Space-filling Curves Research Articles

Search of Fractal Space-Filling Curves with Minimal Dilation

We introduce an algorithm for a search of extremal fractal curves in large curve classes. It heavily uses SAT-solvers—heuristic algorithms that find models for CNF boolean formulas. Our algorithm was implemented and applied to the search of fractal surjective curves \(\gamma :[0,1]\rightarrow [0,1]^d\) with minimal dilation $$\begin{aligned} \sup _{t_1<t_2}\frac{\Vert \gamma (t_2)-\gamma (t_1)\Vert ^d}{t_2-t_1}. \end{aligned}$$We report new results of that search in the case of Euclidean norm. We have found a new curve that we call “YE”, a self-similar (monofractal) plane curve of genus \(5\times 5\) with dilation \(5+{43}/{73}=5.5890\ldots \) In dimension 3 we have found facet-gated bifractals (which we call “Spring”) of genus \(2\times 2\times 2\) with dilation \(<17\). In dimension 4 we obtained that there is a curve with dilation \(<62\). Some lower bounds on the dilation for wider classes of cubically decomposable curves are proven.

SADHANA: A Doubly Linked List-based Multidimensional Adaptive Mesh Refinement Framework for Solving Hyperbolic Conservation Laws with Application to Astrophysical Hydrodynamics and Magnetohydrodynamics

We report the development of a nested block-structured adaptive mesh framework to solve multidimensional, time-dependent hyperbolic equations encountered in astrophysics. An approach based on a tabular list is used to construct variants of Hilbert space-filling curves in an iterative fashion to maintain the connectivity of locally refined mesh configurations using a doubly linked list. Modifications are made to conventional boundaries of computational blocks to aid the adaptive mesh. We also describe a well-defined, computationally efficient data structure to hold self-similar mesh units for this purpose. The flexibility of this code is demonstrated by the performance of various Riemann solvers implemented in this computational framework.

A GtoG Direct Coding Mapping Method for Multi-Type Global Discrete Grids Based on Space Filling Curves

DGGS (Discrete Global Grid System) has many subdivision models and coding methods. Due to the lack of underlying consistency of different DGGS codes, most of them are converted through longitude–latitude, which greatly reduces the interoperability efficiency of different DGGS data and has become one of the bottlenecks in efficient integration of multi-source DGGS data. Therefore, a direct mapping method from one grid code to another (Grid to Grid, GtoG) for multi-type DGGSs is proposed based on three classical DGGSs (triangular, diamond and hexagonal grids) and two commonly used filling curves (Hilbert curve and Z-curve). The mutual conversion rules of different grids expressing spatial point, line and surface data are constructed. Then, the above method is extended to the spherical icosahedral grid framework, and three different region coding mapping rule tables of the basic inside cells, boundary cells and vertex cells are designed. Finally, the experimental results show that, compared with the longitude–latitude conversion method, the average conversion efficiency of spatial point, line and surface data is increased by 2–4 orders of magnitude. This new method greatly improves the interoperability efficiency and provides a feasible solution for the efficient integration of multi-source DGGS data.

In-Memory Distributed Mosaicking for Large-Scale Remote Sensing Applications with Geo-Gridded Data Staging on Alluxio

The unprecedented availability of petascale analysis-ready earth observation data has given rise to a remarkable surge in demand for regional to global environmental studies, which exploit tons of data for temporal–spatial analysis at a much larger scale than ever. Imagery mosaicking, which is critical for forming “One Map” with a continuous view for large-scale climate research, has drawn significant concern. However, despite employing distributed data processing engines such as Spark, large-scale data mosaicking still significantly suffers from a staggering number of remote sensing images which could inevitably lead to discouraging performance. The main ill-posed problem of traditional parallel mosaicking algorithms is inherent in the huge computation demand and incredible heavy data I/O burden resulting from intensively shifting tremendous RS data back and forth between limited local memory and bulk external storage throughout the multiple processing stages. To address these issues, we propose an in-memory Spark-enabled distributed data mosaicking at a large scale with geo-gridded data staging accelerated by Alluxio. It organizes enormous “messy” remote sensing datasets into geo-encoded gird groups and indexes them with multi-dimensional space-filling curves geo-encoding assisted by GeoTrellis. All the buckets of geo-grided remote sensing data groups could be loaded directly from Alluxio with data prefetching and expressed as RDDs implemented concurrently as grid tasks of mosaicking on top of the Spark-enabled cluster. It is worth noticing that an in-memory data orchestration is offered to facilitate in-memory big data staging among multiple mosaicking processing stages to eliminate the tremendous data transferring at a great extent while maintaining a better data locality. As a result, benefiting from parallel processing with distributed data prefetching and in-memory data staging, this is a much more effective approach to facilitate large-scale data mosaicking in the context of big data. Experimental results have demonstrated our approach is much more efficient and scalable than the traditional ways of parallel implementing.

Memory access optimization for particle operations in computational fluid dynamics-discrete element method simulations

Computational Fluid Dynamics – Discrete Element Method is used to model gas-solid systems in several applications in energy, pharmaceutical and petrochemical industries. Computational performance bottlenecks often limit the problem sizes that can be simulated at industrial scale. The data structures used to store several millions of particles in such large-scale simulations have a large memory footprint that does not fit into the processor cache hierarchies on current high-performance-computing platforms, leading to reduced computational performance. This paper specifically addresses this aspect of memory access bottlenecks in industrial scale simulations. The use of space-filling curves to improve memory access patterns is described and their impact on computational performance is quantified in both shared and distributed memory parallelization paradigms. The Morton space filling curve applied to uniform grids and k-dimensional tree partitions are used to reorder the particle data-structure thus improving spatial and temporal locality in memory. The performance impact of these techniques when applied to two benchmark problems, namely the homogeneous-cooling-system and a fluidized-bed, are presented. These optimization techniques lead to approximately two-fold performance improvement in particle focused operations such as neighbor-list creation and data-exchange, with ∼ 1.5 times overall improvement in a fluidization simulation with 1.27 million particles.

Extraction of local structure information of point clouds through space-filling curve for semantic segmentation

• The relative angle feature between the center point in the neighborhood ball and their neighbors can improve the semantic segmentation results. • The space filling curve has a good ability to maintain the local geometric structure of point cloud. • The result of semantic segmentation does not always improve with the increase of the number of neighborhood points. • With the increase of feature vector dimension, the result of semantic segmentation will tend to be stable. The point cloud semantic segmentation network based on point-wise multi-layer perceptron (MLP) has gained extensive applications because of its end-to-end advantages. However, there exist two major limitations for this type of network: (1) In a neighborhood ball, the relative features between the central point and its neighboring point are not adequately exploited. (2) Only the relative features in the neighborhood ball are extracted, but the overall morphological information of the neighborhood ball is ignored. To overcome the limitations, this paper proposes two-fold improvements on local structure information extraction: (1) The relative angular feature is added and combined with other initial features as the input of the model to exploit the relative geometric features between points to the greatest extent. (2) The unordered points in the neighborhood ball are reordered based on the space-filling curve (SFC) and then fed into the MLP to extract the overall structure information of the neighborhood ball. And a semantic segmentation model is developed based on the two proposed feature extraction modules and U-Net, which is evaluated on two public datasets. The experimental result has demonstrated that the two proposed feature extraction modules can effectively extract geometric information in the point cloud and the proposed semantic segmentation model has strong semantic recognition capability for objects with complex morphologies. The mean intersection over union (mIoU) of the proposed model reached 70.6% and 47.8% for the Semantic 3D and Semantic KITTI datasets, respectively. Besides, the proposed model achieves real-time segmentation with only four encoder layers.

New Method to Control the Passband and the Stopband of UWB Ring Resonator Based Bandpass Filters

In this article, resonance frequencies, stopband rejection level, and the passband of ultra-wideband compact ring resonator-based bandpass filters are controlled by a new and understandable method. In this method, the passage and the rejection of resonance frequencies are controlled by the ports' positions. This can be achieved by investigating the electric field distribution of the successive resonant frequencies of the ring resonator considering symmetrical excitation. Then, decide which resonant frequencies will be passed and which will be rejected according to the electric field distribution. For size reduction, the Sierpinski space-filling curve is used. The stopband rejection level is controlled by inserting a bandstop filter with the resultant bandpass filter with the minimum extra area. Band-pass filter passband width is controlled by the attenuation poles' frequency positions that are controlled by inserting open stubs without the need for any further area. For verification, the proposed filter passband is chosen to cover the S-band frequency spectrum. The proposed bandpass filter has an ultra-wide 3 dB passband of 2.11 GHz (2.27 GHz - 4.38 GHz) and a 15 dB ultra-wide stopband of 5.36 GHz (4.64 - 10 GHz). The proposed bandpass filter is fabricated, measured, and the results are in good harmony with their simulated counterparts.

Organizing and visualizing point clouds with continuous levels of detail

Point clouds contain high detail and high accuracy geometry representation of the scanned Earth surface parts. To manage the huge amount of data, the point clouds are traditionally organized on location and map-scale; e.g. in an octree structure, where top-levels of the tree contain few points suitable for small scale overviews and lower levels of the tree contain more points suitable for large scale detailed views. The drawback of this solution is that it is based on discrete levels, causing visual artifacts in the form of data density shocks when creating the commonly used perspective views. This paper presents a method based on an optimized distribution of points over continuous levels, avoiding the visualization shocks. The traditional distribution ratio’s of data amounts over discrete levels of raster or vector data is considered the reference. How to convert this to point clouds with continuous levels (still benefiting from the proven advantages of the data distribution in discrete levels for efficient access at a wide range of scales)? In our solution, for each point a cLoD (continuous Level of Detail) value is computed and added as dimension to the point. A SFC (Space Filling Curve)-based nD data clustering technique can be used to organize the points, so that they can be efficiently queried. It should be noted that also other multi-dimensional indexing and clustering techniques could be applied to realize continuous levels based on the cLoD value. Besides the mathematical foundation of the approach also several implementations are described, varying from a 3D web-browser based solution to an augmented reality point cloud app in a mobile phone. The cLoD enables interactive real-time visualization using perspective views without data density shocks, while supporting continuous zoom-in/out and progressive data streaming between server and client. The described cLoD based approach is generic and supports different types of point clouds: from airborne, terrestrial, mobile and indoor laser scanning, but also from dense matching optical imagery or multi-beam echo soundings.

PCSERVE – ND-POINTCLOUDS RETRIEVAL OVER THE WEB

Abstract. We investigate how PCServe, a web service for disseminating massive point clouds, performs for read-only access (i.e. a visualization application). PCServe is backed by a database model based on Space Filling Curves. By adding a virtual hierarchy of blocks to the database, we can support different visualization applications for retrieval of point cloud data over the web without having to store the data multiple times. This makes expressive access to point clouds over the web possible. We investigate the amount of processing that is needed to create the database model and how well PCServe handles requests from the visualization application. Some suggestions are provided how the current approach can be improved.

4SA: OPTIMIZING SPACE FILLING CURVE BASED GRID CELL INDEXING TO SCALABLY MANAGE REMOTELY SENSED IMAGES IN KEY-VALUE DATABASES

Abstract. State-of-the-art remote sensing image management systems adopt scalable databases and employ sophisticated indexing techniques to perform window and containment queries. Many rely on space-filling curve (SFC) based index techniques designed for key-value databases and are predominantly employable for images that are iso-oriented. Critically, these indexes do not consider the high degree of overlap among images that exists in many data sets and the affiliated storage requirements. Specifically, employing an SFC-based grid cell index approach in consort with ground footprint coverage of the images requires storage of a unique image object identification (IOI) for each image in every grid cell where overlap occurs. Such an approach adversely affects both storage and query response times. In response, this paper presents an optimization technique for an SFC-based grid cell space indexing. The optimization is specifically designed for window and containment queries where the region of interest overlaps with at least a 2 × 2 grid of cells. The technique is based on four cell removal steps, thus called “four step algorithm” (4SA). Each step employs a unique spatial configuration to check for continuous spatial extent. If present, the IOI of the target cell is omitted from further consideration. Analysis and experiments on real world and synthetic image data demonstrated that 4SA improved storage demands by 41.3% – 47.8%. Furthermore, in the performed querying experiments, only 42% of IOI elements needed to be processed, thus yielding a 58% productivity gain. The reduction of IOI elements in querying also impacted the CPU execution time (3.0% – 5.2%). The 4SA also demonstrated data scalability and concurrent user scalability in querying large regions by completing the index searching and concurrent user scalability 1.86% – 3.35% faster than when 4SA was not applied.

A separable and dual data hiding algorithm based on adaptive ternary segmentation coding and ZN-shape space-filling curve

A separable and dual data hiding algorithm based on adaptive ternary segmentation coding and ZN-shape space-filling curve

Numerical methods using two different approximations of space-filling curves for black-box global optimization

In this paper, multi-dimensional global optimization problems are considered, where the objective function is supposed to be Lipschitz continuous, multiextremal, and without a known analytic expression. Two different approximations of Peano-Hilbert curve applied to reduce the problem to a univariate one satisfying the Hölder condition are discussed. The first of them, piecewise-linear approximation, is broadly used in global optimization and not only whereas the second one, non-univalent approximation, is less known. Multi-dimensional geometric algorithms employing these Peano curve approximations are introduced and their convergence conditions are established. Numerical experiments executed on 800 randomly generated test functions taken from the literature show a promising performance of algorithms employing Peano curve approximations w.r.t. their direct competitors.

An unstructured adaptive mesh refinement approach for computational fluid dynamics of reacting flows

A new Direct Numerical Simulation (DNS) code HAMISH with Adaptive Mesh Refinement (AMR) has been developed to simulate compressible reacting flow in a computationally economical manner. The focus is on problems where high gradients of temperature, density and species mass fraction remain localised, for example within the interior structure of flames. The resolution requirements of such local high-gradient regions often determine the global mesh spacing when uniform meshes are used. The numerical framework in HAMISH is based on an unstructured finite-volume approach together with a Runge-Kutta algorithm for time-stepping. The unstructured Cartesian mesh used in HAMISH allows for adaptive refinement and de-refinement of the local mesh depending on the demands of the flow physics and numerical accuracy. The code utilises an octree data structure and a Morton space-filling curve, allowing efficient cell division as well as straightforward parallel domain decomposition. The spatial discretisation scheme uses fourth-order polynomial reconstruction to evaluate the inter-cell fluxes. A third-order three-step explicit Runge-Kutta time-marching algorithm is used together with adaptive time-stepping with proportional-integral-type error control using an embedded Runge-Kutta scheme. The code is fully parallelised through domain decomposition over an arbitrary number of processors and exhibits a high level of parallel efficiency. In this paper the key capabilities of HAMISH are demonstrated based on a number of test cases. These test cases include (a) simulation of 1-D planar laminar premixed flames to demonstrate the capability of AMR in capturing the sharp gradients within the flame; (b) fully developed laminar channel flow, which shows that the code is capable of refining the mesh in the boundary layer next to the wall; (c) 2-D expanding flame under quiescent laminar condition, which demonstrates that the code is capable of dynamic local mesh refinement based on the position of the flame front; (d) 3-D non-reacting Taylor-Green vortex which demonstrates HAMISH can deal with vortical motion typical of turbulent flows; (e) 3-D premixed turbulent flame propagation under isotropic homogeneous decaying turbulence to demonstrate that the code can be employed for production level DNS of turbulent reacting flows.

Hagrid: using Hilbert and Gosper curves to gridify scatterplots

A common enhancement of scatterplots represents points as small multiples, glyphs, or thumbnail images. As this encoding often results in overlaps, a general strategy is to alter the position of the data points, for instance, to a grid-like structure. Previous approaches rely on solving expensive optimization problems or on dividing the space that alter the global structure of the scatterplot. To find a good balance between efficiency and neighborhood and layout preservation, we propose Hagrid, a technique that uses space-filling curves (SFCs) to “gridify” a scatterplot without employing expensive collision detection and handling mechanisms. Using SFCs ensures that the points are plotted close to their original position, retaining approximately the same global structure. The resulting scatterplot is mapped onto a rectangular or hexagonal grid, using Hilbert and Gosper curves. We discuss and evaluate the theoretic runtime of our approach and quantitatively compare our approach to three state-of-the-art gridifying approaches, DGrid, Small multiples with gaps SMWG, and CorrelatedMultiples CMDS, in an evaluation comprising 339 scatterplots. Here, we compute several quality measures for neighborhood preservation together with an analysis of the actual runtimes. The main results show that, compared to the best other technique, Hagrid is faster by a factor of four, while achieving similar or even better quality of the gridified layout. Due to its computational efficiency, our approach also allows novel applications of gridifying approaches in interactive settings, such as removing local overlap upon hovering over a scatterplot.Graphical abstract

Endocytic trafficking promotes vacuolar enlargements for fast cell expansion rates in plants.

The vacuole has a space-filling function, allowing a particularly rapid plant cell expansion with very little increase in cytosolic content (Löfke et al., 2015; Scheuring et al., 2016; Dünser et al., 2019). Despite its importance for cell size determination in plants, very little is known about the mechanisms that define vacuolar size. Here, we show that the cellular and vacuolar size expansions are coordinated. By developing a pharmacological tool, we enabled the investigation of membrane delivery to the vacuole during cellular expansion. Our data reveal that endocytic membrane sorting from the plasma membrane to the vacuole is enhanced in the course of rapid root cell expansion. While this 'compromise' mechanism may theoretically at first decelerate cell surface enlargements, it fuels vacuolar expansion and, thereby, ensures the coordinated augmentation of vacuolar occupancy in dynamically expanding plant cells.

A Fast kNN Algorithm Using Multiple Space-Filling Curves.

The paper considers a time-efficient implementation of the k nearest neighbours (kNN) algorithm. A well-known approach for accelerating the kNN algorithm is to utilise dimensionality reduction methods based on the use of space-filling curves. In this paper, we take this approach further and propose an algorithm that employs multiple space-filling curves and is faster (with comparable quality) compared with the kNN algorithm, which uses kd-trees to determine the nearest neighbours. A specific method for constructing multiple Peano curves is outlined, and statements are given about the preservation of object proximity information in the course of dimensionality reduction. An experimental comparison with known kNN implementations using kd-trees was performed using test and real-life data.

Continuous Maps Admitting No Tangent Lines: A Centennial of Besicovitch Functions

One of the most influential examples in analysis is a Weierstrass function from to that is continuous but differentiable at no point. However this map, as well most of the others among the myriad similar examples, still admits vertical tangent lines. The examples of continuous maps that admit no tangent line in any direction are also known; however, all currently existing presentations of such maps are not easily accessible due to their very complicated descriptions and hard-to-follow proofs of their desired properties. The goal of this article is to present in an accessible way two such examples. The first—a coordinate of the classical Peano space-filling curve—is simpler, but admits one-sided vertical tangent lines at some points. The second is a variation of a function from a 1924 paper of Besicovitch, which is continuous but admits no one-sided tangent line in any direction. The proofs of nondifferentiability of these two examples will be facilitated by a simple yet general lemma that also implies nondifferentiability of other similar maps, including those of Takagi and van der Waerden.

Ultrasound simulation of blood with different red blood cell aggregations and concentrations

Considerable progress of ultrasound simulation on blood has enhanced the characterizing of red blood cell (RBC) aggregation. A novel simulation method aims at modeling the blood with different RBC aggregations and concentrations is proposed. The modeling process is as follows: (i) A three-dimensional scatterer model is first built by a mapping with a Hilbert space-filling curve from the one-dimensional scatterer distribution. (ii) To illustrate the relationship between the model parameters and the RBC aggregation level, a variety of blood samples are prepared and scanned to acquire their radiofrequency signals in-vitro. (iii) The model parameters are determined by matching the Nakagami-distribution characteristics of envelope signals simulated from the model with those measured from the blood samples. Nakagami metrics m estimated from 15 kinds of blood samples (hematocrits of 20%, 40%, 60% and plasma concentrations of 15%, 30%, 45%, 60%, 75%) are compared with metrics estimated by their corresponding models (each with different eligible parameters). Results show that for the three hematocrit levels, the mean and standard deviation of the root-mean-squared deviations of m are 0.27 ± 0.0026, 0.16 ± 0.0021, 0.12 ± 0.0018 respectively. The proposed simulation model provides a viable data source to evaluate the performance of the ultrasound-based methods for quantifying RBC aggregation.

Ultra-Compact UWB Power Divider for 5G Sub-6 GHz Wireless Communication System

This paper presents ultra-compact ultra-wideband three-section power dividers for 5G sub-6 GHz applications. The n-section power divider design approach is based on the even/odd mode analysis. Hilbert's space-filling curve is used to shrink the dividers’ branches’ length. Hence, the size is reduced to 89.6 % for the resultant 1-to-2 power divider as compared to the conventional one. Both conventional and ultra-compact 1-to-2 dividers are used to develop 1-to-4 power dividers. The size is reduced to 62 % for the 1-to4 power divider as compared to the conventional one. Both conventional and compact 1-to-4 power dividers are fabricated on an FR4 substrate with a thickness of 1.6 mm and a relative dielectric constant of 4.3. Measured results of the proposed power divider show that an equal power split and good isolation among the output ports are achieved over the specified 5G frequency bands.

A three-dimensional timbre model via Peano curves

Creating a formal model for timbre is one of the most compelling open questions in music research. In contrast to more traditional perceptually-oriented approaches, often aimed at sound analysis, we introduce a three-dimensional geometric model of timbre, specifically designed for sound synthesis. The proposed model relies on the properties of space-filling curves for multidimensional scaling, and represents via three parameters, any static combination of sinusoidal partials with an additional noisiness component. We detail the construction of the model and its properties and discuss future implications for music research.