Space-filling Curves Research Articles

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.

Minimizing Heat Transfer Resistance in an Integrated Methanol Steam Reformer Designed Using Space-Filling Curves

Conventional packed bed methanol steam reformers producing hydrogen for PEM fuel cells have an inherent trade-off problem between simultaneously attaining high methanol conversion and low carbon monoxide production, owing to high thermal resistance in the packed bed. A sophisticated reformer design using Hilbert’s space-filling curves is proposed and demonstrated using CFD modeling and simulation. The reactor comprises a flow-through domain divided into two interdigitating but nonmixing continua that are separated by a thin wall, providing high interfacial area for heat transfer and surface reactions. The interfacial wall is modeled as coated with catalyst for performing reforming on one side and fuel cell anode exhaust hydrogen combustion on the other side. About 84% methanol conversion and CO production less than 2% was achieved due to remarkable reduction in heat transfer resistance. External mass transfer controlled the reactor operation, while reaction kinetics exhibited nontrivial influence on reactor performance.

Shallow Deep Learning using Space-filling Curves for Malware Classification

The incidents of malware attacks are continually increasing at a rapid rate, thanks to the lucrative potential in schemes such as ransomware, credential stealing Trojans and cryptominers. Their explosive growth is compounded by the ease with which variants can be created from original strains. As a result, anti-virus organisations are struggling to keep up, with some reporting upwards of 14 million samples processed per month. These sheer volumes have caused a shift towards machine learning and artificial intelligence in an effort to alleviate the manual burden of analysis and classification. This research presents a novel framework for the classification of malware into distinct family classes through computer vision and deep learning. In the proposed framework, malware binaries are represented in an abstract form as images mapped through mathematical constructs known as space-filling curves. Convolutional neural networks were constructed and applied to the malware images to build predictive models for classification. The models were optimised using an auto-tuning function for the hyper parameters, which included Bayesian Optimisation, Random search and HyperBand, providing an exhaustive search on the hyper parameters. On a training dataset of 13k malware samples from 23 distinct families, the models yielded an average score of 95% for precision, recall and f1-score. The final deep learning model was validated for robustness against a dataset of more recent variants, comprising 12,816 samples from 16 malware families, returning classification scores of 95%, 86% and 90% for precision, recall and f1-score. The final model was demonstrated to outperform a similar benchmark model considerably. The results show the potential of the deep learning framework as a viable solution to the classification of malware, without the need for manually intensive feature generation or invasive processing techniques.

Image-based malware classification hybrid framework based on space-filling curves

There exists a never-ending “arms race” between malware analysts and adversarial malicious code developers as malevolent programs evolve and countermeasures are developed to detect and eradicate them. Malware has become more complex in its intent and capabilities over time, which has prompted the need for constant improvement in detection and defence methods. Of particular concern are the anti-analysis obfuscation techniques, such as packing and encryption, that are employed by malware developers to evade detection and thwart the analysis process. In such cases, malware is generally impervious to basic analysis methods and so analysts must use more invasive techniques to extract signatures for classification, which are inevitably not scalable due to their complexity. In this article, we present a hybrid framework for malware classification designed to overcome the challenges incurred by current approaches. The framework incorporates novel static and dynamic malware analysis methods, where static malware executables and dynamic process memory dumps are converted to images mapped through space-filling curves, from which visual features are extracted for classification. The framework is less invasive than traditional analysis methods in that there is no reverse engineering required, nor does it suffer from the obfuscation limitations of static analysis. On a dataset of 13,599 obfuscated and non-obfuscated malware samples from 23 families, the framework outperformed both static and dynamic standalone methods with precision, recall and accuracy scores of 97.6%, 97.6% and 97.6% respectively.

Dynamic load balancing with over decomposition in plasma plume simulations

An electric propulsion plasma plume simulation employs the Particle-in-Cell (PIC) model for heavy species (i.e. ions and neutral atoms) and yields a highly non-uniform particle distribution at steady state. For parallel simulations, partitioning the computational domain uniformly and evenly assigning the subdomains to MPI processes would never lead to properly balanced loads across MPI processes. For this reason, a patch-based dynamic load balancing method with over-decomposition and Hilbert space-filling curve has been implemented into the Thermophysics Universal Research Framework (TURF). The data transfer is accomplished by utilizing TURF's serialization and deserialization feature, in which the tree hierarchical object structure is copied into a contiguous memory block of data, and the tree hierarchy is reconstructed after the data transfer. This generalized dynamic load balancing approach allows for the same routine to be used for any models besides the PIC model in TURF (e.g. fluid, continuum kinetic, hybrid models, etc). This paper first introduces individual pieces of TURF that enable dynamic load balancing when they are combined. Then, the dynamic load balancing method is demonstrated in three different particle simulations, showing performance gains when the number of subdomains is larger than that of MPI processes. The benefit is even more notable in simulations with highly non-uniform particle distributions at steady state, commonly seen in plasma plume simulations.

Tame Topology over Definable Uniform Structures

A visceral structure on M is given by a definable base for a uniform topology on its universe M in which all basic open sets are infinite and any infinite definable subset X⊆M has nonempty interior. Assuming only viscerality, we show that the definable sets in M satisfy some desirable topological tameness conditions. For example, any definable function f:M→M has a finite set of discontinuities; any definable function f:Mn→Mm is continuous on a nonnempty open set; and assuming definable finite choice, we obtain a cell decomposition result for definable sets. Under an additional topological assumption (“no space-filling functions”), we prove that the natural notion of topological dimension is invariant under definable bijections. These results generalize some of the theorems proved by Simon and Walsberg, who assumed dp-minimality in addition to viscerality. In the final section, we construct new examples of visceral structures.

Temporal Merge Tree Maps: A Topology-Based Static Visualization for Temporal Scalar Data.

Creating a static visualization for a time-dependent scalar field is a non-trivial task, yet very insightful as it shows the dynamics in one picture. Existing approaches are based on a linearization of the domain or on feature tracking. Domain linearizations use space-filling curves to place all sample points into a 1D domain, thereby breaking up individual features. Feature tracking methods explicitly respect feature continuity in space and time, but generally neglect the data context in which those features live. We present a feature-based linearization of the spatial domain that keeps features together and preserves their context by involving all data samples. We use augmented merge trees to linearize the domain and show that our linearized function has the same merge tree as the original data. A greedy optimization scheme aligns the trees over time providing temporal continuity. This leads to a static 2D visualization with one temporal dimension, and all spatial dimensions compressed into one. We compare our method against other domain linearizations as well as feature-tracking approaches, and apply it to several real-world data sets.

Active Remodeling of Chromatin and Implications for In Vivo Folding.

Building on the observation that chromatin compaction can be locally modulated by activity, we propose a model of in vivo chromatin as an active polymer and study its large scale conformations. In particular, we study an active mechanochemical model of chromosomal folding based on the interplay among polymer elasticity, confinement, topological constraints, and fluctuating active stresses arising from the ATP-dependent action of a variety of chromatin-associated protein machines and chromatin-remodeling proteins and their stochastic turnover. We find that activity drives the chromatin to a nonequilibrium steady state; the statistics of conformations in this nonequilibrium steady state are consistent with recent measurements on intrachromosomal contact probabilities and chromosomal compaction. The contact exponents at steady state show a systematic variation with changes in the nature of activity and the rates of turnover. The steady state configuration of the active chromatin in two dimensions resembles a space-filling Peano curve, which might have implications for the optimization of genome information storage.

Fractal Design for Advancing the Performance of Chemoresistive Sensors.

The rapid advancement of internet of things (IoT)-enabled applications along with connected automation in sensing technologies is the heart of future intelligent systems. The probable applications have significant implications, from chemical process monitoring to agriculture, mining, space, wearable electronics, industrial manufacturing, smart cities, and point-of-care (PoC) diagnostics. Advancing sensor performance such as sensitivity to detect trace amounts (ppb-ppm) of analytes (gas/VOCs), selectivity, portability, and low cost is critical for many of these applications. These advancements are mainly achieved by selecting and optimizing sensing materials by their surface functionalization and/or structural optimization to achieve favorable transport characteristics or chemical binding/reaction sites. Surprisingly, the sensor geometry, shapes, and patterns were not considered as critical parameters, and most of these sensors were designed by following simple planar and interdigitated electrode geometry. In this study, we introduce a new bioinspired fractal approach to design chemoresistive sensors with fractal geometry, which grasp the architecture of fern leaves represented by the geometric group of space-filling curves of fractal patterns. These fractal sensors were printed by an extrusion process on a flexible substrate (PET) using specially formulated graphene ink as a sensing material, which provided significant enhancement of the active surface area to volume ratio and allowed high-resolution fractal patterning along with a reduced current transportation path. To demonstrate the advantages and influence of fractal geometry on sensor performance, here, three different kinds of sensors were fabricated based on different fractal geometrics (Sierpinski, Peano, and Hilbert), and the sensing performance was explored toward different VOC analytes (e.g., ethanol, methanol, and acetone). Among all these fractal-designed sensors including interdigitate sensors, the Hilbert-designed printed sensor shows enhanced sensing properties in terms of fast response time (6 s for 30 ppm), response value (14%), enhanced detection range (5-100 ppm), high selectivity, and low interference to humidity (up to RH 80%) for ethanol at room temperature (20 °C). Moreover, a significant improvement of this sensor performance was observed by applying the mechanical deformation (positive bending) technique. The practical application of this sensor was successfully demonstrated by monitoring food spoilage using a commercial box of strawberries as a model. Based on these presented results, this biofractal biomimetic VOC sensor is demonstrated for a prospective application in food monitoring.

Using Hilbert curves to organize, sample, and sonify solar data

How many ways can we explore the Sun? We have images in many wavelengths and squiggly lines of many parameters that we can use to characterize the Sun. We know that while the Sun is blindingly bright to the naked eye, it also has regions that are dark in some wavelengths of light. All of those classifications are based on vision. Hearing is another sense that can be used to explore solar data. Some data, such as the sunspot number or the extreme ultraviolet spectral irradiance, can be readily sonified by converting the data values to musical pitches. Images are more difficult. Using a raster scan algorithm to convert a full-disk image of the Sun to a stream of pixel values creates variations that are dominated by the pattern of moving on and off the limb of the Sun. A sonification of such a raster scan will contain discontinuities at the limbs that mask the information contained in the image. As an alternative, Hilbert curves are continuous space-filling curves that map a linear variable onto the two-dimensional coordinates of an image. We have investigated using Hilbert curves as a way to sample and analyze solar images. Reading the image along a Hilbert curve keeps most neighborhoods close together as the resolution (i.e., the order of the Hilbert curve) increases. It also removes most of the detector size periodicities and may reveal larger-scale features. We present several examples of sonified solar data, including sunspot number, extreme ultraviolet (EUV) spectral irradiances, an EUV image, and a sequence of EUV images during a filament eruption.

Hilbert curve vs Hilbert space: exploiting fractal 2D covering to increase tensor network efficiency

We present a novel mapping for studying 2D many-body quantum systems by solving an effective, one-dimensional long-range model in place of the original two-dimensional short-range one. In particular, we address the problem of choosing an efficient mapping from the 2D lattice to a 1D chain that optimally preserves the locality of interactions within the TN structure. By using Matrix Product States (MPS) and Tree Tensor Network (TTN) algorithms, we compute the ground state of the 2D quantum Ising model in transverse field with lattice size up to 64×64, comparing the results obtained from different mappings based on two space-filling curves, the snake curve and the Hilbert curve. We show that the locality-preserving properties of the Hilbert curve leads to a clear improvement of numerical precision, especially for large sizes, and turns out to provide the best performances for the simulation of 2D lattice systems via 1D TN structures.

Space-Filling Curve Resistor on Ultra-Thin Polyetherimide Foil for Strain Impervious Temperature Sensing.

Monitoring process parameters in the manufacture of composite structures is key to ensuring product quality and safety. Ideally, this can be done by sensors that are embedded during production and can remain as devices to monitor structural health. Extremely thin foil-based sensors weaken the finished workpiece very little. Under ideal conditions, the foil substrate bonds with the resin in the autoclaving process, as is the case when polyetherimide is used. Here, we present a temperature sensor as part of an 8 µm thick multi-sensor node foil for monitoring processing conditions during the production and structural health during the lifetime of a construction. A metallic thin film conductor was shaped in the form of a space-filling curve to suppress the influences of resistance changes due to strain, which could otherwise interfere with the measurement of the temperature. FEM simulations as well as experiments confirm that this type of sensor is completely insensitive to the direction of strain and sufficiently insensitive to the amount of strain, so that mechanical strains that can occur in the composite curing process practically do not interfere with the temperature measurement. The temperature sensor is combined with a capacitive sensor for curing monitoring based on impedance measurement and a half-bridge strain gauge sensor element. All three types are made of the same materials and are manufactured together in one process flow. This is the key to cost-effective distributed sensor arrays that can be embedded during production and remain in the workpiece, thus ensuring not only the quality of the initial product but also the operational reliability during the service life of light-weight composite constructions.

Boosting memory access locality of the Spectral Element Method with Hilbert space-filling curves

We propose an algorithm based on Hilbert space-filling curves to reorder mesh elements in memory for use with the Spectral Element Method, aiming to attain fewer cache misses, better locality of data reference and faster execution. We present a technique to numerically simulate acoustic wave propagation in 2D domains using the Spectral Element Method, and discuss computational performance aspects of this procedure. We reorder mesh-related data via Hilbert curves to achieve sizable reductions in execution time under several mesh configurations in shared-memory systems. Our experiments show that the Hilbert curve approach works well with meshes of several granularities and also with small and large variations in element sizes, achieving reductions between 9% and 25% in execution time when compared with three other ordering schemes.

Wide-range routing method for lunar exploration rovers using multi-objective optimization

This paper presents a wide-range routing method for multiple rovers deployed for a lunar polar exploration mission. In the mission scenario considered in this paper, multiple exploration points are divided into several sets of points to be assigned to multiple rovers. Subsequently, routes between the exploration points are generated for each rover. This assignment and routing problems are well known as the multiple Traveling Salesman Problem (mTSP). The scenario aims to minimize two objective functions: the sum and standard deviation of operation time among the multiple rovers. This problem is then subject to the multi-objective mTSP (MOmTSP). Therefore, the proposed method is composed of a three-step procedure for solving the MOmTSP. First, the k-means algorithm divides an exploration area into several regions for the number of rovers. Subsequently, a routing algorithm with the space-filling curve and 2-opt algorithm determines the order of exploration points in each region. A local path planner then generates a feasible path between two exploration points. The simulation results using the proposed method with 80 cases are statistically analyzed. We confirm that the proposed method solves the multiple optimization problems by finding the routes that can equalize the rover's operation time and minimize their deviations.

Selective encryption and optimization based watermarking for robust transmission of landslide images

In this paper, a secure watermarking algorithm is proposed to efficiently enhance invisibility and robustness simultaneously. First, the combination of transform domain scheme is used to identify the appropriate coefficient for data embedding. To offer a good balance between robustness and invisibility, we computed optimal embedding factors by using the fusion of particle swarm optimisation (PSO) and the Firefly algorithm. Subsequently, the watermark is scrambled by using a step space-filling curve, and the scrambled watermark is embedded into the wavelet domain of the cover image via the optimal embedding factor. Furthermore, selective encryption is applied to the original host image so that the proposed algorithm can also provide an additional level of security at low cost. We achieved average scores of peak signal-to-noise ratio (PSNR), normalised coefficient (NC), number of changing pixel rate (NPCR) and unified averaged changed intensity (UACI) as 55.1471 dB, 0.9956, 0.9801 and 0.3471, respectively.

Acoustic space filling curve metamaterials for grazing flow in Jet engine inlets

Acoustic metamaterials research has grown exponentially in the past 10 years driven by the advances in manufacturing and an increased understanding of damaging environment noise. 2020 was the first noise reduction target as set by Advisory Council for Aircraft Research and Innovation in Europe with a relative 50% decrease. This was missed by current Jet engine noise control technology; however, metamaterials offer an encouraging alternative. Space Filling Curves (SFC) have the potential to provide a lightweight, thin, high performance acoustic liner. SFC have a history in mathematical geometry dating back to the 1890's but are a comparatively new addition to acoustics. They are designed with a sub-wavelength curled cross-section creating a maze-like pattern which slows acoustic wave propagation through the liner enabling characteristics such as negative refraction and low frequency attenuation. This paper contains a comparison of some of the most promising SFC metamaterial acoustic liner designs, in terms of the fundamental theory of the design category and a discussion of the reflection, absorption and transmission characteristics in terms of a grazing flow conditions. Computer simulation and impedance tube based experimental testing compares the designs. The paper concludes with future application for aeroacoustics with particular focus on the engine inlet.

Three phases of multiple SLE driven by non-colliding Dyson’s Brownian motions

The present paper is concerned with properties of multiple Schramm–Loewner evolutions (SLEs) labelled by a parameter κ ∈ (0, 8]. Specifically, we consider the solution of the multiple Loewner equation driven by a time change of Dyson’s Brownian motions in the non-colliding regime. Although it is often considered that several properties of the solution can be studied by means of commutation relations of SLEs and the absolute continuity, this method is available only in the case that the curves generated by commuting SLEs are separated. Beyond this restriction, it is not even obvious that the solution of the multiple Loewner equation generates multiple curves. To overcome this difficulty, we employ the coupling of Gaussian free fields and multiple SLEs. Consequently, we prove the longstanding conjecture that the solution indeed generates multiple continuous curves. Furthermore, these multiple curves are (i) simple disjoint curves when κ ∈ (0, 4], (ii) intersecting curves when κ ∈ (4, 8), and (iii) space-filling curves when κ = 8.

Deterministic sampling from uniform distributions with Sierpiński space-filling curves

In this paper the problem of sampling from uniform probability distributions is approached by means of space-filling curves, a topological concept that has found a number of important applications in recent years. Departing from the theoretical fact that they are surjective but not necessarily injective, the investigation focused upon the structure of the distributions obtained when their domains are swept in a uniform and discrete manner, and the corresponding values used to build histograms, that are approximations of their true PDFs. This work concentrates on the real interval [0,1] and the Sierpiński space-filling curve was chosen because of its favorable computational properties. In order to validate the results, the Kullback–Leibler and other divergence measures are used when comparing the obtained distributions in several levels of granularity with other already established sampling methods. In truth, the generation of uniform random numbers is a deterministic simulation of randomness using numerical operations. In this fashion, sequences resulting from this sort of process are not truly random. Despite this, and to be coherent with the literature, the expression “random number” will be used along the text to mean “pseudo-random number”.

Space-Filling Curves for Real-Space Electronic Structure Calculations

Hamiltonian matrices for Kohn-Sham calculations implemented in real space are often large (millions by millions) but very sparse. This poses challenges and opportunities for iterative eigensolvers, which often require a large number of matrix-vector multiplications. As a consequence, an efficient parallel sparse matrix-vector multiplication algorithm is desired. Here, we investigate the benefits of using Hilbert space-filling curves (SFCs) in domain partitioning. We show that the use of Hilbert SFCs in grid-point partitioning brings better locality of the grid points, improves balance of communication, and reduces communication overhead. We also demonstrate an extension of Hilbert SFCs coupled with blockwise operations. The use of blockwise operations helps exploit the vector-processing units in contemporary computational platforms. We illustrate speedup and scalability improvements for an iterative eigensolver based on the Chebyshev-filtered subspace iteration method. Using blockwise Hilbert SFCs, we solve the Kohn-Sham problem for silicon nanocrystals up to 10 nm in diameter, which contain over 26,000 atoms. We illustrate how the density of states of silicon nanocrystals evolves to the bulk limit, where Van Hove singularities are clearly apparent.

A Novel Hilbert Curve for Cache-Locality Preserving Loops

Modern microprocessors offer a rich memory hierarchy including various levels of cache and registers. Some of these memories (like main memory, L3 cache) are big but slow and shared among all cores. Others (registers, L1 cache) are fast and exclusively assigned to a single core but small. Only if the data accesses have a high locality, we can avoid excessive data transfers between the memory hierarchy. In this paper we consider fundamental algorithms like matrix multiplication, K-Means, Cholesky decomposition as well as the algorithm by Floyd and Warshall typically operating in two or three nested loops. We propose to traverse these loops whenever possible not in the canonical order but in an order defined by a space-filling curve. This traversal order dramatically improves data locality over a wide granularity allowing not only to efficiently support a cache of a single, known size (cache conscious) but also a hierarchy of various caches where the effective size available to our algorithms may even be unknown (cache oblivious). We propose a new space-filling curve called Fast Unrestricted (FUR) Hilbert with the following advantages: (1) we overcome the usual limitation to square-like grid sizes where the side-length is a power of 2 or 3. Instead, our approach allows arbitrary loop boundaries for all variables. (2) FUR-Hilbert is non-recursive with a guaranteed constant worst case time complexity per loop iteration (in contrast to O(log(grid-size)) for previous methods). (3) Our non-recursive approach makes the application of our cache-oblivious loops in any host algorithm as easy as conventional loops and facilitates automatic optimization by the compiler. (4) We demonstrate that crucial algorithms like Cholesky decomposition as well as the algorithm by Floyd and Warshall by can be efficiently supported. (5) Extensive experiments on runtime efficiency, cache usage and energy consumption demonstrate the profit of our approach. We believe that future compilers could translate nested loops into cache-oblivious loops either fully automatic or by a user-guided analysis of the data dependency.