Computational Geometry Techniques Research Articles

Detection of Individual Corn Crop and Canopy Delineation from Unmanned Aerial Vehicle Imagery

Precise monitoring of individual crop growth and health status is crucial for precision agriculture practices. However, traditional inspection methods are time-consuming, labor-intensive, prone to human error, and may not provide the comprehensive coverage required for the detailed analysis of crop variability across an entire field. This research addresses the need for efficient and high-resolution crop monitoring by leveraging Unmanned Aerial Vehicle (UAV) imagery and advanced computational techniques. The primary goal was to develop a methodology for the precise identification, extraction, and monitoring of individual corn crops throughout their growth cycle. This involved integrating UAV-derived data with image processing, computational geometry, and machine learning techniques. Bi-weekly UAV imagery was captured at altitudes of 40 m and 70 m from 30 April to 11 August, covering the entire growth cycle of the corn crop from planting to harvest. A time-series Canopy Height Model (CHM) was generated by analyzing the differences between the Digital Terrain Model (DTM) and the Digital Surface Model (DSM) derived from the UAV data. To ensure the accuracy of the elevation data, the DSM was validated against Ground Control Points (GCPs), adhering to standard practices in remote sensing data verification. Local spatial analysis and image processing techniques were employed to determine the local maximum height of each crop. Subsequently, a Voronoi data model was developed to delineate individual crop canopies, successfully identifying 13,000 out of 13,050 corn crops in the study area. To enhance accuracy in canopy size delineation, vegetation indices were incorporated into the Voronoi model segmentation, refining the initial canopy area estimates by eliminating interference from soil and shadows. The proposed methodology enables the precise estimation and monitoring of crop canopy size, height, biomass reduction, lodging, and stunted growth over time by incorporating advanced image processing techniques and integrating metrics for quantitative assessment of fields. Additionally, machine learning models were employed to determine relationships between the canopy sizes, crop height, and normalized difference vegetation index, with Polynomial Regression recording an R-squared of 11% compared to other models. This work contributes to the scientific community by demonstrating the potential of integrating UAV technology, computational geometry, and machine learning for accurate and efficient crop monitoring at the individual plant level.

Counting \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 8 black holes as algebraic varieties

We calculate the helicity trace index B14 for \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 8 pure D-brane black holes using various techniques of computational algebraic geometry and find perfect agreement with the existing results in the literature. For these black holes, microstate counting is equivalent to finding the number of supersymmetric vacua of a multi-variable supersymmetric quantum mechanics which in turn is equivalent to solving a set of multi-variable polynomial equations modulo gauge symmetries. We explore four different techniques to solve a set of polynomial equations, namely Newton Polytopes, Homotopy continuation, Monodromy and Hilbert series. The first three methods rely on a mixture of symbolic and high precision numerics whereas the Hilbert series is symbolic and admit a gauge invariant analysis. Furthermore, exploiting various exchange symmetries, we show that quartic and higher order terms are absent in the potential, which if present would have spoiled the counting. Incorporating recent developments in algebraic geometry focusing on computational algorithms, we have extended the scope of one of the authors previous works [1, 2] and presented a new perspective for the black hole microstate counting problem. This further establishes the pure D-brane system as a consistent model, bringing us a step closer to \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 2 black hole microstate counting.

Voronoi discretization to improve the meshless local Petrov–Galerkin method in 3D-computational fracture mechanics

Purpose1) The OE-MLPG penalty meshfree method is developed to solve cracked structure.(2) Smartening the numerical meshfree method by combining the particle swarm optimization (PSO) optimization algorithms and Voronoi computational geometric algorithm. (3). Selection of base functions, finding optimal penalty factor and distribution of appropriate nodal points to the accuracy of calculation in the meshless local Petrov–Galekrin (MLPG) meshless method.Design/methodology/approachUsing appropriate shape functions and distribution of nodal points in local domains and sub-domains and choosing an approximation or interpolation method has an effective role in the application of meshless methods for the analysis of computational fracture mechanics problems, especially problems with geometric discontinuity and cracks. In this research, computational geometry technique, based on the Voronoi diagram (VD) and Delaunay triangulation and PSO algorithm, are used to distribute nodal points in the sub-domain of analysis (crack line and around it on the crack plane).FindingsBy doing this process, the problems caused by too closeness of nodal points in computationally sensitive areas that exist in general methods of nodal point distribution are also solved. Comparing the effect of the number of sentences of basic functions and their order in the definition of shape functions, performing the mono-objective PSO algorithm to find the penalty factor, the coefficient, convergence, arrangement of nodal points during the three stages of VD implementation and the accuracy of the answers found indicates, the efficiency of V-E-MLPG method with Ns = 7 and ß = 0.0037–0.0075 to estimation of 3D-stress intensity factors (3D-SIFs) in computational fracture mechanics.Originality/valueThe present manuscript is a continuation of the studies (Ref. [33]) carried out by the authors, about; feasibility assessment, improvement and solution of challenges, introduction of more capacities and capabilities of the numerical MLPG method have been used. In order to validate the modeling and accuracy of calculations, the results have been compared with the findings of reference article [34] and [35].

A novel model for representing a plane target and finding the worst-case coverage in wireless sensor network based on Clifford algebra

Wireless ad hoc sensor networks have recently emerged as a premier research topic. They have great long-term economic potential and ability to transform our lives and pose many new system building challenges. Sensor networks also pose a number of new conceptual and optimization problems. Most of researches in wireless sensor networks are focused in obtaining better target coverage in order to reduce energy and cost of the network. The problem of planar target analysis is one of the crucial problems that should be considered while studying coverage problem of sensor networks. By combining computational geometry and graph theoretic techniques, specifically the Voronoi diagram and graph search algorithms, this paper introduces a novel sensor network coverage model that deals with plane target problem based on Clifford algebra which is a powerful tool that is coordinate free. Also, the calculations of the node coverage rate for the plane target in the sensor network using Clifford algebra are presented. Then, the maximum clearance path (worst-case coverage) of the sensor network for a plane target is proposed. The optimality and reliability of the proposed algorithm have been proved using simulation. Also, a comparison between the breach weight of the point target and the plane target is provided.

An extension to Voro++ for multithreaded computation of Voronoi cells

Voro++ is a software library written in C++ for computing the Voronoi tessellation, a technique in computational geometry that is widely used for analyzing systems of particles. Voro++ was released in 2009 and is based on computing the Voronoi cell for each particle individually. Here, we take advantage of modern computer hardware, and extend the original serial version to allow for multithreaded computation of Voronoi cells via the OpenMP application programming interface. We test the performance of the code, and demonstrate that it can achieve parallel efficiencies greater than 95% in many cases. The multithreaded extension follows standard OpenMP programming paradigms, allowing it to be incorporated into other programs. We provide an example of this using the VoroTop software library, performing a multithreaded Voronoi cell topology analysis of up to 102.4 million particles. Program summaryProgram title:Voro++CPC Library link to program files:https://doi.org/10.17632/tddc4w4zkk.1Developer's repository link:https://github.com/chr1shr/voroLicensing provisions: BSD 3-clause (with LBNL modification)Programming language: C++External routines/libraries: OpenMPNature of problem: Multithreaded computation of the Voronoi tessellation in two and three dimensionsSolution method: The Voro++ library is built around several C++ classes that can be incorporated into other programs. The two largest components are the container... classes that spatially sort input particles into a grid-based data structure, allowing for efficient searches of nearby particles, and the voronoicell... classes that represent a single Voronoi cell as an arbitrary convex polygon or polyhedron. The Voronoi cell for each particle is built by considering a sequence of plane cuts based on neighboring particles, after which many different statistics (e.g. volume, centroid, number of vertices) can be computed. Since each Voronoi cell is calculated individually, the Voronoi cells can be computed using multithreading via OpenMP.

Enriched MQ-H-RPIM method optimized by M-PSO algorithm and Voronoi discretization to estimation of the 3D-SIF's in fracture mechanics problems

The RPIM is one of the meshless methods that by enriching and extending Base Functions in the standard form of MQ-RPIM, can be developed and solving the Fracture Mechanics problems. In this research, with using the Hermite-RBFs and optimizing the Shape Parameters in the RBFs by the Multi-Objective PSO algorithm, the Enriched-MQ-RPIM method has been converted to the OE-MQ-HRPIM method. The appropriate distribution of nodal points in the analysis domain has also been done with the Voronoi computational geometry technique. A cracked structure will be used to discretize the analysis domain and calculation of the displacement and stress fields to estimate the 3D-SIFs in Mode-I under Uni-axial Cyclic Loading. The research findings, indicate that the OE-MQ-HRPIM 's answers are closer to the experimental results compared to the MQ-RPIM, OE-MLPG Penalty and E-PIM methods. Although in this method the size of the matrices, the convergence speed and the number of unknown coefficients from n+m+l to 2n+m+l are increased, but the answers are calculated with higher density nodal point in the Sub-Local domains and fewer nodal points in the domain analysis .Also Combining Second-Order sentences in the Khayyam-Pascal pyramid (Ns=7) to defining PBFs in OE-MQ-HRPIM, will lead to more accurate answers.

The Use of Computational Geometry Techniques to Resolve the Issues of Coverage and Connectivity in Wireless Sensor Networks.

Wireless Sensor Networks (WSNs) enhance the ability to sense and control the physical environment in various applications. The functionality of WSNs depends on various aspects like the localization of nodes, the strategies of node deployment, and a lifetime of nodes and routing techniques, etc. Coverage is an essential part of WSNs wherein the targeted area is covered by at least one node. Computational Geometry (CG) -based techniques significantly improve the coverage and connectivity of WSNs. This paper is a step towards employing some of the popular techniques in WSNs in a productive manner. Furthermore, this paper attempts to survey the existing research conducted using Computational Geometry-based methods in WSNs. In order to address coverage and connectivity issues in WSNs, the use of the Voronoi Diagram, Delaunay Triangulation, Voronoi Tessellation, and the Convex Hull have played a prominent role. Finally, the paper concludes by discussing various research challenges and proposed solutions using Computational Geometry-based techniques.

Particle shape characterizations for energetic materials by computational geometry and stereology method

Particle shapes significantly affect viscosity and flow behavior of energetic materials, and therefore affect their packability and processability. This study presents a computational geometry framework for automatically quantifying two-dimensional (2D) and three-dimensional (3D) particle shapes of energetic materials. A specimen by mixing three typical energetic materials including HMX (octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine), RDX (1,3,5-Trinitroperhydro-1,3,5-triazine) and AP (Ammonium Perchlorate) particles is used in this study. This specimen is scanned by high-resolution X-ray computed tomography (X-ray CT), yielding a volumetric image. An improved watershed analysis algorithm is used to process the volumetric image to identify individual 3D particles. The stereology sampling method is used to obtain 2D projections of 3D particles. Computational geometry techniques are developed by this study to analyze 2D particle projections and 3D particle geometries to compute seven commonly used shape descriptors, including convexity, circularity, intercept sphericity, area sphericity, diameter sphericity, circle ratio sphericity, and surface area sphericity. Results show that those different shape descriptors of energetic materials can be divided into three groups based on their numerical ranges. This study also evaluates the effectiveness and accuracy of 2D shape descriptors for quantifying the true 3D shapes. The inconsistent characterization results between 2D and 3D shape descriptors suggest that researchers should be cautious when using 2D images to characterize 3D particle shapes of energetic materials. The computational geometry framework and particle shape analysis results presented in this study can be potentially useful in numerical modeling, experimental analysis, and theoretical investigation for energetic materials.

Heterogeneous thermal conductance of nanoparticle-fluid interfaces: An atomistic nodal approach.

The Interfacial Thermal Conductance (ITC) is a fundamental property of materials and has particular relevance at the nanoscale. The ITC quantifies the thermal resistance between materials of different compositions or between fluids in contact with materials. Furthermore, the ITC determines the rate of cooling/heating of the materials and the temperature drop across the interface. Here, we propose a method to compute local ITCs and temperature drops of nanoparticle-fluid interfaces. Our approach resolves the ITC at the atomic level using the atomic coordinates of the nanomaterial as nodes to compute local thermal transport properties. We obtain high-resolution descriptions of the interfacial thermal transport by combining the atomistic nodal approach, computational geometry techniques, and "computational farming" using non-equilibrium molecular dynamics simulations. We use our method to investigate the ITC of nanoparticle-fluid interfaces as a function of the nanoparticle size and geometry, targeting experimentally relevant structures of gold nanoparticles: capped octagonal rods, cuboctahedrons, decahedrons, rhombic dodecahedrons, cubes, icosahedrons, truncated octahedrons, octahedrons, and spheres. We show that the ITC of these very different geometries varies significantly in different regions of the nanoparticle, increasing generally in the order face < edge < vertex. We show that the ITC of these complex geometries can be accurately described in terms of the local coordination number of the atoms in the nanoparticle surface. Nanoparticle geometries with lower surface coordination numbers feature higher ITCs, and the ITC generally increases with the decreasing particle size.

Improved algorithm for minimum zone of roundness error evaluation using alternating exchange approach

Based on computational geometry techniques, an improved algorithm for the minimum zone of roundness error evaluation using an alternating exchange method is presented. A minimum zone fitting function was created to enhance the roundness error evaluation. The function uses three candidate points to determine the initial solution: the expected centre, the mean circle radius, and the corresponding zone half-width. The best solution function is designed to use the initial solution as the input to determine the optimum solution for the minimum zone circle (MZC). The proposed algorithm was validated using data available in the literature. The roundness error evaluation comparison results demonstrate that the proposed method accurately detects both the centre error magnitude and MZC and overcomes the insufficiency of using selected colinear points for four selected points.

No Fine-Tuning, No Cry: Robust SVD for Compressing Deep Networks.

A common technique for compressing a neural network is to compute the k-rank approximation of the matrix via SVD that corresponds to a fully connected layer (or embedding layer). Here, d is the number of input neurons in the layer, n is the number in the next one, and is stored in memory instead of . Then, a fine-tuning step is used to improve this initial compression. However, end users may not have the required computation resources, time, or budget to run this fine-tuning stage. Furthermore, the original training set may not be available. In this paper, we provide an algorithm for compressing neural networks using a similar initial compression time (to common techniques) but without the fine-tuning step. The main idea is replacing the k-rank approximation with , for , which is known to be less sensitive to outliers but much harder to compute. Our main technical result is a practical and provable approximation algorithm to compute it for any , based on modern techniques in computational geometry. Extensive experimental results on the GLUE benchmark for compressing the networks BERT, DistilBERT, XLNet, and RoBERTa confirm this theoretical advantage.

A computational algebraic geometry technique for determining nonlinear normal modes of structural systems

A computational algebraic geometry technique is developed for determining nonlinear normal modes (NNMs) of multi-degree-of-freedom (MDOF) nonlinear dynamic systems. Specifically, resorting to computational algebraic geometry concepts and tools, such as Gröbner basis, it is shown that the complete solution set of a nonlinear algebraic system of coupled multivariate polynomial equations is determined exactly, even for a relatively large number of unknowns. Herein, these unknowns represent expansion coefficients for approximating NNMs of nonlinear dynamic systems. In this regard, the exact nature of the determined coefficient vector, which is one the significant advantages of the technique, implies increased confidence in approximating NNMs more accurately by utilizing higher-order expansions with additional terms. Several numerical examples referring to various MDOF nonlinear dynamic systems are considered for demonstrating the efficacy and potential of the solution technique. Also, note that although the technique is developed herein by relying on the classical Shaw–Pierre definition of NNMs, this is not necessarily a restriction of the methodology. In fact, the proposed computational algebraic geometry solution framework can be potentially used, in a rather straightforward manner, in conjunction with alternative NNM definitions and problem formulations requiring the solution of nonlinear algebraic equations of polynomial form.

Extracting the topology of urban airspace through graph abstraction

With the rapid adoption of the operational concept of Unmanned Aerial Systems (UAS) in urban airspace, there is an increased focus on the spatial extent that Unmanned Aircraft (UA) can safely navigate. Urban airspace is full of intrinsic risks associated with several factors, including but not limited to unpredictable microclimates, radio-frequency disturbances and electro-magnetic interferences. For safe and efficient use of airspace, it is imperative to proactively assess the availability of airspace volumes to accommodate large-scale operations. In this paper, we highlight that urban airspace is a unique environment where highly obstructed (complex topology) and sparsely structured (simple topology) regions coexist. The presence of static obstacles and required separation from those obstacles create particular spatial structures, such as narrow passages, islands and funnels, throughout the airspace. Considering that not only throughput capacity of airspace but also complexity of path planning is highly dependent on the topology of a navigable space, we propose a new airspace assessment approach to extract the topology of airspace using computational geometry techniques. We focus on identifying geospatial locations suitable for high density UAS operations as well as examining spatial contiguity between those locations in highly urbanized areas. In doing so, we define segment as an airspace region that is wide enough for multiple vehicles to simultaneously pass through. Low-altitude urban airspace is abstracted into a compact and informative graph structure called segment graph that summarizes segment connectivity in the horizontal and vertical dimension. The proposed method shows that the geometry of low-altitude airspace is fundamentally different from that of conventional high-altitude airspace and can provide a useful basis for airspace design and management within low-altitude urban environments.

Evaluation of plant spatial placement by TF POLYM Software

Food production intensity is a key issue in the modern world. Due to the development of the possibility of increasing the production area, the technology is constantly being improved and supplemented to increase their efficiency. The basis of any cultivation is sowing, which should provide everyone with a suitable living space. It allows the plant to effectively use the applied fertilization, chemical protection or mechanical treatments. The article develops the options for determining the living space of plants. The computational algorithm uses the Voronoi diagram belonging to the computational geometry technique, and then the Delphi software, the TfPolyM computer application, which was developed at the Faculty of Engineering of the Slovak University of Agriculture in Nitra. Based on the digitization of photos, the software is able to calculate the living space of the plant, and also allows for a precise assessment of the uniformity and quality of planting.

Image Processing-based Method for Automatic Design of Patient-Specific Cranial Implant for Additive Manufacturing

Decompressive craniectomy (DC) is a surgical procedure where a portion of the skull (flap) is removed to relieve the built-up pressure from the patient’s brain due to swelling of the brain tissue after a traumatic injury to the head. Subsequently, another surgical procedure called cranioplasty is carried out to fix an implant or bone flap in patients who have undergone DC. In this paper, an automatic design methodology for additive manufacturing of a PSCI (patient-specific cranial implant) has been proposed. The input is the DICOM digital data from a CT scan and the output is the STL file geometry of the cranial implant. The proposed method has been tested and validated using real de-identified DICOM data, and the resultant implant was 3D printed and fit to the skull of a cadaver. The key contribution made in this paper is the complete automation of the design of a PSCI based on the skull’s unique geometry using a combination of image-processing and computational geometry techniques. Another important characteristic of the proposed method is that medical professionals need not have any technical expertise in additive manufacturing or part design for generating a PSCI.

Digital Image Segmentation Using Delaunay Triangulation Algorithm

Image segmentation is the process of partitioning a digital image into multiple segments according to some homogeneity criterion. Different approaches are suited to different types of images and the quality of output of a particular algorithm is difficult to measure quantitatively due to the fact that there may be much “correct” segmentation for a single image. Although the field encounters several challenges, this research work seeks to present a segmentation algorithm for gray intensity images. Image features extraction is first performed to obtain the approximate, if not exact, representation of a given image. Image enhancement was also carried out by exploiting the principle of histogram equalization which is one of the well-known image enhancement techniques. The proposed approach is based on the Delaunay (DT), one of the techniques in computational geometry, which generates clusters of intensity values using information from the vertices of the external boundary of triangulation. This makes it possible to produce segmented image regions. The algorithm was tested on sampled images before and after enhancement and the production compared. Our algorithm successfully extracts cogent features from a given image and made use of the Delaunay Triangulation to carry out segmentation. The method is robust, precise and independent of translation, rotation and scaling and that makes the result quite good for image segmentation. Image segmentation based on Delaunay triangulation technique seem to be a promising future research area with respect to its applicability to underwater mine and submarine detection, better target recognition, medical imaging as well as to automated computer vision.

Three-dimensional particle shape characterizations from half particle geometries

Many three-dimensional (3D) imaging techniques, such as 3D laser scanner, stereophotography, and structured light techniques, can only capture upper half particle geometries in the camera view, and lower half particle geometries are occluded from the camera. This research aims to evaluate the accuracy of using half particle geometries to compute 3D particle shape descriptors. Computational geometry techniques are developed to compute eight commonly used particle shape descriptors automatically. Five coarse sand specimens are scanned by X-ray computed tomography and structured light to generate full 3D and half particle geometries, respectively. The convexity, diameter sphericity, and surface area sphericity computed from half particles overestimate these values computed from full 3D particles, which are corrected by statistical equations. The circularity, volume sphericity, intercept sphericity, sphere ratio sphericity, and roundness computed from half particles agreed excellently with these values computed from 3D particles.

Realistic soil particle generation based on limited morphological information by probability-based spherical harmonics

Three-dimensional imaging techniques, such as X-ray computed tomography, have been used to scan realistic particle geometries. However, these techniques are labor intensive, time-consuming, and costly to obtain a large number of particles. Therefore, it is desirable if computers can be taught to generate realistic particles based on given morphological properties. This paper develops a particle generation technique by integrating spherical harmonics and probability functions. This technique only requires morphological information from one particle to generate a large number of particles and eliminates the need for scanning many particles for particle generation. The spherical harmonics coefficients of this particle are analog of the morphological gene. The probability function is used to add variances to spherical harmonics coefficients to simulate gene mutation. A dimensionless factor is developed to control degrees of gene mutation. The effectiveness and accuracy of the proposed technique are verified by particle shape descriptors computed by the computational geometry techniques.

Aligning Points to Lines: Provable Approximations

We suggest a new optimization technique for minimizing the sum $\sum_{i=1}^n f_i(x)$ of $n$ non-convex real functions that satisfy a property that we call piecewise log-Lipschitz. This is by forging links between techniques in computational geometry, combinatorics and convex optimization. As an example application, we provide the first constant-factor approximation algorithms whose running-time is polynomial in $n$ for the fundamental problem of \emph{Points-to-Lines alignment}: Given $n$ points $p_1,\cdots,p_n$ and $n$ lines $\ell_1,\cdots,\ell_n$ on the plane and $z>0$, compute the matching $\pi:[n]\to[n]$ and alignment (rotation matrix $R$ and a translation vector $t$) that minimize the sum of Euclidean distances $\sum_{i=1}^n \mathrm{dist}(Rp_i-t,\ell_{\pi(i)})^z$ between each point to its corresponding line. This problem is non-trivial even if $z=1$ and the matching $\pi$ is given. If $\pi$ is given, the running time of our algorithms is $O(n^3)$, and even near-linear in $n$ using core-sets that support: streaming, dynamic, and distributed parallel computations in poly-logarithmic update time. Generalizations for handling e.g. outliers or pseudo-distances such as $M$-estimators for the problem are also provided. Experimental results and open source code show that our provable algorithms improve existing heuristics also in practice. A companion demonstration video in the context of Augmented Reality shows how such algorithms may be used in real-time systems.

A Method for Removing PET/CT Imaging Artifact Using Combination of Standard Deviation and Computational Geometry Technique

Positron emission tomography/Computed tomography is a non-intrusive imaging methodology, which is clinically utilized for both finding and getting to treatment reaction in oncology, cardiology, as well as neurology. Many different artifacts can occur during Positron emission tomography (PET) imaging, these artifacts are expected essentially to metallic inserts, respiratory movement, utilization of contrast media also image truncation. Artifacts are relatively common in PET/CT imaging and may potentially degrade image quality and interfere with accurate radiological reporting and diagnosis. Improving the recognition of PET/CT artifacts may assist imaging practitioners to avoid or limit their effect on image quality and interpretation. This paper proposed a statistical method of standard deviation to figure the global threshold for binarizing image and computational geometry to reach the output. This combination would produce good results for removing the artifacts from pancreatic PET/CT images