Convex Hull Algorithm Research Articles

Pipe assembly planning algorithm by imitating human imaginal thinking

PurposeThe purpose of this paper is to develop a computationally efficient and generally applicable measure for a pipe routing problem which decides the pipe paths and affects the pipe assembly feasibility. By imitating human thinking in pipe assembly planning, human's experience and intuition are quantified and applied in the pipe routing algorithm.Design/methodology/approachHuman's imaginal thinking is simulated with procedures of knowledge representation, pattern recognition, and logical deduction; the algorithm transforms the physical obstacles and constraints into 3D pipe routing space and then into 2D planar projection, by using convex hull algorithm onto the projection, the shortest pipe route is found efficiently.FindingsA novel pipe assembly planning algorithm by imitating human imaginal thinking is presented, which effectively solved the problem of conceiving the shortest pipe route in 3D space with obstacles and constraints.Practical implicationsThe application of the algorithm in assembly planning of an aircraft engine pipe system is demonstrated. The algorithm can also be used in similar pipe planning problems such as industrial plant pipe planning and submarine pipe system design.Originality/valueHuman's imaginal thinking is introduced into pipe routing algorithm for the first time. By using graphics as the media to transfer the pipe routing information, human's experience and intuition in pipe assembly planning are quantified and computationally applicable.

Method of orienting curves for determining the convex hull of a finite set of points in the plane

In this article, we present an efficient algorithm to determine the convex hull of a finite planar set using the idea of the Method of Orienting Curves (introduced by Phu in Zur Lösung einer regulären Aufgabenklasse der optimalen Steuerung in Großen mittels Orientierungskurven, Optimization, 18 (1987), pp. 65–81, for solving optimal control problems with state constraints). The convex hull is determined by parts of orienting lines and a final line. Two advantages of this algorithm over some variations of Graham's convex hull algorithm are presented.

Stochastic Dynamic Programming Applied to Hydrothermal Power Systems Operation Planning Based on the Convex Hull Algorithm

This paper presents a new approach for the expected cost-to-go functions modeling used in the stochastic dynamic programming (SDP) algorithm. The SDP technique is applied to the long-term operation planning of electrical power systems. Using state space discretization, the Convex Hull algorithm is used for constructing a series of hyperplanes that composes a convex set. These planes represent a piecewise linear approximation for the expected cost-to-go functions. The mean operational costs for using the proposed methodology were compared with those from the deterministic dual dynamic problem in a case study, considering a single inflow scenario. This sensitivity analysis shows the convergence of both methods and is used to determine the minimum discretization level. Additionally, the applicability of the proposed methodology for two hydroplants in a cascade is demonstrated. With proper adaptations, this work can be extended to a complete hydrothermal system.

Evaluation of Algorithms for the Determination of Color Gamut Boundaries

Several techniques for the computation of gamut bound- aries have been presented in the past. In this article we take an in-depth look at some of the gamut boundary descriptors used when performing today's gamut mapping algorithms. We present a method for evaluating the mismatch introduced when using a de- scriptor to approximate the boundary of a device gamut. First, a visually verified reference gamut boundary is created by triangulat- ing the gamut surface using a device profile or a device character- ization model. The different gamut boundary descriptor techniques are then used to construct gamut boundaries based on several sets of simulated measurement data from the device. These boundaries are then compared against the reference gamut by utilizing a novel voxel based approach. Results from experiments using several gamut boundary descriptors are presented and analyzed statisti- cally. The modified convex hull algorithm proposed by Balasubramian and Dalal performs well for all the different data sets. © 2010 Society for Imaging Science and technology. DOI: 10.2352/J.ImagingSci.Technol.2010.54.5.050502

A Fast Convex Hull Algorithm for Binary Image

Convex hull is widely used in computer graphic, image processing, CAD/CAM and pattern recognition. In this work, we derive some new convex hull properties and then propose a fast algorithm based on these new properties to extract convex hull of the object in binary image. It is achieved by computing the extreme points, dividing the binary image into several regions, scanning the regions existing vertices dynamically, calculating the monotone segments, and merging these calculated segments. Theoretical analyses show that the proposed algorithm has low complexities of time and space. Povzetek: Predstavljen je nov algoritem za obdelavo binarnih slik.

Methods to quantify the pedodiversity of 1 km 2 areas — results from southern African drylands

When describing soil variability of a given area, pedodiversity measures are used as an integrative index for soil information and comparative diversity analyses. This paper compares different methods to derive pedodiversity indices based on a consistent data set of field and laboratory soil data of 16 dryland areas (1 km 2 each) in southern Africa. Three approaches to quantify pedodiversity indices are compared: i) Taxonomic classification using WRB soil units, ii) Parametric classification using ‘soil eco-types’ — a strictly variable-based classification method, and iii) Variable space pedodiversity with direct use of variable values for the calculation of ‘environmental envelopes’ as an approach with convex hull algorithms. The meaning of all approaches was tested by correlating the pedodiversity indices with the biodiversity of the identical areas (diversity of higher plants). Irrespective of the methodological approach, correlations between the pedodiversity and the statistically significant biodiversity became apparent ( p < 0.015…< 0.0001). Highest values of r 2 = 0.83 occur in the variable space approach calculated based on four variables. In conclusion, the variable space method provides a powerful tool for creating a diversity measure which is neither affected by class size and limits nor by autocorrelation patterns.

FLOATING-POINT ARITHMETIC FOR COMPUTATIONAL GEOMETRY PROBLEMS WITH UNCERTAIN DATA

It has been suggested in the literature that ordinary finite-precision floating-point arithmetic is inadequate for geometric computation, and that researchers in numerical analysis may believe that the difficulties of error in geometric computation can be overcome by simple approaches. It is the purpose of this paper to show that these suggestions, based on an example showing failure of a certain algorithm for computing planar convex hulls, are misleading, and why this is so.It is first shown how the now-classical backward error analysis can be applied in the area of computational geometry. This analysis is relevant in the context of uncertain data, which may well be the practical context for computational-geometry algorithms such as, say, those for computing convex hulls. The exposition will illustrate the fact that the backward error analysis does not pretend to overcome the problem of finite precision: it merely provides a way to distinguish those algorithms that overcome the problem to whatever extent it is possible to do so.It is then shown that often the situation in computational geometry is exactly parallel to other areas, such as the numerical solution of linear equations, or the algebraic eigenvalue problem. Indeed, the example mentioned can be viewed simply as an example of the use of an unstable algorithm, for a problem for which computational geometry has already discovered provably stable algorithms.Finally, the paper discusses the implications of these analyses for applications in three-dimensional solid modeling. This is done by considering a problem defined in terms of a simple extension of the planar convex-hull algorithm, namely, the verification of the well-formedness of extruded objects. A brief discussion concerning more difficult problems in solid modeling is also included.

New Serial and Parallel Algorithms for Finding Convex Hull Based on Clusters, Domains and Directions from Single to Multitude

An Isomorphic Fundamental Theorem of the Convex Hull Construction is given and proved. A representative serial algorithm convex hull with half-dividing and recurrence is commented as compared example. A more efficient new serial algorithm to find a convex hull based on a dynamical maximum base line pitch is given; its general characters are: 1) find out the outside-most points as the initial apexes, 2) divide the original distributed domain of a given 2D point set into four sub-domain at most, 3) construct a current apex with a maximum pitch to its base line in every sub-domain. Another new improved serial convex algorithm based on a minimum lever half line pitch coiling with 4-domains and 4-derections in all sub-domains is advanced; its isomorphic new characteristics are: 1) take a new pattern of 1-clusters, 4-domains and 4-directions, 2) construct a current apex with a minimum pitch from its lever half line in every sub-domain, 3) the computational time for finding a current apex is less. Further, a more efficient new parallel algorithm for finding convex hull based on 2-clusters, 2-domains and 4-directions is created; its isomorphic new characteristics are: 1) take pattern of 2-clusters, 2-domains and 4-directions, 2) have great isomorphic new potentialities to construct a better, newer and more efficient parallel algorithm for finding convex hull with m-Clusters, n-Domains and p-Directions (m &gt; 2, n &gt; 2, p &gt; 2).

Polyhedral approximation and practical convex hull algorithm for certain classes of voxel sets

In this paper we introduce an algorithm for the creation of polyhedral approximations for certain kinds of digital objects in a three-dimensional space. The objects are sets of voxels represented as strongly connected subsets of an abstract cell complex. The proposed algorithm generates the convex hull of a given object and modifies the hull afterwards by recursive repetitions of generating convex hulls of subsets of the given voxel set or subsets of the background voxels. The result of this method is a polyhedron which separates object voxels from background voxels. The objects processed by this algorithm and also the background voxel components inside the convex hull of the objects are restricted to have genus 0. The second aim of this paper is to present some practical improvements to the discussed convex hull algorithm to reduce computation time.

Approximating a real number by a rational number with a limited denominator: A geometric approach

We consider the problem of determining the rational number which best approximates the real number a and such that its denominator belongs to an interval [ b , b ′ ] . There is a related geometric problem consisting in finding the integer point lying in the vertical domain D of the form { ( x , y ) ∈ R 2 ∣ b ≤ x ≤ b ′ } such that the straight line passing through the origin and through this point best approximates the straight line L of slope a passing through the origin. The computation of this point is interlinked with the computation of both the convex hulls of the integer points located above and below the straight line L respectively and lying in the vertical domain D . In the literature, many general convex hull algorithms exist, as the gift wrapping algorithm for example. However, we focus on two interesting approaches to compute these convex hulls which are especially appropriated in our special configuration. The first one mainly uses number theory and runs in O ( log ( b ′ ) ) time. The other is in line with computational geometry as the method proposed in 1999 by Balza-Gomez et al. [H. Balza-Gomez, J.-M. Moreau, D. Michelucci, Convex hull of grid points below a line or a convex curve, in: DGCI ’99: Proceedings of the 8th International Conference on Discrete Geometry for Computer Imagery, Springer, Marne-la-Vallée, France, 1999, pp. 361–374] which runs in O ( log ( b ′ − b ) ) time. We propose a new method for the computation of these convex hulls which combines number theory and computational geometry. Our method preserves the optimal time complexity and is the first being output sensitive. Indeed, we compute the convex hulls in time linear in their vertex number. Moreover, the resulting algorithm is very simple and so is suitable for implementation.

An Improved Level Set for Liver Segmentation and Perfusion Analysis in MRIs

Determining liver segmentation accurately from MRIs is the primary and crucial step for any automated liver perfusion analysis, which provides important information about the blood supply to the liver. Although implicit contour extraction methods, such as level set methods (LSMs) and active contours, are often used to segment livers, the results are not always satisfactory due to the presence of artifacts and low-gradient response on the liver boundary. In this paper, we propose a multiple-initialization, multiple-step LSM to overcome the leakage and over-segmentation problems. The multiple-initialization curves are first evolved separately using the fast marching methods and LSMs, which are then combined with a convex hull algorithm to obtain a rough liver contour. Finally, the contour is evolved again using global level set smoothing to determine a precise liver boundary. Experimental results on 12 abdominal MRI series showed that the proposed approach obtained better liver segmentation results, so that a refined liver perfusion curve without respiration affection can be obtained by using a modified chamfer matching algorithm and the perfusion curve is evaluated by radiologists.

Gift-wrapping based preimage computation algorithm

Based on a classical convex hull algorithm called gift-wrapping, the purpose of the paper is to provide a new algorithm for computing the vertices of a polytope called preimage—roughly the set of naive digital planes containing a finite subset S of Z 3 . The vertices of the upper hemisphere, the ones of the lower hemisphere and at last the equatorial vertices are computed independently. The principle of the algorithm is based on duality and especially on the fact that the vertices of the preimage correspond to faces of the input set S or of its chords set S ⊖ S ∪ { ( 0 , 0 , 1 ) } . It allows to go from one vertex to another by gift-wrapping until the whole region of interest has been explored.

Local Hull-Based Surface Construction of Volumetric Data From Silhouettes

The marching cubes (MC) is a general method which can construct a surface of an object from its volumetric data generated using a shape from silhouette method. Although MC is efficient and straightforward to implement, a MC surface may have discontinuity even though the volumetric data is continuous. This is because surface construction is more sensitive to image noise than the construction of volumetric data. To address this problem, we propose a surface construction algorithm which aggregates local surfaces constructed by the 3-D convex hull algorithm. Thus, the proposed method initially classifies local convexities from imperfect MC vertices based on sliced volumetric data. Experimental results show that continuous surfaces are obtained from imperfect silhouette images of both convex and nonconvex objects.

Some Tours are More Equal than Others: The Convex-Hull Model Revisited with Lessons for Testing Models of the Traveling Salesperson Problem

To explain human performance on the Traveling Salesperson problem (TSP), MacGregor, Ormerod, and Chronicle (2000) proposed that humans construct solutions according to the steps described by their convex-hull algorithm. Focusing on tour length as the dependent variable, and using only random or semirandom point sets, the authors claimed empirical support for their model. In this paper we argue that the empirical tests performed by MacGregor et al. do not constitute support for the model, because they instantiate what Meehl (1997) coined "weak tests" (i.e., tests with a high probability of yielding confi rmation even if the model is false). To perform "strong" tests of the model, we implemented the algorithm in a computer program and compared its performance to that of humans on six point sets. The comparison reveals substantial and systematic differences in the shapes of the tours produced by the algorithm and human participants, for fi ve of the six point sets. The methodological lesson for testing TSP models is twofold: (1) Include qualitative measures (such as tour shape) as a dependent variable, and (2) use point sets for which the model makes “risky” predictions.

SU-GG-T-56: Planning Target Volume Adaptation Using Convex Hull of CTVs for Cervix Cancer Patients

Purpose: Adaptive IMRT planning studies have shown that intra‐ and inter‐fractional organ motion can significantly decrease the coverage of the clinical target volume (CTV) for cervix cancer patients. Adaptive mid‐treatment re‐planning was effective in minimizing CTV underdosage, but this strategy only worked for patients with consistent bladder filling. This study investigated planning target volume (PTV) adaptation using a convex hull (CH) technique for patients whose bladder filling is variable over the course of radiation treatment.Method and Materials: Nine cervix cancer patients with variable bladder filling during the treatment were selected from 35 patients enrolled in an adaptive IMRT planning study. All patients were MR‐scanned once for planning and weekly during treatment. CTV and organs at risk (OAR) were contoured on each MR scan fused to planning CT with bony alignment. Contours were converted to 3D surface meshes for deformation analysis considering tissuebiomechanics. Two MR scans with full/empty bladder were identified and an adaptive PTV was found using a convex hull algorithm in the research planning software. IMRT plans were generated on the CH PTV and accumulated dose on deformed organs were simulated. Results: The average volume of CH PTV was 16% larger than that of small margin (3mm) PTV (477vs411cc). Most of the volume increase was found in the area of rectum where the concavity of the CTV was largest. The CH PTV had a non‐uniform margin around CTV representing patient‐specific CTV motion and deformation. The CH plan maintained target coverage but increased dose at OARs compared to adaptive mid‐treatment re‐planning. Discussion: Adaptive planning strategy using convex hull of CTVs demonstrates good target coverage for cervix cancer patients with variable bladder filling. Refindment of the CH to spare OARs might allow a reduction in critical organdose while still maintaining target coverage.

A 3D Convex Hull algorithm for modelling medical data in a virtual environment

Computer-based surgical simulation systems are one of the most recent technologies in Virtual Reality (VR). These intelligent systems have become the training method and the tool to acquire valuable information for many medical students. They allow the students to interact with the anatomical structures by modelling and displaying virtual objects on the computer screen. Since a real object does not have a deterministic shape, it is impossible to derive a geometric equation for modelling purposes. Thus, alternative approaches are based on Convex Hull (CH) algorithms to construct the convex envelope of any object and to simulate a realistic environment with exact Collision Detection (CD) between objects during the training on a surgical operation. In this paper, a hybrid approach to generate the CH is developed and presented. The new algorithm is validated by performing a comparison with three conventional methods, namely, the Gift Wrapping, the QuickHull and the Chan algorithm. The evaluation is achieved by generating the convex envelopes of 3D wrist and knee bones. The results show the improvement associated with the proposed approach.

A new algorithm for computing the convex hull of a planar point set

When the edges of a convex polygon are traversed along one direction, the interior of the convex polygon is always on the same side of the edges. Based on this characteristic of convex polygons, a new algorithm for computing the convex hull of a simple polygon is proposed in this paper, which is then extended to a new algorithm for computing the convex hull of a planar point set. First, the extreme points of the planar point set are found, and the subsets of point candidate for vertex of the convex hull between extreme points are obtained. Then, the ordered convex hull point sequences between extreme points are constructed separately and concatenated by removing redundant extreme points to get the convex hull. The time complexity of the new planar convex hull algorithm is O(nlogh), which is equal to the time complexity of the best output-sensitive planar convex hull algorithms. Compared with the algorithm having the same complexity, the new algorithm is much faster.

A PRIORI FILTRATION OF POINTS FOR FINDING CONVEX HULL

Convex hull is the minimum area convex polygon containing the planar set. By now there are quite many convex hull algorithms (Graham Scan, Jarvis March, QuickHull, Incremental, Divide‐and‐Conquer, Marriage‐before‐Conquest, Monotone Chain, Brute Force). The main attention while choosing the algorithm is paid to the running time. In order to raise the efficiency of all the algorithms an idea of a priori filtration of points is given in this article. Besides, two new algorithms have been created and presented. The experiment research has shown a very good efficiency of these algorithms.

Evaluation of nondominated solution sets for k-objective optimization problems: An exact method and approximations

Integrated Preference Functional (IPF) is a set functional that, given a discrete set of points for a multiple objective optimization problem, assigns a numerical value to that point set. This value provides a quantitative measure for comparing different sets of points generated by solution procedures for difficult multiple objective optimization problems. We introduced the IPF for bi-criteria optimization problems in [Carlyle, W.M., Fowler, J.W., Gel, E., Kim, B., 2003. Quantitative comparison of approximate solution sets for bi-criteria optimization problems. Decision Sciences 34 (1), 63–82]. As indicated in that paper, the computational effort to obtain IPF is negligible for bi-criteria problems. For three or more objective function cases, however, the exact calculation of IPF is computationally demanding, since this requires k (⩾3) dimensional integration. In this paper, we suggest a theoretical framework for obtaining IPF for k (⩾3) objectives. The exact method includes solving two main sub-problems: (1) finding the optimality region of weights for all potentially optimal points, and (2) computing volumes of k dimensional convex polytopes. Several different algorithms for both sub-problems can be found in the literature. We use existing methods from computational geometry (i.e., triangulation and convex hull algorithms) to develop a reasonable exact method for obtaining IPF. We have also experimented with a Monte Carlo approximation method and compared the results to those with the exact IPF method.

A New Active Convex Hull Model for Image Regions

This paper presents a new active convex hull model with the following advantages: invariant with respect to the number of pixels to be enveloped; the number of time iterations is invariant, with respect to the image size; time-cheap for large image regions. The model is based on the geometric heat differential equations, derived from parabolic equation, and parameterized by arc length. To prevent the active contour from intruding into concavities and evolve it to the proper convex hull we use a vector field given as a difference between normal and tangent forces. The vector field is also used to segment an image to shells, such that a single region is present in each shell. A penalty function is developed to stop evolvement of those arc segments, whose vectors encountered boundary points of an image region. Based on the model a discrete algorithm is designed and coded by Mathematica 5.2. A condition is developed, with respect to the image size, to guarantee stable convergence of the active contour to the convex hull of the desired region. To validate the advantages and contributions a set of experiments is performed using synthetic, groundwater and medical images of different size and modalities. The paper concludes with a discussion and comparison of the active convex hull model with set of existing convex hull algorithms.