Subdivision: From Stationary to Non-stationary scheme.

Slides:



Advertisements
Similar presentations
Bicubic G1 interpolation of arbitrary quad meshes using a 4-split
Advertisements

Lecture Notes #11 Curves and Surfaces II
Flexible smoothing with B-splines and Penalties or P-splines P-splines = B-splines + Penalization Applications : Generalized Linear and non linear Modelling.
Interpolation By Radial Basis Function ( RBF ) By: Reihane Khajepiri, Narges Gorji Supervisor: Dr.Rabiei 1.
MATH 685/ CSI 700/ OR 682 Lecture Notes
Selected from presentations by Jim Ramsay, McGill University, Hongliang Fei, and Brian Quanz Basis Basics.
Basis Expansion and Regularization Presenter: Hongliang Fei Brian Quanz Brian Quanz Date: July 03, 2008.
Classic Subdivision Schemes. Schemes Catmull-Clark (1978) Doo-Sabin (1978) Loop (1987) Butterfly (1990) Kobbelt (1996) Mid-edge (1996 / 1997)
CS CS 175 – Week 9 B-Splines Blossoming, Bézier Splines.
CS Peter Schröder Subdivision I: The Basic Ideas.
Notes Assignment questions… cs533d-winter-2005.
Surfaces Chiew-Lan Tai. Surfaces 2 Reading Required Hills Section Hearn & Baker, sections 8.11, 8.13 Recommended Sections 2.1.4, , 3D Computer.
Curve-Fitting Regression
Paul Heckbert Computer Science Department Carnegie Mellon University
Normal based subdivision scheme for curve and surface design 杨勋年
Basic Concepts and Definitions Vector and Function Space. A finite or an infinite dimensional linear vector/function space described with set of non-unique.
11/08/00 Dinesh Manocha, COMP258 Subdivision Curves & Surfaces Work of G. de Rham on Corner Cutting in 40’s and 50’s Work of Catmull/Clark and Doo/Sabin.
Fat Curves and Representation of Planar Figures L.M. Mestetskii Department of Information Technologies, Tver’ State University, Tver, Russia Computers.
Scott Schaefer Joe Warren A Factored, Interpolatory Subdivision for Surfaces of Revolution Rice University.
ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 24 Regression Analysis-Chapter 17.
1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.
CS Subdivision I: The Univariate Setting Peter Schröder.
Subdivision Analysis via JSR We already know the z-transform formulation of schemes: To check if the scheme generates a continuous limit curve ( the scheme.
Introduction to Subdivision Surfaces. Subdivision Curves and Surfaces 4 Subdivision curves –The basic concepts of subdivision. 4 Subdivision surfaces.
Mathematics and Computation in Imaging Science and Information Processing July-December, 2003 Organized by Institute of Mathematical Sciences and Center.
ENG4BF3 Medical Image Processing
Unitary Extension Principle: Ten Years After Zuowei Shen Department of Mathematics National University of Singapore.
Curve Modeling Bézier Curves
11/19/02 (c) 2002, University of Wisconsin, CS 559 Last Time Many, many modeling techniques –Polygon meshes –Parametric instancing –Hierarchical modeling.
Extrapolation Models for Convergence Acceleration and Function ’ s Extension David Levin Tel-Aviv University MAIA Erice 2013.
Review of Interpolation. A method of constructing a function that crosses through a discrete set of known data points.
Introduction to Subdivision surfaces Martin Reimers CMA, University of Oslo.
1 Dr. Scott Schaefer Coons Patches and Gregory Patches.
Exponential Quasi-interpolatory Subdivision Scheme Yeon Ju Lee and Jungho Yoon Department of Mathematics, Ewha W. University Seoul, Korea.
Wavelets and their applications in CG&CAGD Speaker: Qianqian Hu Date: Mar. 28, 2007.
2011 COURSE IN NEUROINFORMATICS MARINE BIOLOGICAL LABORATORY WOODS HOLE, MA Introduction to Spline Models or Advanced Connect-the-Dots Uri Eden BU Department.
The Lifting Scheme: a custom-design construction of biorthogonal wavelets Sweldens95, Sweldens 98 (appeared in SIAM Journal on Mathematical Analysis)
Computer Programming (TKK-2144) 13/14 Semester 1 Instructor: Rama Oktavian Office Hr.: M.13-15, W Th , F
Ship Computer Aided Design MR 422. Geometry of Curves 1.Introduction 2.Mathematical Curve Definitions 3.Analytic Properties of Curves 4.Fairness of Curves.
Lee Byung-Gook Dongseo Univ.
using Radial Basis Function Interpolation
A Note on Subdivision Kwan Pyo Ko Dongseo University
Application: Multiresolution Curves Jyun-Ming Chen Spring 2001.
1. Systems of Linear Equations and Matrices (8 Lectures) 1.1 Introduction to Systems of Linear Equations 1.2 Gaussian Elimination 1.3 Matrices and Matrix.
Splines Sang Il Park Sejong University. Particle Motion A curve in 3-dimensional space World coordinates.
Data Visualization Fall The Data as a Quantity Quantities can be classified in two categories: Intrinsically continuous (scientific visualization,
Computer vision. Applications and Algorithms in CV Tutorial 3: Multi scale signal representation Pyramids DFT - Discrete Fourier transform.
Interpolation Local Interpolation Methods –IDW – Inverse Distance Weighting –Natural Neighbor –Spline – Radial Basis Functions –Kriging – Geostatistical.
Subdivision Schemes. Center for Graphics and Geometric Computing, Technion What is Subdivision?  Subdivision is a process in which a poly-line/mesh is.
Wavelets Chapter 7 Serkan ERGUN. 1.Introduction Wavelets are mathematical tools for hierarchically decomposing functions. Regardless of whether the function.
Lecture 24: Surface Representation
Evgeny Lipovetsky School of Computer Sciences, Tel-Aviv University
Multiresolution Analysis (Chapter 7)
Lecture 22: B Spline Curve Properties
Computer Graphics Lecture 38
Wavelets : Introduction and Examples
Introduction to Graphics Modeling
Constructing Objects in Computer Graphics By Andries van Dam©
© University of Wisconsin, CS559 Spring 2004
The Variety of Subdivision Schemes
Chapter XVII Parametric Curves and Surfaces
ECIV 720 A Advanced Structural Mechanics and Analysis
Generalization of (2n+4)-point approximating subdivision scheme
Today’s class Multiple Variable Linear Regression
Mesh Parameterization: Theory and Practice
A Family of Subdivision Scheme with Cubic Precision
Mask of interpolatory symmetric subdivision schemes
Coons Patches and Gregory Patches
ANSWER THE FOLLOWING BRIEFLY BUT COMPREHENSIVELY.
Spline representation. ❖ A spline is a flexible strip used to produce a smooth curve through a designated set of points. ❖ Mathematically describe such.
Presentation transcript:

Subdivision: From Stationary to Non-stationary scheme. Jungho Yoon Department of Mathematics Ewha W. University

Data Type 2006.01.09 KMMCS 동서대학교

Sampling/Reconstruction How to Sample/Re-sample ? - From Continuous object to a finite point set How to handle the sampled data - From a finite sampled data to a continuous representation Error between the reconstructed shape and the original shape 2006.01.09 KMMCS 동서대학교

Subdivision Scheme A simple local averaging rule to build curves and surfaces in computer graphics A progress scheme with naturally built-in Multiresolution Structure One of the most im portant tool in Wavelet Theory 2006.01.09 KMMCS 동서대학교

Approximation Methods Polynomial Interpolation Fourier Series Spline Radial Basis Function (Moving) Least Square Subdivision Wavelets 2006.01.09 KMMCS 동서대학교

Example Consider the function with the data on 2006.01.09 KMMCS 동서대학교

Polynomial Interpolation 2006.01.09 KMMCS 동서대학교

Shifts of One Basis Function Approximation by shifts of one basis function : How to choose ? 2006.01.09 KMMCS 동서대학교

Gaussian Interpolation 2006.01.09 KMMCS 동서대학교

Stationary and Non-stationary Subdivision Scheme Stationary and Non-stationary

Chainkin’s Algorithm : corner cutting 2006.01.09 KMMCS 동서대학교

Deslauriers-Dubuc Algorithm 2006.01.09 KMMCS 동서대학교

Subdivision Non-stationary Butterfly Scheme 2006.01.09 KMMCS 동서대학교

Subdivision Scheme Types ► Stationary or Nonstationary ► Interpolating or Approximating ► Curve or Surface ► Triangular or Quadrilateral 2006.01.09 KMMCS 동서대학교

Subdivision Scheme Formulation 2006.01.09 KMMCS 동서대학교

Subdivision Scheme Stationary Scheme, i.e., Curve scheme (which consists of two rules) 2006.01.09 KMMCS 동서대학교

Subdivision : The Limit Function : the limit function of the subdivision Let Then is called the basic limit funtion. In particular, satisfies the two scale relation 2006.01.09 KMMCS 동서대학교

Basic Limit Function : B-splines B_1 spline Cubic spline 2006.01.09 KMMCS 동서대학교

Basic Limit Function : DD-scheme 2006.01.09 KMMCS 동서대학교

Basic Issues Convergence Smoothness Accuracy (Approximation Order) 2006.01.09 KMMCS 동서대학교

Bm-spline subdivision scheme Laurent polynomial : Smoothness Cm-1 with minimal support. Approximation order is two for all m. 2006.01.09 KMMCS 동서대학교

Interpolatory Subdivision The general form 4-point interpolatory scheme : The Smoothness is C1 in some range of w. The Approximation order is 4 with w=1/16. 2006.01.09 KMMCS 동서대학교

Interpolatory Scheme 2006.01.09 KMMCS 동서대학교

Goal Construct a new scheme which combines the advantages of the aforementioned schemes, while overcoming their drawbacks. Construct Biorthogonal Wavelets This large family of Subdivision Schemes includes the DD interpolatory scheme and B-splines up to degree 4. 2006.01.09 KMMCS 동서대학교

Reprod. Polynomials < L Case 1 : L is Even, i.e., L=2N 2006.01.09 KMMCS 동서대학교

Reprod. Polynomials < L Case 2 : L is Odd, i.e., L=2N+1 2006.01.09 KMMCS 동서대학교

Stencils of Masks 2006.01.09 KMMCS 동서대학교

Quasi-interpolatory subdivision General case L Mask set Sm. Range of tension 1 O=[v, 1-v] (* If v=1/4, quad spline) E= [1-v, v] C1 1/4 2 O=[v, 1-2v, v] (* If v= 1/8, cubic spline) E= [1/2, 1/2] C2 1/8 3 O=[-1/16,9/16,9/16,-1/16] E= [-v, 4v,1-6v,4v,-v] 0.0208<v<0.0404 4 O=[-v,–77/2048+5v,385/512-10v, 385/1024+10v,-55/512-5v,35/2048+v] E(i)=O(7-i) for i=1:6 C3 -0.0106<v<-0.0012 5 O=[3,–25,150,150,–25,3]/256] E=[-v,6v,–15v,1+20v,-15v,6v,-v] -0.0084<v<-0.0046 2006.01.09 KMMCS 동서대학교

Quasi-interpolatory subdivision Comparison Cubic B-spline 4-pts interpolatory scheme SL Where L=4 (4-5)-scheme Support of limit ftn [-2, 2] [-3, 3] [-4, 4] Maximal Smoothness C2 C1 C3 Approximation Order 2 4 2006.01.09 KMMCS 동서대학교

Quasi-interpolatory subdivision Basic limit functions for the case L=4 2006.01.09 KMMCS 동서대학교

Example 2006.01.09 KMMCS 동서대학교

Example 2006.01.09 KMMCS 동서대학교

Laurent Polynomial 2006.01.09 KMMCS 동서대학교

Smoothness 2006.01.09 KMMCS 동서대학교

Smoothness : Comparison 2006.01.09 KMMCS 동서대학교

Biorthogonal Wavelets Let and be dual each other if The corresponding wavelet functions are constructed by 2006.01.09 KMMCS 동서대학교

Symmetric Biorthogonal Wavelets 2006.01.09 KMMCS 동서대학교

Symmetric Biorthogonal Wavelets 2006.01.09 KMMCS 동서대학교

Nonstationary Subdivision Varying masks depending on the levels, i.e., 2006.01.09 KMMCS 동서대학교

Advantages Design Flexibility Higher Accuracy than the Scheme based on Polynomial 2006.01.09 KMMCS 동서대학교

Nonstationary Subdivision Smoothness Accuracy Scheme (Quasi-Interpolatory) Non-Stationary Wavelets Schemes for Surface 2006.01.09 KMMCS 동서대학교

Current Project Construct a new compactly supported biorthogonal wavelet systems based on Exponential B-splines Application to Signal process and Medical Imaging (MRI or CT data) Wavelets on special points such GCL points for Numerical PDE 2006.01.09 KMMCS 동서대학교

Thank You ! and Have a Good Tme in Busan! 2006.01.09 KMMCS 동서대학교

Hope to see you in 2006.01.09 KMMCS 동서대학교