Journal articles on the topic 'Irregular polygonal objects'

Introduction. Optimization placement problems are NP-hard. In most cases related to cutting and packing problems, heuristic approaches are used. The development of analytical methods for mathematical modeling of the problems is of paramount important for expanding the class of placement problems that can be solved optimally using state of the art NLP-solvers. The problem of placing two irregular two-dimensional objects in a convex polygonal region of the minimum size, which is a convex polygonal hull of the minimum area or perimeter, is considered. Continuous rotations and translations of non-overlapping objects are allowed. To solve the problem of optimal compaction of a pair of objects, two algorithms are proposed. The first is a sequentially search for local extrema on all feasible subdomains using a solution tree. The second algorithm searches for a locally optimal extremum on a single subdomain using a "good" feasible starting point. Purpose of the paper. Show how to construct a minimal convex polygonal hull for two continuously moving irregular objects bounded by circular arcs and line segments. Results. A mathematical model is constructed in the form of a nonlinear programming problem using the phi-function technique. Two algorithms are proposed for solving the problem of placing a pair of objects in order to minimize the area and perimeter of the enclosing polygonal area. The results of computational experiments are presented. Conclusions. The construction of a minimal convex polygonal hull for a pair of two-dimensional objects having an arbitrary spatial shape and allowing continuous rotations and translations makes it possible to speed up the process of finding feasible solutions for the problem of placing a large number of objects with complex geometry. Keywords: convex polygonal hull, irregular objects, phi-function technique, nonlinear optimization.

This paper presents an automated approach to the extraction of building footprints from airborne LiDAR data based on energy minimization. Automated 3D building reconstruction in complex urban scenes has been a long-standing challenge in photogrammetry and computer vision. Building footprints constitute a fundamental component of a 3D building model and they are useful for a variety of applications. Airborne LiDAR provides large-scale elevation representation of urban scene and as such is an important data source for object reconstruction in spatial information systems. However, LiDAR points on building edges often exhibit a jagged pattern, partially due to either occlusion from neighbouring objects, such as overhanging trees, or to the nature of the data itself, including unavoidable noise and irregular point distributions. The explicit 3D reconstruction may thus result in irregular or incomplete building polygons. In the presented work, a vertex-driven Douglas-Peucker method is developed to generate polygonal hypotheses from points forming initial building outlines. The energy function is adopted to examine and evaluate each hypothesis and the optimal polygon is determined through energy minimization. The energy minimization also plays a key role in bridging gaps, where the building outlines are ambiguous due to insufficient LiDAR points. In formulating the energy function, hard constraints such as parallelism and perpendicularity of building edges are imposed, and local and global adjustments are applied. The developed approach has been extensively tested and evaluated on datasets with varying point cloud density over different terrain types. Results are presented and analysed. The successful reconstruction of building footprints, of varying structural complexity, along with a quantitative assessment employing accurate reference data, demonstrate the practical potential of the proposed approach.

<p><strong>Abstract.</strong> In many Cultural Heritage cases study, where usually the shapes to be reproduced have complex and irregular geometries there is a growing demand to create and use very high-resolution polygonal models that represent the real objects with great accuracy and level of detail. In archaeological field, for example, it is fundamental to create a virtual reconstruction that is as close as possible to the reality, because the digging operations are often destructive and it is necessary to track and preserve for the future as much as possible. This requirement leads to three relevant problems: 1) the elaboration of high-poly and high-resolution models, 2) the management of these models, 3) their sharing and access. In this paper, both a scientific approach for realizing the accurate and detailed models and a system to manage, share and visualize these models on the web will be shown. Moreover, a multi-texture digital model elaboration is proposed for the correct definition of the geometrical and texture resolution in relation to the survey level of detail, the physical size of the object and the project requirement. In particular, a web-system that allows the sharing of very high-resolution models, with multi-textures support will be presented.</p>

Abstract This paper proposes an improved modelling approach for tessellating regular polygons in such a way that it is environmentally sustainable. In this paper, tessellation of polygons that have been innovated through the formed motifs, is an innovation from the traditional tessellations of objects and animals. The main contribution of this work is the simplification and innovating new patterns from the existing regular polygons, in which only three polygons (triangle, square and hexagon) that can free be tessellated are used, compared to using irregular polygons or other objects. This is achieved by reducing the size of each polygon to smallest value and tessellating each of the reduced figure to the right or to left to obtain a two different designs of one unit called motif. These motifs are then combined together to form a pattern. In this innovation it is found that the proposed model is superior than tessellating ordinary regular polygon, because more designs are obtained, more colours may be obtained or introduced to give meaningful tiles or patterns. In particular Tessellations can be found in many areas of life. Art, architecture, hobbies, clothing design, including traditional wears and many other areas hold examples of tessellations found in our everyday surroundings.

Sector plots are fixed-angle “sector-shaped” samples of objects, typically in small irregular polygons, with a central ray extending from sector vertex or pivot point to polygon boundary. If the number of objects in the sector plot is very high, a subsample may be used. The sector angle can be reduced, or small fixed-area “subplots” can be installed along the central ray. The simplicity of subplot establishment is offset by additional computations that are avoided in the full sector plots. Specifically, to compute ray means or totals, trees in the plots must be weighted by the distance from the pivot point. When several rays are sampled in the same polygon, the ray totals or means should be weighted by the squared number of plots along the ray. Finally, polygon edge overlap with plots may also need correction. Different sampling strategies to avoid bias and increase efficiency are discussed. The sampling design is new to forestry and may have novel statistical applications such as subsampling dense natural regeneration.

Abstract Stochastic imaging has become an important tool for risk assessment and has successfully been applied to oil field management. This procedure aims at generating several possible and equiprobable 3D models of subsurface structures that enhance the available data set. Among these stochastic simulation techniques, object-based approaches consist of defining and distributing objects reproducing underground geobodies. A technical challenge still remains in object-based simulation. Due to advances in deep water drilling technology, new hydrocarbon exploration has been opened along the Atlantic margins. In these turbidite oil fields, segments of meandering channels can be observed on high-resolution seismic horizons. However, no present object-based simulation technique can reproduce exactly such known segments of channel. An improved object-based approach is proposed to simulate meandering turbidite channels conditioned on well observations and such seismic data. The only approaches dealing with meandering channels are process-based as opposed to structure-imitating. They are based on the reproduction of continental river evolution through time. Unfortunately, such process-based approaches cannot be used for stochastic imaging as they are based on equations reflecting meandering river processes and not turbiditic phenomena. Moreover, they incoporate neither shape constraints (such as channel dimensions and sinuosity) nor location constraints, such as well data. Last, these methods generally require hydraulic parameters that are not available from oil field study. The proposed approach aims at stochastically generating meandering channels with specified geometry that can be constrained to pass through well-observations. The method relies on the definition of geometrical parameters that characterize the shape of the expected channels such as dimensions, directions and sinuosity. The meandering channel object is modelled via a flexible parametric shape. The object is defined by a polygonal center-line (called backbone) that supports several sections. Channel sinuosity and local channel profiles are controlled by the backbone and, respectively the sections. Channel generation is performed within a 2D domain, D representing the channel-belt area. The proposed approach proceeds in two main steps. The first step consists in generating a channel center-line (C) defined by an equation v=Z(u) within the domain D. The geometry of this line is simulated using a geostatistical simulation technique that allows the generation of controlled but irregular center-lines conditioned on data points. During the second step, a vector field enabling the curve (C) to be transformed into a meandering curve (C’) is estimated. This vector field acts as a transform that specifies the third degree of channel sinuosity, in other words, the meandering parts of the loops. This field is parameterized by geometrical parameters such as curvature and tangent vectors along the curve (C) and the a priori maximum amplitude of the meander loops of the curve (C’). To make channel objects pass through conditioning points, adjustment vectors are computed at these locations and are interpolated along the curves. Synthetic datasets have been built to check if a priori parameters such as tortuosity are reproduced, and if the simulations are equiprobable. From this dataset, hundred simulations have been generated and enable one to verify that these two conditions are satisfied. Equiprobability is however not always satisfied from data points that are very close and located in a multivalued part of a meander : preferential orientation of the loops may indeed be observed. Solving this issue will be the focus of future works. Nevertheless, the results presented in this paper show that the approach provides satisfying simulations in any other configurations. This approach is moreover well-suited for petroleum reservoir characterization because it only needs specification of geometrical parameters such as dimension and sinuosity that can be inferred from the channel parts seen on seismic horizons or analogues.

This paper addresses the irregular strip packing problem, a particular two-dimensional cutting and packing problem in which convex/nonconvex shapes (polygons) have to be packed onto a single rectangular object. We propose an approach that prescribes the integration of a metaheuristic engine (i.e., genetic algorithm) and a placement rule (i.e., greedy bottom-left). Moreover, a shrinking algorithm is encapsulated into the metaheuristic engine to improve good quality solutions. To accomplish this task, we propose a no-fit polygon based heuristic that shifts polygons closer to each other. Computational experiments performed on standard benchmark problems, as well as practical case studies developed in the ambit of a large textile industry, are also reported and discussed here in order to testify the potentialities of proposed approach.

Semantic and instance segmentation methods are commonly used to build extraction from high-resolution images. The semantic segmentation method involves assigning a class label to each pixel in the image, thus ignoring the geometry of the building rooftop, which results in irregular shapes of the rooftop edges. As for instance segmentation, there is a strong assumption within this method that there exists only one outline polygon along the rooftop boundary. In this paper, we present a novel method to sequentially delineate exterior and interior contours of rooftops with holes from VHR aerial images, where most of the buildings have holes, by integrating semantic segmentation and polygon delineation. Specifically, semantic segmentation from the Mask R-CNN is used as a prior for hole detection. Then, the holes are used as objects for generating the internal contours of the rooftop. The external and internal contours of the rooftop are inferred separately using a convolutional recurrent neural network. Experimental results showed that the proposed method can effectively delineate the rooftops with both one and multiple polygons and outperform state-of-the-art methods in terms of the visual results and six statistical indicators, including IoU, OA, F1, BoundF, RE and Hd.

By the time middle school students start a prealgebra course, they should have explored a variety of familiar two-dimensional and three-dimensional shapes and should have been exposed to the concepts of perimeter, area, and volume. They know that they can assign numerical values to some attributes of a shape, such as length and surface area. However, my classroom experience confirms the statement that although “students may have developed an initial understanding of area…, many will need additional experiences in measuring directly to deepen their understanding of the area of two-dimensional shapes” (NCTM 2000, p. 242). In addition, the students' previous practice with area is usually with polygons, circles, or a combination of both. However, many real-life objects cannot be described or approximated with simple geometric shapes or with combinations of shapes. Therefore, this activity, which asks students to estimate the area of irregular shapes using finer and finer grids, is not only novel but also a way to apply mathematics to real life.

The outline is one of the easiest and immediate methods to expresses people’s image of object, records and acquires the inspiration and the design that will be lost during a wink. The 3D modeling based on the outline adopts the computers’ strong power of technologies to sculpt. It demands not only grasping the NURBS and the polygon modeling technology, but also the sound sculpting base and good understanding of the space structure. To the 3D modeling method based on the irregular outline, this paper analyzes an a-ka role model, come up with the modeling method of 3D geometry reference, and make the a-ka model with this method. At last the model through the fit designing program is finished.

Abstract. Deep learning methods based on Fully convolution networks (FCNs) have shown an impressive progress in building outline delineation from very high resolution (VHR) remote sensing (RS) imagery. Common issues still exist in extracting precise building shapes and outlines, often resulting in irregular edges and over smoothed corners. In this paper, we use PolyMapper, a recently introduced deep-learning framework that is able to predict object outlines in a vector representation directly. We have introduced two main modifications to this baseline method. First, we introduce EffcientNet as backbone feature encoder to our network, which uses compound coefficient to scale up all dimensions of depth/width/resolution uniformly, to improve the processing speed with fewer parameters. Second, we integrate a boundary refinement block (BRB) to strengthen the boundary feature learning and to further improve the accuracy of corner prediction. The results demonstrate that the end-to-end learnable model is capable of delineating polygons of building outlines that closely approximate the structure of reference labels. Experiments on the crowdAI building instance segmentation datasets show that our model outperforms PolyMapper in all COCO metrics, for instance showing a 0.13 higher mean Average Precision (AP) value and a 0.60 higher mean Average Recall value. Also qualitative results show that our method segments building instances of various shapes more accurately.

Abstract Sector sampling is designed to sample objects in clusters or small, irregularly shaped polygons, such as variable retention patches and associated harvested areas. A number of statistical aspects of sector sampling were examined by using real data and a resampling framework. When the sector angle is selected at random, the probability of sampling each tree is the same; thus, a simple expansion factor method is all that is required to calculate tract totals and mean tree values. Standard variance formulas can then be used. For unit area estimates (such as basal area per hectare) a ratio-of-means estimator balances the areas in different-sized sectors. However, both the ratio-of-means mean and variance may be underestimated. An empirical correction to the biased variance estimator was derived. Alternatively, an unbiased and also more efficient unit area estimate can be made using a random point sector angle selection with a mean-of-ratios method. In this case standard variance formulas can again be used. A systematic sector arrangement reduced variance under an expansion factor but did not reduce variance using a ratio-of-means approach where sector area was already considered.

<p><strong>Abstract.</strong> This paper presents an experience of 3D modeling starting from laser scanning data and following two alternative approaches: the first one, called “Advanced 3D modeling”, based on an original meshing algorithm, while the second make use of Revit BIM software.</p><p>The case study in Palace Ettoreo in Sacile (Pordenone, Italy), constructed in Renaissance Venetian style in the 16th century: it has a trapezoid plan and is developed on three floors, with the ground one endowing a portico on two façades.</p><p>The palace has been surveyed by two terrestrial laser scanners: a Riegl Z420i for 5 external scans and a FARO Photon 120 for 53 internal scans; also a topographic surveying of 270 targets have been carried out. The final TLS cloud has 1,4 billions of points.</p><p>The Advanced 3D modeling has produced a “smart” mesh, allowing also to model the elements with deformations (out of plumb, bulges and troughs). Moreover, this model drastically reduce the stored data: the whole palace is modeled by 111.496 polygons only.</p><p>The modeling with Revit follows the classical flowchart where the principal architectonical elements are gradually composed: this HBIM process has required a strong manual work in exploiting the available parametric objects and/or in the definition of new objects.</p><p>Comparing the two models with respect the points cloud, both have evidenced advantages and limitations: therefore, the best solution is a process involving their combination. At the beginning, the Advanced 3D modeling is performed onto the points cloud, so well exploiting the segmentation tools and the smart meshing of the surfaces preserving any geometrical irregularity. Such obtained model allows metrical and morphological evaluation on the various structural and architectonical elements. Afterwards, this very light model becomes the entry data for the modeling in BIM environment, where also the shape of irregular elements are so imported.</p>

This paper proposes an improved modelling approach for tessellating regular polygons in such a way that it is environmentally sustainable. In this paper, tessellation of polygons that have been innovated through the formed motifs, is an innovation from the traditional tessellations of objects and animals. The main contribution of this work is the simplification and innovating new patterns from the existing regular polygons, in which only three polygons (triangle, square and hexagon) that can freely be tessellated are used, compared to using irregular polygons or other objects. This is achieved by reducing the size of each polygon to smallest value and tessellating each of the reduced figure to the right or to left to obtain a two different designs of one unit called motif. These motifs are then combined together to form a pattern. In this innovation it is found that the proposed model is superior than tessellating ordinary regular polygon, because more designs are obtained, more colours may be obtained or introduced to give meaningful tiles or patterns. In particular Tessellations can be found in many areas of life. Art, architecture, hobbies, clothing design, including traditional wears and many other areas hold examples of tessellations found in our everyday surroundings.

Abstract This paper presents a method to overlay laser scan data from at least two different epochs using B-splines. Here, an approach is described in which the laser scan points are directly taken as control points for the B-spline curves or surfaces. In combination with a second part of the publication, the presented technique should be able to detect small deformations compared to the object size based on the curvature parameter. However, the method does not aim at detecting object movements, rotations or translations. Before the epoch-wise laser scan acquisition of an object, a sampling rate adjustment according to the object properties has to be performed. By means of a defining algorithm, control polygons with different numbers of points are defined from the point clouds, whose points are available in scanning order, however not in a uniform grid structure, but an irregular distributed form. As a basis for a bidirectional grid structure of tensor product B-spline surfaces a knot insertion algorithm (Boehm’s algorithm) is to be applied. While the previous process steps are implemented for both B-spline curves and surfaces, the following process steps are only available for B-spline curves so far. In a correlation analysis where the ranks of the curvatures of the B-spline curve points are used, one shift position of the curve can be generated in both epochs. This forms the foundation for the determination of identical curve points, which are iteratively improved to provide the basis for a Helmert transformation to epoch 0. The rotation and translation parameters calculated with the Helmert transformation are only used to map the B-spline curve of an epoch n to epoch 0. A determination of an object rotation is not possible, because the laser scans are not georeferenced. The transformation applied allows the representation of coordinate differences between the laser scan data of different epochs. An implementation of the process steps is available for B-spline curves so far and has to be completed for B-spline surfaces. The finalization of the implementation for B-spline surfaces will be done in a second part of the publication. The calculation process of the standard deviations of the laser scan data up to the standard deviations of the curve and surface points for detecting deformations by means of a significance test will also be listed in more detail in the second part of the publication.