Longest Edge Research Articles

On Dealing with Minima at the Border of a Simplicial Feasible Area in Simplicial Branch and Bound

AbstractWe consider a simplicial branch and bound Global Optimization algorithm, where the search region is a simplex. Apart from using longest edge bisection, a simplicial partition set can be reduced due to monotonicity of the objective function. If there is a direction in which the objective function is monotone over a simplex, depending on whether the facets that may contain the minimum are at the border of the search region, we can remove the simplex completely, or reduce it to some of its border facets. Our research question deals with finding monotone directions and labeling facets of a simplex as border after longest edge bisection and reduction due to monotonicity. Experimental results are shown over a set of global optimization problems where the feasible set is defined as a simplex, and a global minimum point is located at a face of the simplicial feasible area.

The longest edge in discrete and continuous long-range percolation

The longest edge in discrete and continuous long-range percolation

Infragalenic triangle as a gateway to dorsal midbrain and posteromedial thalamic lesions: descriptive and quantitative analysis of microsurgical anatomy.

Anatomical triangles provide neurosurgeons with the specificity required to access deep targets, supplementing more general instructions, such as craniotomy and approach. The infragalenic triangle (IGT), bordered by the basal vein of Rosenthal (BVR), precentral cerebellar vein (PCV), and the quadrangular lobule of the cerebellum, is one of a system of anatomical triangles recently introduced to guide dissection to brainstem cavernous malformations and has not been described in detail. This study aimed to quantitatively analyze the anatomical parameters of the IGT and present key nuances for its microsurgical use. A midline supracerebellar infratentorial (SCIT) approach through a torcular craniotomy was performed on 5 cadaveric heads, and the IGT was identified in each specimen bilaterally. Anatomical measurements were obtained with point coordinates collected using neuronavigation. Three cadaveric brains were used to illustrate relevant brainstem anatomy, and 3D virtual modeling was used to simulate various perspectives of the IGT through different approach angles. In addition, 2 illustrative surgical cases are presented. The longest edge of the IGT was the lateral edge formed by the BVR (mean ± SD length 19.1 ± 2.3 mm), and the shortest edge was the medial edge formed by the PCV (13.9 ± 3.6 mm). The mean surface area of the IGT was 110 ± 34.2 mm2 in the standard exposure. Full expansion of all 3 edges (arachnoid dissection, mobilization, and retraction) resulted in a mean area of 226.0 ± 48.8 mm2 and a 2.5-times increase in surface area exposure of deep structures (e.g., brainstem and thalamus). Thus, almost the entire tectal plate and its relevant safe entry zones can be exposed through an expanded unilateral IGT except for the contralateral inferior colliculus, access to which is usually hindered by PCV tributaries. Exposure of bilateral IGTs may be required to resect larger midline lesions to increase surgical maneuverability or to access the contralateral pulvinar. The IGT provides a safe access route to the dorsal midbrain and reliable intraoperative guidance in the deep and complex anatomy of the posterior tentorial incisura. Its potential for expansion makes it a versatile anatomical corridor not only for intrinsic brainstem lesions but also for tumors and vascular malformations of the pineal region, dorsal midbrain, and posteromedial thalamus.

PRUNING MINIMUM SPANNING TREES AND CUTTING LONGEST EDGES TO CONNECT A GIVEN NUMBER OF NODES BY MINIMIZING TOTAL EDGE LENGTH

Background. Whereas in many tasks of designing efficient telecommunication networks, the number of network nodes is limited, the initial choice of nodes is wider. There are more possible locations than factually active tools to be settled to those locations to further satisfy consumers. This induces an available node constraint problem.
 Objective. Given an initial set of planar nodes, the problem is to build a minimum spanning tree connecting a given number of the nodes, which is less than the cardinality of the initial set. Therefore, the available node constraint problem aims at building an optimally minimum spanning tree to connect a given number of planar nodes being less than an initial number of nodes by minimizing the tree length.
 Methods. The initial set of nodes is triangulated. This gives a set of edges, whose lengths are calculated and used as graph weights. A minimum spanning tree is built over this graph. The desired number of nodes is reached by pruning the minimum spanning tree connecting the initial number of nodes, where free edges whose weights are the largest are iteratively removed from the tree. The other approach, the cutting method, removes longest edges off the initial minimum spanning tree, regardless of whether they are free or not.
 Results. Unlike the pruning method, the method of cutting longest edges may result in a minimum spanning tree connecting fewer nodes than the desired number. However, the cutting method often outputs a shorter tree, especially when the edge length varies much. Besides, the cutting method is slower due to it iteratively rebuilds a minimum spanning tree.
 Conclusions. The problem is initially solved by the pruning method. Then the cutting method is used and its solution is compared to the solution by the pruning method. The best tree is shorter. A tradeoff for the nodes and tree length is possible.

New variants of perfect non-crossing matchings

Given a set of points in the plane, we are interested in matching them with straight line segments. We focus on perfect (all points are matched) non-crossing (no two edges intersect) matchings. Apart from the well known MinMax variation, where the length of the longest edge is minimized, we extend work by looking into different optimization variants such as MaxMin, MinMin, and MaxMax. We consider both the monochromatic and bichromatic versions of these problems and by employing diverse techniques we provide efficient algorithms for various input point configurations.

Similarity Classes of the Longest-Edge Trisection of Triangles

This paper studies the triangle similarity classes obtained by iterative application of the longest-edge trisection of triangles. The longest-edge trisection (3T-LE) of a triangle is obtained by joining the two points which divide the longest edge in three equal parts with the opposite vertex. This partition, as well as the longest-edge bisection (2T-LE), does not degenerate, which means that there is a positive lower bound to the minimum angle generated. However, unlike what happens with the 2T-LE, the number of similarity classes appearing by the iterative application of the 3T-LE to a single initial triangle is not finite in general. There are only three exceptions to this fact: the right triangle with its sides in the ratio 1:2:3 and the other two triangles in its orbit. This result, although of a combinatorial nature, is proved here with the machinery of discrete dynamics in a triangle shape space with hyperbolic metric. It is also shown that for a point with an infinite orbit, infinite points of the orbit are in three circles with centers at the points with finite orbits.

The longest edge of the one-dimensional soft random geometric graph with boundaries

The object of study is a soft random geometric graph with vertices given by a Poisson point process on a line and edges between vertices present with probability that has a polynomial decay in the distance between them. Various aspects of such models related to connectivity structures have been studied extensively. In this article, we study the random graph from the perspective of extreme value theory and focus on the occurrence of single long edges. The model we investigate has non-periodic boundary and is parameterized by a positive constant α, which is the power for the polynomial decay of the probabilities determining the presence of an edge. As a main result, we provide a precise description of the magnitude of the longest edge in terms of asymptotic behavior in distribution. Thereby we illustrate a crucial dependence on the power α and we recover a phase transition which coincides with exactly the same phases in Benjamini and Berger[ 2 ].

THE INDUSTRY OF COMPLEX II MALYI RAKOVETS IV SITE IN TRANSCARPATHIA

The site is located on an isolated hill of the Vyrhorlat-Hutyn volcanic mountain massif of the right bank of the Tysa River, near the village Maly Rakovets, 7 km north of the Korolevo site. At the site, due to the highest location level, there is a reduction in the capacity of Pleistocene deposits. This caused variability in the chronological interpretation of the materials. In the initial description of the stratigraphic context, the layer is fixed in the lower part of the loam soil and at the border with the Pryluky horizon.
 During the complex study, 2899 artifacts from the collection were processed. The main raw material is local Carpathian III obsidian (94.69 %), which occurs near the site in erosion washes on the slopes of the hill.
 In the technological sense, the industry can be described as non-Levallois, non-blades, non-facetted, the indices of which are: IF large = 17.5; IF strict = 3.8; ILam = 3. There is in the collection various cores: radial, convergent, parallel, Kombewa. Technology is based on the two methods: centripetal, as well as a fairly simple technique for obtaining blanks, which is presented by parallel and unidirectional cores, with a clear predominance of transverse proportions of products. The process of core reduction was quite intensive, a hard hammer was used. Production mostly aimed at receiving massive flakes of lateral proportions with naturally back, often crusted (43 %). Such blanks were prioritized for the manufacture of tools — 60 %.
 There are few tools, but among them a group of single-edged scrapers, the most prominent of which are transverse ones, is highlighted. They show specific design features, namely the retouching of the longest edge of the blank with a stepped retouch of the half-Quina type and the presence of accommodation elements. The technical and typological characteristics of the II complex are quite similar to the industry of the Korolevo site layer II, and allow to attribute it to the circle of industries of the Charentian / Quina type.

A parallel adaptive finite-element approach for 3-D realistic controlled-source electromagnetic problems using hierarchical tetrahedral grids

SUMMARY To effectively and efficiently interpret or invert controlled-source electromagnetic (CSEM) data which are recorded in areas with the kind of complex geological environments and arbitrary topography that are typical, 3-D CSEM forward modelling software that can quickly solve large-scale problems, provide accurate electromagnetic responses for complex geo-electrical models and can be easily incorporated into inversion algorithms are required. We have developed a parallel goal-oriented adaptive mesh refinement finite-element approach for frequency-domain 3-D CSEM forward modelling with hierarchical tetrahedral grids that can offer accurate electromagnetic responses for large-scale complex models and that can efficiently serve for inversion. The approach uses the goal-oriented adaptive vector finite element method to solve the total electric field vector equation. The geo-electrical model is discretized by unstructured tetrahedral grids which can deal with complex underground geological models with arbitrary surface topography. Different from previous adaptive finite element software working on unstructured tetrahedral grids, we have utilized a novel mesh refinement technique named the longest edge bisection method to generate hierarchically refined grids. As the refined grids are nested into the coarse grids, the refinement technique can precisely map the electrical parameters of inversion grids onto the forward modelling grids so that the extra numerical errors generated by the inconsistency of electrical parameters between inversion grids and forward modelling grids are eliminated. In addition, we use the parallel domain-decomposition technique to further accelerate the computations, and the flexible generalized minimum residual solver (FGMRES) with an auxiliary Maxwell solver pre-conditioner to solve the final large-scale system of linear equations. In the end, we validate the performance of the proposed scheme using two synthetic models and one realistic model. We demonstrate that accurate electromagnetic fields can be obtained by comparison with the analytic solutions and that the code is highly scalable for large-scale problems with millions or even hundreds of millions of unknowns. For the synthetic 3-D model and the realistic model with complex geometry, our solutions match well with the results calculated by an existing 3-D CSEM forward modelling code. Both synthetic and realistic examples demonstrate that our newly developed code is an effective, efficient forward modelling engine for interpreting CSEM field data acquired in areas of complex geology and topography.

GUBS: Graph-Based Unsupervised Brain Segmentation in MRI Images.

Brain segmentation in magnetic resonance imaging (MRI) images is the process of isolating the brain from non-brain tissues to simplify the further analysis, such as detecting pathology or calculating volumes. This paper proposes a Graph-based Unsupervised Brain Segmentation (GUBS) that processes 3D MRI images and segments them into brain, non-brain tissues, and backgrounds. GUBS first constructs an adjacency graph from a preprocessed MRI image, weights it by the difference between voxel intensities, and computes its minimum spanning tree (MST). It then uses domain knowledge about the different regions of MRIs to sample representative points from the brain, non-brain, and background regions of the MRI image. The adjacency graph nodes corresponding to sampled points in each region are identified and used as the terminal nodes for paths connecting the regions in the MST. GUBS then computes a subgraph of the MST by first removing the longest edge of the path connecting the terminal nodes in the brain and other regions, followed by removing the longest edge of the path connecting non-brain and background regions. This process results in three labeled, connected components, whose labels are used to segment the brain, non-brain tissues, and the background. GUBS was tested by segmenting 3D T1 weighted MRI images from three publicly available data sets. GUBS shows comparable results to the state-of-the-art methods in terms of performance. However, many competing methods rely on having labeled data available for training. Labeling is a time-intensive and costly process, and a big advantage of GUBS is that it does not require labels.

Drawing Graphs as Spanners

We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph G, the goal is to construct a straight-line drawing \(\Gamma \) of G in the plane such that, for any two vertices u and v of G, the ratio between the minimum length of any path from u to v and the Euclidean distance between u and v is small. The maximum such ratio, over all pairs of vertices of G, is the spanning ratio of \(\Gamma \). First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio 1, a proper straight-line drawing with spanning ratio 1, and a planar straight-line drawing with spanning ratio 1 are NP-complete, \(\exists \mathbb R\)-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a vertex. Second, we show that moving from spanning ratio 1 to spanning ratio \(1+\epsilon \) allows us to draw every graph. Namely, we prove that, for every \(\epsilon >0\), every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than \(1+\epsilon \). Third, our drawings with spanning ratio smaller than \(1+\epsilon \) have large edge-length ratio, that is, the ratio between the length of the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line drawing with constant spanning ratio.

Minimum spanning tree‐based cluster analysis: A new algorithm for determining inconsistent edges

AbstractIn recent years, graph‐based data clustering algorithms have become popular as they perform connectivity‐based rather than centroid‐based partitioning. Methods related to minimum spanning tree (MST)‐based data clustering are types of graph‐based algorithms that can recognize arbitrary shapes of clusters by eliminating inconsistent edges from MST graphs. In all MST‐based data clustering algorithms, definition of inconsistent edges is the main problem that needs to be addressed. The longest edges in MST graphs are considered as inconsistent edges under ideal conditions. Nevertheless, outliers often exist in real‐world tasks, which makes the longest edges inaccurate cluster separation indicators. In this paper, we propose a new data clustering algorithm using MST and a critical distance method. The proposed algorithm solves the main issue of MST‐based data clustering, namely identifying and removing inconsistent edges to obtain clusters even in the event that the dataset contains some outliers. It begins by constructing the MST over a given weighted graph based on Euclidean distance and then splits up the graph into clusters by eliminating inconsistent edges using critical distance as a threshold. Integration of the advantages of both MST and critical distance methodology to obtain optimal clusters is the main contribution of this work. The conducted experimental analysis and results using different datasets prove that our proposed clustering algorithm yields better overall performance compared with the most common data clustering algorithms. Taking the Liver and Tumor datasets as an example, the proposed algorithm outperforms all other clustering algorithms with clustering accuracy equal to 0.579 and 0.660, respectively.

Species Detection and Segmentation of Multi-specimen Historical Herbaria

Historically, herbarium specimens have provided users with documented occurrences of plants in specific locations over time. Herbarium collections have therefore been the basis of systematic botany for centuries (Younis et al. 2020). According to the latest summary report based on the data from Index Herbariorum, there are around 3400 active herbaria in the world containing 397 million specimens that are spread across 182 countries (Thiers 2021). Exponential growth in high quality image capturing devices induced by the enormous amount of uncovered collections has further led to rising interest in large scale digitization initiatives across the world (Le Bras et al. 2017). As herbarium specimens are increasingly becoming digitised and accessible in online repositories, an important need has also emerged to develop automated tools to process and enrich these collections to facilitate better access to the preserved archives. This rising number of digitised herbarium sheets provides an opportunity to employ computer-based image processing techniques, such as deep learning, to automatically identify species and higher taxa (Carranza-Rojas and Joly 2018, Carranza-Rojas et al. 2017, Younis et al. 2020) or to extract other useful information from the herbaria sheets, such as detecting handwritten text, color bars, scales and barcodes. The species identification task works well for herbarium sheets that have only one species in a page. However, there are many herbarium books that have multiple species on the same page (as shown in Fig. 1) for which the complexity of the identification problem increases tremendously. It also involves a great deal of time and effort if they are to be enriched manually. In this work, we propose a pipeline that can automatically detect, identify, and enrich plant species in multi-specimen herbaria. The core idea of the pipeline is to detect unique plant species and handwritten text around the plant species and map the text to the correct plant species. As shown in Fig. 2, the proposed pipeline begins with the pre-processing of the images. The images are rotated and aligned such that the longest edge is maintained as its height. In the case of herbarium books, the pages are detected and morphological transformations are performed to reduce occlusions (Thirukokaranam Chandrasekar and Verstockt 2020). A YOLOv3 (You Only Look Once version 3) object detection model (Zhao and Li 2020) is trained from scratch to detect plants and text. The model was trained on a dataset of single species herbarium sheets with a mosaic augmentation technique to extend the plants model to detect multiple species. The first results of the training shows impressive results although it could be further improved with more labelled data. We also plan to train an object segmentation model and contrast its performance with the plant detection model for multi-specimen herbarium sheets. After detecting both the plants and the text, the text will be recognized with a state-of-the-art handwritten text recognition (HTR) model. The recognized text can then be matched with a database of specimens, to identify each detected specimen. Furthermore, additional textual metadata (e.g. date, locality, collector's name, institution) visible on the sheet will be recognized and used to enrich the collection.

Planar bichromatic bottleneck spanning trees

Given a set $P$ of red and blue points in the plane, a planar bichromatic spanning tree of $P$ is a geometric spanning tree of $P$, such that each edge connects a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree $T$, such that the length of the longest edge (i.e., bottleneck) in $T$ is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time $(8\sqrt{2})$-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck $\lambda$ can be converted to a planar bichromatic spanning tree of bottleneck at most $8\sqrt{2}\lambda$.

Computing the Exact Number of Similarity Classes in the Longest Edge Bisection of Tetrahedra

Showing whether the longest-edge (LE) bisection of tetrahedra meshes degenerates the stability condition or not is still an open problem. Some reasons, in part, are due to the cost for achieving the computation of similarity classes of millions of tetrahedra. We prove the existence of tetrahedra where the LE bisection introduces, at most, 37 similarity classes. This family of new tetrahedra was roughly pointed out by Adler in 1983. However, as far as we know, there has been no evidence confirming its existence. We also introduce a new data structure and algorithm for computing the number of similarity tetrahedral classes based on integer arithmetic, storing only the square of edges. The algorithm lets us perform compact and efficient high-level similarity class computations with a cost that is only dependent on the number of similarity classes.

Edge-Rich Interconnected Graphene Mesh Electrode with High Electrochemical Reactivity Applicable for Glucose Detection.

The development of graphene structures with controlled edges is greatly desired for understanding heterogeneous electrochemical (EC) transfer and boosting EC applications of graphene-based electrodes. We herein report a facile, scalable, and robust method to produce graphene mesh (GM) electrodes with tailorable edge lengths. Specifically, the GMs were fabricated at 850 °C under a vacuum level of 0.6 Pa using catalytic nickel templates obtained based on a crack lithography. As the edge lengths of the GM electrodes increased from 5.48 to 24.04 m, their electron transfer rates linearly increased from 0.08 to 0.16 cm∙s−1, which are considerably greater than that (0.056 ± 0.007 cm∙s−1) of basal graphene structures (defined as zero edge length electrodes). To illustrate the EC sensing potentiality of the GM, a high-sensitivity glucose detection was conducted on the graphene/Ni hybrid mesh with the longest edge length. At a detection potential of 0.6 V, the edge-rich graphene/Ni hybrid mesh sensor exhibited a wide linear response range from 10.0 μM to 2.5 mM with a limit of detection of 1.8 μM and a high sensitivity of 1118.9 μA∙mM−1∙cm−2. Our findings suggest that edge-rich GMs can be valuable platforms in various graphene applications such as graphene-based EC sensors with controlled and improved performance.

Clustering with Local Density Peaks-Based Minimum Spanning Tree

Clustering analysis has been widely used in statistics, machine learning, pattern recognition, image processing, and so on. It is a great challenge for most existing clustering algorithms to discover clusters with arbitrary shapes. Clustering algorithms based on Minimum spanning tree (MST) are able to discover clusters with arbitrary shapes, but they are time consuming and susceptible to noise points. In this paper, we employ local density peaks (LDP) to represent the whole data set and define a shared neighbors-based distance between local density peaks to better measure the dissimilarity between objects on manifold data. On the basis of local density peaks and the new distance, we propose a novel MST-based clustering algorithm called LDP-MST. It first uses local density peaks to construct MST and then repeatedly cuts the longest edge until a given number of clusters are found. The experimental results on synthetic data sets and real data sets show that our algorithm is competent with state-of-the-art methods when discovering clusters with complex structures.

Parallel graph-grammar-based algorithm for the longest-edge refinement of triangular meshes and the pollution simulations in Lesser Poland area

In this paper, we propose parallel graph-grammar-based algorithm for the longest-edge refinements and the pollution simulations in Lesser Poland area. We introduce graph-grammar productions for Rivara’s longest-edged algorithm for the local refinement of unstructured triangular meshes. We utilize the hyper-graph to represent the computational mesh and the graph-grammar productions to express the longest-edge mesh refinement algorithm. The parallelism in the original Rivara’s longest edge refinement algorithm is obtained by processing different longest edge refinement paths in different three ads. Our graph-grammar-based algorithm allows for additional parallelization within a single longest-edge refinement path. The graph-grammar-based algorithm automatically guarantees the validity and conformity of the generated mesh; it prevents the generation of duplicated nodes and edges, elongated elements with Jacobians converging to zero, and removes all the hanging nodes automatically from the mesh. We test the algorithm on generating a surface mesh based on a topographic data of Lesser Poland area. The graph-grammar productions also generate the layers of prismatic three-dimensional elements on top of the triangular mesh, and they break each prismatic element into three tetrahedral elements. Next, we propose graph-grammar productions generating element matrices and right-hand-side vectors for each tetrahedral element. We utilize the Streamline Upwind Petrov–Galerkin (SUPG) stabilization for the pollution propagation simulations in Lesser Poland area. We use the advection–diffusion-reaction model, the Crank–Nicolson time integration scheme, and the graph-grammar-based interface to the GMRES solver.

Characterization of alphabaculovirus: HearNPV-IIPR05 isolate infecting Helicoverpa armigera (Hubner) larvae

BackgroundThe alphabaculoviruses are lethal pathogens of lepidopteran caterpillars including a polyphagous and globally recognized pest, Helicoverpa armigera (Hubner) infesting economically important agriculture crops worldwide. The biological and molecular characterizations of indigenous nucleopolyhedrovirus of the genus Alphabaculovirus isolated from H. armigera in chickpea fields are described.ResultsThe virulence of virus isolate was tested in 3rd instar H. armigera larvae, and LC50 (median lethal concentration) was estimated to be 2.69 × 104 OBs ml−1. The ST50 (median survival time) was 4 days post-inoculation, when the 3rd instar H. armigera larvae were inoculated by OB (occlusion body) concentration equivalent to LC90. An average incubation period of the virus isolate in 3rd instar ranged between 4 and 6 days post-inoculation. The OBs of a virus isolate appeared irregular in shape and variable in size with diameter ranging from 0.57 to 1.46 μm on the longest edge and average of 1.071 ± 0.068 μm (mean ± SE). On the basis of phylogenetic analysis of polh, pif-1, and lef-8 genes, the isolate was found to be a member of the genus Alphabaculovirus. The isolate showed a genetic affinity with species of group II Alphabaculoviruses and appeared to be a group II NPV.ConclusionsOn the basis of molecular phylogeny and associated host insect, this indigenous isolate was designated as HearNPV-IIPR05 isolate, which could be a potential candidate for the biological control of H. armigera infesting legumes and other commercial crops.

A sampling type discernment approach towards reconstruction of a point set in R2

In this paper, we propose a Delaunay triangulation based algorithmic framework for the 2D point set type discernment and its reconstruction. The point set discernment deals with determining whether the given point set S consists of only the points sampled along the boundary of an object (referred to as boundary sample) or S is distributed across the object (referred to as dot-pattern). In general, the existing approaches deal with the reconstruction of an a priori known input type - boundary sample or dot-pattern. Our approach works on both sets of input data type by first identifying the input type that a given point set belongs to. To distinguish the point set type, we introduce the notion of Petal-Ratio(PR) which captures the ratio of the longest edge to the smallest edge emanating from a vertex in Delaunay triangulation. Further, we employ the Petal-Ratio as the essence to design simple reconstruction algorithms pertaining to each input type that unveil the shape of the object hidden in the given point set. The reconstruction algorithms start with computing the Delaunay triangulation of the given point set and for each vertex, the Petal-Ratio is calculated. Following this, the edges are removed based on the value of the PR for every vertex to arrive at the boundaries. Theoretical analysis of determining appropriate values of PRs under ϵ- sampling and minimal reach sampling models have been presented. Unlike many other algorithms, our approach is non-parametric and non-feature specific that can capture features like disconnected components, multiple holes even with the presence of outliers, self-intersection (only for boundary sample) and open curves (only for boundary sample). Moreover, our algorithms take only one pass to reconstruct the hole boundaries as well as outer boundary irrespective of the object features like the number of holes and the number of components. The comparative analysis shows that our algorithms perform equally well or better than their counterparts for the objects with contrasting features. We also demonstrate that the proposed idea can be easily extended to surface reconstruction.