Daubechies Wavelets and Interpolating Scaling Functions and Application on PDEs
|
|
- Egbert Richard
- 5 years ago
- Views:
Transcription
1 Daubechies Wavelets and Interpolating Scaling Functions and Application on PDEs R. Schneider F. Krüger TUB - Technical University of Berlin November 22, 2007 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 1 / 1
2 Outline Outline R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 2 / 1
3 Daubechies Wavelets R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 3 / 1
4 Daubechies Wavelets Daubechies Wavelets Conditions that fulfill the Daubechies wavelets: 1. They perform an orthonormal basis. 2. They have p vanishing moments for a given p N. 3. They have the minimal support of all functions that fulfill the first two conditions. 4. They are rather smooth. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 4 / 1
5 Fourier Space Daubechies Wavelets The Fourier transform of a function f is defined by ˆf(ω) = f(x)e iωx dx R Given a scaling function ϕ and its corresponding wavelet ψ with its refinement equations ϕ = k ψ = k h k ϕ 1 k g k ϕ 1 k. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 5 / 1
6 Daubechies Wavelets In Fourier space it gets to 2 ˆϕ(ω) = 2 h(ω 2 ) ˆϕ(ω 2 ) (1) 2 ˆψ(ω) = 2 g(ω 2 ) ˆϕ(ω 2 ) with h(ω) = k Z h k e ikω g(ω) = k Z g k e ikω. By succesive application of (??) we get h(2 k ω) ˆϕ(ω) = ˆϕ(0) 2 if ˆϕ is continuos at 0. k=1 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 6 / 1
7 Daubechies Wavelets Computation of the Daubechies Wavelets There is no analytical formula for the Daubechies scaling functions and wavelets. But you can compute exactly the filters h and g. They are tabulated in the internet, for example at wikipedia. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 7 / 1
8 Daubechies Wavelets Computation of the Daubechies Wavelets With the filters you can use the formulae in the Fourier space. ˆϕ(ω) = k=1 h(2 k ω) 2 ˆψ(ω) = 1 2 g( ω 2 ) ˆϕ(ω 2 ) with h(ω) = h k e ikω k Z g(ω) = g k e ikω. k Z R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 8 / 1
9 ϕ Daubechies Wavelets 1.4 p = p = p = 4 p = Figure: Daubechies scaling functions ϕ with p = 2, 3, 4 and 5. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 9 / 1
10 ψ Daubechies Wavelets 2 p = 2 2 p = p = 4 p = Figure: Daubechies wavelets ψ with p = 2, 3, 4 and 5. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs10 / 1
11 Vanishing Moments Daubechies Wavelets A very important property of the Daubechies wavelets are the vanishing moments x k ψ(x)dx = 0, k = 0,..., p 1. R That also means that every polynomial of degree at most p 1 is in V 0 in a compact subset K of R. For example is f = 1 K V 0 K. The series ( n k= n ϕ k(x)) n N is converging pointwise to the unity. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs11 / 1
12 Partition of Unity Daubechies Wavelets Partition of Unity In the picture is shown the function 10 f = k=0 with ϕ the Daubechies scaling function of order p = 2. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs12 / 1 ϕ k
13 Interpolating Scaling Functions R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs13 / 1
14 Interpolating Scaling Functions An interpolating scaling function φ fulfills φ(k) = δ k, k Z. Since the Daubechies scaling functions ϕ k are orthonormal we get an interpolating scaling function φ (Interpolet) by folding the Daubechies function with itself. φ(t) = ϕ(u)ϕ(u t)du = ϕ ϕ(t), where R ϕ(t) =ϕ( t). R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs14 / 1
15 Interpolating Scaling Functions The interpolation property can be easily seen: φ(k) = ϕ(u)ϕ(u k)du = ϕ, ϕ k = δ k. R For the filter h I corresponding to φ we get h I k = (h h) k. To hold the interpolating property for j Z we omit the normalisation factor: φ j k (x) = φ(2j x k). R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs15 / 1
16 Interpolating Scaling Functions p = 2 p = p = p = Figure: Interpolating scaling function φ for p = 2, 3, 4 and 5. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs16 / 1
17 Interpolation Projection Interpolating Scaling Functions With Interpolets we have a very simple projection P J onto V J : P J f = k Z f(x k )φ J k, x k = 2 J k This projection is called interpolation projection or collocation projection. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs17 / 1
18 Computing Integrals R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs18 / 1
19 Moments Computing Integrals We want to compute the moments M k = R x k ϕ(x)dx = 2p 1 0 x k ϕ(x)dx. By using the refinement equation we get M k = 2 h l x k ϕ(2x l)dx l R 2 = 2 k+1 h l (x + l) k ϕ(x)dx 2 = 2 k+1 l l R k h l j=0 ( ) k l j M k j. j R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs19 / 1
20 Computing Integrals Using the fact that M 0 = 1 we get the recursion formula M k = 2p 1 1 2(2 k 1) l=0 k h l j=1 ( ) k l j M k j for k > 0. j With the M k we can easily compute x k, ϕ j l = x k ϕ j l (x)dx R =2 jk j/2 (x + l) k ϕ(x)dx R =2 jk j/2 k m=0 ( ) k l m M k m. m R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs20 / 1
21 Computing Integrals Therefore for any polynomial π the integral π, ϕ j l = π(x)ϕ j l (x)dx is exactly computable. For computing R f, ϕ j l where you know the function values of some nodes of f, you can do a polynomial interpolation with a polynomial π f and integrate π f (x)ϕ j l (x)dx f(x)ϕ j l (x)dx. R R R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs21 / 1
22 Computing Dϕ j, Dϕ j k Computing Integrals ϕ satisfies a refinement equation, too: Dϕ = k h k Dϕ 1 k. We can use this to compute a k = Dϕ, Dϕ k. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs22 / 1
23 Computing Integrals a k = l =4 l =4 l h l h m Dϕ 1 l, Dϕ1 2k+m m h l h m 2k Dϕ, Dϕ m l m (h l h l+m 2k )a l m If you set H k,l = 4h l m h l+m 2k you have the eigen equation Ha = a i.e. you have to compute an eigenvector for the eigenvalue 1. It can be shown that this eigenvector is unique up to normalisation. To find the correct normalisation of the a ks we use the equation k 2 a k = 2, which holds for p 3. k R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs23 / 1
24 Computing Integrals For a j k = Dϕj, Dϕ j k you have a j k = 22j Dϕ, Dϕ k. Remark: It holds a j k = φj k (0). R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs24 / 1
25 Computing ϕ, φ k Computing Integrals With the same trick as before you can compute b k = ϕ, φ k = h l h I m ϕ 1 l, φ1 2k+m l m =2 1/2 h l h I m 2k ϕ, φ m l l m =2 1/2 (h l h I l+m 2k )b l. l m R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs25 / 1
26 Computing Integrals With H k,l = 2 1/2 m hi l+m 2k we have the equation Hb = b, i.e. we search an eigenvector of H for the eigenvalue 1. Here you also have to find a further information to achieve a unique solution. For b j k = ϕj, φ j k we get b j k = 2 j/2 b k. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs26 / 1
27 Multivariate MRAs R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs27 / 1
28 Multivariate MRAs Until now we only considered one dimensional functions. If ϕ is a scaling function we define a d-dimensional scaling function Φ as follows. Φ(x 1,..., x d ) = d ϕ(x i ) i=1 For k = (k 1,..., k d ) Z d and j Z we define Then with Φ j k (x 1,..., x d ) = d ϕ j k i (x i ). i=1 V j = span{φ j k k Zd } = d i=1 V j we get a MRA for L 2 (R d ). R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs28 / 1
29 Wavelet Spaces Multivariate MRAs The wavelet spaces are a little bit more complicated. With W j 0 = V j and W j 1 = W j we have V J = = d i=1 V J d (V J 1 W J 1 ) i=1 = d e {0,1} d i=1 W J 1 e i. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs29 / 1
30 Multivariate MRAs That means that in d dimensions we have 2 d 1 different wavelets. For example, for d = 3, we have ψ 1 =ϕ ϕ ψ, ψ 2 =ϕ ψ ϕ, ψ 3 =ϕ ψ ψ, ψ 4 =ψ ϕ ϕ, ψ 5 =ψ ϕ ψ, ψ 6 =ψ ψ ϕ, ψ 7 =ψ ψ ψ. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs30 / 1
31 Multivariate MRAs Isotropic and Anisotropic Structures If we pass to more levels, we can do it in two ways. 1. isotrop: V J = d J 1 V 0 i=1 j=0 e {0,1} d e 0 d i=1 W j e i 2. anisotrop: With the notation W 1 = V 0 V J = J 1 J 1 j 1 = 1 j 2 = 1 J 1 d j d = 1 i=1 W j i R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs31 / 1
32 Isotropic Decomposition Multivariate MRAs V 3 V 3 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs32 / 1
33 Isotropic Decomposition Multivariate MRAs V 2 W 2 W 2 W 2 V 2 V 2 W 2 V 2 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs33 / 1
34 Isotropic Decomposition Multivariate MRAs V 2 W 2 W 2 W 2 V 1 W 1 W 1 W 1 W 2 V 2 V 1 V 1 W 1 V 1 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs34 / 1
35 Isotropic Decomposition Multivariate MRAs V 2 W 2 W 2 W 2 V 1 W 1 W 1 W 1 V 0 W 0 W 0 W 0 V 0 V 0 W 0 V 0 W 1 V 1 W 2 V 2 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs35 / 1
36 Anisotropic Decomposition Multivariate MRAs V 3 V 3 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs36 / 1
37 Anisotropic Decomposition Multivariate MRAs V 2 W 2 W 2 W 2 V 2 V 2 W 2 V 2 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs37 / 1
38 Anisotropic Decomposition Multivariate MRAs V 1 W 2 W 1 W 2 W 2 W 2 V 1 W 1 W 1 W 1 W 2 W 1 V 1 V 1 W 1 V 1 W 2 V 1 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs38 / 1
39 Anisotropic Decomposition Multivariate MRAs V 0 W 2 W 0 W 2 W 1 W 2 W 2 W 2 V 0 W 1 W 0 W 1 W 1 W 1 W 2 W 1 V 0 W 0 W 0 W 0 V 0 V 0 W 0 V 0 W 1 W 0 W 1 V 0 W 2 W 0 W 2 V 0 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs39 / 1
40 Galerkin and Collocation Schemes R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs40 / 1
41 Galerkin Scheme Galerkin and Collocation Schemes Given an equation Au f = 0 (2) for a given function f and operator A. We are looking for the solution u in a Hilbert space H. (??) is equivalent to Au f, v = 0 for all v H. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs41 / 1
42 Galerkin Scheme Galerkin and Collocation Schemes We want to approximate the solution u in a N-dimensional subspace of H and denote the approximated solution by u N. Let {v 1,..., v N } be an orthonormal basis for V N. We are getting the equations Au N f, v i = 0, i = {1,..., n}. We make the ansatz u N = N k=1 λ kv k. We define the matrix A and the vector f by A k,l = Av k, v l, f k = f, v k. Then we have to solve the linear system Aλ = f. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs42 / 1
43 Collocation Scheme Galerkin and Collocation Schemes We make the ansatz u N = k u(x k )v k. Here the v k dont have to be orthonormal. Here are the Interpolets φ j k senseful. This leads to the equations By defining A and f by we get the linear system (Au N f)(x k ) = 0, k = 1,..., N. A k,l = (Av l )(x k ), f k = f(x k ) Au = f, with u k u(k). R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs43 / 1
44 References References S. Mallat, A Wavelet Tour of Signal Processing, 2nd. ed., Academic Press, 1999 A. Cohen, Numerical Analysis of Wavelet Methods, Elsevier Science B.V., 2003 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs44 / 1
Erasing Haar Coefficients
Recap Haar simple and fast wavelet transform Limitations not smooth enough: blocky How to improve? classical approach: basis functions Lifting: transforms 1 Erasing Haar Coefficients 2 Page 1 Classical
More informationHighly Symmetric Bi-frames for Triangle Surface Multiresolution Processing
Highly Symmetric Bi-frames for Triangle Surface Multiresolution Processing Qingtang Jiang and Dale K. Pounds Abstract In this paper we investigate the construction of dyadic affine (wavelet) bi-frames
More informationEdge Detections Using Box Spline Tight Frames
Edge Detections Using Box Spline Tight Frames Ming-Jun Lai 1) Abstract. We present a piece of numerical evidence that box spline tight frames are very useful for edge detection of images. Comparsion with
More informationRegularity Analysis of Non Uniform Data
Regularity Analysis of Non Uniform Data Christine Potier and Christine Vercken Abstract. A particular class of wavelet, derivatives of B-splines, leads to fast and ecient algorithms for contours detection
More information3D B Spline Interval Wavelet Moments for 3D Objects
Journal of Information & Computational Science 10:5 (2013) 1377 1389 March 20, 2013 Available at http://www.joics.com 3D B Spline Interval Wavelet Moments for 3D Objects Li Cui a,, Ying Li b a School of
More informationInverse and Implicit functions
CHAPTER 3 Inverse and Implicit functions. Inverse Functions and Coordinate Changes Let U R d be a domain. Theorem. (Inverse function theorem). If ϕ : U R d is differentiable at a and Dϕ a is invertible,
More informationMatrix-valued 4-point Spline and 3-point Non-spline Interpolatory Curve Subdivision Schemes
Matrix-valued 4-point Spline and -point Non-spline Interpolatory Curve Subdivision Schemes Charles K. Chui, Qingtang Jiang Department of Mathematics and Computer Science University of Missouri St. Louis
More informationSecond-G eneration Wavelet Collocation Method for the Solution of Partial Differential Equations
Journal of Computational Physics 165, 660 693 (2000) doi:10.1006/jcph.2000.6638, available online at http://www.idealibrary.com on Second-G eneration Wavelet Collocation Method for the Solution of Partial
More informationA dimension adaptive sparse grid combination technique for machine learning
A dimension adaptive sparse grid combination technique for machine learning Jochen Garcke Abstract We introduce a dimension adaptive sparse grid combination technique for the machine learning problems
More informationMultiscale 3-D texture segmentation: Isotropic Representations
Multiscale 3-D texture segmentation: Isotropic Representations Saurabh Jain Department of Mathematics University of Houston March 28th 2008 2008 Spring Southeastern Meeting Special Session on Wavelets,
More informationFilters and Fourier Transforms
Filters and Fourier Transforms NOTE: Before reading these notes, see 15-462 Basic Raster notes: http://www.cs.cmu.edu/afs/cs/academic/class/15462/web/notes/notes.html OUTLINE: The Cost of Filtering Fourier
More informationInterpolation and Splines
Interpolation and Splines Anna Gryboś October 23, 27 1 Problem setting Many of physical phenomenona are described by the functions that we don t know exactly. Often we can calculate or measure the values
More informationSparse Grids. One-Dimensional Multilevel Basis
T. Gerstner, M. Griebel. Sparse Grids. In Encyclopedia of Quantitative Finance, R. Cont (ed.), Wiley, 2008. Sparse Grids The sparse grid method is a general numerical discretization technique for multivariate
More informationGalerkin Projections Between Finite Element Spaces
Galerkin Projections Between Finite Element Spaces Ross A. Thompson Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements
More informationChapter 5: Basis Expansion and Regularization
Chapter 5: Basis Expansion and Regularization DD3364 April 1, 2012 Introduction Main idea Moving beyond linearity Augment the vector of inputs X with additional variables. These are transformations of
More informationThe Pre-Image Problem in Kernel Methods
The Pre-Image Problem in Kernel Methods James Kwok Ivor Tsang Department of Computer Science Hong Kong University of Science and Technology Hong Kong The Pre-Image Problem in Kernel Methods ICML-2003 1
More informationHarmonic Spline Series Representation of Scaling Functions
Harmonic Spline Series Representation of Scaling Functions Thierry Blu and Michael Unser Biomedical Imaging Group, STI/BIO-E, BM 4.34 Swiss Federal Institute of Technology, Lausanne CH-5 Lausanne-EPFL,
More informationDivide and Conquer Kernel Ridge Regression
Divide and Conquer Kernel Ridge Regression Yuchen Zhang John Duchi Martin Wainwright University of California, Berkeley COLT 2013 Yuchen Zhang (UC Berkeley) Divide and Conquer KRR COLT 2013 1 / 15 Problem
More informationContents. Implementing the QR factorization The algebraic eigenvalue problem. Applied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
More informationAn Investigation into Iterative Methods for Solving Elliptic PDE s Andrew M Brown Computer Science/Maths Session (2000/2001)
An Investigation into Iterative Methods for Solving Elliptic PDE s Andrew M Brown Computer Science/Maths Session (000/001) Summary The objectives of this project were as follows: 1) Investigate iterative
More informationBernstein-Bezier Splines on the Unit Sphere. Victoria Baramidze. Department of Mathematics. Western Illinois University
Bernstein-Bezier Splines on the Unit Sphere Victoria Baramidze Department of Mathematics Western Illinois University ABSTRACT I will introduce scattered data fitting problems on the sphere and discuss
More informationINTRODUCTION TO FINITE ELEMENT METHODS
INTRODUCTION TO FINITE ELEMENT METHODS LONG CHEN Finite element methods are based on the variational formulation of partial differential equations which only need to compute the gradient of a function.
More informationMatrix-Nullspace Wavelet Construction. Subendran Sivalingam
Matrix-Nullspace Wavelet Construction Subendran Sivalingam September 30, 1994 Abstract Wavelets have been a very popular topic of conversation in many scientific and engineering gatherings these days.
More informationECG782: Multidimensional Digital Signal Processing
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu ECG782: Multidimensional Digital Signal Processing Spring 2014 TTh 14:30-15:45 CBC C313 Lecture 06 Image Structures 13/02/06 http://www.ee.unlv.edu/~b1morris/ecg782/
More informationAdaptive Filters Algorithms (Part 2)
Adaptive Filters Algorithms (Part 2) Gerhard Schmidt Christian-Albrechts-Universität zu Kiel Faculty of Engineering Electrical Engineering and Information Technology Digital Signal Processing and System
More informationA ray-based IPDG method for high-frequency time-domain acoustic wave propagation in inhomogeneous media
A ray-based IPDG method for high-frequency time-domain acoustic wave propagation in inhomogeneous media Eric T. Chung Chi Yeung Lam Jianliang Qian arxiv:1704.06916v1 [math.na] 23 Apr 2017 October 15, 2018
More informationFinite Element Methods
Chapter 5 Finite Element Methods 5.1 Finite Element Spaces Remark 5.1 Mesh cells, faces, edges, vertices. A mesh cell is a compact polyhedron in R d, d {2,3}, whose interior is not empty. The boundary
More informationFrom Fourier Transform to Wavelets
From Fourier Transform to Wavelets Otto Seppälä April . TRANSFORMS.. BASIS FUNCTIONS... SOME POSSIBLE BASIS FUNCTION CONDITIONS... Orthogonality... Redundancy...3. Compact support.. FOURIER TRANSFORMS
More informationNew Basis Functions and Their Applications to PDEs
Copyright c 2007 ICCES ICCES, vol.3, no.4, pp.169-175, 2007 New Basis Functions and Their Applications to PDEs Haiyan Tian 1, Sergiy Reustkiy 2 and C.S. Chen 1 Summary We introduce a new type of basis
More informationPoint Lattices in Computer Graphics and Visualization how signal processing may help computer graphics
Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Dimitri Van De Ville Ecole Polytechnique Fédérale de Lausanne Biomedical Imaging Group dimitri.vandeville@epfl.ch
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 informationAdaptive boundary element methods in industrial applications
Adaptive boundary element methods in industrial applications Günther Of, Olaf Steinbach, Wolfgang Wendland Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th,
More informationCommutative filters for LES on unstructured meshes
Center for Turbulence Research Annual Research Briefs 1999 389 Commutative filters for LES on unstructured meshes By Alison L. Marsden AND Oleg V. Vasilyev 1 Motivation and objectives Application of large
More informationSubdivision Curves and Surfaces
Subdivision Surfaces or How to Generate a Smooth Mesh?? Subdivision Curves and Surfaces Subdivision given polyline(2d)/mesh(3d) recursively modify & add vertices to achieve smooth curve/surface Each iteration
More informationInterpolation by Spline Functions
Interpolation by Spline Functions Com S 477/577 Sep 0 007 High-degree polynomials tend to have large oscillations which are not the characteristics of the original data. To yield smooth interpolating curves
More informationBasic Finite Element Method
Basic Finite Element Method The purpose of this lecture is to show a systematic method of developing a computer program to find the numerical solution for a partial differential equation of fourth order
More informationSpace Filling Curves and Hierarchical Basis. Klaus Speer
Space Filling Curves and Hierarchical Basis Klaus Speer Abstract Real world phenomena can be best described using differential equations. After linearisation we have to deal with huge linear systems of
More informationQuadratic and cubic b-splines by generalizing higher-order voronoi diagrams
Quadratic and cubic b-splines by generalizing higher-order voronoi diagrams Yuanxin Liu and Jack Snoeyink Joshua Levine April 18, 2007 Computer Science and Engineering, The Ohio State University 1 / 24
More informationInterpolation with complex B-splines
Interpolation with complex B-splines Brigitte Forster Fakultät für Informatik und Mathematik Universität Passau Brigitte Forster (Passau) Bernried 2016 1 / 32 The first part of a story in three chapters
More informationEdge Detection by Multi-Dimensional Wavelets
Edge Detection by Multi-Dimensional Wavelets Marlana Anderson, Chris Brasfield, Pierre Gremaud, Demetrio Labate, Katherine Maschmeyer, Kevin McGoff, Julie Siloti August 1, 2007 Abstract It is well known
More informationPerspective projection and Transformations
Perspective projection and Transformations The pinhole camera The pinhole camera P = (X,,) p = (x,y) O λ = 0 Q λ = O λ = 1 Q λ = P =-1 Q λ X = 0 + λ X 0, 0 + λ 0, 0 + λ 0 = (λx, λ, λ) The pinhole camera
More informationA spectral boundary element method
Boundary Elements XXVII 165 A spectral boundary element method A. Calaon, R. Adey & J. Baynham Wessex Institute of Technology, Southampton, UK Abstract The Boundary Element Method (BEM) is not local and
More informationDon t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary?
Don t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case?
More informationApplication of Daubechies Wavelets for Image Compression
Application of Daubechies Wavelets for Image Compression Heydari. Aghile 1,*, Naseri.Roghaye 2 1 Department of Math., Payame Noor University, Mashad, IRAN, Email Address a_heidari@pnu.ac.ir, Funded by
More informationRemark. Jacobs University Visualization and Computer Graphics Lab : ESM4A - Numerical Methods 331
Remark Reconsidering the motivating example, we observe that the derivatives are typically not given by the problem specification. However, they can be estimated in a pre-processing step. A good estimate
More informationSurfaces and Integral Curves
MODULE 1: MATHEMATICAL PRELIMINARIES 16 Lecture 3 Surfaces and Integral Curves In Lecture 3, we recall some geometrical concepts that are essential for understanding the nature of solutions of partial
More informationConvex or non-convex: which is better?
Sparsity Amplified Ivan Selesnick Electrical and Computer Engineering Tandon School of Engineering New York University Brooklyn, New York March 7 / 4 Convex or non-convex: which is better? (for sparse-regularized
More informationApplications of Wavelets and Framelets
Applications of Wavelets and Framelets Bin Han Department of Mathematical and Statistical Sciences University of Alberta, Edmonton, Canada Present at 2017 International Undergraduate Summer Enrichment
More informationScientific Computing: Interpolation
Scientific Computing: Interpolation Aleksandar Donev Courant Institute, NYU donev@courant.nyu.edu Course MATH-GA.243 or CSCI-GA.22, Fall 25 October 22nd, 25 A. Donev (Courant Institute) Lecture VIII /22/25
More information3. Lifting Scheme of Wavelet Transform
3. Lifting Scheme of Wavelet Transform 3. Introduction The Wim Sweldens 76 developed the lifting scheme for the construction of biorthogonal wavelets. The main feature of the lifting scheme is that all
More informationContents. I The Basic Framework for Stationary Problems 1
page v Preface xiii I The Basic Framework for Stationary Problems 1 1 Some model PDEs 3 1.1 Laplace s equation; elliptic BVPs... 3 1.1.1 Physical experiments modeled by Laplace s equation... 5 1.2 Other
More informationManifolds (Relates to text Sec. 36)
22M:132 Fall 07 J. Simon Manifolds (Relates to text Sec. 36) Introduction. Manifolds are one of the most important classes of topological spaces (the other is function spaces). Much of your work in subsequent
More informationWavelet transforms generated by splines
Wavelet transforms generated by splines Amir Z. Averbuch Valery A. Zheludev School of Computer Science Tel Aviv University Tel Aviv 69978, Israel Abstract In this paper we design a new family of biorthogonal
More informationWavelet Neural Networks
Wavelet Neural Networks and their application in the study of dynamical systems David Veitch Dissertation submitted for the MSc in Data Analysis, Networks and Nonlinear Dynamics. Department of Mathematics
More informationNumerical Methods for PDEs : Video 11: 1D FiniteFebruary Difference 15, Mappings Theory / 15
22.520 Numerical Methods for PDEs : Video 11: 1D Finite Difference Mappings Theory and Matlab February 15, 2015 22.520 Numerical Methods for PDEs : Video 11: 1D FiniteFebruary Difference 15, Mappings 2015
More informationEvaluation of Loop Subdivision Surfaces
Evaluation of Loop Subdivision Surfaces Jos Stam Alias wavefront, Inc. 8 Third Ave, 8th Floor, Seattle, WA 980, U.S.A. jstam@aw.sgi.com Abstract This paper describes a technique to evaluate Loop subdivision
More informationCS 534: Computer Vision Segmentation II Graph Cuts and Image Segmentation
CS 534: Computer Vision Segmentation II Graph Cuts and Image Segmentation Spring 2005 Ahmed Elgammal Dept of Computer Science CS 534 Segmentation II - 1 Outlines What is Graph cuts Graph-based clustering
More informationAlgebraic Flux Correction Schemes for High-Order B-Spline Based Finite Element Approximations
Algebraic Flux Correction Schemes for High-Order B-Spline Based Finite Element Approximations Numerical Analysis group Matthias Möller m.moller@tudelft.nl 7th DIAM onderwijs- en onderzoeksdag, 5th November
More informationTensor products in a wavelet setting
Chapter 8 Tensor products in a wavelet setting In Chapter 7 we defined tensor products in terms of vectors, and we saw that the tensor product of two vectors is in fact a matrix. The same construction
More informationMULTI-RESOLUTION STATISTICAL ANALYSIS ON GRAPH STRUCTURED DATA IN NEUROIMAGING
MULTI-RESOLUTION STATISTICAL ANALYSIS ON GRAPH STRUCTURED DATA IN NEUROIMAGING, Vikas Singh, Moo Chung, Nagesh Adluru, Barbara B. Bendlin, Sterling C. Johnson University of Wisconsin Madison Apr. 19, 2015
More informationSampling and interpolation for biomedical imaging: Part I
Sampling and interpolation for biomedical imaging: Part I Michael Unser Biomedical Imaging Group EPFL, Lausanne Switzerland ISBI 2006, Tutorial, Washington DC, April 2006 INTRODUCTION Fundamental issue
More informationAdaptive Isogeometric Analysis by Local h-refinement with T-splines
Adaptive Isogeometric Analysis by Local h-refinement with T-splines Michael Dörfel 1, Bert Jüttler 2, Bernd Simeon 1 1 TU Munich, Germany 2 JKU Linz, Austria SIMAI, Minisymposium M13 Outline Preliminaries:
More informationSolving a Two Dimensional Unsteady-State. Flow Problem by Meshless Method
Applied Mathematical Sciences, Vol. 7, 203, no. 49, 242-2428 HIKARI Ltd, www.m-hikari.com Solving a Two Dimensional Unsteady-State Flow Problem by Meshless Method A. Koomsubsiri * and D. Sukawat Department
More informationMulti-Resolution Approximate Inverses
Multi-Resolution Approximate Inverses by Robert Edward Bridson A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Master of Mathematics in Computer
More informationPROGRAMMING OF MULTIGRID METHODS
PROGRAMMING OF MULTIGRID METHODS LONG CHEN In this note, we explain the implementation detail of multigrid methods. We will use the approach by space decomposition and subspace correction method; see Chapter:
More informationBlock-based Thiele-like blending rational interpolation
Journal of Computational and Applied Mathematics 195 (2006) 312 325 www.elsevier.com/locate/cam Block-based Thiele-like blending rational interpolation Qian-Jin Zhao a, Jieqing Tan b, a School of Computer
More informationTwo Algorithms for Adaptive Approximation of Bivariate Functions by Piecewise Linear Polynomials on Triangulations
Two Algorithms for Adaptive Approximation of Bivariate Functions by Piecewise Linear Polynomials on Triangulations Nira Dyn School of Mathematical Sciences Tel Aviv University, Israel First algorithm from
More informationA Wavelet Tour of Signal Processing The Sparse Way
A Wavelet Tour of Signal Processing The Sparse Way Stephane Mallat with contributions from Gabriel Peyre AMSTERDAM BOSTON HEIDELBERG LONDON NEWYORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY»TOKYO
More informationRipplet: A New Transform for Image Processing
Ripplet: A New Transform for Image Processing Jun Xu, Lei Yang and Dapeng Wu 1 Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA Abstract Efficient representation
More informationA C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions
A C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions Nira Dyn School of Mathematical Sciences Tel Aviv University Michael S. Floater Department of Informatics University of
More informationAn Intuitive Framework for Real-Time Freeform Modeling
An Intuitive Framework for Real-Time Freeform Modeling Leif Kobbelt Shape Deformation Complex shapes Complex deformations User Interaction Very limited user interface 2D screen & mouse Intuitive metaphor
More informationMultilevel quasi-interpolation on a sparse grid with the Gaussian
Multilevel quasi-interpolation on a sparse grid with the Gaussian Fuat Usta 1 and Jeremy Levesley 2 1 Department of Mathematics, Duzce University, Konuralp Campus, 81620, Duzce, Turkey, fuatusta@duzce.edu.tr,
More informationDigital Image Processing. Chapter 7: Wavelets and Multiresolution Processing ( )
Digital Image Processing Chapter 7: Wavelets and Multiresolution Processing (7.4 7.6) 7.4 Fast Wavelet Transform Fast wavelet transform (FWT) = Mallat s herringbone algorithm Mallat, S. [1989a]. "A Theory
More informationA Hybrid Geometric+Algebraic Multigrid Method with Semi-Iterative Smoothers
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 013; 00:1 18 Published online in Wiley InterScience www.interscience.wiley.com). A Hybrid Geometric+Algebraic Multigrid Method with
More informationImplementation of PML in the Depth-oriented Extended Forward Modeling
Implementation of PML in the Depth-oriented Extended Forward Modeling Lei Fu, William W. Symes The Rice Inversion Project (TRIP) April 19, 2013 Lei Fu, William W. Symes (TRIP) PML in Extended modeling
More informationLECTURE 8: SMOOTH SUBMANIFOLDS
LECTURE 8: SMOOTH SUBMANIFOLDS 1. Smooth submanifolds Let M be a smooth manifold of dimension n. What object can be called a smooth submanifold of M? (Recall: what is a vector subspace W of a vector space
More informationFinite Element Methods for the Poisson Equation and its Applications
Finite Element Methods for the Poisson Equation and its Applications Charles Crook July 30, 2013 Abstract The finite element method is a fast computational method that also has a solid mathematical theory
More informationWAVELET TRANSFORM BASED FEATURE DETECTION
WAVELET TRANSFORM BASED FEATURE DETECTION David Bařina Doctoral Degree Programme (1), DCGM, FIT BUT E-mail: ibarina@fit.vutbr.cz Supervised by: Pavel Zemčík E-mail: zemcik@fit.vutbr.cz ABSTRACT This paper
More informationPartition of unity algorithm for two-dimensional interpolation using compactly supported radial basis functions
Communications in Applied and Industrial Mathematics, ISSN 2038-0909, e-431 DOI: 10.1685/journal.caim.431 Partition of unity algorithm for two-dimensional interpolation using compactly supported radial
More informationDomain Decomposition and hp-adaptive Finite Elements
Domain Decomposition and hp-adaptive Finite Elements Randolph E. Bank 1 and Hieu Nguyen 1 Department of Mathematics, University of California, San Diego, La Jolla, CA 9093-011, USA, rbank@ucsd.edu. Department
More informationRobust Classification of MR Brain Images Based on Multiscale Geometric Analysis
Robust Classification of MR Brain Images Based on Multiscale Geometric Analysis Sudeb Das and Malay Kumar Kundu Machine Intelligence Unit, Indian Statistical Institute, Kolkata 700108, India to.sudeb@gmail.com,
More informationContourlets: Construction and Properties
Contourlets: Construction and Properties Minh N. Do Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign www.ifp.uiuc.edu/ minhdo minhdo@uiuc.edu Joint work with
More informationA Weighted Kernel PCA Approach to Graph-Based Image Segmentation
A Weighted Kernel PCA Approach to Graph-Based Image Segmentation Carlos Alzate Johan A. K. Suykens ESAT-SCD-SISTA Katholieke Universiteit Leuven Leuven, Belgium January 25, 2007 International Conference
More informationCOMPUTABILITY THEORY AND RECURSIVELY ENUMERABLE SETS
COMPUTABILITY THEORY AND RECURSIVELY ENUMERABLE SETS JOSHUA LENERS Abstract. An algorithm is function from ω to ω defined by a finite set of instructions to transform a given input x to the desired output
More informationFinite Difference Calculus
Chapter 2 Finite Difference Calculus In this chapter we review the calculus of finite differences. The topic is classic and covered in many places. The Taylor series is fundamental to most analysis. A
More informationA Survey on Wavelet Applications in Data Mining
A Survey on Wavelet Applications in Data Mining Tao Li Department of Computer Science Univ. of Rochester Rochester, NY 14627 taoli@cs.rochester.edu Qi Li Dept. of Computer & Information Sciences Univ.
More informationJacobian: Velocities and Static Forces 1/4
Jacobian: Velocities and Static Forces /4 Models of Robot Manipulation - EE 54 - Department of Electrical Engineering - University of Washington Kinematics Relations - Joint & Cartesian Spaces A robot
More informationAlgorithms of Scientific Computing
Algorithms of Scientific Computing Overview and General Remarks Michael Bader Technical University of Munich Summer 2017 Classification of the Lecture Who is Who? Students of Informatics: Informatics Bachelor
More informationWavelet-Galerkin Solutions of One and Two Dimensional Partial Differential Equations
VOL 3, NO0 Oct, 202 ISSN 2079-8407 2009-202 CIS Journal All rights reserved http://wwwcisjournalorg Wavelet-Galerkin Solutions of One and Two Dimensional Partial Differential Equations Sabina, 2 Vinod
More informationA C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions
A C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions Nira Dyn Michael S. Floater Kai Hormann Abstract. We present a new four-point subdivision scheme that generates C 2 curves.
More informationComputation of interpolatory splines via triadic subdivision
Computation of interpolatory splines via triadic subdivision Valery A. Zheludev and Amir Z. Averbuch Abstract We present an algorithm for computation of interpolatory splines of arbitrary order at triadic
More informationfor the Stokes Equations in Three Dimensional Complex Geometries Lexing Ying
An Efficient and High-Order Accurate Boundary Integral Solver for the Stokes Equations in Three Dimensional Complex Geometries by Lexing Ying A dissertation submitted in partial fulfillment of the requirements
More informationLocality-Sensitive Codes from Shift-Invariant Kernels Maxim Raginsky (Duke) and Svetlana Lazebnik (UNC)
Locality-Sensitive Codes from Shift-Invariant Kernels Maxim Raginsky (Duke) and Svetlana Lazebnik (UNC) Goal We want to design a binary encoding of data such that similar data points (similarity measures
More informationImplicit Matrix Representations of Rational Bézier Curves and Surfaces
Implicit Matrix Representations of Rational Bézier Curves and Surfaces Laurent Busé INRIA Sophia Antipolis, France Laurent.Buse@inria.fr GD/SPM Conference, Denver, USA November 11, 2013 Overall motivation
More informationOptic Flow and Basics Towards Horn-Schunck 1
Optic Flow and Basics Towards Horn-Schunck 1 Lecture 7 See Section 4.1 and Beginning of 4.2 in Reinhard Klette: Concise Computer Vision Springer-Verlag, London, 2014 1 See last slide for copyright information.
More informationRelationship between Fourier Space and Image Space. Academic Resource Center
Relationship between Fourier Space and Image Space Academic Resource Center Presentation Outline What is an image? Noise Why do we transform images? What is the Fourier Transform? Examples of images in
More informationBOX SPLINE WAVELET FRAMES FOR IMAGE EDGE ANALYSIS
BOX SPLINE WAVELET FRAMES FOR IMAGE EDGE ANALYSIS WEIHONG GUO AND MING-JUN LAI Abstract. We present a new box spline wavelet frame and apply it for image edge analysis. The wavelet frame is constructed
More informationNon-Linear Finite Element Methods in Solid Mechanics Attilio Frangi, Politecnico di Milano, February 3, 2017, Lesson 1
Non-Linear Finite Element Methods in Solid Mechanics Attilio Frangi, attilio.frangi@polimi.it Politecnico di Milano, February 3, 2017, Lesson 1 1 Politecnico di Milano, February 3, 2017, Lesson 1 2 Outline
More informationPoint-Set Topology II
Point-Set Topology II Charles Staats September 14, 2010 1 More on Quotients Universal Property of Quotients. Let X be a topological space with equivalence relation. Suppose that f : X Y is continuous and
More informationIntegral Equation Methods for Vortex Dominated Flows, a High-order Conservative Eulerian Approach
Integral Equation Methods for Vortex Dominated Flows, a High-order Conservative Eulerian Approach J. Bevan, UIUC ICERM/HKUST Fast Integral Equation Methods January 5, 2016 Vorticity and Circulation Γ =
More information