Implicit Surface Reconstruction from 3D Scattered Points Based on Variational Level Set Method
|
|
- Sharleen Dean
- 6 years ago
- Views:
Transcription
1 Implicit Surface econstruction from D Scattered Points Based on Variational Level Set Method Hanbo Liu Department ofshenzhen graduate school, Harbin Institute oftechnology, Shenzhen, 58055, China liu_hanbo@hit.edu.cn Xin Wang Department ofshenzhen graduate school, Harbin Institute oftechnology, Shenzhen, 58055, China wang_xin@hit.edu.cn Wenyi Qiang Space Control and Inertial Technology esearch Center, Harbin Institute oftechnology, Harbin, 50080, China qiangwy@hit.edu.cn Abstract-In this paper we propose a novel variational form ulation for arbitrary surface reconstruction from D scattered points. An implicit surface is adopted for its more advantages, such as continuity and differentiability, easy estimation of points inside or outside the shape, and convenient implementation for these complicated set operations. The presented new energy functional in this paper considers more factors on properties ofthe scattered points (including distances and normal vectors), smoothness and the constraint of the signed distance function. The gradient flow which minimizes the total energy functional updates and drives the motion ofthe zero level set interface to the desired surface. The derived formulations were applied to certain D surface reconstruction with good results. I. INTODUCTION Constructing a computer model from an existing object is a common problem in everse Engineering. The sampled data might be scanned with a device like the laser range scanner, or be measured on the object with a mechanical probe. Sometimes, not only the spatial location of points, but also normal vectors can be obtained. The problem ofautomatically reconstructing a topologically consistent and geometrically accurate CAD model of complex object from these sampled data widely occurs in manufacturing industry. Obviously solving this problem depends heavily and only on the dense and uniform sampled data itself. And our research aims to provide a compact and topologically consistent representation ofthe complex object and usable and accurate reconstruction method from dense samples. The level set method is a theoretical and numerical method for implicit surfaces. It initially was introduced in the area of fluid dynamics establishing a connection between the family of evolving curves and the family of evolving surfaces []. The level set method was devised by Osher and Sethian as a simple and versatile method for computing and analyzing the motion of an interface under a velocity field in two or three dimensions [][]. The velocity can depend on position, time, the geometry of the interface and the external physics. The level set method is widely used in D image processing and segmentation [4]. The implicit surface methods for reconstruction have received considerable attention in the past few years by researchers, such as Hoppe, Carr, Turk and Ohtake et al. Carr fit a radial basis function (BF) to the signed distance function and used BF to polygonize sampled points to create a triangulated surface [5], but this algorithm is difficult to implement. Subsequently Ohtake proposed MPU method [6]. And Zhao et al. first combined variational and PDE formulation with level set method for D surface reconstruction from unorganized data [7][8]. But they not make us ofthe measurable normal vector. In this paper we mainly propose a novel energy functional additionally involving factors of distance and unit normal vector, constraint ofsigned distance function and smoothness. In section II the implicit surface representation is presented. In section III we introduce our energy functional and the corresponding gradient flow. In section IV concerned details of the numerical implementation are presented. Finally in section V gives the results of reconstructed surface by our variational level set method. II. IMPLICIT SUFACE EPESENTATION We consider the following reconstruction problem: let an unorganized set of points S = {X i }: E associated with normal vectors V = {n i }: E and position vectors P = {Pi = (Xi' Y i, Zi ):} E.And the points S are sampled from a boundary surface r of a three-dimensional object in domain Q in.so we can utilize the level set theory, for a given open region Q with smooth boundary surface r,there exists a level set function (X,Y, z), which at any time is Lipschitz continuous, and do not depend on any underlying parameterization, satisfying /08/$ IEEE
2 φ( xyz,, ) < 0 for( xyz,, ) Ω φ( xyz,, ) = 0 for( xyz,, ) φ( xyz,, ) > 0 for( xyz,, ) Ω. So the coordinates ( xyz,, ) of the points on the surface are defined by the equation φ ( xyz,, ) = 0. The surface remains the zero level set of φ even over time. Moreover, the implicit surface is convenient implementation for complicated set operations. Using convention, the outward unit normal n and the mean curvature k of are given by (). n = and k = () Where φ = φ + φ + φ. x y z Taking the derivatives of φ ( xyz,, ) = 0 with respect to time t gives (), which is a first order Hamilton-Jacobi equation. + = + v n φ = 0 () t t t Where is proportional to the surface normal, x affects t φ only in the direction of the normal, which motion in any direction is merely a change in the parameterization. v n is the velocity normal to the surface. This PDE is solved on a fixed grid in the domain Ω.one of the advantages of implicit surface representation is that the topology of the surface is allowed to change as the surface evolves, thus making it easy to represent complex surfaces that can merge or split and also with holes. III. VAIATIONAL FOMULATION The main idea of the level set method is simple: using energy minimization techniques a gradient flow derived as the steepest descent of the energy functional is obtained and then implemented in the level set space. So the minima correspond to desired surface. In this paper we construct a novel energy functional E( x ). Given a set of scattered D points S = { x} N i i=, for each xi with properties of normal vectors ni and position vectors p i, define the energy functional E( x) to be minimized as Ex ( ) = En + Ed + Esmooth. (4) Furthermore, if f ( x ) is a minimum of the energy functional E( x ), then ' E ( f( x); v ) = 0 for all v. The idea behind (5) is to minimize the difference between the desired normal and the normal of the surface in nature. Normal vectors are coupled with the surface, thus the normal vectors drag the surface while their values are modified during processing. c () En = ( nx ( ) Nx ( )) ds (5) Where N( x) is the extension of the desired normal nx ( i ) at point xi to the whole domain Ω, nx ( ) is the surface normal vector, ds is the surface area. If is the zero level set of φ ( x), then the energy functional corresponding to the normal can be reformulated as En = ( n ( x) N( x)) ds (6) = ( nx ( ) Nx ( )) δφ ( ) dx Where δ ( x) is the one dimensional Delta function and δφ ( ) dx is the surface area element at the zero level set of φ ( x). So (6) can extend the integration surface to a D computation domain containing the zero level set of φ ( x). Let F( x, φ, φ) = ( N( x) ) δ ( φ), then the condition for a minimum given by the Gateaux derivative is: F F F F ( ) ( ) ( ) = 0 (7) x x y y z z So the gradient flow obtained from (7) is φt_ n = ( ( ) N( x)) (8) = ( k N( x)) Similarly equation (9) deals with the position difference between the desired and the reconstructed surface. The energy functional corresponding to distance is defined as: E = d ( x) ds = d ( x) δφ ( ) dx Where d (9) d( x) = dist( x, S) = min dist( x, p ), dist( x, y) = x y. i N So the steepest descent flow is: φt_ d = d ( ) φ + ( d ) = dk φ + ( d). i (0) Es is the smoothing term, and we choose a small positive ω as the smoothing constant. Es = ωds = ωδ( φ( x)) ( x) dx. () Which gets the Well-known mean curvature motion (). φt_ s = ω ( ) φ = ωk () If object with more edges and corners, we change the mean curvature motion to constant speed motion. φt_ s = ω φ () Combing the energy functional (5), (9) and (), the overall PDE motion is expressed as (4). φt = φt_ d + φt_ s + φt_ n = dk φ + ( d) + ωk φ + ( k N( x)) (4) = ( d ) φ + ( d + ω) k N( x) A B C Where ω = ω+. The first term A is corresponding to the
3 advection motion, the middle B is corresponding to motion by mean curvature and the last C corresponding to motion in the normal direction. And this total motion leads the energy functional E( x) decrease until to the equilibrium. PDEs are solved on discrete fixed grids in some domains. IV. NUMEICAL IMPLEMENTATION A. The Signed Distance Function The signed distance of a point x from a closed, oriented manifold S is defined as the Euclidean distance from x to the closest point on S, with a positive sign if x lies outside S and a negative sign otherwise. So computes the distance function to an arbitrary data set on rectangular fixed grids. dist( x, ( t)), x Ω( t) d( x, t) = 0, x ( t) (5) dist( x, ( t)), x \ Ω( t) We use the same numerical approximation to δ ( x) as defined in [0] in this paper. 0, x > α δα ( x) = π x [ + cos( )] x α α α And in (5) N( x ) is defined as nx ( i) if x Bε ( xi) N( x) = nx ( ) otherwise (6) Where Bε ( x i ) a ball with radius ε centered at x i and covers the nearest gird. B. Numerical Scheme This family of PDEs and the numerical scheme for solving them on discrete grids use the traditional level set method [][9]. The spatial derivatives of A and C terms in (4) are approximated by the upwind finite difference scheme, while the spatial derivatives of B term are approximated by the central difference scheme. And the temporal partial derivative φ t is approximated by the forward difference. So the approximation of (4) by the above difference scheme can be expressed as m+ m m φ = φ + τ( φ ) i, j, k i, j, k i, j, k m Where L( φi, j, k) is the approximation of the right hand side in (4) by the above spatial difference schemes. The curvature k usually used is computed by φxφyy φxφyφxy + φyφxx + φxφzz φxφzφxz + φz φxx k = φφ y zz φφφ y z yz + φφ z yy +. In addition, when solving the PDE (4), we must make a point of the spatial and time step. According to the density of D scattered points and K-nearest radius of each point, we decide the spatial and time step. At the same time, the time step and spatial step must satisfy the CFL condition for stable evolution. C. Initialization and e-initialization A good initial surface guess is crucial to the efficiency of the variational level set method. We use the traditional and common method that build (7) an offset of the distance function to the scattered points as the initial surface in initialization process. d( x, t) ε (7) e-initialization has been widely used as a numerical remedy in traditional level set method. In practice, the evolving level set function can deviate greatly from its value as signed distance after several iterations especially when the time step is not chosen small enough for fast evolution. It is well known that a signed distance function must satisfy φ =. Conversely, any function φ satisfying φ = is the signed distance function plus a constant. Illumined by D image re-initialization [], we additionally introduce a penal term for avoiding the deviation of the level set function φ from a signed distance function in D surface reconstruction. F( φ) = ( ) dx (8) Ω Also to minimize the F( φ ), we derive the evolving motion as =Δφ div( ) = div[(- ) ]. (9) t Where Δ is the Laplacian operator. If φ >, this term makes φ more even and therefore reduce the contrary, if φ < will increase φ. φ. On the V. ESULTS Fig. shows the constructed grids for an easy sphere model. According to the red scattered points, we can computer the signed distances and determine the domain of effective grids for initial surface. Fig. illustrates the reconstructed surfaces by the presented variational level set method in this paper. The dots mark the D scattered points all measured directly from real D objects. And the reconstructed surfaces are smooth and express the original shapes well with good precision. esults demonstrate that our method can be applied to arbitrary D scattered points without reference to surface topology and geometrical features.
4 geometrical features. Fig.. Grids and the initial distance to the scattered points EFEENCES [] S. Osher and J.A. Sethian, Fronts Propagating with Curvature Dependent Speed: Algorithm based on the Hamilton-Jacobin Formulation, Computational Physics (79), pp.-49, 988. [] J.A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, New York, 999. [] S. Osher and. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York, 00. [4] S. Osher and. Fedkiw, Level Set Methods: an Overview and Some ecent esults, Journal of Computational Physics (69), pp , 00. [5] J. C. Carr,. K. Beatson, J. B. Cherrie, et al, econstruction and epresentation of D Object with adial Basis Functions, Proceedings of SIGGAPH00, Los Angeles, pp ,00. [6] Y. Ohtake, A. Belyaev and H. Seidel, A Multi-scale Approach to D Scattered Data Interpolation with Compactly Supported Basis Functions, Proceedings of SMI00, Seoul, pp.5-6, 00. [7]. H.K. Zhao, S. Osher, B. Merriman, and M. Kang, Implicit and Non-parametric Shape econstruction from Unorganized Data Using Variational Level Set Method, Computer Vision and Image Understanding, 80(), pp.95-9, 000. [8]. H.K. Zhao, S. Osher and. Fedkiw, Fast Surface econstruction Using the Level Set Method, proceedings of the IEEE Workshop on Variational and Level Set Methods in Computer Vision, Vancouver, pp.94-0, 00. [9] D. Peng, B. Merriman, S. Osher, et al, A PDE-Based Fast Local Level Set Method, Journal of Computational Physics (55), pp.40-48, 999. [0] H.K. Zhao, B. Merriman and S. Osher, A Variational level set approach to multiphase motion, Journal of Computational Physics (7), pp.79-95, 996. [] C.M. Li, C.Y. Xu, C.F. Gui and M.D. Fox, Level Set Evolution Without e-initialization: A New Variational Formulation, proceedings of Computer Vision and Pattern ecognition, pp , 005. Fig.. econstructed surfaces and the initial D scattered points VI. CONCLUSIONS AND FUTUE WOK In this paper we propose a new variational formulation for arbitrary surface reconstruction from D scattered points. The implicit surface representation is adopted because of its advantages, such as continuity and differentiability, easy to identify whether a point lies inside or outside the shape, and convenient implementation for complicated set operations. Furthermore, the novel energy functional in this paper considers more factors on properties of the scattered points (including distances and normal vectors), smoothness and the constraint of the signed distance function. The gradient flow which minimizes the total energy functional updates and drives the motion of the zero level set interface to the desired surface. The derived formulation and method constructs the implicit surface on fixed rectangular grids and has been applied to certain D surface reconstruction with good results. In future work we will pay more attention on efficiency and speed-up of our variational method for D surface reconstruction from large scattered points with more
5
The Level Set Method. Lecture Notes, MIT J / 2.097J / 6.339J Numerical Methods for Partial Differential Equations
The Level Set Method Lecture Notes, MIT 16.920J / 2.097J / 6.339J Numerical Methods for Partial Differential Equations Per-Olof Persson persson@mit.edu March 7, 2005 1 Evolving Curves and Surfaces Evolving
More informationweighted minimal surface model for surface reconstruction from scattered points, curves, and/or pieces of surfaces.
weighted minimal surface model for surface reconstruction from scattered points, curves, and/or pieces of surfaces. joint work with (S. Osher, R. Fedkiw and M. Kang) Desired properties for surface reconstruction:
More informationLevel Set Methods and Fast Marching Methods
Level Set Methods and Fast Marching Methods I.Lyulina Scientific Computing Group May, 2002 Overview Existing Techniques for Tracking Interfaces Basic Ideas of Level Set Method and Fast Marching Method
More informationAutomated Segmentation Using a Fast Implementation of the Chan-Vese Models
Automated Segmentation Using a Fast Implementation of the Chan-Vese Models Huan Xu, and Xiao-Feng Wang,,3 Intelligent Computation Lab, Hefei Institute of Intelligent Machines, Chinese Academy of Science,
More informationPARAMETRIC SHAPE AND TOPOLOGY OPTIMIZATION WITH RADIAL BASIS FUNCTIONS
PARAMETRIC SHAPE AND TOPOLOGY OPTIMIZATION WITH RADIAL BASIS FUNCTIONS Michael Yu Wang 1 and Shengyin Wang 1 Department of Automation and Computer-Aided Engineering The Chinese University of Hong Kong
More informationLevel set methods Formulation of Interface Propagation Boundary Value PDE Initial Value PDE Motion in an externally generated velocity field
Level Set Methods Overview Level set methods Formulation of Interface Propagation Boundary Value PDE Initial Value PDE Motion in an externally generated velocity field Convection Upwind ddifferencingi
More informationAn Active Contour Model without Edges
An Active Contour Model without Edges Tony Chan and Luminita Vese Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095-1555 chan,lvese@math.ucla.edu
More informationUnstructured Mesh Generation for Implicit Moving Geometries and Level Set Applications
Unstructured Mesh Generation for Implicit Moving Geometries and Level Set Applications Per-Olof Persson (persson@mit.edu) Department of Mathematics Massachusetts Institute of Technology http://www.mit.edu/
More informationThe Level Set Method applied to Structural Topology Optimization
The Level Set Method applied to Structural Topology Optimization Dr Peter Dunning 22-Jan-2013 Structural Optimization Sizing Optimization Shape Optimization Increasing: No. design variables Opportunity
More informationBACK AND FORTH ERROR COMPENSATION AND CORRECTION METHODS FOR REMOVING ERRORS INDUCED BY UNEVEN GRADIENTS OF THE LEVEL SET FUNCTION
BACK AND FORTH ERROR COMPENSATION AND CORRECTION METHODS FOR REMOVING ERRORS INDUCED BY UNEVEN GRADIENTS OF THE LEVEL SET FUNCTION TODD F. DUPONT AND YINGJIE LIU Abstract. We propose a method that significantly
More informationTheoretical Background for OpenLSTO v0.1: Open Source Level Set Topology Optimization. M2DO Lab 1,2. 1 Cardiff University
Theoretical Background for OpenLSTO v0.1: Open Source Level Set Topology Optimization M2DO Lab 1,2 1 Cardiff University 2 University of California, San Diego November 2017 A brief description of theory
More informationA Toolbox of Level Set Methods
A Toolbox of Level Set Methods Ian Mitchell Department of Computer Science University of British Columbia http://www.cs.ubc.ca/~mitchell mitchell@cs.ubc.ca research supported by the Natural Science and
More informationDr. Ulas Bagci
Lecture 9: Deformable Models and Segmentation CAP-Computer Vision Lecture 9-Deformable Models and Segmentation Dr. Ulas Bagci bagci@ucf.edu Lecture 9: Deformable Models and Segmentation Motivation A limitation
More informationMath 690N - Final Report
Math 690N - Final Report Yuanhong Li May 05, 008 Accurate tracking of a discontinuous, thin and evolving turbulent flame front has been a challenging subject in modelling a premixed turbulent combustion.
More informationOutline. Level Set Methods. For Inverse Obstacle Problems 4. Introduction. Introduction. Martin Burger
For Inverse Obstacle Problems Martin Burger Outline Introduction Optimal Geometries Inverse Obstacle Problems & Shape Optimization Sensitivity Analysis based on Gradient Flows Numerical Methods University
More informationCollege of Engineering, Trivandrum.
Analysis of CT Liver Images Using Level Sets with Bayesian Analysis-A Hybrid Approach Sajith A.G 1, Dr. Hariharan.S 2 1 Research Scholar, 2 Professor, Department of Electrical&Electronics Engineering College
More informationMedical Image Segmentation using Level Sets
Medical Image Segmentation using Level Sets Technical Report #CS-8-1 Tenn Francis Chen Abstract Segmentation is a vital aspect of medical imaging. It aids in the visualization of medical data and diagnostics
More informationFast 3D surface reconstruction from point clouds using graph-based fronts propagation
Fast 3D surface reconstruction from point clouds using graph-based fronts propagation Abdallah El Chakik, Xavier Desquesnes, Abderrahim Elmoataz UCBN, GREYC - UMR CNRS 6972, 6.Bvd Marechal Juin, 14050
More informationLevel-set and ALE Based Topology Optimization Using Nonlinear Programming
10 th World Congress on Structural and Multidisciplinary Optimization May 19-24, 2013, Orlando, Florida, USA Level-set and ALE Based Topology Optimization Using Nonlinear Programming Shintaro Yamasaki
More informationSurface Reconstruction. Gianpaolo Palma
Surface Reconstruction Gianpaolo Palma Surface reconstruction Input Point cloud With or without normals Examples: multi-view stereo, union of range scan vertices Range scans Each scan is a triangular mesh
More informationCS205b/CME306. Lecture 9
CS205b/CME306 Lecture 9 1 Convection Supplementary Reading: Osher and Fedkiw, Sections 3.3 and 3.5; Leveque, Sections 6.7, 8.3, 10.2, 10.4. For a reference on Newton polynomial interpolation via divided
More informationIntroduction to Computer Graphics. Modeling (3) April 27, 2017 Kenshi Takayama
Introduction to Computer Graphics Modeling (3) April 27, 2017 Kenshi Takayama Solid modeling 2 Solid models Thin shapes represented by single polygons Unorientable Clear definition of inside & outside
More informationSurface Reconstruction
Eurographics Symposium on Geometry Processing (2006) Surface Reconstruction 2009.12.29 Some methods for surface reconstruction Classification 1. Based on Delaunay triangulation(or Voronoi diagram) Alpha
More informationGeometric Modeling and Processing
Geometric Modeling and Processing Tutorial of 3DIM&PVT 2011 (Hangzhou, China) May 16, 2011 6. Mesh Simplification Problems High resolution meshes becoming increasingly available 3D active scanners Computer
More informationAn explicit feature control approach in structural topology optimization
th World Congress on Structural and Multidisciplinary Optimisation 07 th -2 th, June 205, Sydney Australia An explicit feature control approach in structural topology optimization Weisheng Zhang, Xu Guo
More informationScanning Real World Objects without Worries 3D Reconstruction
Scanning Real World Objects without Worries 3D Reconstruction 1. Overview Feng Li 308262 Kuan Tian 308263 This document is written for the 3D reconstruction part in the course Scanning real world objects
More informationSTATISTICS AND ANALYSIS OF SHAPE
Control and Cybernetics vol. 36 (2007) No. 2 Book review: STATISTICS AND ANALYSIS OF SHAPE by H. Krim, A. Yezzi, Jr., eds. There are numerous definitions of a notion of shape of an object. These definitions
More informationThe p-laplacian on Graphs with Applications in Image Processing and Classification
The p-laplacian on Graphs with Applications in Image Processing and Classification Abderrahim Elmoataz 1,2, Matthieu Toutain 1, Daniel Tenbrinck 3 1 UMR6072 GREYC, Université de Caen Normandie 2 Université
More informationGeometric Modeling in Graphics
Geometric Modeling in Graphics Part 10: Surface reconstruction Martin Samuelčík www.sccg.sk/~samuelcik samuelcik@sccg.sk Curve, surface reconstruction Finding compact connected orientable 2-manifold surface
More informationSegmentation of medical images using three-dimensional active shape models
Segmentation of medical images using three-dimensional active shape models Josephson, Klas; Ericsson, Anders; Karlsson, Johan Published in: Image Analysis (Lecture Notes in Computer Science) DOI: 10.1007/11499145_73
More informationMultiple Motion and Occlusion Segmentation with a Multiphase Level Set Method
Multiple Motion and Occlusion Segmentation with a Multiphase Level Set Method Yonggang Shi, Janusz Konrad, W. Clem Karl Department of Electrical and Computer Engineering Boston University, Boston, MA 02215
More informationShape fitting and non convex data analysis
Shape fitting and non convex data analysis Petra Surynková, Zbyněk Šír Faculty of Mathematics and Physics, Charles University in Prague Sokolovská 83, 186 7 Praha 8, Czech Republic email: petra.surynkova@mff.cuni.cz,
More informationA new Eulerian computational method for the propagation of short acoustic and electromagnetic pulses
A new Eulerian computational method for the propagation of short acoustic and electromagnetic pulses J. Steinhoff, M. Fan & L. Wang. Abstract A new method is described to compute short acoustic or electromagnetic
More informationLevel Sets Methods in Imaging Science
Level Sets Methods in Imaging Science Dr. Corina S. Drapaca csd12@psu.edu Pennsylvania State University University Park, PA 16802, USA Level Sets Methods in Imaging Science p.1/36 Textbooks S.Osher, R.Fedkiw,
More informationVariational Methods II
Mathematical Foundations of Computer Graphics and Vision Variational Methods II Luca Ballan Institute of Visual Computing Last Lecture If we have a topological vector space with an inner product and functionals
More information03 - Reconstruction. Acknowledgements: Olga Sorkine-Hornung. CSCI-GA Geometric Modeling - Spring 17 - Daniele Panozzo
3 - Reconstruction Acknowledgements: Olga Sorkine-Hornung Geometry Acquisition Pipeline Scanning: results in range images Registration: bring all range images to one coordinate system Stitching/ reconstruction:
More informationNumerical Methods for (Time-Dependent) HJ PDEs
Numerical Methods for (Time-Dependent) HJ PDEs Ian Mitchell Department of Computer Science The University of British Columbia research supported by National Science and Engineering Research Council of
More informationcoding of various parts showing different features, the possibility of rotation or of hiding covering parts of the object's surface to gain an insight
Three-Dimensional Object Reconstruction from Layered Spatial Data Michael Dangl and Robert Sablatnig Vienna University of Technology, Institute of Computer Aided Automation, Pattern Recognition and Image
More informationInternational Conference on Materials Engineering and Information Technology Applications (MEITA 2015)
International Conference on Materials Engineering and Information Technology Applications (MEITA 05) An Adaptive Image Segmentation Method Based on the Level Set Zhang Aili,3,a, Li Sijia,b, Liu Tuanning,3,c,
More informationActive Contours Using a Constraint-Based Implicit Representation
To appear in Proceedings Computer Vision and Pattern Recognition, IEEE Computer Society Press, June 2005 Active Contours Using a Constraint-Based Implicit Representation Bryan S. Morse 1, Weiming Liu 1,
More informationFast marching methods
1 Fast marching methods Lecture 3 Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009 Metric discretization 2 Approach I:
More informationImplicit Active Model using Radial Basis Function Interpolated Level Sets
Implicit Active Model using Radial Basis Function Interpolated Level Sets Xianghua Xie and Majid Mirmehdi Department of Computer Science University of Bristol, Bristol BS8 1UB, England. {xie,majid}@cs.bris.ac.uk
More informationConverting Level Set Gradients to Shape Gradients
Converting Level Set Gradients to Shape Gradients Siqi Chen 1, Guillaume Charpiat 2, and Richard J. Radke 1 1 Department of ECSE, Rensselaer Polytechnic Institute, Troy, NY, USA chens@rpi.edu, rjradke@ecse.rpi.edu
More informationLecture 2 Unstructured Mesh Generation
Lecture 2 Unstructured Mesh Generation MIT 16.930 Advanced Topics in Numerical Methods for Partial Differential Equations Per-Olof Persson (persson@mit.edu) February 13, 2006 1 Mesh Generation Given a
More informationIEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 5, MAY
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 5, MAY 2008 645 A Real-Time Algorithm for the Approximation of Level-Set-Based Curve Evolution Yonggang Shi, Member, IEEE, and William Clem Karl, Senior
More informationExtract Object Boundaries in Noisy Images using Level Set. Literature Survey
Extract Object Boundaries in Noisy Images using Level Set by: Quming Zhou Literature Survey Submitted to Professor Brian Evans EE381K Multidimensional Digital Signal Processing March 15, 003 Abstract Finding
More informationEdge Detection and Active Contours
Edge Detection and Active Contours Elsa Angelini, Florence Tupin Department TSI, Telecom ParisTech Name.surname@telecom-paristech.fr 2011 Outline Introduction Edge Detection Active Contours Introduction
More informationLevel lines based disocclusion
Level lines based disocclusion Simon Masnou Jean-Michel Morel CEREMADE CMLA Université Paris-IX Dauphine Ecole Normale Supérieure de Cachan 75775 Paris Cedex 16, France 94235 Cachan Cedex, France Abstract
More informationApproximation of 3D-Parametric Functions by Bicubic B-spline Functions
International Journal of Mathematical Modelling & Computations Vol. 02, No. 03, 2012, 211-220 Approximation of 3D-Parametric Functions by Bicubic B-spline Functions M. Amirfakhrian a, a Department of Mathematics,
More informationThe Immersed Interface Method
The Immersed Interface Method Numerical Solutions of PDEs Involving Interfaces and Irregular Domains Zhiiin Li Kazufumi Ito North Carolina State University Raleigh, North Carolina Society for Industrial
More informationAPPLICATION OF ALGORITHMS FOR AUTOMATIC GENERATION OF HEXAHEDRAL FINITE ELEMENT MESHES
MESTRADO EM ENGENHARIA MECÂNICA November 2014 APPLICATION OF ALGORITHMS FOR AUTOMATIC GENERATION OF HEXAHEDRAL FINITE ELEMENT MESHES Luís Miguel Rodrigues Reis Abstract. The accuracy of a finite element
More informationSpline Curves. Spline Curves. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1
Spline Curves Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1 Problem: In the previous chapter, we have seen that interpolating polynomials, especially those of high degree, tend to produce strong
More informationMeshless Modeling, Animating, and Simulating Point-Based Geometry
Meshless Modeling, Animating, and Simulating Point-Based Geometry Xiaohu Guo SUNY @ Stony Brook Email: xguo@cs.sunysb.edu http://www.cs.sunysb.edu/~xguo Graphics Primitives - Points The emergence of points
More informationActive Geodesics: Region-based Active Contour Segmentation with a Global Edge-based Constraint
Active Geodesics: Region-based Active Contour Segmentation with a Global Edge-based Constraint Vikram Appia Anthony Yezzi Georgia Institute of Technology, Atlanta, GA, USA. Abstract We present an active
More informationimplicit surfaces, approximate implicitization, B-splines, A- patches, surface fitting
24. KONFERENCE O GEOMETRII A POČÍTAČOVÉ GRAFICE ZBYNĚK ŠÍR FITTING OF PIECEWISE POLYNOMIAL IMPLICIT SURFACES Abstrakt In our contribution we discuss the possibility of an efficient fitting of piecewise
More informationGEOMETRICAL CONSTRAINTS IN THE LEVEL SET METHOD FOR SHAPE AND TOPOLOGY OPTIMIZATION
1 GEOMETRICAL CONSTRAINTS IN THE LEVEL SET METHOD FOR SHAPE AND TOPOLOGY OPTIMIZATION Grégoire ALLAIRE CMAP, Ecole Polytechnique Results obtained in collaboration with F. Jouve (LJLL, Paris 7), G. Michailidis
More informationCS 565 Computer Vision. Nazar Khan PUCIT Lectures 15 and 16: Optic Flow
CS 565 Computer Vision Nazar Khan PUCIT Lectures 15 and 16: Optic Flow Introduction Basic Problem given: image sequence f(x, y, z), where (x, y) specifies the location and z denotes time wanted: displacement
More informationSOLVING PARTIAL DIFFERENTIAL EQUATIONS ON POINT CLOUDS
SOLVING PARTIAL DIFFERENTIAL EQUATIONS ON POINT CLOUDS JIAN LIANG AND HONGKAI ZHAO Abstract. In this paper we present a general framework for solving partial differential equations on manifolds represented
More information3D MORPHISM & IMPLICIT SURFACES
3D MORPHISM & IMPLICIT SURFACES ROMAIN BALP AND CHARLEY PAULUS Abstract. The purpose of this paper is to present a framework based on implicit surfaces that allows to visualize dynamic shapes, and see
More informationSegmentation Using Active Contour Model and Level Set Method Applied to Medical Images
Segmentation Using Active Contour Model and Level Set Method Applied to Medical Images Dr. K.Bikshalu R.Srikanth Assistant Professor, Dept. of ECE, KUCE&T, KU, Warangal, Telangana, India kalagaddaashu@gmail.com
More informationSurface Curvature Estimation for Edge Spinning Algorithm *
Surface Curvature Estimation for Edge Spinning Algorithm * Martin Cermak and Vaclav Skala University of West Bohemia in Pilsen Department of Computer Science and Engineering Czech Republic {cermakm skala}@kiv.zcu.cz
More informationIsophote-Based Interpolation
Isophote-Based Interpolation Bryan S. Morse and Duane Schwartzwald Department of Computer Science, Brigham Young University 3361 TMCB, Provo, UT 84602 {morse,duane}@cs.byu.edu Abstract Standard methods
More informationCOUPLING THE LEVEL SET METHOD AND THE TOPOLOGICAL GRADIENT IN STRUCTURAL OPTIMIZATION
COUPLING THE LEVEL SET METHOD AND THE TOPOLOGICAL GRADIENT IN STRUCTURAL OPTIMIZATION Grégoire ALLAIRE Centre de Mathématiques Appliquées (UMR 7641) Ecole Polytechnique, 91128 Palaiseau, France gregoire.allaire@polytechnique.fr
More informationGeometric and Solid Modeling. Problems
Geometric and Solid Modeling Problems Define a Solid Define Representation Schemes Devise Data Structures Construct Solids Page 1 Mathematical Models Points Curves Surfaces Solids A shape is a set of Points
More informationProvably Good Moving Least Squares
Provably Good Moving Least Squares Ravikrishna Kolluri Computer Science Division University of California at Berkeley 1 Problem Definition Given a set of samples on a closed surface build a representation
More informationA Multi-scale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis Functions
Shape Modeling International 2003 Seoul, Korea A Multi-scale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis Functions Yutaa Ohtae Alexander Belyaev Hans-Peter Seidel Objective
More informationA NEW LEVEL SET METHOD FOR MOTION IN NORMAL DIRECTION BASED ON A FORWARD-BACKWARD DIFFUSION FORMULATION
A NEW LEVEL SET METHOD FOR MOTION IN NORMAL DIRECTION BASED ON A FORWARD-BACKWARD DIFFUSION FORMULATION KAROL MIKULA AND MARIO OHLBERGER Abstract. We introduce a new level set method for motion in normal
More informationBackground for Surface Integration
Background for urface Integration 1 urface Integrals We have seen in previous work how to define and compute line integrals in R 2. You should remember the basic surface integrals that we will need to
More informationTopology-Adaptive Modeling of Objects Using Surface Evolutions Based on 3D Mathematical Morphology
Systems and Computers in Japan, Vol. 33, No. 9, 2002 Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J84-D-II, No. 5, May 2001, pp. 778 788 Topology-Adaptive Modeling of Objects Using Surface
More informationFast Surface Reconstruction Using the Level Set Method
Fast Surface Reconstruction Using the Level Set Method Hong-Kai Zhao Stanley Osher y Ronald Fedkiw z Abstract In this paper we describe new formulations and develop fast algorithms for implicit surface
More informationLevel Set Evolution without Reinitilization
Level Set Evolution without Reinitilization Outline Parametric active contour (snake) models. Concepts of Level set method and geometric active contours. A level set formulation without reinitialization.
More informationVariational Level Set Formulation and Filtering Techniques on CT Images
Variational Level Set Formulation and Filtering Techniques on CT Images Shweta Gupta Assistant Professor, Dept. of Electronics and Communication Dronacharya College of Engineering, Khentawas, Farrukhnagar,
More informationChemnitz Scientific Computing Preprints
Roman Unger Obstacle Description with Radial Basis Functions for Contact Problems in Elasticity CSC/09-01 Chemnitz Scientific Computing Preprints Impressum: Chemnitz Scientific Computing Preprints ISSN
More informationDevelopment of a Maxwell Equation Solver for Application to Two Fluid Plasma Models. C. Aberle, A. Hakim, and U. Shumlak
Development of a Maxwell Equation Solver for Application to Two Fluid Plasma Models C. Aberle, A. Hakim, and U. Shumlak Aerospace and Astronautics University of Washington, Seattle American Physical Society
More informationAPPENDIX: DETAILS ABOUT THE DISTANCE TRANSFORM
APPENDIX: DETAILS ABOUT THE DISTANCE TRANSFORM To speed up the closest-point distance computation, 3D Euclidean Distance Transform (DT) can be used in the proposed method. A DT is a uniform discretization
More informationSurface reconstruction based on a dynamical system
EUROGRAPHICS 2002 / G. Drettakis and H.-P. Seidel (Guest Editors) Volume 21 (2002), Number 3 Surface reconstruction based on a dynamical system N.N. Abstract We present an efficient algorithm that computes
More informationFast Marching and Geodesic Methods. Some Applications
Fast Marching and Geodesic Methods. Some Applications Laurent D. COHEN Directeur de Recherche CNRS CEREMADE, UMR CNRS 7534 Université Paris-9 Dauphine Place du Maréchal de Lattre de Tassigny 75016 Paris,
More informationModule 1 Lecture Notes 2. Optimization Problem and Model Formulation
Optimization Methods: Introduction and Basic concepts 1 Module 1 Lecture Notes 2 Optimization Problem and Model Formulation Introduction In the previous lecture we studied the evolution of optimization
More informationAn Efficient Solution to the Eikonal Equation on Parametric Manifolds
An Efficient Solution to the Eikonal Equation on Parametric Manifolds Alon Spira Ron Kimmel Department of Computer Science Technion Israel Institute of Technology Technion City, Haifa 32000, Israel {salon,ron}@cs.technion.ac.il
More informationElastic Bands: Connecting Path Planning and Control
Elastic Bands: Connecting Path Planning and Control Sean Quinlan and Oussama Khatib Robotics Laboratory Computer Science Department Stanford University Abstract Elastic bands are proposed as the basis
More informationIan Mitchell. Department of Computer Science The University of British Columbia
CPSC 542D: Level Set Methods Dynamic Implicit Surfaces and the Hamilton-Jacobi Equation or What Water Simulation, Robot Path Planning and Aircraft Collision Avoidance Have in Common Ian Mitchell Department
More informationA MESH EVOLUTION ALGORITHM BASED ON THE LEVEL SET METHOD FOR GEOMETRY AND TOPOLOGY OPTIMIZATION
A MESH EVOLUTION ALGORITHM BASED ON THE LEVEL SET METHOD FOR GEOMETRY AND TOPOLOGY OPTIMIZATION G. ALLAIRE 1 C. DAPOGNY 2,3, P. FREY 2 1 Centre de Mathématiques Appliquées (UMR 7641), École Polytechnique
More informationThe Pennsylvania State University. The Graduate School. Eberly College of Science A COMPUTATIONAL STUDY OF ROBUSTNESS IN
The Pennsylvania State University The Graduate School Eberly College of Science A COMPUTATIONAL STUDY OF ROBUSTNESS IN LEVEL SET SURFACE RECONSTRUCTION A Thesis in Mathematics by Matthew S. Baran 2012
More informationThree-dimensional segmentation of bones from CT and MRI using fast level sets
Three-dimensional segmentation of bones from CT and MRI using fast level sets Jakub Krátký and Jan Kybic Center for Machine perception, Faculty of Electrical Engineering, Czech Technical University, Prague,
More informationCS354 Computer Graphics Surface Representation IV. Qixing Huang March 7th 2018
CS354 Computer Graphics Surface Representation IV Qixing Huang March 7th 2018 Today s Topic Subdivision surfaces Implicit surface representation Subdivision Surfaces Building complex models We can extend
More informationOutline. Reconstruction of 3D Meshes from Point Clouds. Motivation. Problem Statement. Applications. Challenges
Reconstruction of 3D Meshes from Point Clouds Ming Zhang Patrick Min cs598b, Geometric Modeling for Computer Graphics Feb. 17, 2000 Outline - problem statement - motivation - applications - challenges
More informationSpacetime-coherent Geometry Reconstruction from Multiple Video Streams
Spacetime-coherent Geometry Reconstruction from Multiple Video Streams Marcus Magnor and Bastian Goldlücke MPI Informatik Saarbrücken, Germany magnor@mpi-sb.mpg.de Abstract By reconstructing time-varying
More informationOPTIMAL DAMPING OF A MEMBRANE AND TOPOLOGICAL SHAPE OPTIMIZATION
OPTIMAL DAMPING OF A MEMBRANE AND TOPOLOGICAL SHAPE OPTIMIZATION TONI LASSILA ABSTRACT. We consider a shape optimization problem of finding the optimal damping set of a two-dimensional membrane such that
More informationFairing Scalar Fields by Variational Modeling of Contours
Fairing Scalar Fields by Variational Modeling of Contours Martin Bertram University of Kaiserslautern, Germany Abstract Volume rendering and isosurface extraction from three-dimensional scalar fields are
More informationFoetus Ultrasound Medical Image Segmentation via Variational Level Set Algorithm
2012 Third International Conference on Intelligent Systems Modelling and Simulation Foetus Ultrasound Medical Image Segmentation via Variational Level Set Algorithm M.Y. Choong M.C. Seng S.S. Yang A. Kiring
More informationSurface Tension Approximation in Semi-Lagrangian Level Set Based Fluid Simulations for Computer Graphics
Surface Tension Approximation in Semi-Lagrangian Level Set Based Fluid Simulations for Computer Graphics Israel Pineda and Oubong Gwun Chonbuk National University israel_pineda_arias@yahoo.com, obgwun@jbnu.ac.kr
More informationSegmentation. Namrata Vaswani,
Segmentation Namrata Vaswani, namrata@iastate.edu Read Sections 5.1,5.2,5.3 of [1] Edge detection and filtering : Canny edge detection algorithm to get a contour of the object boundary Hough transform:
More informationParticle based T-Spline Level Set Evolution for 3D object reconstruction with Range and Volume Constraints
Particle based T-Spline Level Set Evolution for 3D object reconstruction with Range and Volume Constraints Robert Feichtinger Johannes Kepler University Linz, Austria robert.feichtinger@jku.at Huaiping
More informationAn introduction to mesh generation Part IV : elliptic meshing
Elliptic An introduction to mesh generation Part IV : elliptic meshing Department of Civil Engineering, Université catholique de Louvain, Belgium Elliptic Curvilinear Meshes Basic concept A curvilinear
More informationChapter 15 Vector Calculus
Chapter 15 Vector Calculus 151 Vector Fields 152 Line Integrals 153 Fundamental Theorem and Independence of Path 153 Conservative Fields and Potential Functions 154 Green s Theorem 155 urface Integrals
More informationSIZE PRESERVING MESH GENERATION IN ADAPTIVITY PROCESSES
Congreso de Métodos Numéricos en Ingeniería 25-28 junio 2013, Bilbao, España c SEMNI, 2013 SIZE PRESERVING MESH GENERATION IN ADAPTIVITY PROCESSES Eloi Ruiz-Gironés 1, Xevi Roca 2 and Josep Sarrate 1 1:
More informationDroplet collisions using a Level Set method: comparisons between simulation and experiments
Computational Methods in Multiphase Flow III 63 Droplet collisions using a Level Set method: comparisons between simulation and experiments S. Tanguy, T. Ménard & A. Berlemont CNRS-UMR6614-CORIA, Rouen
More informationMATH 234. Excercises on Integration in Several Variables. I. Double Integrals
MATH 234 Excercises on Integration in everal Variables I. Double Integrals Problem 1. D = {(x, y) : y x 1, 0 y 1}. Compute D ex3 da. Problem 2. Find the volume of the solid bounded above by the plane 3x
More informationProcessing 3D Surface Data
Processing 3D Surface Data Computer Animation and Visualisation Lecture 15 Institute for Perception, Action & Behaviour School of Informatics 3D Surfaces 1 3D surface data... where from? Iso-surfacing
More information1.2 Numerical Solutions of Flow Problems
1.2 Numerical Solutions of Flow Problems DIFFERENTIAL EQUATIONS OF MOTION FOR A SIMPLIFIED FLOW PROBLEM Continuity equation for incompressible flow: 0 Momentum (Navier-Stokes) equations for a Newtonian
More information