2) which are applied to an ensemble of 105 CMB realization maps of the CDM model and to the Planck CMB maps. In our analysis we use general Hermite expansions of the up to the 12th order, where the coefficients are explicitly given in terms of cumulants. Assuming hierarchical ordering of the cumulants, we obtain the perturbative expansions generalizing the second order expansions of Matsubara to arbitrary order in the standard deviation for and v0, where the perturbative expansion coefficients are explicitly given in terms of complete Bell polynomials. The comparison of the Hermite expansions and the perturbative expansions is performed for the CDM map sample and the Planck data. We confirm the weak level of non-Gaussianity (1–2)σ of the foreground corrected masked Planck 2015 maps.

Probing the statistical properties of CMB B-mode polarization through Minkowski functionals

The detection of the magnetic type B-mode polarization is the main goal of future cosmic microwave background (CMB) experiments. In the standard model, the B-mode map is a strong non-gaussian field due to the CMB lensing component. Besides the two-point correlation function, the other statistics are also very important to dig the information of the polarization map. In this paper, we employ the Minkowski functionals to study the morphological properties of the lensed B-mode maps. We find that the deviations from Gaussianity are very significant for both full and partial-sky surveys. As an application of the analysis, we investigate the morphological imprints of the foreground residuals in the B-mode map. We find that even for very tiny foreground residuals, the effects on the map can be detected by the Minkowski functional analysis. Therefore, it provides a complementary way to investigate the foreground contaminations in the CMB studies.

A lattice Boltzmann study on the impact of the geometrical properties of porous media on the steady state relative permeabilities on two-phase immiscible flows

In the current paper, the effect of the geometrical characteristics of 2-D porous media on the relative permeability in immiscible two-phase flows is studied. The generation of the different artificial porous media is performed using a Boolean model based on a random distribution of overlapping circles/ellipses, the size and shape of which are chosen to satisfy the specific Minkowski functionals (i.e. volume fraction, solid line contour length, connectivity). The study aims to identify how each different Minkowski functional affects the relative permeability of each phase at various saturations of the non-wetting phase. A 2-D multi-relaxation time (MRT) lattice Boltzmann model (LBM) that can handle high density ratios is employed in the simulation. The relationship between the driving forces G and the relative permeabilities of the two phases for every artificial structure is quantified. It is found that for high non-wetting phase saturations (fully connected flow), a non-linear relationship exists between the non-wetting phase flow rate and the driving force, whilst this relationship becomes linear at higher magnitudes of the latter. The force magnitude required to approach the linear region is highly influenced by the pore size distribution and the connectivity of the solid phase. For lower non-wetting phase saturation values, its relative permeability in the linear regime decreases as the fraction of small pores in the structure increases and the non-wetting phase flow becomes disconnected. A strong influence of the solid phase connectivity is also observed.

Average methods and their applications in differential geometry I

In Minkowski geometry the metric features are based on a compact convex body containing the origin in its interior. This body works as a unit ball and its boundary is formed by the unit vectors. Using one-homogeneous extension we have a so-called Minkowski functional to measure the length of vectors. The half of its square is called the energy function. Under some regularity conditions we can introduce an averaged Euclidean inner product by integrating the Hessian matrix of the energy function on the Minkowskian unit sphere. Changing the origin in the interior of the body we have a collection of Minkowskian unit balls together with Minkowski functionals depending on the base points. It is a kind of special Finsler manifolds called a Funk space. Using the previous method we can associate a Riemannian metric as the collection of the averaged Euclidean inner products belonging to different base points. We investigate this procedure in case of Finsler manifolds in general. Central objects of the associated Riemannian structure will be expressed in terms of the canonical data of the Finsler space. We take one more step forward. Randers spaces will be introduced by averaging of the vertical derivatives of the Finslerian fundamental function. The construction will have a crucial role when we apply the general results to Funk spaces together with some contributions to Brickell’s conjecture on Finsler manifolds with vanishing curvature tensor Q .

Direct Minkowski Functional analysis of large redshift surveys: a new high-speed code tested on the luminous red galaxy Sloan Digital Sky Survey-DR7 catalogue

As deeper galaxy catalogues are soon to come, it becomes even more importantto measure large-scale fluctuations in the catalogues with robust statistics that cover all moments of the galaxy distribution.In this paper we reinforce a direct analysis of galaxy data by employing the Germ-Grain method to calculate thefamily of Minkowski Functionals. We introduce a new code, suitable for the analysis of large data sets without smoothingand without the construction of excursion sets. We provide new tools to measure correlation properties, putting emphasis onexplicitly isolating non-Gaussian correlations with the help of integral-geometric relations. As a first application we present the analysis of large-scale fluctuations in the luminous red galaxy sample of Sloan Digital Sky Survey data release 7 data. We findsignificant deviations from the $\Lambda$ cold dark matter mock catalogues on samples as large as $500h^{-1}$Mpc (more than $3\sigma$)and slight deviations of around $2\sigma$ on $700h^{-1}$Mpc, and we investigate possible sources of these deviations.

Probing the CMB cold spot through local Minkowski functionals

Both the Wilkinson Microwave Anisotropy Probe (WMAP) and Planck missions have reported an extremely cold spot (CS) centered at Galactic coordinate (l = 209°, b = −57°) in the cosmic microwave background map. We study the local non-Gaussianity of the CS by defining local Minkowski functionals. We find that the third Minkowski functional ν2 is quite sensitive to the non-Gaussianity caused by the CS. Compared with random Gaussian simulations, the WMAP CS deviates from Gaussianity at more than a 99% confidence level with a scale of R ∼ 10°. Meanwhile, we find that cosmic texture provides an excellent explanation for these anomalies related to the WMAP CS, which could be further tested by future polarization data.

Predicting the Biomechanical Strength of Proximal Femur Specimens with Minkowski Functionals and Support Vector Regression.

Regional trabecular bone quality estimation for purposes of femoral bone strength prediction is important for improving the clinical assessment of osteoporotic fracture risk. In this study, we explore the ability of 3D Minkowski Functionals derived from multi-detector computed tomography (MDCT) images of proximal femur specimens in predicting their corresponding biomechanical strength. MDCT scans were acquired for 50 proximal femur specimens harvested from human cadavers. An automated volume of interest (VOI)-fitting algorithm was used to define a consistent volume in the femoral head of each specimen. In these VOIs, the trabecular bone micro-architecture was characterized by statistical moments of its BMD distribution and by topological features derived from Minkowski Functionals. A linear multi-regression analysis and a support vector regression (SVR) algorithm with a linear kernel were used to predict the failure load (FL) from the feature sets; the predicted FL was compared to the true FL determined through biomechanical testing. The prediction performance was measured by the root mean square error (RMSE) for each feature set. The best prediction result was obtained from the Minkowski Functional surface used in combination with SVR, which had the lowest prediction error (RMSE = 0.939 ± 0.345) and which was significantly lower than mean BMD (RMSE = 1.075 ± 0.279, p<0.005). Our results indicate that the biomechanical strength prediction can be significantly improved in proximal femur specimens with Minkowski Functionals extracted from on MDCT images used in conjunction with support vector regression.

Classification of small lesions on dynamic breast MRI: Integrating dimension reduction and out-of-sample extension into CADx methodology

While dimension reduction has been previously explored in computer aided diagnosis (CADx) as an alternative to feature selection, previous implementations of its integration into CADx do not ensure strict separation between training and test data required for the machine learning task. This compromises the integrity of the independent test set, which serves as the basis for evaluating classifier performance. We propose, implement and evaluate an improved CADx methodology where strict separation is maintained. This is achieved by subjecting the training data alone to dimension reduction; the test data is subsequently processed with out-of-sample extension methods. Our approach is demonstrated in the research context of classifying small diagnostically challenging lesions annotated on dynamic breast magnetic resonance imaging (MRI) studies. The lesions were dynamically characterized through topological feature vectors derived from Minkowski functionals. These feature vectors were then subject to dimension reduction with different linear and non-linear algorithms applied in conjunction with out-of-sample extension techniques. This was followed by classification through supervised learning with support vector regression. Area under the receiver-operating characteristic curve (AUC) was evaluated as the metric of classifier performance. Of the feature vectors investigated, the best performance was observed with Minkowski functional 'perimeter' while comparable performance was observed with 'area'. Of the dimension reduction algorithms tested with 'perimeter', the best performance was observed with Sammon's mapping (0.84±0.10) while comparable performance was achieved with exploratory observation machine (0.82±0.09) and principal component analysis (0.80±0.10). The results reported in this study with the proposed CADx methodology present a significant improvement over previous results reported with such small lesions on dynamic breast MRI. In particular, non-linear algorithms for dimension reduction exhibited better classification performance than linear approaches, when integrated into our CADx methodology. We also note that while dimension reduction techniques may not necessarily provide an improvement in classification performance over feature selection, they do allow for a higher degree of feature compaction.

Classification of small lesions in dynamic breast MRI: eliminating the need for precise lesion segmentation through spatio-temporal analysis of contrast enhancement

Characterizing the dignity of breast lesions as benign or malignant is specifically difficult for small lesions; they don't exhibit typical characteristics of malignancy and are harder to segment since margins are harder to visualize. Previous attempts at using dynamic or morphologic criteria to classify small lesions (mean lesion diameter of about 1 cm) have not yielded satisfactory results. The goal of this work was to improve the classification performance in such small diagnostically challenging lesions while concurrently eliminating the need for precise lesion segmentation. To this end, we introduce a method for topological characterization of lesion enhancement patterns over time. Three Minkowski Functionals were extracted from all five post-contrast images of sixty annotated lesions on dynamic breast MRI exams. For each Minkowski Functional, topological features extracted from each post-contrast image of the lesions were combined into a high-dimensional texture feature vector. These feature vectors were classified in a machine learning task with support vector regression. For comparison, conventional Haralick texture features derived from gray-level co-occurrence matrices (GLCM) were also used. A new method for extracting thresholded GLCM features was also introduced and investigated here. The best classification performance was observed with Minkowski Functionals area and perimeter, thresholded GLCM features f8 and f9, and conventional GLCM features f4 and f6. However, both Minkowski Functionals and thresholded GLCM achieved such results without lesion segmentation while the performance of GLCM features significantly deteriorated when lesions were not segmented (p < 0.05). This suggests that such advanced spatio-temporal characterization can improve the classification performance achieved in such small lesions, while simultaneously eliminating the need for precise segmentation.

Assessing Methods for Characterising Local and Global Structural and Biomechanical Properties of the Trabecular Bone Network

We apply noval techniques, the Scaling Index Method (SIM), which reveals local topology of the structure, and the Minkowski Functionals (MF), which provide four global topological characteristics, to assess strength of the trabecular network of the human bone. We compare capabilities of these methods with the standard analysis, biomechanical Finite Element Method (FEM) and morphological parameters, in prediction of bone strength and fracture risk. Our study is based on a sample of 151 specimens taken from the trabecular part of human thoracic and lumbar vertebrae in vitro, visualised using µCT imaging (isotropic resolution 26µm) and tested by uniaxial compression. The sample of donors is heterogeneous, consisting of 58 male and 54 female cadavers with a mean age of 80 ±10 years. To estimate the predictive power of the methods, we correlate texture measures derived from µCT images with the maximum compressive strength (MCS) as obtained in biomechanical tests. A linear regression analysis reveals that the failure load estimated by FEM shows the highest correlation with MCS (Pearson's correlation coefficient r=0.76). None of the methods in current study is superior to the FEM: morphometric parameters give r<0.5, global topological characteristics show r=0.73 for the first Minkowski Functional MF₁, which coincides with bone volume fraction BV/TV and r=0.61 for the second Minkowski functional MF₂, which coincides with bone surface BS. Although scaling indices provided by SIM correlate only moderately with MCS (r=0.55), texture measures based on the nonlinear combination of local (SIM) and global (MF) topological characteristics demonstrate high correlation with experimental MCS (r=0.74) and with failure load estimated by FEM (r=0.95). Additional advantage of the proposed texture measures is possibility to reveal the role of the topologically different trabecular structure elements for the bone strength.