scholarly journals Approximation and Analytical Studies of Inter-clustering Performances of Space-Filling Curves

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Ho-Kwok Dai ◽  
Hung-Chi Su
Keyword(s):  
Random Walk ◽  
Dimensional Space ◽  
Hilbert Curve ◽  
Space Filling ◽  
Space Filling Curve ◽  
Space Filling Curves ◽  
Order Curve ◽  
Cluster Distance ◽  
The Mean ◽  
Curve Family

International audience A discrete space-filling curve provides a linear traversal/indexing of a multi-dimensional grid space.This paper presents an application of random walk to the study of inter-clustering of space-filling curves and an analytical study on the inter-clustering performances of 2-dimensional Hilbert and z-order curve families.Two underlying measures are employed: the mean inter-cluster distance over all inter-cluster gaps and the mean total inter-cluster distance over all subgrids.We show how approximating the mean inter-cluster distance statistics of continuous multi-dimensional space-filling curves fits into the formalism of random walk, and derive the exact formulas for the two statistics for both curve families.The excellent agreement in the approximate and true mean inter-cluster distance statistics suggests that the random walk may furnish an effective model to develop approximations to clustering and locality statistics for space-filling curves.Based upon the analytical results, the asymptotic comparisons indicate that z-order curve family performs better than Hilbert curve family with respect to both statistics.

scholarly journals The Effect of Space-filling Curves on the Efficiency of Hand Gesture Recognition Based on sEMG Signals

2021 ◽  
Vol 12 (1) ◽  
pp. 23-31
Author(s):  
Panagiotis Tsinganos ◽  
Bruno Cornelis ◽  
Jan Cornelis ◽  
Bart Jansen ◽  
Athanassios Skodras
Keyword(s):  
Computer Vision ◽  
Gesture Recognition ◽  
Hand Gesture Recognition ◽  
Hand Gesture ◽  
Network Architectures ◽  
Hilbert Curve ◽  
Space Filling ◽  
Space Filling Curve ◽  
Space Filling Curves ◽  
Semg Signals

Over the past few years, Deep learning (DL) has revolutionized the field of data analysis. Not only are the algorithmic paradigms changed, but also the performance in various classification and prediction tasks has been significantly improved with respect to the state-of-the-art, especially in the area of computer vision. The progress made in computer vision has produced a spillover in many other domains, such as biomedical engineering. Some recent works are directed towards surface electromyography (sEMG) based hand gesture recognition, often addressed as an image classification problem and solved using tools such as Convolutional Neural Networks (CNN). This paper extends our previous work on the application of the Hilbert space-filling curve for the generation of image representations from multi-electrode sEMG signals, by investigating how the Hilbert curve compares to the Peano- and Z-order space-filling curves. The proposed space-filling mapping methods are evaluated on a variety of network architectures and in some cases yield a classification improvement of at least 3%, when used to structure the inputs before feeding them into the original network architectures.

scholarly journals A Parallel N-Dimensional Space-Filling Curve Library and Its Application in Massive Point Cloud Management

2018 ◽  
Vol 7 (8) ◽  
pp. 327 ◽  
Author(s):  
Xuefeng Guan ◽  
Peter van Oosterom ◽  
Bo Cheng
Keyword(s):  
Point Cloud ◽  
Parallel Implementation ◽  
Dimensional Space ◽  
Space Filling ◽  
Oracle Database ◽  
Space Filling Curve ◽  
Object Oriented Design ◽  
Massive Point ◽  
Locality Preservation ◽  
Cloud Management

Because of their locality preservation properties, Space-Filling Curves (SFC) have been widely used in massive point dataset management. However, the completeness, universality, and scalability of current SFC implementations are still not well resolved. To address this problem, a generic n-dimensional (nD) SFC library is proposed and validated in massive multiscale nD points management. The library supports two well-known types of SFCs (Morton and Hilbert) with an object-oriented design, and provides common interfaces for encoding, decoding, and nD box query. Parallel implementation permits effective exploitation of underlying multicore resources. During massive point cloud management, all xyz points are attached an additional random level of detail (LOD) value l. A unique 4D SFC key is generated from each xyzl with this library, and then only the keys are stored as flat records in an Oracle Index Organized Table (IOT). The key-only schema benefits both data compression and multiscale clustering. Experiments show that the proposed nD SFC library provides complete functions and robust scalability for massive points management. When loading 23 billion Light Detection and Ranging (LiDAR) points into an Oracle database, the parallel mode takes about 10 h and the loading speed is estimated four times faster than sequential loading. Furthermore, 4D queries using the Hilbert keys take about 1~5 s and scale well with the dataset size.

scholarly journals Tag-systems for the Hilbert curve

2007 ◽  
Vol Vol. 9 no. 2 ◽  
Author(s):  
Patrice Séébold
Keyword(s):  
Hilbert Space ◽  
Hilbert Curve ◽  
Space Filling ◽  
Space Filling Curve ◽  
Finite Approximations ◽  
Infinite Word ◽  
International Audience ◽  
Filling Curve

International audience Hilbert words correspond to finite approximations of the Hilbert space filling curve. The Hilbert infinite word H is obtained as the limit of these words. It gives a description of the Hilbert (infinite) curve. We give a uniform tag-system to generate automatically H and, by showing that it is almost cube-free, we prove that it cannot be obtained by simply iterating a morphism.

On the Behaviour of p-Adic Scaled Space Filling Curve Indices for High-Dimensional Data

2020 ◽  
Author(s):  
Patrick Erik Bradley ◽  
Markus Wilhelm Jahn
Keyword(s):  
Gray Code ◽  
High Dimensional ◽  
Affine Transformations ◽  
Hilbert Curve ◽  
Space Filling ◽  
Real World Data ◽  
Space Filling Curve ◽  
Indexing Method ◽  
Filling Curve ◽  
Hilbert Curves

Abstract Space filling curves are widely used in computer science. In particular, Hilbert curves and their generalizations to higher dimension are used as an indexing method because of their nice locality properties. This article generalizes this concept to the systematic construction of $p$-adic versions of Hilbert curves based on special affine transformations of the $p$-adic Gray code and develops a scaled indexing method for data taken from high-dimensional spaces based on these new curves, which with increasing dimension is shown to be less space consuming than the optimal standard static Hilbert curve index. A measure is derived, which allows to assess the local sparsity of a dataset, and is tested on some real-world data.

scholarly journals CLUSTERING AND INDEXING HISTORIC VESSEL MOVEMENT DATA WITH SPACE FILLING CURVES

2018 ◽  
Vol XLII-4 ◽  
pp. 417-424 ◽  
Author(s):  
M. Meijers ◽  
P. van Oosterom
Keyword(s):  
Integrated Approach ◽  
Space Time ◽  
Database Management System ◽  
Identification System ◽  
Time Interval ◽  
Storage Space ◽  
Space Filling ◽  
Space Filling Curve ◽  
Space Filling Curves ◽  
Integrated Index

<p><strong>Abstract.</strong> This paper reports on the result of an on-going study using Space Filling Curves (SFCs) for indexing and clustering vessel movement message data (obtained via the Automated Identification System, AIS) inside a geographical Database Management System (Geo-DBMS). With AIS, vessels transmit their positions in intervals ranging from 2 seconds to 3 minutes. Every 6 minutes voyage related information is broadcast.</p><p> Relevant AIS messages contain a position, timestamp and vessel identifier. This information can be stored in a DBMS as separate columns with different types (as 2D point plus time plus identifier), or in an integrated column (as higher dimensional 4D point which is encoded as the position on a space filling curve, that we will call the SFC-key). Subsequently, indexing based on this SFC-key column can replace separate indexes (where this one integrated index will need less storage space than separate indexes). Moreover, this integrated index allows a good clustering (physical ordering of the table). Also, in an approach with separate indexes for location, time and object identifier the query optimizer inside a DBMS has to estimate which index is most selective for a given query. It is not possible to use two indexes at the same time &amp;ndash; e.g. in case of a space-time query. An approach with one multi-dimensional integrated index does not have this problem. It results in faster query responses when specifying multiple selection criteria; i.e. both search geometry and time interval.</p><p> We explain the steps needed to make this SFC approach available <i>fully inside</i> a DBMS (to avoid expensive data transfer to external programs during use). The SFC approach makes it possible to better cluster the (spatio-temporal) data compared to an approach with separate indexes. Moreover, we show experiments (with 723,853,597 AIS position report messages spanning 3 months, Sep&amp;ndash;Dec 2016, using data for Europe, both on-sea and inland water ways) to compare an approach based on one multi-dimensional integrated index (using a SFC) with non-integrated approach. We analyze loading time (including SFC encoding) and storage requirements, together with the speed of execution of queries and granularity of answers.</p><p> Conclusion is that time spend on query execution in case of space-time queries where both dimensions are selective using the integrated SFC approach outperforms the non-integrated approach (typically a factor 2&amp;ndash;6). Also, the SFC approach saves considerably on storage space (less space needed for indexes). Lastly, we propose some future improvements to get some better query performance using the SFC approach (e.g. IOT, range-glueing and nD-histogram).</p>

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