Parallel Line Segments Research Articles

Continuity-sets of pullback random attractors for discrete porous media equations with colored noise

For a random non-autonomous porous media lattice system driven by nonlinear colored noise, we prove the unique existence and local compactness of a pullback random attractor. We then mainly study the continuity-set (the set of all points of continuity) of the pullback random attractor on the time-sample plane with respect to the Hausdorff metric. With some calculations, we find that the continuity-set has four geometrical numerical features:(1) The continuity-set is residual and dense in the time-sample plane.(2) The continuity-set is composed of infinitely many diagonal rays (with slope one).(3) The continuity-set is dense in any non-diagonal line (with an inclined angle not being 45∘).(4) Restrictions of the continuity-set on diamonds are composed of parallel line segments with increasing numbers.

Geometry and Topology Reconstruction of BIM Wall Objects from Photogrammetric Meshes and Laser Point Clouds

As the foundation for digitalization, building information modeling (BIM) technology has been widely used in the field of architecture, engineering, construction, and facility management (AEC/FM). Unmanned aerial vehicle (UAV) oblique photogrammetry and laser scanning have become increasingly popular data acquisition techniques for surveying buildings and providing original data for BIM modeling. However, the geometric and topological reconstruction of solid walls, which are among the most important architectural structures in BIM, is still a challenging undertaking. Due to noise and missing data in 3D point clouds, current research mostly focuses on segmenting wall planar surfaces from unstructured 3D point clouds and fitting the plane parameters without considering the thickness or 3D shape of the wall. Point clouds acquired only from the indoor space are insufficient for modeling exterior walls. It is also important to maintain the topological relationships between wall objects to meet the needs of complex BIM modeling. Therefore, in this study, a geometry and topology modeling method is proposed for solid walls in BIM based on photogrammetric meshes and laser point clouds. The method uses a kinetic space-partitioning algorithm to generate the building footprint and indoor floor plan. It classifies interior and exterior wall segments and infers parallel line segments to extract wall centerlines. The topological relationships are reconstructed and maintained to build wall objects with consistency. Experimental results on two datasets, including both photogrammetric meshes and indoor laser point clouds, exhibit more than 90% completeness and correctness, as well as centimeter-level accuracy of the wall surfaces.

Wavenumber-Division Multiplexing in Line-of-Sight Holographic MIMO Communications

Starting from first principles of wave propagation, we consider a multiple-input multiple-output (MIMO) representation of a communication system between two spatially-continuous volumes. This is the concept of holographic MIMO communications. The analysis takes into account the electromagnetic interference, generated by external sources, and the constraint on the physical radiated power. The electromagnetic MIMO model is particularized for a pair of parallel line segments in line-of-sight conditions. Inspired by orthogonal-frequency division-multiplexing, we assume that the spatially-continuous transmit currents and received fields are represented using the Fourier basis functions. In doing so, a wavenumber-division multiplexing (WDM) scheme is obtained, which is not optimal but can be efficiently implemented. The interplay among the different system parameters (e.g., transmission range, wavelength, and sizes of source and receiver) in terms of number of communication modes and level of interference among them is studied with conventional tools of linear systems theory. Due to the non-finite support (in the spatial domain) of the electromagnetic channel, WDM cannot provide non-interfering communication modes. The interference decreases as the receiver size grows, and goes to zero only asymptotically. Different digital processing architectures, operating in the wavenumber domain, are thus used to deal with the interference. The simplest implementation provides the same spectral efficiency of a singular-value decomposition architecture with water-filling when the receiver size is comparable to the transmission range. The developed framework is also used to represent a communication scheme that performs only an integration over short spatial segments. This is equivalent to a classical MIMO system with uniform linear arrays made of electrically small dipoles. Numerical comparisons show that better performance than WDM can be achieved only when a higher number of radio-frequency chains is used.

An unsupervised framework for extracting multilane roads from OpenStreetMap

Multilane roads are a set of approximately parallel line segments representing the same road in large-scale vector maps. They must be extracted first in cartographic generalization. There are numerous multilane roads in the easily accessible OpenStreetMap (OSM) dataset. For this dataset, polygon-based methods have achieved state-of-the-art performance. However, traditional polygon-based methods usually rely on manually labeled data, which means they are time-consuming and labor-intensive. To address this problem, an unsupervised framework for extracting multilane roads is proposed in this study. Road segments were first grouped to form the road polygons. A set of shape descriptors was formulated to reduce the dimensions of individual road polygons into conceptual points. Next, dimensional shape descriptors were standardized using logarithmic standardization. The density peaks clustering (DPC) algorithm was employed to classify these points. Then, cluster tags were identified manually to recognize which clusters represent multilane polygons. Finally, post-processing learning from the concept of assimilation is proposed to fill holes and remove islands. Experiments were conducted to extract multilane roads with datasets from three cities: Wuhan, Beijing and Munich. The experimental results show that the proposed framework effectively extracted multilane roads without any labels with accuracy levels comparable to those of supervised methods.

Stratified β$\beta$‐numbers and traveling salesman in Carnot groups

We introduce a modified version of Jones's β $\beta$ -numbers for Carnot groups which we call stratified β $\beta$ -numbers. We show that an analogue of Jones's traveling salesman theorem on 1-rectifiability of sets holds for any Carnot group if we replace previous notions of β $\beta$ -numbers in Carnot groups with stratified β $\beta$ -numbers. As we generalize both directions of the traveling salesman theorem, we get a characterization of subsets of Carnot groups that lie on finite length rectifiable curves. Our proof expands upon previous analysis of the Hebisch–Sikora norm for Carnot groups. In particular, we find new estimates on the drift between almost parallel line segments that take advantage of the stratified functions β $\beta$ and also develop a Taylor expansion technique of the norm. We also give an example of a Carnot group for which a traveling salesman theorem based on the unmodified β $\beta$ -numbers must exhibit a gap between the necessary and sufficient directions.

Middle school students’ reasoning with regards to parallelism and perpendicularity of line segments

This study investigates middle school students’ reasoning about parallelism and perpendicularity of two line segments. Data were collected from 83 middle school students through an identification task consisting of various examples and non-examples of the parallelism and perpendicularity of two line segments. One-to-one interviews were also conducted with fifteen students to support students’ written reasons. The results indicated that middle school students identify parallel/perpendicular line segments by using three types of reasoning: visual reasoning, attribute reasoning, and language-based reasoning. With the influence of the concept images shaped by prototypical and intuitive examples and linguistic factors, some students either focused on non-critical attributes of the concepts as the critical ones or omitted critical attributes of the concepts. Hence, they either identified examples of parallel line segments and perpendicular line segments as non-examples or considered non-examples of the concepts as examples.

A Benchmark Sentinel-1 SAR Dataset for Airport Detection

Airports are important targets in both military and civilian tasks. Synthetic aperture radar (SAR) based airport detection has received increasing attention in recent years. However, due to the high cost of the SAR imaging and annotation process, there is no publicly available SAR dataset for airport detection, which leads to the fact that deep learning methods have not been applied to airport detection and no unified evaluation benchmark for traditional methods. To provide a benchmark for airport detection studies in SAR images, a large SAR airport dataset (SAD) is presented in this paper. It contains 624 SAR images from Sentinel 1B, covering 104 instances of airports with different scales, orientations and shapes, which can realistically reflect the real world. Experiments with four deep learning-based methods and three traditional methods on this dataset demonstrate its effectiveness and challenge. It serves for the development of state-of-the-art airport detection algorithms or other related tasks. In addition, we found that airport runways in SAR images always have some parallel line segments. Inspired by this, the fusion of deep features and line segments is achieved by designing a line segment detector branch, which further improves the accuracy of airport detection.

A Sweep-Line Algorithm and Its Application to Spiral Pocketing.(Dept.M)

This paper presents an efficient line-offset algorithm for general polygonal shapes with islands. A developed sweep-line algorithm (SL) is introduced to find all sell-intersection points accurately and quickly. The previous work is limited to handle polygons that having no line-segments in parallel to $Weep-line directions. An invalid loop detection and removing (ILDR) algorithm is proposed. The invalid loops detection algorithm divides the polygon al self-intersection points into a set of small polygons, and re-polygonized them. The polygons are checked for direction; invalid polygons are always having inverse direction with the boundary polygon. The proposed algorithm has been implemented in Visual C++ and applied to offset point scquence curves, which contain several islands.

Analysis and modelling of rampart line antennas

A new, efficient, and accurate analytical model is introduced for estimating the beam direction of rampart line antennas with microstrip radiators that can change beam direction with feeding frequency. The model takes its origins from 1-D array theory, in which the bends represent the radiating elements, and the interconnecting lines represent the phase shifters. A mathematical expression accounting for the equivalent length of the bends for the correct estimation of the distance between the radiators is introduced. The model was tested for various substrate relative permittivity values ranging from 2.9 to 6.45. This model can be used for arbitrary rampart line configurations by providing the equivalent dipole direction, depending on the orientation of the bends. An alternative geometry was proposed to reduce the coupling between the parallel line segments. A prototype operating in a bandwidth of 1 GHz around 7.8 GHz was realized on a Rogers RO4350B high-frequency laminate with a thickness of 1.524 mm. Both the radiation pattern and the reflection coefficient were measured and show a good agreement with the proposed analytical model and commercial full-wave simulation tool results.

Mathematically optimized trajectory for terrestrial close-range photogrammetric 3D reconstruction of forest stands

Terrestrial close-range photogrammetry offers a low-cost method of three-dimensional (3D) reconstruction of forest stands that provides automatically processable 3D data that can be used to evaluate inventory parameters of forest stands and individual trees. However, fundamental methodological problems in image acquisition and processing remain. This study enhances the methodology of photogrammetric Structure from Motion reconstruction of forest stands by determining the best photographer's trajectory for image acquisition. The study comprises 1) mathematical optimization of the route in a square grid using integer programming, 2) evaluation of point clouds derived from sequences of real photographs, simulating different trajectories, and 3) verification on real trajectories. In a forest research plot, we established a 1 m square grid of 625 (i.e., 25 × 25) photographic positions, and at each position, we captured 16 photographs in uniformly spaced directions. We adopted real tree positions and diameters, and the coordinates of the photographic positions, including orientation angles of captured images, were recorded. We then formulated an integer programming optimization model to find the most efficient trajectory that provided coverage of all sides of all trees with sufficient counts of images. Subsequently, we used the 10,000 captured images to produce image subsets simulating image sequences acquired during the photographer's movement along 84 different systematic trajectories of seven patterns based on either parallel lines or concentric orbits. 3D point clouds derived from the simulated image sequences were evaluated for their suitability for automatic tree detection and estimation of diameters at breast height. The results of the integer programming model indicated that the optimal trajectory consisted of parallel line segments if the camera is pointed forward – in the travel direction, or concentric orbits if the camera is pointed to a side – perpendicular to the travel direction. With point clouds derived from the images of the simulated trajectories, the best diameter estimates on automatically detected trees were achieved with trajectories consisting of parallel lines in two perpendicular directions where each line was passed in both opposite directions. For efficient image acquisition, resulting in point clouds of reasonable quality with low counts of images, a trajectory consisting of concentric orbits, including the plot perimeter with the camera pointed towards the plot center, proved to be the best. Results of simulated trajectories were verified with the photogrammetric reconstruction of the forest stand based on real trajectories for six patterns. The mathematical optimization was consistent with the results of the experiment, which indicated that mathematical optimization may represent a valid tool for planning trajectories for photogrammetric 3D reconstruction of scenes in general.

Largest and smallest area triangles on imprecise points

Assume we are given a set of parallel line segments in the plane, and we wish to place a point on each line segment such that the resulting point set maximizes or minimizes the area of the largest or smallest triangle in the set. We analyze the complexity of the four resulting computational problems, and we show that three of them admit polynomial-time algorithms, while the fourth is NP-hard. Specifically, we show that maximizing the largest triangle can be done in O(n2) time (or in O(nlog⁡n) time for unit segments); minimizing the largest triangle can be done in O(n4) time; maximizing the smallest triangle is NP-hard; but minimizing the smallest triangle can be done in O(n2) time. We also discuss to what extent our results can be generalized to polygons with k>3 sides.

Automated mapping of buildings through classification of DSM-based ortho-images and cartographic enhancement

Urban areas are changing rapidly. In order to document the urban realities in topographic databases and geographic information systems efficient methods are required. Vector data of buildings are of special importance. A methodology for an automated generation of cartographically enhanced data is presented and applied to two test sites at Vaihingen, Germany. The steps of the workflow are described in detail. The examples use imagery of a large-format aerial camera to map different types of buildings. First, land cover maps are generated by means of supervised classification using two sets of attributes (basic attributes and attribute profile, basic attributes and dispersion). After the enhancement of the extracted buildings their outlines have straight, orthogonal, and parallel line segments created by least squares adjustment. The assessment of the geometric accuracy used 264 well-defined building corners and two types of references (land cover map, ortho-image). The obtained average standard deviation of the coordinates was σx,y = 1.0 m. The additional use of an attribute profile did not improve upon the geometric accuracy that was obtained by means of five attributes (height above ground, normalized difference vegetation index, standard deviation of the elevations in the 5 × 5 pixels window, intensity value of the near-infrared band, and standard deviation of intensities in the 5 × 5 surrounding at a pixel of the near-infrared band). The experiences with the developed software reveal that a graphical output of intermediate results is helpful to obtain complete and reliable results at complex building structures.

A LOWER BOUND OF TOPOLOGICAL HAUSDORFF DIMENSION OF FRACTAL SQUARES

Given an integer [Formula: see text] and a digit set [Formula: see text], there is a self-similar set [Formula: see text] satisfying the set equation [Formula: see text]. This set [Formula: see text] is called a fractal square. By studying the line segments contained in [Formula: see text], we give a lower estimate of the topological Hausdorff dimension of fractal squares. Moreover, we compute the topological Hausdorff dimension of fractal squares whose nontrivial connected components are parallel line segments, and introduce the Latin fractal squares to investigate the question when the topological Hausdorff dimension of a fractal square coincides with its Hausdorff dimension.

Joint Estimation of Camera Orientation and Vanishing Points from an Image Sequence in a Non-Manhattan World

A widely used approach for estimating camera orientation is to use the points at infinity, i.e., the vanishing points (VPs). Enforcement of the orthogonal constraint between the VPs, known as the Manhattan world constraint, enables an estimation of the drift-free camera orientation to be achieved. However, in practical applications, this approach is neither effective (because of noisy parallel line segments) nor performable in non-Manhattan world scenes. To overcome these limitations, we propose a novel method that jointly estimates the VPs and camera orientation based on sequential Bayesian filtering. The proposed method does not require the Manhattan world assumption, and can perform a highly accurate estimation of camera orientation. In order to enhance the robustness of the joint estimation, we propose a keyframe-based feature management technique that removes false positives from parallel line clusters and detects new parallel line sets using geometric properties such as the orthogonality and rotational dependence for a VP, a line, and the camera rotation. In addition, we propose a 3-line camera rotation estimation method that does not require the Manhattan world assumption. The 3-line method is applied to the RANSAC-based outlier rejection technique to eliminate outlier measurements; therefore, the proposed method achieves accurate and robust estimation of the camera orientation and VPs in general scenes with non-orthogonal parallel lines. We demonstrate the superiority of the proposed method by conducting an extensive evaluation using synthetic and real datasets and by comparison with other state-of-the-art methods.

Generating Topographic Map Data from Classification Results

The use of classification results as topographic map data requires cartographic enhancement and checking of the geometric accuracy. Urban areas are of special interest. The conversion of the classification result into topographic map data of high thematic and geometric quality is subject of this contribution. After reviewing the existing literature on this topic, a methodology is presented. The extraction of point clouds belonging to line segments is solved by the Hough transform. The mathematics for deriving polygons of orthogonal, parallel and general line segments by least squares adjustment is presented. A unique solution for polylines, where the Hough parameters are optimized, is also given. By means of two data sets land cover maps of six classes were produced and then enhanced by the proposed method. The classification used the decision tree method applying a variety of attributes including object heights derived from imagery. The cartographic enhancement is carried out with two different levels of quality. The user’s accuracies for the classes “impervious surface” and “building” were above 85% in the “Level 1” map of Example 1. The geometric accuracy of building corners at the “Level 2” maps is assessed by means of reference data derived from ortho-images. The obtained root mean square errors (RMSE) of the generated coordinates (x, y) were RMSEx = 1.2 m and RMSEy = 0.7 m (Example 1) and RMSEx = 0.8 m and RMSEy = 1.0 m (Example 2) using 31 and 62 check points, respectively. All processing for Level 1 (raster data) could be carried out with a high degree of automation. Level 2 maps (vector data) were compiled for the classes “building” and “road and parking lot”. For urban areas with numerous classes and of large size, universal algorithms are necessary to produce vector data fully automatically. The recent progress in sensors and machine learning methods will support the generation of topographic map data of high thematic and geometric accuracy.

Harbor Detection in Polarimetric SAR Images Based on the Characteristics of Parallel Curves

This letter proposes a harbor detection method by extracting specific paralleled contours of harbors in polarimetric synthetic aperture radar (SAR) images. First, land and sea are partitioned by the region-based level set segmentation algorithm which is applied to the extraction of the volume scattering component. Then, parallel curves are modeled by a series of parallel line segments. They are detected by finding out and connecting paralleled line segments given certain criteria. The detected parallel curves are merged according to a distance measurement so that the harbor contour is finally detected. RADARSAT-2 polarimetric SAR data covering an area in Singapore are used to test the harbor detection method. Experimental results show that the method improves sea–land segmentation accuracy and extracts all parallel curves along the coast correctly. The method enables an accurate detection of harbors.

Quantitative Evaluation of Metal Artifacts on CT Images on the Basis of Statistics of Extremes

It is well-known that metal artifacts have a harmful effect on the image quality of computed tomography (CT) images. However, the physical property remains still unknown. In this study, we investigated the relationship between metal artifacts and tube currents using statistics of extremes. A commercially available phantom for measuring CT dose index 160 mm in diameter was prepared and a brass rod 13 mm in diameter was placed at the centerline of the phantom. This phantom was used as a target object to evaluate metal artifacts and was scanned using an area detector CT scanner with various tube currents under a constant tube voltage of 120 kV. Sixty parallel line segments with a length of 100 pixels were placed to cross metal artifacts on CT images and the largest difference between two adjacent CT values in each of 60 CT value profiles of these line segments was employed as a feature variable for measuring metal artifacts; these feature variables were analyzed on the basis of extreme value theory. The CT value variation induced by metal artifacts was statistically characterized by Gumbel distribution, which was one of the extreme value distributions; namely, metal artifacts have the same statistical characteristic as streak artifacts. Therefore, Gumbel evaluation method makes it possible to analyze not only streak artifacts but also metal artifacts. Furthermore, the location parameter in Gumbel distribution was shown to be in inverse proportion to the square root of a tube current. This result suggested that metal artifacts have the same dose dependence as image noises.

On shape-preserving capability of cubic L1 spline fits

Cubic L1 spline fits have shown some favorable shape-preserving property for geometric data. To quantify the shape-preserving capability, we consider the basic shape of two parallel line segments in a given window. When one line segment is sufficiently longer than the other, the spline fit can preserve its linear shape in at least half of the window. We propose to use the minimum of such length difference as a shape-preserving metric because it represents the extra information that the spline fits need to preserve the shape. We analytically calculate this metric in a 3-node window for second-derivative-based, first-derivative-based and function-value-based spline fits. In a 5-node window, we compute this metric numerically. In both cases, the shape-preserving metric is rather small, which explains the observed strong shape-preserving capability of spline fits. Moreover, the function-value-based spline fits are indicated to preserve shape better than the other two types of spline fits. This study initiates a quantitative research on shape preservation of L1 spline fits.

PARALLEL LINE SEGMENTS

In this article I will address some questions about a mathematical problem that my friend Patrizio Frederic, a researcher in statistics at the University of Modena, proposed to me. Given some parallel line segments, is there at least straight line that passes through all of them? If there were many lines that solve the problem, can I choose a best one among all of them? I will fully address the first question. As for the second question, I will illustrate it with some experimental examples and suggest an outline for future explorations.

Fast image matching algorithm based on affine invariants

Feature-based image matching algorithms play an indispensable role in automatic target recognition (ATR). In this work, a fast image matching algorithm (FIMA) is proposed which utilizes the geometry feature of extended centroid (EC) to build affine invariants. Based on affine invariants of the length ratio of two parallel line segments, FIMA overcomes the invalidation problem of the state-of-the-art algorithms based on affine geometry features, and increases the feature diversity of different targets, thus reducing misjudgment rate during recognizing targets. However, it is found that FIMA suffers from the parallelogram contour problem and the coincidence invalidation. An advanced FIMA is designed to cope with these problems. Experiments prove that the proposed algorithms have better robustness for Gaussian noise, gray-scale change, contrast change, illumination and small three-dimensional rotation. Compared with the latest fast image matching algorithms based on geometry features, FIMA reaches the speedup of approximate 1.75 times. Thus, FIMA would be more suitable for actual ATR applications.