Planarity Constraints Research Articles

Solver-free heuristics to retrieve feasible points for offshore wind farm collection system

A set of solver-free heuristics for the offshore wind collection system problem is presented. Currently, algorithms of this type are not able to cope with relevant constraints when explicitly minimizing for investment cost. The first algorithm is a two-step decision process of which the output is a tree network satisfying cable thermal limits, but vulnerable to violate planarity constraints. A cable crossings repair heuristic is then introduced in order to fix infeasible points from the first heuristic. Finally, the negative cycle cancelling refining heuristic takes over to improve feasible points. The latter iteratively swaps connections, intending to find cycles with negative costs to reduce investment expenditure. The applicability of the workflow is empirically demonstrated through a set of large-scale real-world offshore farms. Results indicate that: (i) feasible points can be retrieved in a computing time of seconds; and (ii) warm starting can help global optimization solvers to converge significantly faster for hard problems.

Positive planar satisfiability problems under 3-connectivity constraints

A 3-SAT problem is called positive and planar if all the literals are positive and the clause-variable incidence graph (i.e., SAT graph) is planar. The NAE 3-SAT and 1-in-3-SAT are two variants of 3-SAT that remain NP-complete even when they are positive. The positive 1-in-3-SAT problem remains NP-complete under planarity constraint, but planar NAE 3-SAT is solvable in O(n1.5log⁡n) time, where n is the number of vertices. In this paper we prove that a positive planar NAE 3-SAT is always satisfiable when the underlying SAT graph is 3-connected, and a satisfiable assignment can be obtained in linear time. We also show that without 3-connectivity constraint, existence of a linear-time algorithm for positive planar NAE 3-SAT problem is unlikely as it would imply a linear-time algorithm for finding a spanning 2-matching in a planar subcubic graph. We then prove that positive planar 1-in-3-SAT remains NP-complete under the 3-connectivity constraint, even when each variable appears in at most 4 clauses. However, we show that the 3-connected planar 1-in-3-SAT is always satisfiable when each variable appears in an even number of clauses.

Intuitive and efficient roof modeling for reconstruction and synthesis

We propose a novel and flexible roof modeling approach that can be used for constructing planar 3D polygon roof meshes. Our method uses a graph structure to encode roof topology and enforces the roof validity by optimizing a simple but effective planarity metric we propose. This approach is significantly more efficient than using general purpose 3D modeling tools such as 3ds Max or SketchUp, and more powerful and expressive than specialized tools such as the straight skeleton. Our optimization-based formulation is also flexible and can accommodate different styles and user preferences for roof modeling. We showcase two applications. The first application is an interactive roof editing framework that can be used for roof design or roof reconstruction from aerial images. We highlight the efficiency and generality of our approach by constructing a mesh-image paired dataset consisting of 2539 roofs. Our second application is a generative model to synthesize new roof meshes from scratch. We use our novel dataset to combine machine learning and our roof optimization techniques, by using transformers and graph convolutional networks to model roof topology, and our roof optimization methods to enforce the planarity constraint.

SCALAR: Simultaneous Calibration of 2-D Laser and Robot Kinematic Parameters Using Planarity and Distance Constraints

In this paper, we propose SCALAR, a calibration method to simultaneously calibrate the kinematic parameters of a 6-DoF robot and the extrinsic parameters of a 2D Laser Range Finder (LRF) attached to the robot's flange. The calibration setup requires only a flat plate with two small holes carved on it at a known distance from each other, and a sharp tool-tip attached to the robot's flange. The calibration is formulated as a nonlinear optimization problem where the laser and the tool-tip are used to provide planar and distance constraints, and the optimization problem is solved using Levenberg-Marquardt algorithm. We demonstrate through experiments that SCALAR can reduce the mean and the maximum tool position error from 0.44 mm to 0.19 mm and from 1.41 mm to 0.50 mm, respectively.

BENDING THE DOMING EFFECT IN STRUCTURE FROM MOTION RECONSTRUCTIONS THROUGH BUNDLE ADJUSTMENT

Abstract. Structure from Motion techniques provides low-cost and flexible methods that can be adopted in arial surveying to collect topographic data with accurate results. Nevertheless, the so-called “doming effect”, due to unfortunate acquisition conditions or unreliable modeling of radial distortion, has been recognized as a critical issue that disrupts the quality of the attained 3D reconstruction. In this paper we propose a novel method, that works effectively in the presence of a nearly flat soil, to tackle a posteriori the doming effect: an automatic ground detection method is used to capture the doming deformation flawing the reconstruction, which in turn is wrapped to the correct geometry by iteratively enforcing a planarity constraint through a Bundle Adjustment framework. Experiments on real word datasets demonstrate promising results.

Planar growth generates scale-free networks

In this paper we introduce a model of spatial network growth in which nodes are placed at randomly selected locations on a unit square in R 2 , forming new connections to old nodes subject to the constraint that edges do not cross. The resulting network has a power law degree distribution, high clustering and the small world property. We argue that these characteristics are a consequence of the two defining features of the network formation procedure; growth and planarity conservation. We demonstrate that the model can be understood as a variant of random Apollonian growth and further propose a one parameter family of models with the Random Apollonian Network and the Deterministic Apollonian Network as extreme cases and our model as a midpoint between them. We then relax the planarity constraint by allowing edge crossings with some probability and find a smooth crossover from power law to exponential degree distributions when this probability is increased.

Camera Self-Calibration with Varying Intrinsic Parameters by an Unknown Three-Dimensional Scene

In the present paper, we will propose a new and robust method of camera self-calibration having varying intrinsic parameters from a sequence of images of an unknown 3D object. The projection of two points of the 3D scene in the image planes is used to determine the projection matrices. The present method is based on the formulation of a non linear cost function from the determination of a relationship between two points of the scene with their opposite relative to the axis of abscise and their projections in the image planes. The resolution of this function with genetic algorithm enables us to estimate the intrinsic parameters of different cameras. The important of our approach reside in the use of a single pair of images which provides fewer equations, simplifies the mathematical complexities and minimizes the execution time of the application, the use of the data of the first image only without the use of the data of the second image, the use of any camera which makes the intrinsic parameters variable not constant and the use of a 3D scene reduces the planarity constraints. The experimental results on synthetic and real data prove the performance and robustness of our approach.

Multiscale Embedded Gene Co-expression Network Analysis.

Gene co-expression network analysis has been shown effective in identifying functional co-expressed gene modules associated with complex human diseases. However, existing techniques to construct co-expression networks require some critical prior information such as predefined number of clusters, numerical thresholds for defining co-expression/interaction, or do not naturally reproduce the hallmarks of complex systems such as the scale-free degree distribution of small-worldness. Previously, a graph filtering technique called Planar Maximally Filtered Graph (PMFG) has been applied to many real-world data sets such as financial stock prices and gene expression to extract meaningful and relevant interactions. However, PMFG is not suitable for large-scale genomic data due to several drawbacks, such as the high computation complexity O(|V|3), the presence of false-positives due to the maximal planarity constraint, and the inadequacy of the clustering framework. Here, we developed a new co-expression network analysis framework called Multiscale Embedded Gene Co-expression Network Analysis (MEGENA) by: i) introducing quality control of co-expression similarities, ii) parallelizing embedded network construction, and iii) developing a novel clustering technique to identify multi-scale clustering structures in Planar Filtered Networks (PFNs). We applied MEGENA to a series of simulated data and the gene expression data in breast carcinoma and lung adenocarcinoma from The Cancer Genome Atlas (TCGA). MEGENA showed improved performance over well-established clustering methods and co-expression network construction approaches. MEGENA revealed not only meaningful multi-scale organizations of co-expressed gene clusters but also novel targets in breast carcinoma and lung adenocarcinoma.

There are Plane Spanners of Degree 4 and Moderate Stretch Factor

Let $${\fancyscript{E}}$$E be the complete Euclidean graph on a set of points embedded in the plane. Given a constant $$t \ge 1$$t?1, a spanning subgraph $$G$$G of $${\fancyscript{E}}$$E is said to be a $$t$$t-spanner, or simply a spanner, if for any pair of nodes $$u,v$$u,v in $${\fancyscript{E}}$$E the distance between $$u$$u and $$v$$v in $$G$$G is at most $$t$$t times their distance in $${\fancyscript{E}}$$E. The constant $$t$$t is referred to as the stretch factor. A spanner is plane if its edges do not cross. This paper considers the question: "What is the smallest maximum degree that can always be achieved for a plane spanner of $${\fancyscript{E}}$$E?" Without the planarity constraint, it is known that the answer is 3 which is thus the best known lower bound on the degree of any plane spanner. With the planarity requirement, the best known upper bound on the maximum degree is 6, the last in a long sequence of results improving the upper bound. In this paper, we show that the complete Euclidean graph always contains a plane spanner of maximum degree 4 and make a big step toward closing the question. The stretch factor of the spanner is bounded by $$156.82$$156.82. Our construction leads to an efficient algorithm for obtaining the spanner from Chew's $$L_1$$L1-Delaunay triangulation.

Camera self-calibration with varying intrinsic parameters by an unknown three-dimensional scene

This work proposes a method of camera self-calibration having varying intrinsic parameters from a sequence of images of an unknown 3D object. The projection of two points of the 3D scene in the image planes is used with fundamental matrices to determine the projection matrices. The present approach is based on the formulation of a nonlinear cost function from the determination of a relationship between two points of the scene and their projections in the image planes. The resolution of this function enables us to estimate the intrinsic parameters of different cameras. The strong point of the present approach is clearly seen in the minimization of the three constraints of a self-calibration system (a pair of images, 3D scene, any camera): The use of a single pair of images provides fewer equations, which minimizes the execution time of the program, the use of a 3D scene reduces the planarity constraints, and the use of any camera eliminates the constraints of cameras having constant parameters. The experiment results on synthetic and real data are presented to demonstrate the performance of the present approach in terms of accuracy, simplicity, stability, and convergence.

Calibration of double stripe 3D laser scanner systems using planarity and orthogonality constraints

In this study, 3D scanning systems that utilize a pair of laser stripes are studied. Three types of scanning systems are implemented to scan environments, rough surfaces of near planar objects and small 3D objects. These scanners make use of double laser stripes to minimize the undesired effect of occlusions. Calibration of these scanning systems is crucially important for the alignment of 3D points which are reconstructed from different stripes. In this paper, the main focus is on the calibration problem, following a treatment on the pre-processing of stripe projections using dynamic programming and localization of 2D image points with sub-pixel accuracy. The 3D points corresponding to laser stripes are used in an optimization procedure that imposes geometrical constraints such as coplanarities and orthogonalities. It is shown that, calibration procedure proposed here, significantly improves the alignment of 3D points scanned using two laser stripes.

A novel plane extraction approach using supervised learning

This paper presents a novel approach for the classification of planar surfaces in an unorganized point clouds. A feature-based planner surface detection method is proposed which classifies a point cloud data into planar and non-planar points by learning a classification model from an example set of planes. The algorithm performs segmentation of the scene by applying a graph partitioning approach with improved representation of association among graph nodes. The planarity estimation of the points in a scene segment is then achieved by classifying input points as planar points which satisfy planarity constraint imposed by the learned model. The resultant planes have potential application in solving simultaneous localization and mapping problem for navigation of an unmanned-air vehicle. The proposed method is validated on real and synthetic scenes. The real data consist of five datasets recorded by capturing three-dimensional(3D) point clouds when a RGBD camera is moved in five different indoor scenes. A set of synthetic 3D scenes are constructed containing planar and non-planar structures. The synthetic data are contaminated with Gaussian and random structure noise. The results of the empirical evaluation on both the real and the simulated data suggest that the method provides a generalized solution for plane detection even in the presence of the noise and non-planar objects in the scene. Furthermore, a comparative study has been performed between multiple plane extraction methods.

Simultaneous Calibration of ALS Systems and Alignment of Multiview LiDAR Scans of Urban Areas

Tasks such as city modeling or urban planning require the registration, alignment, and comparison of multiview and/or multitemporal remote sensing data. Airborne laser scanning (ALS) is one of the established techniques to deliver these data. Regrettably, direct georeferencing of ALS measurements usually leads to considerable displacements that limit connectivity and/or comparability of overlapping point clouds. Most reasons for this effect can be found in the impreciseness of the positioning and orientation sensors and their misalignment to the laser scanner. Typically, these sensors are comprised of a global navigation satellite system receiver and an inertial measurement unit. This paper presents a method for the automatic self-calibration of such ALS systems and the alignment of the acquired laser point clouds. Although applicable to classical nadir configurations, a novelty of our approach is the consideration of multiple data sets that were recorded with an oblique forward-looking full-waveform laser scanner. A combination of a region-growing approach with a random-sample-consensus segmentation method is used to extract planar shapes. Matching objects in overlapping data sets are identified with regard to several geometric attributes. A new methodology is presented to transfer the planarity constraints into systems of linear equations to determine both the boresight parameters and the data alignment. In addition to system calibration and data registration, the presented workflow results in merged 3-D point clouds that contain information concerning rooftops and all building facades. This database represents a solid basis and reference for applications such as change detection.

Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation

Symmetry has been shown to be a very effective a priori constraint in solving a 3D shape recovery problem. Symmetry is useful in 3D recovery because it is a form of redundancy. There are, however, some fundamental limits to the effectiveness of symmetry. Specifically, given two arbitrary curves in a single 2D image, one can always find a 3D mirror-symmetric interpretation of these curves under quite general assumptions. The symmetric interpretation is unique under a perspective projection and there is a one parameter family of symmetric interpretations under an orthographic projection. We formally state and prove this observation for the case of one-to-one and many-to-many point correspondences. We conclude by discussing the role of degenerate views, higher-order features in determining the point correspondences, as well as the role of the planarity constraint. When the correspondence of features is known and/or curves can be assumed to be planar, 3D symmetry becomes non-accidental in the sense that a 2D image of a 3D asymmetric shape obtained from a random viewing direction will not allow for 3D symmetric interpretations.

A MP2 and DFT study of the aromatic character of polyphosphaphospholes. Is the pyramidality the only factor to take into consideration?

A comprehensive MP2/6-311 + G(d,p) and B3LYP/6-311 + G(d,p) study of the aromatic character of phospholes, P(n)(CH)(4-n )PH with n = 0-4 was conducted. For this purpose, the structures for these compounds were optimized at both theoretical levels and different magnetic properties (magnetic susceptibility anisotropy, χ(anis), and the nucleus-independent chemical shifts, NICS) were evaluated. For comparison, these magnetic properties were also calculated in the optimized structures with planarity constraints. We have also applied the ACID (anisotropy of the current-induced density) method in this analysis. The main conclusions are the aromatic character of these compounds, the relationship between aromaticity and planarity and the importance of other factors in this aromaticity.

Reconstruction of 3D symmetrical shapes by using planarity and compactness constraints

Reconstruction of 3D symmetrical shapes by using planarity and compactness constraints

A Self-Organizing Algorithm for Modeling Protein Loops

Protein loops, the flexible short segments connecting two stable secondary structural units in proteins, play a critical role in protein structure and function. Constructing chemically sensible conformations of protein loops that seamlessly bridge the gap between the anchor points without introducing any steric collisions remains an open challenge. A variety of algorithms have been developed to tackle the loop closure problem, ranging from inverse kinematics to knowledge-based approaches that utilize pre-existing fragments extracted from known protein structures. However, many of these approaches focus on the generation of conformations that mainly satisfy the fixed end point condition, leaving the steric constraints to be resolved in subsequent post-processing steps. In the present work, we describe a simple solution that simultaneously satisfies not only the end point and steric conditions, but also chirality and planarity constraints. Starting from random initial atomic coordinates, each individual conformation is generated independently by using a simple alternating scheme of pairwise distance adjustments of randomly chosen atoms, followed by fast geometric matching of the conformationally rigid components of the constituent amino acids. The method is conceptually simple, numerically stable and computationally efficient. Very importantly, additional constraints, such as those derived from NMR experiments, hydrogen bonds or salt bridges, can be incorporated into the algorithm in a straightforward and inexpensive way, making the method ideal for solving more complex multi-loop problems. The remarkable performance and robustness of the algorithm are demonstrated on a set of protein loops of length 4, 8, and 12 that have been used in previous studies.

Perception of Parallelepipeds: Perkins's Law

Two experiments were carried out to study the perception of parallelepipeds. In the first, the subjects were shown images of parallelepipeds and were asked to judge the 3-D orientations of the faces of the parallelepipeds, as well as the shapes of the faces. These two types of judgments were found to be inconsistent. Specifically, the parallelepipeds reconstructed from judgments of 3-D orientations of the faces were systematically different from the parallelepipeds reconstructed from judgments of the shape of the faces. In the second experiment, the subjects were asked to choose which reconstruction was closer to their percept. In most trials the subjects chose the 3-D parallelepiped reconstructed from judgments of the shapes of the faces. These results suggest that the percept of the shape of a 3-D object is not based on the judgments of the 3-D orientations of the object's surfaces. Instead, the 3-D shape percept is based on simplicity constraints. A new computational model is presented, which generalizes Perkins's law (Perkins 1972). Instead of orthogonality, the new model uses mirror-symmetry and planarity constraints, in conjunction with maximum 3-D compactness and minimum surface-area constraints. The parallelepipeds recovered by the model are very close to the parallelepipeds perceived by the subjects.

3D Slit Scanning with Planar Constraints★

Abstract We present a planarity constraint and a novel three‐dimensional (3D) point reconstruction algorithm for a multiview laser range slit scanner. The constraint is based on the fact that all observed points on a projected laser line lie on the same plane of laser light in 3D. The parameters of the plane of laser light linearly parametrize a homography between a pair of images of the laser points. This homography can be recovered from point correspondences derived from epipolar geometry. The use of the planar constraint reduces outliers in the reconstruction and allows for the reconstruction of points seen in only one view. We derive an optimal reconstruction of points subject to the planar constraint and compare the accuracy to the suboptimal approach in prior work. We also construct a catadioptric stereo rig with high quality optical components to remove error due to camera synchronization and non‐uniform laser projection. The reconstruction results are compared to prior work that uses inexpensive optics and two cameras.

Altering the allowed/forbidden gap in cyclobutene electrocyclic reactions: experimental and theoretical evaluations of the effect of planarity constraints.

The allowed conrotatory cyclobutene ring-opening has a distinctly nonplanar carbon skeleton. Classic experiments by Brauman and Archie, and by Freedman et al., placed the allowed/forbidden gap at greater than 15 kcal/mol. Wolfgang Roth proposed that a system forced to planarity might have a smaller preference for the conrotatory mode than unconstrained systems. Such systems have now been studied theoretically and experimentally, and results that confirm Roth's postulate are presented here. The experiments were performed in Bochum, and the calculations were carried out in Osaka and Los Angeles. As the cyclobutene ring-opening transition structure approaches planarity, the energy gap between allowed conrotatory and the forbidden disrotatory pathways decreases. For the ring-opening of a cyclobutene fused to norbornene, the energy gap between the forbidden and the allowed transition state is only 4.1 kcal/mol by CASSCF and 8.0 kcal/mol by CAS-MP2 as compared to 13.4 and 19.2 kcal/mol, respectively, for the parent cyclobutene. Experimental studies of 3,4-dimethylcyclobutenes fused to various ring systems are reported, and a trend is found toward a reduced allowed/forbidden gap as the planarity of the cyclobutene is enforced.