Geometric Diversity Research Articles

3D Data Integration in the Voxel Domain

Abstract. The interest in 3D data integration is growing as the concept of Spatial Digital Twins gains traction among local governments and stakeholders. Spatial data is typically organized into standardised themes such as buildings, transport, and vegetation, or as ad hoc data products in either 2D or 3D dimensions. However, integrating these data sets into a cohesive 3D model presents several challenges. The foundational data sets may vary in terms of accuracy, resolution, representation, and actuality due to differences in creation standards and procedures. Consequently, inconsistencies often arise within the 3D models, making them difficult to identify and rectify. Many of these data integration issues can be traced back to the 3D vector representation. Vector 3D models exhibit significant geometric diversity, which complicates the computational operations required for examining and validating the models. In this paper, we propose a novel voxel-based approach for integrating 3D data. Voxels, analogous to pixels in 2D raster images, inherit several key properties. We employ a dedicated voxel data structure and a set of operations tailored for this purpose. First, we voxelise vector datasets, converting continuous geometric information into discrete voxel representations. Next, we merge voxel layers using a specialized voxel overlay operation. The resulting 3D model adheres to most validity and integrity requirements. To highlight the flexibility of our voxel data structure, we transformed it into point clouds for visualisation in ArcGIS Pro, enabling the addition of 3D base maps and GIS layers. We then published these datasets to Scene Services, granting public access. This allows users to easily explore the 3D scenes through a web browser, making complex geographic data more accessible. By leveraging voxels, our approach facilitates efficient and accurate 3D data integration, making it a valuable contribution to the field.

Cage-by-cage supramolecular polymerization via panel-decorated triphenylphosphine organic cage

Significant efforts have been dedicated to designing porous organic cage compounds with geometric complexity and topological diversity. However, the use of these cage molecules as premade building units for constructing infinite cage-based superstructures remains unexplored. Here, we report the use of a panel-decorated phosphine organic cage as a special monomer to achieve supramolecular polymerization, resulting in cage-by-cage noncovalent polymers through the synergy of metal-coordination and intercage π–π dimerization. At a monomer concentration of 122 mM, the average degree of polymerization reaches 17, corresponding to a molecular weight of 26 kDa. The obtained cage-based supramolecular polymers can further hierarchically self-assemble into vesicular morphologies or one-dimensional nanofiber architectures. Selective control over the cosolvents can regulate their structural hierarchy and assembled morphology. This approach paves a new way for the construction of cage-based hierarchical assemblies and materials.

Lane Reduction and Queue Length Relationship on Arterial and Collector Roads

Geometric diversity of roads for an example is the narrowing of road segments. The purpose of this research is to determine whether lane reduction leads to significant queue lengths, observe the behaviour of drivers on those road segments, and compare the actual queue length to the queue length generated by PTV Vissim. The data used in this study consists of direct observations during peak hours in the morning, afternoon, and evening, along with simulations using PTV Vissim. The queue results obtained from PTV Vissim were validated using two types of tests. The research findings indicate a significant relationship between queue length and lane reduction during peak hours on workdays. However, during the evening peak hours on weekends, the queue length is not significantly affected by lane reduction. As a potential solution, widening each road segment in PTV Vissim was tested, which increased road capacity and slightly reduced the queues that occurred.

A Deformable Constraint Transport Network for Optimal Aortic Segmentation From CT Images.

Aortic segmentation from computed tomography (CT) is crucial for facilitating aortic intervention, as it enables clinicians to visualize aortic anatomy for diagnosis and measurement. However, aortic segmentation faces the challenge of variable geometry in space, as the geometric diversity of different diseases and the geometric transformations that occur between raw and measured images. Existing constraint-based methods can potentially solve the challenge, but they are hindered by two key issues: inaccurate definition of properties and inappropriate topology of transformation in space. In this paper, we propose a deformable constraint transport network (DCTN). The DCTN adaptively extracts aortic features to define intra-image constrained properties and guides topological implementation in space to constrain inter-image geometric transformation between raw and curved planar reformation (CPR) images. The DCTN contains a deformable attention extractor, a geometry-aware decoder and an optimal transport guider. The extractor generates variable patches that preserve semantic integrity and long-range dependency in long-sequence images. The decoder enhances the perception of geometric texture and semantic features, particularly for low-intensity aortic coarctation and false lumen, which removes background interference. The guider explores the geometric discrepancies between raw and CPR images, constructs probability distributions of discrepancies, and matches them with inter-image transformation to guide geometric topology in space. Experimental studies on 267 aortic subjects and four public datasets show the superiority of our DCTN over 23 methods. The results demonstrate DCTN's advantages in aortic segmentation for different types of aortic disease, for different aortic segments, and in the measurement of clinical indexes.

Novel Knit Structure with Adjustable Tensile Behaviour Based on Combined Weft/Warp Technology

Weft-knitted fabrics are used in a wide range of applications. The tensile behaviour, the geometric diversity and the materials to be used are essential structural parameters that are also economically important. To improve this behaviour, the new approach is to integrate warp-stitch threads into a weft-knitted fabric using conventional weft-knitting machine technology, thus significantly increasing the range of binding possibilities for the adjustment of elasticity in wale direction. The characteristics that can be achieved in this way open up new areas of application, for example in compression textiles.

MeshCL: Towards robust 3D mesh analysis via contrastive learning

Recent years have witnessed the success of deep learning in 3D mesh analysis, including classification and segmentation. Previous works mainly rely on supervised learning, which trains models on the labeled 3D mesh data with the cross-entropy loss. However, in the supervised training, the very limited amount of 3D mesh data restricts the feature’s discriminative power and the model’s robustness. To address or at least alleviate this issue, we propose to regularize the learned representations by associating the data points in feature space via contrastive learning. To achieve this, we first explore mesh samples with diverse geometric structures, and then apply contrastive regularization into the positive and negative pairs sampled from the feature space. In this way, our method, MeshCL, can learn robust feature representations invariant to the geometric deformation for 3D manifold mesh. In detail, to augment the geometric diversity, we examine different data augmentation techniques for mesh data. Furthermore, considering the geometric characteristics of 3D mesh, we develop more efficient positive/negative pair sampling strategies for the dense prediction task. To verify the effectiveness of MeshCL, we study two typical mesh analysis tasks, classification and segmentation. Experimental results demonstrate that MeshCL not only achieves state-of-the-art performance on the two tasks mentioned above but also significantly improves the model’s robustness.

A computational investigation of shock wave train in diverse duct geometries using machine learning-adopted k-ω turbulence model

This research paper presents a thorough examination of shock wave train phenomena in various duct structures, utilizing a machine learning-optimized k-ω model. The focus of the study is to apply machine learning techniques to adapt the constant coefficients of the k-ω turbulence model, improving its accuracy and computational efficiency in capturing the dynamics of shock waves. An important aspect of this investigation is the impact of diverging sections within the duct, specifically how changes in the divergence angle, while maintaining a constant ratio of the exit area to the throat area, affect compressible flow parameters, shock wave positions, and other related characteristics of shock trains. The study systematically explores these effects and provides fresh insights into the behavior of shock wave trains under different divergence conditions. In a departure from the usual practices in supersonic research, this study compares the phenomena of shock wave trains in rectangular and circular duct geometries. The findings contribute valuable data and analyses to a field that has traditionally focused on rectangular configurations. The results indicate that the shape of the duct has a significant influence on the shock wave train, emphasizing the importance of considering geometric diversity in the study of supersonic flows and shock wave phenomena. This research sets the stage for further investigations into the interaction between duct geometry, shock wave propagation, and the dynamics of compressible flows.

Digital core: study of textural heterogeneities in a rock

Relevance. When real fluid flows through channels of limited dimensions, it experiences a constant loss of mechanical energy (hydraulic resistance) due to the action of viscous friction. Hydraulic drag has two components: drag along the length of the flow and local drag. The drag along the length of the flow arises because the fluid encounters a force per unit area perpendicular to the direction of the flow. Local drag results from the viscous interaction between the fluid and the channel walls. Local hydraulic resistances include sudden and gradual constrictions/expansions (diffusers/confusions) of the fluid channel, angular and choke gaps. The value of the hydraulic resistance coefficient depends on the type of channel geometry, conditions of fluid entry (wettability of the rock texture) and on the flow regime. When fluid flows through channels of different cross-section, as the channel narrows, the flow velocity increases and the pressure, based on Bernoulli's equation, decreases. Objects. Polymictic sandstones of the Tyumen Formation. Methods. The developed algorithms for identifying areas of contraction/expansion (confusers/diffusers) in the rock texture (polymictic sandstone) are based on neural network reconstruction of core data. The study of geometric diversity of geological rock images is based on generalization of real spatial forms in combination with the axiom of hyperplanes. A methodical approach for finding distribution of inertial exciters (confusers/diffusers) affecting the character of multiphase flow front changes in the rock texture is based on the metrics for estimating possible distributions. Results. The paper introduces the initial stages of core digital transformation data. It was determined that the main problem of geological data digital projection on a large scale is the need to take into account the systems of inertial effects, both in the available volume of the rock and in the extrapolated space of it. In order to adequately represent the character of multiphase fluid flow for different scales, the authors have developed the algorithms for segmentation of geological images of fluid space with further identification of distribution laws of inertial exciters (confusers/diffusers) in the rock texture. For training and testing of the developed algorithms we used the computer tomography images of core material and detailed resolution images of rock sections. The source code of neural network algorithms was written in the Python programming language with the additional use of non-commercial libraries. The algorithms were tested on real data, confirming by experimental studies in the laboratory of digital research in oil and gas as part of the technological project "Digital Core" (Industrial University of Tyumen, Tyumen), the laboratory of the V.I. Shpilman Research and Analytical Center for Rational Subsoil Use (Khanty-Mansiysk), Core Research Laboratory (University of Tyumen, Tyumen).

Parametric generative schemes with geometric constraints for encoding and synthesizing airfoils

The modern aerodynamic optimization has a strong demand for parametric methods with high levels of intuitiveness, flexibility, and representative accuracy, which cannot be fully achieved through traditional airfoil parametric techniques. In this paper, two deep learning-based generative schemes are proposed to effectively capture the complexity of the design space while satisfying specific constraints. 1. Soft-constrained scheme: a Conditional Variational Autoencoder-based model that directly incorporates geometric constraints as part of the network. 2. Hard-constrained scheme: a Variational Autoencoder-based model for generating diverse airfoils, with projection onto given constraints using Free Form Deformation-based techniques. According to the statistical results, the reconstructed airfoils are both accurate and smooth, without any need for additional filters. The soft-constrained scheme generates airfoils that exhibit slight deviations from the expected geometric constraints, yet still converge to the reference airfoil in both geometry space and objective space with some degree of distribution bias. In contrast, the hard-constrained scheme produces airfoils with a wider range of geometric diversity while strictly adhering to the geometric constraints. The corresponding distribution in the objective space is also more diverse, with isotropic uniformity around the reference point and no significant bias. These proposed airfoil parametric methods can break through the boundaries of training data in the objective space, providing higher quality samples for random sampling and improving the efficiency of optimization design.

Kernel-Free Quadratic Surface Regression for Multi-Class Classification.

For multi-class classification problems, a new kernel-free nonlinear classifier is presented, called the hard quadratic surface least squares regression (HQSLSR). It combines the benefits of the least squares loss function and quadratic kernel-free trick. The optimization problem of HQSLSR is convex and unconstrained, making it easy to solve. Further, to improve the generalization ability of HQSLSR, a softened version (SQSLSR) is proposed by introducing an ε-dragging technique, which can enlarge the between-class distance. The optimization problem of SQSLSR is solved by designing an alteration iteration algorithm. The convergence, interpretability and computational complexity of our methods are addressed in a theoretical analysis. The visualization results on five artificial datasets demonstrate that the obtained regression function in each category has geometric diversity and the advantage of the ε-dragging technique. Furthermore, experimental results on benchmark datasets show that our methods perform comparably to some state-of-the-art classifiers.

ГРАФОАНАЛІТИЧНЕ ПРОФІЛЮВАННЯ СПРЯЖЕНИХ КРИВОЛІНІЙНИХ ПОВЕРХОНЬ РІЖУЧОГО ІНСТРУМЕНТУ

In the research, graphic-analytical methods for profiling the cutting tool for processing the helical surface of the drill groove were developed. The basis of these studies was the definition of the characteristics of conjugated surfaces, as the locus of the bases of the normal, which are directed from the points of the surface of revolution to a given helical surface. The methods were based on the position of the differential geometry about the common normal at the points of tangent conjugacy of the surface. Graphic-analytical profiling is a fairly wide and developed branch of science, based on the one hand on the theoretical and practical achievements of applied geometry, which makes it possible to use it, and on the other hand, it is a powerful analytical tool. Formation is an approximate representation of any set of objects, phenomena of the external world through geometric diversity and relationships between them. Conjugation of curvilinear surfaces of the cutting tool for processing parts in weapons and military equipment is one of the most important topics. The article proposes to optimize the process by creating universal tools, which may include the creation of a kinematic method that will be used to design products with specified parameters. The use of graphic-analytical formation of curved surfaces as a design tool in the "manual" mode, in which all graphic formations of conjugate curved surfaces, their interconnection and interdependence will be laid. Of great importance in the design of mating curvilinear surfaces of a cutting tool is the exact reproduction of their shape as the basis for reliable and long-term operation of future real products in weapons and military equipment. This article aims to take a certain step in this direction regarding the creation of mating curved surfaces for the design of cutting tools, processing of products in weapons and military equipment. Keywords: Graph-analytical formation, kinematic method, surface conjugation, kinematic pairs, parameters, shaping, curved surfaces, drill groove.

Improving the Modeling of Extracellular Ligand Binding Pockets in RosettaGPCR for Conformational Selection.

G protein-coupled receptors (GPCRs) are the largest class of drug targets and undergo substantial conformational changes in response to ligand binding. Despite recent progress in GPCR structure determination, static snapshots fail to reflect the conformational space of putative binding pocket geometries to which small molecule ligands can bind. In comparative modeling of GPCRs in the absence of a ligand, often a shrinking of the orthosteric binding pocket is observed. However, the exact prediction of the flexible orthosteric binding site is crucial for adequate structure-based drug discovery. In order to improve ligand docking and guide virtual screening experiments in computer-aided drug discovery, we developed RosettaGPCRPocketSize. The algorithm creates a conformational ensemble of biophysically realistic conformations of the GPCR binding pocket between the TM bundle, which is consistent with a knowledge base of expected pocket geometries. Specifically, tetrahedral volume restraints are defined based on information about critical residues in the orthosteric binding site and their experimentally observed range of Cα-Cα-distances. The output of RosettaGPCRPocketSize is an ensemble of binding pocket geometries that are filtered by energy to ensure biophysically probable arrangements, which can be used for docking simulations. In a benchmark set, pocket shrinkage observed in the default RosettaGPCR was reduced by up to 80% and the binding pocket volume range and geometric diversity were increased. Compared to models from four different GPCR homology model databases (RosettaGPCR, GPCR-Tasser, GPCR-SSFE, and GPCRdb), the here-created models showed more accurate volumes of the orthosteric pocket when evaluated with respect to the crystallographic reference structure. Furthermore, RosettaGPCRPocketSize was able to generate an improved realistic pocket distribution. However, while being superior to other homology models, the accuracy of generated model pockets was comparable to AlphaFold2 models. Furthermore, in a docking benchmark using small-molecule ligands with a higher molecular weight between 400 and 700 Da, a higher success rate in creating native-like binding poses was observed. In summary, RosettaGPCRPocketSize can generate GPCR models with realistic orthosteric pocket volumes, which are useful for structure-based drug discovery applications.

Sparing old-growth maximises conservation outcomes within selectively logged Amazonian rainforest

Timber extraction threatens a vast area of tropical ecosystems, making it vital to design productive harvesting operations that limit biodiversity declines. Contrasting management options span a continuum from less-intensive, land-sharing logging applied over a larger area to land-sparing operations that combine intensive harvesting with the preservation of old-growth forest. Combining company-reported extraction rates with dung beetle surveys along an Amazonian logging gradient, we explore how individual species' abundances, geometric mean population sizes, functional diversity, and trait characteristics vary across simulated logging concessions and production targets. We substantially extend previous studies by evaluating 8000 mixed-harvesting scenarios and by assessing the profitability of contrasting practices. Simply maximising old-growth protection delivers the highest species' abundances and population sizes for species negatively affected by logging. Maximising old-growth also supports communities with a functional trait dissimilarity (FDis, RaoQ) and functional structure of nesting guilds, biomass, pronotum volume, front leg area, and front:back leg ratio traits that closely resembles old-growth forest. Functional evenness (FEve), richness (FRic), and divergence (FDiv) did not vary across logging strategies. Some 3 % of mixed approaches outperform extreme sparing (which maximises old-growth retention through intensive logging) but still involve substantial sparing, enabled by intensified logging elsewhere. However more-extensive business-as-usual harvesting is up to 90 % more profitable than extreme sparing, suggesting active policy mechanisms, standards, or regulations would be needed to make spatially-concentrated logging operations (which benefit biodiversity) more commercially attractive. Old-growth sparing appears key to limiting biodiversity declines within tropical timber concessions, but would require payments to compensate for reduced profits.

Comparing the ice nucleation properties of the kaolin minerals kaolinite and halloysite

Abstract. Heterogeneous ice nucleation on dust particles in the atmosphere is a key mechanism for ice formation in clouds. However, the conditions of a particle surface for efficient ice nucleation are poorly understood. In this study, we present results of immersion freezing experiments using differential scanning calorimetry on emulsified mineral dust suspensions, involving the two chemically identical, but morphologically different, kaolin minerals of kaolinite and halloysite. Kaolinite occurs in a platy morphology, while halloysites form predominantly tubular structures. We investigated six different halloysite and two different kaolinite samples. Our results show that, on average, the halloysite samples not only exhibit a higher ice nucleation (IN) activity than the kaolinite samples but also a higher diversity in terms of freezing onset temperatures and heterogeneously frozen fraction. Repeating the freezing experiments after shortly milling the samples led to a decrease in freezing onset temperatures and in the heterogeneously frozen fraction of the halloysite samples, bringing their IN activity closer to that of the kaolinites. To interpret these findings, the freezing experiments were complemented by dynamic vapor sorption (DVS), BET (Brunauer–Emmett–Teller) surface area measurements, pore ice melting experiments with slurries, and transmission electron microscopy (TEM) before and after milling. These measurements demonstrate an increase in surface area and the destruction of tubes by milling and provide evidence for the influence of the tubular structure of the halloysites on their IN activity. We identify the OH–Al–O–Si–OH functionalized edges as being the most likely site for ice nucleation, as the high geometric diversity of the edges best accounts for the high diversity in IN activity of halloysites. We hypothesize that the stacking of layers and the number of stacks in halloysite tubes and kaolinite platelets affect the freezing temperature, with thicker stacks having the potential to freeze water at higher temperatures. The notion that the edges constitute the IN-active part of kaolin minerals is further supported by comparing kaolin minerals with montmorillonites and feldspars, all of which exhibit enhanced IN activity in the presence of ammonia and ammonium-containing solutions. As OH–Al–O–Si–OH functionalized edge surfaces are the only surface type that kaolin particles have in common with montmorillonites and feldspars, the common feature of IN activity enhancement in ammoniated solutions can only be explained by ice nucleation occurring at the edges of kaolin minerals.

ARC–MOF: A Diverse Database of Metal-Organic Frameworks with DFT-Derived Partial Atomic Charges and Descriptors for Machine Learning

Metal–organic frameworks (MOFs) are a class of crystalline materials composed of metal nodes or clusters connected via semi-rigid organic linkers. Owing to their high-surface area, porosity, and tunability, MOFs have received significant attention for numerous applications such as gas separation and storage. Atomistic simulations and data-driven methods [e.g., machine learning (ML)] have been successfully employed to screen large databases and successfully develop new experimentally synthesized and validated MOFs for CO2 capture. To enable data-driven materials discovery for any application, the first (and arguably most crucial) step is database curation. This work introduces the ab initio REPEAT charge MOF (ARC–MOF) database. This is a database of ∼280,000 MOFs which have been either experimentally characterized or computationally generated, spanning all publicly available MOF databases. A key feature of ARC–MOF is that it contains density functional theory-derived electrostatic potential fitted partial atomic charges for each MOF. Additionally, ARC–MOF contains pre-computed descriptors for out-of-the-box ML applications. An in-depth analysis of the diversity of ARC–MOF with respect to the currently mapped design space of MOFs was performed─a critical, yet commonly overlooked aspect of previously reported MOF databases. Using this analysis, balanced subsets from ARC–MOF for various ML purposes have been identified, with a case study of the effect of training set on the ML performance. Other chemical and geometric diversity analyses are presented, with an analysis on the effect of the charge-assignment method on atomistic simulation of the gas uptake in MOFs.

A Spacecraft Equipment Layout Optimization Method for Diverse and Competitive Design

The spacecraft equipment layout optimization design (SELOD) problems with complicated performance constraints and diversity are studied in this paper. The previous literature uses the gradient-based algorithm to obtain optimized non-overlap layout schemes from randomly initialized cases effectively. However, these local optimal solutions are too difficult to jump out of their current relative geometry relationships, significantly limiting their further improvement in performance indicators. Therefore, considering the geometric diversity of layout schemes is put forward to alleviate this limitation. First, similarity measures, including modified cosine similarity and gaussian kernel function similarity, are introduced into the layout optimization process. Then the optimization produces a set of feasible layout candidates with the most remarkable difference in geometric distribution and the most representative schemes are sampled. Finally, these feasible geometric solutions are used as initial solutions to optimize the physical performance indicators of the spacecraft, and diversified layout schemes of spacecraft equipment are generated for the engineering practice. The validity and effectiveness of the proposed methodology are demonstrated by two SELOD applications.

Nanostructured intermetallics: from rational synthesis to energy electrocatalysis

Intermetallics are a large family of structurally ordered alloys that combines a metal element with other metal/metalloid elements with a clearly defined stoichiometric ratio. Intermetallics possess abundant crystal structures and atomic packing motifs, giving rise to a great variety of electronic configurations and surface adsorption properties. The wide electronic and geometric diversity makes intermetallics a highly promising population for discovering advanced materials for various catalytic applications. This review presents recent advances in the reaction synthesis of intermetallic materials at the nanoscale and their energy-related electrocatalytic applications. Initially, we introduce general principles for the formation of stable intermetallic structures. Subsequently, we elaborate on common synthetic strategies of nanostructured intermetallics, such as thermal annealing, wet-chemical methods, metallothermic reduction, and template-directed synthesis. Furthermore, we discuss the wide employment of these intermetallic nanocatalysts in many different kinds of electrocatalytic applications, as well as highlight the theoretical and experimental evidence for establishing a reasonable relationship between atomic arrangement and catalytic activity. Finally, we propose some perspectives for future developments of intermetallic preparation and catalytic applications.

Chiral Gold Clusters with Crosslinking Ligands: Geometric Structures and Chiroptical Activities

AbstractThe geometric diversity of ligand‐protected metal clusters has greatly provoked the interest in the chirality of gold clusters bearing crosslinking ligands, which are a subject of active investigations. This review introduces recent geometric and computational studies on chiral gold clusters whose metal core is covered by intramolecular crosslinking moieties. Following an overview of the structural constructions, the geometric features of the ligand shells and metal cores of chiral gold clusters are elucidated, where an emphasis is placed on the importance of helical patterns of the crosslinking ligands arranged around the gold cores. In addition, insights into the atomic‐level origin of chiroptical activity, which cannot be accessed solely through experiments, are presented from computational studies using numerical and theoretical approaches, thereby providing a novel methodology based on transition‐moment and transition‐density analyses.

Land Cover Classification in Foggy Conditions: Toward Robust Models

Robust semantic labeling of high-resolution remote sensing images in foggy conditions is crucial for automatic monitoring of land covers. This remains a challenging task owing to the low inter-class differentiation yet high intra-class variance and geometric size diversity. Although conventional Convolutional Neural Networks have demonstrated state of the art performance in semantic segmentation, most networks are primarily concerned with standard accuracy, while the influence on robustness is rarely explored. This letter proposes a reliable framework which is evaluated across various severity levels of fog corruptions. Utilizing HRNet as the backbone to maintain high-resolution representations, we develop a multimodal fusion module to exploit the complementary information of lidar and multispectral data. Based on the evaluation experiment on fog corrupted ISPRS 2D datasets, our model demonstrates promising performance with an average mIoU on the clean along with the corrupted datasets exceeding 80% and 56% respectively.

Visualization and Quantification of Geometric Diversity in Metal-Organic Frameworks.

With ever-growing numbers of metal–organic framework (MOF) materials being reported, new computational approaches are required for a quantitative understanding of structure–property correlations in MOFs. Here, we show how structural coarse-graining and embedding (“unsupervised learning”) schemes can together give new insights into the geometric diversity of MOF structures. Based on a curated data set of 1262 reported experimental structures, we automatically generate coarse-grained and rescaled representations which we couple to a kernel-based similarity metric and to widely used embedding schemes. This approach allows us to visualize the breadth of geometric diversity within individual topologies and to quantify the distributions of local and global similarities across the structural space of MOFs. The methodology is implemented in an openly available Python package and is expected to be useful in future high-throughput studies.