Morton Order Research Articles

A location privacy protection method based on cipher‐text retrieval in cloud environment

In order to improve the efficiency of location query and the effect of privacy protection in cloud environment, a location privacy protection method based on cipher-text retrieval was proposed. Firstly, the spatial area is divided into several square grids, and the big Morton code of each location is calculated, the spatial area in the grid is divided again, and the small Morton code of each location is calculated. Secondly, the location data is encrypted by the combination of public key encryption and scrambling encryption, and then outsourced to the cloud server. When querying the location data of the interest points, the query request is sent to the data owner. After obtaining the query trapdoor and the private key, the query trapdoor is sent to the location data server for query. On the server side, the interest points in the region are found according to the cipher-text matching, and the cipher-text top-k sorting is implemented. Finally, the cipher-text query results are decrypted into plaintext by the private key. Experimental results show that the proposed method ensures the location data privacy, and improves query efficiency. At the same time, the balance between privacy protection security and query quality is achieved.

A GPU-based multilevel additive schwarz preconditioner for cloth and deformable body simulation

In this paper, we wish to push the limit of real-time cloth and deformable body simulation to a higher level with 50K to 500K vertices, based on the development of a novel GPU-based multilevel additive Schwarz (MAS) pre-conditioner. Similar to other preconditioners under the MAS framework, our preconditioner naturally adopts multilevel and domain decomposition concepts. But contrary to previous works, we advocate the use of small, non-overlapping domains that can well explore the parallel computing power on a GPU. Based on this idea, we investigate and invent a series of algorithms for our preconditioner, including multilevel domain construction using Morton codes, low-cost matrix precomputation by one-way Gauss-Jordan elimination, and conflict-free symmetric-matrix-vector multiplication in runtime preconditioning. The experiment shows that our preconditioner is effective, fast, cheap to precompute and scalable with respect to stiffness and problem size. It is compatible with many linear and nonlinear solvers used in cloth and deformable body simulation with dynamic contacts, such as PCG, accelerated gradient descent and L-BFGS. On a GPU, our preconditioner speeds up a PCG solver by approximately a factor of four, and its CPU version outperforms a number of competitors, including ILU0 and ILUT.

A parallel modular computing environment for three-dimensional multiresolution simulations of compressible flows

Numerical investigation of compressible flows faces two main challenges. In order to accurately describe the flow characteristics, high-resolution nonlinear numerical schemes are needed to capture discontinuities and resolve wide convective, acoustic and interfacial scale ranges. The simulation of realistic three-dimensional (3D) problems with state-of-the-art finite-volume method (FVM) based on approximate Riemann solvers with weighted nonlinear reconstruction schemes requires the usage of high-performance computing (HPC) architectures. Efficient compression algorithms reduce computational and memory load. Fully adaptive multiresolution (MR) algorithms with LTS have proven their potential for such applications. While modern central processing units (CPUs) requires multiple levels of parallelism to achieve peak performance, the fine-grained MR mesh adaptivity results in challenging compute/communication patterns. Moreover, local time stepping (LTS) incurs strong data dependencies which challenge a parallelization strategy.We address these challenges with a block-based MR algorithm, where arbitrary cuts in the underlying octree are possible. This allows for a parallelization on distributed-memory machines via the Message Passing Interface (MPI). We obtain neighbor relations by a simple bit-logic in a modified Morton Order. The block-based concept allows for a modular setup of the source code framework in which the building blocks of the algorithm, such as the choice of the Riemann solver or the reconstruction stencil, are interchangeable without loss of parallel performance. We present the capabilities of the modular framework with a range of test cases and scaling analysis with effective resolutions beyond one billion cells using O(104) cores.

An Intelligent Detection Method for Different Types of Outliers in Multibeam Bathymetric Point Cloud

Bathymetric Multibeam echo sounder systems (MBESs) are the most effective and reliable way to survey the earth’s seafloor. However, multibeam bathymetric data inevitably contain different types of outliers due to measurement characteristics and complex underwater environments. The traditional automatic approach to eliminate outliers may lead to more than one of the questions of reliability, limitation, and efficiency, respectively. This paper offers an algorithm aiming to detect different types of outliers by considering their characteristic of distribution and distance of them, rapidly. First, a coding octree based on Morton code is built to guarantee perfect efficiency and space division. Second, coarse outlier removal is performed by octree-based voxelized representation of the bathymetric data, and outliers far away from the seafloor will be detected and eliminated by connected component labeling. Third, fine outlier removal is employed to delete outliers connected closely to the seafloor by the improved morphological method based on the combination of the k-d tree and octree. Experimental results show that the proposed algorithm can achieve promising results. The percent of outliers that are detected by the hand-edit method that is regarded as a reference result is 5.06%. 4.70% of points in a total number of 3645541 points are detected in our method. Compared with other classical filtering methods, the intelligent method for detecting different types of the outlier from coarse to fine attains favorable performance in a reasonable time, avoiding over-filtering, and demonstrates high reliability for multibeam bathymetric point cloud.

PointSCNet: Point Cloud Structure and Correlation Learning Based on Space-Filling Curve-Guided Sampling

Geometrical structures and the internal local region relationship, such as symmetry, regular array, junction, etc., are essential for understanding a 3D shape. This paper proposes a point cloud feature extraction network named PointSCNet, to capture the geometrical structure information and local region correlation information of a point cloud. The PointSCNet consists of three main modules: the space-filling curve-guided sampling module, the information fusion module, and the channel-spatial attention module. The space-filling curve-guided sampling module uses Z-order curve coding to sample points that contain geometrical correlation. The information fusion module uses a correlation tensor and a set of skip connections to fuse the structure and correlation information. The channel-spatial attention module enhances the representation of key points and crucial feature channels to refine the network. The proposed PointSCNet is evaluated on shape classification and part segmentation tasks. The experimental results demonstrate that the PointSCNet outperforms or is on par with state-of-the-art methods by learning the structure and correlation of point clouds effectively.

Layer-Wise Geometry Aggregation Framework for Lossless LiDAR Point Cloud Compression

Point cloud compression is critical to deploy 3D applications like autonomous driving. However, LiDAR point clouds contain many disconnected regions, where redundant bits for unoccupied 3D space and weak correlations between points make it a troublesome problem to achieve efficient compression. This paper aims to aggregate LiDAR point clouds to get compact representations with full consideration of the point distribution characteristics. Specifically, we propose a novel Layer-wise Geometry Aggregation (LGA) framework for LiDAR point cloud lossless geometry compression, which adaptively partitions point clouds into three layers based on the content properties, including a ground layer, an object layer, and a noise layer. The aggregation algorithms are delicately designed for each layer. Firstly, the ground layer is fitted to a Gaussian Mixture Model, which can uniformly represent ground points using much fewer model parameters than adopting the original 3D coordinates. Then, the object layer is tightly packed to reduce the space between objects effectively, and a dense layout for points can benefit compression efficiency. Finally, in the noise layer, the difference between neighbor points is reduced by reordering using Morton Code, and the reduced residuals can help saving bit consumption. Experimental results demonstrate that the proposed LGA significantly outperforms competitive methods without prior knowledge by 12.05~23.37% compression ratio gains. Furthermore, the enhanced LGA with prior knowledge shows consistent performance gains than the latest reference software. Additional results also validate the robustness and stability of our proposed scheme with acceptable time complexity.

Systematic Review of Indexing Spatial Skyline Queries for Decision Support

Residing in the data age, researchers inferred that huge amount of geo-tagged data is available and identified the importance of Spatial Skyline queries. Spatial or geographic location in conjunction with textual relevance plays a key role in searching Point of Interest (POI) of the user. Efficient indexing techniques like R-Tree, Quad Tree, Z-order curve and variants of these trees are widely available in terms of spatial context. Inverted file is the popular indexing technique for textual data. As Spatial skyline query aims at analyzing both spatial and skyline dominance, there is a necessity for a hybrid indexing technique. This article presents the review of spatial skyline queries evaluation that include a range of indexing techniques which concentrates on disk access, I/O time, CPU time. The investigation and analysis of studies related to skyline queries based upon the indexing model and research gaps are presented in this review.

Novel approaches on bulk‐loading of large scale spatial datasets

SummaryIndex structures that are bulk‐loaded adaptively based on the query workload have been the subject of many database research, primarily in conventional databases holding one‐dimensional data. Similar problems are also valid for spatial databases, and novel approaches are needed in these systems. In this study, we first consider the adaptive indexing approaches in a broad sense and then specifically investigate some possible forms of adaptivity on bulk‐loading of large spatial datasets while retaining some core mechanics of well‐known spatial data structures. In our first design, we examined how the internal configuration of the R‐tree spatial index could be better for various workload properties, such as the workload's average aspect ratio. Our experiments observed that Adaptive Sort‐Tile‐Recursive (STR)‐tree performs better than the original STR‐tree. Our second design, the main contribution of this paper, focused on incremental indexing, another form of adaptivity. We concentrate on the data‐to‐insight latency and loaded spatial data faster within this multi‐stage process. We used the Partitioned B‐tree loaded with Morton codes of spatial objects to achieve this improvement by compromising spatial proximity. Finally, the response time of the first query in our design has been better than using the ordinary STR‐tree index by a factor of 34%.

A New Fast Matching Method for Dummy K-anonymous Location Privacy Protection in Location Based Services

This paper proposes a dummy k-anonymous location privacy protection based on a fast-matching method that adopts the space coordinate transformation algorithm. First, the 2-D coordinates are converted to binary Morton code. With the fast matching method, non-adjacent position points distributed in different grids are selected as candidate sets of dummy positions. Then, the semantic similarity of place name information of position points in the candidate sets is calculated using the edit distance, and k-1 position points with the smallest semantic similarity are selected as dummy positions. While satisfying the semantic l-diversity and physical dispersion, this method can improve the generation efficiency of dummy locations and further improve the quality of location service.

POD: A Parallel Outlier Detection Algorithm Using Weighted kNN

Outlier detection algorithms based on k nearest neighbors ( kNN) can effectively find outliers from massive data, but most algorithms are difficult to adapt to high-dimensional data sets. In order to highlight the importance of attributes in k nearest neighbors, we propose a weighted kNN query method, which uses the Z-order curve to find the kNN. The method first applies information entropy to calculate each attribute weight, and then uses the Z-order curve to encode high-dimensional data into Z-value. The weighted kNN of each object are searched according to its Z-value. Meanwhile, a novel outlier detection algorithm is presented based on the minimum distance and average distance between each object and its weighted kNN. On this basis, we propose a parallel outlier detection algorithm called POD to improve the efficiency of the outlier detection. Finally, we implement and evaluate POD algorithm on a 10-nodes Hadoop cluster, on which synthetic and UCI standard data are tested. Experimental results show that POD achieves high performance in terms of effectiveness, scalability and extensibility.

VECTOR AND RASTER DATA LAYERED FUSION AND 3D VISUALIZATION

Abstract. Although contemporary geospatial science has made great progress, spatial data fusion of vector and raster data is still a problem in the geoinformation science environment. In order to solve the problem, this paper proposes a method which merges vector and raster data. Firstly, the row and column numbers of the raster data, and the X, Y values of the vector data are represented by Morton code in the C++ environment, respectively. Secondly, we establish the the raster data table and the vector data table in the Oracle database to store the vector data and the raster data. Third, this paper uses the minimum selection bounding box method to extract the top data of the building model. Finally, we divide the vector and raster data into four steps to obtain the fusion data table, and we call the fusion data in the database for 3D visualization. This method compresses the size of data of the original data, and simultaneously divides the data into three levels, which not only solves the problem of data duplication storage and unorganized storage, but also can realize vector data storage and the raster data storage in the same database at the same time. Thus, the fusion original orthophoto data contains the gray values of building roofs and the elevation data, which can improve the availability of vector data and the raster data in the 3D Visualization application.

Utilizing extended geocodes for handling massive three-dimensional point cloud data

Point clouds have become a primitive and fundamental material for manifold spatial representations. It can precisely render real-world environments as high-density points which include three-dimensional (3D) coordinates (x, y & z) and other features (color, intensity, and so on). Accordingly, various applications, including robot navigation and self-driving, make use of point clouds not only to detect near objects but to comprehend overall geospatial surroundings. However, it is challenging to exploit the point clouds in terms of spatial query processing in traditional database systems because of its enormous volume and nonstructural formats. In this paper, we propose an efficient method for the manipulation of 3D point cloud based on a Discrete Global Grid System (DGGS). As DGGS represents the Earth as hierarchical sequences of equal area/volume tessellations, it provides an accurate partitioning to integrate and analyze big geospatial data, unlike a base64 geohash representation. This study extends our previous DGGS-based encoding/decoding work to process 3D range queries with more than 64 bits for precise 3D coordinates of point clouds. In particular, we apply PH-tree as a multi-resolution tessellation storage and indexing structure for 3D bounding box queries. The experimental results show that our query processing significantly outperforms the baseline with a linear quadtree. Also, we present the encoding/decoding efficiency of converting large Morton codes from geographic coordinates by using the combination of bit interleaving and lookup tables.

An Efficient Privacy-Preserving Multi-Keyword Query Scheme in Location Based Services

With the proliferation of location-aware mobile devices and the prevalence of wireless communications, location-based services (LBS) have attracted much particular attention in recent years. For flexibility and cost savings, the LBS provider outsources the LBS data to the cloud in order to serve the increasing number of mobile users. To guarantee users’ privacy and data confidentiality, some excellent works have been proposed which focus on secure query over the location server. However, these existing works have two limitations. On the one hand, they cannot preserve users’ location and query content privacy simultaneously. On the other hand, they fail to support multi-keyword queries. In this article, aiming at a multi-keywords query in LBS, we propose a novel efficient and privacy-preserving multi-keyword query scheme (PPMQ) over the outsourced cloud, which satisfies the requirements of the location and query content privacy protection, query efficiency, the confidentiality of LBS data and scalability regarding the data users. To improve the efficiency of our proposed scheme, we utilize the linear quad-tree technique to build a grid system to represent the location information in the query condition as well as a searchable index. To protect the location privacy, we combine decimal Morton code and public-key cryptography techniques to build a searchable index or to generate a trapdoor. To enable the cloud server to perform a secure multi-keyword query, we systematically construct a privacy-preserving query scheme with bilinear pairing-based cryptography. In particular, our proposed scheme is scalable and very suitable for multi-user environments due to the flexible user registration and revocation mechanisms. Furthermore, a detailed security analysis shows that the proposed scheme can ensure the confidentiality of LBS data, and protect the location and query content privacy. Extensive experiments are conducted on a real LBS dataset, and the simulation results confirm the security and efficiency of our scheme.

Application of CUDA-Accelerated GO/PO Method in Calculation of Electromagnetic Scattering From Coated Targets

In this study, fast calculations for the electromagnetic (EM) scattering of complex targets coated by radar absorbing materials have been reported. The Geometric Optics and Physical Optics (GO/PO) methods are combined with the modified surface reflection coefficient method, to solve the problem of EM scattering. Moreover, two acceleration techniques have been employed to improve the efficiency of computations. Firstly, the neighbor search technique is adopted to accelerate the ray tracing process in GO/PO. In this technique, an Octree structure is applied to divide the space into multiple subspaces, namely sub-nodes, where the efficiency of ray tracing can be improved through Morton code transform. In order to further reduce the computational time, parallel acceleration technique, based on GPU platform within the Compute Unified Device Architecture (CUDA) framework is introduced. Our proposed method has been verified from the nice correlation of simulated and multilevel fast multipole method (MLFMM) and CPU-based GO/PO methods. The runtime is compared with that of GO/PO method in serial model, responsible for a good speedup ratio. Finally, the influence of coating thicknesses and types of material on the EM scattering characteristics of the coated targets are also analyzed.

Extended algorithm to construct a quadtree from Freeman chain code in four directions

This paper introduces improvements to the algorithm that was proposed in 2001 by Chen and Chen. The algorithm constructs a quadtree directly from Freeman chain code in four directions. We have improved the algorithm in two ways: Firstly, a time efficient solution using the space filling Z-order curve is proposed for a self-intersection case that was not considered by Chen and Chen. Secondly, the algorithm is expanded to handle geometric objects containing holes. The computational efficiency of the extended algorithm was confirmed by the experiments.

Coconut: sortable summarizations for scalable indexes over static and streaming data series

Many modern applications produce massive streams of data series that need to be analyzed, requiring efficient similarity search operations. However, the state-of-the-art data series indexes that are used for this purpose do not scale well for massive datasets in terms of performance, or storage costs. We pinpoint the problem to the fact that existing summarizations of data series used for indexing cannot be sorted while keeping similar data series close to each other in the sorted order. To address this problem, we present Coconut, the first data series index based on sortable summarizations and the first efficient solution for indexing and querying streaming series. The first innovation in Coconut is an inverted, sortable data series summarization that organizes data series based on a z-order curve, keeping similar series close to each other in the sorted order. As a result, Coconut is able to use bulk loading and updating techniques that rely on sorting to quickly build and maintain a contiguous index using large sequential disk I/Os. We then explore prefix-based and median-based splitting policies for bottom-up bulk loading, showing that median-based splitting outperforms the state of the art, ensuring that all nodes are densely populated. Finally, we explore the impact of sortable summarizations on variable-sized window queries, showing that they can be supported in the presence of updates through efficient merging of temporal partitions. Overall, we show analytically and empirically that Coconut dominates the state-of-the-art data series indexes in terms of construction speed, query speed, and storage costs.

OMiCroN – Oblique Multipass Hierarchy Creation while Navigating

Rendering large point clouds ordinarily requires building a hierarchical data structure for accessing the points that best represent the object for a given viewing frustum and level-of-detail. The building of such data structures frequently represents a large portion of the cost of the rendering pipeline both in terms of time and space complexity, especially when rendering is done for inspection purposes only. This problem has been addressed in the past by incremental construction approaches, but these either result in low quality hierarchies or in longer construction times. In this work we present OMiCroN – Oblique Multipass Hierarchy Creation while Navigating – which is the first algorithm capable of immediately displaying partial renders of the geometry, provided the cloud is made available sorted in Morton order. OMiCroN is fast, being capable of building the entire data structure in memory spending an amount of time that is comparable to that of just reading the cloud from disk. Thus, there is no need for storing an expensive hierarchy, nor for delaying the rendering until the whole hierarchy is read from disk. In fact, a pipeline coupling OMiCroN with an incremental sorting algorithm running in parallel can start rendering as soon as the first sorted prefix is produced, making this setup very convenient for streamed viewing.

A task and data balanced distributed photon mapping method

Photon mapping is an important global illumination method of simulating some visual effects, such as caustic. However, photon mapping is difficult to be parallelized efficiently, due to unbalanced task distribution and memory management. In this paper, we present a combined data-parallel and task-parallel approach to improve performance across multiple rendering clients in a distributed system. The shading points (receivers) from pixels are divided into groups. We associate the photons with each receiver group by computing a tree cut of the photon hierarchy, thus both the data and the task are tightly coupled. Furthermore, we organize both the photon hierarchy and the receivers with Morton code for efficient tree construction. In the end, our method balances both the illumination task of receivers and related photons, resulting in better load balancing and higher performance without any quality degradation.

Five Practices for Supporting Inquiry in Analysis

This article describes an inquiry-oriented real analysis classroom in which students were guided to discover mathematics for themselves. To support student inquiry, the framework of “five practices” from K-12 education was used. To illustrate this framework, two case examples are given from actual discussions that took place in this classroom. These examples focus on student construction of sequences of functions, both the space-filling Z-order curve and the Cantor staircase function. The article closes with practical guidance for other instructors who would use the five practices in their higher-level mathematics courses.

A multi-dimensional Morton-ordered block storage for mode-oblivious tensor computations

Computation on tensors, treated as multidimensional arrays, revolve around generalized basic linear algebra subroutines (BLAS). We propose a novel data structure in which tensors are blocked and blocks are stored in an order determined by Morton order. This is not only proposed for efficiency reasons, but also to induce efficient performance regardless of which mode a generalized BLAS call is invoked for; we coin the term mode-oblivious to describe data structures and algorithms that induce such behavior. Experiments on one of the most bandwidth-bound generalized BLAS kernel, the tensor–vector multiplication, not only demonstrate superior performance over two state-of-the-art variants by up to 18%, but additionally show that the proposed data structure induces a 71% less sample standard deviation for tensor–vector multiplication across d modes, where d varies from 2 to 10. Finally, we show our data structure naturally expands to other tensor kernels and demonstrate up to 38% higher performance for the higher-order power method.