Optimal Data Structures for Spherical Multiresolution
|
|
- Emery Ford
- 5 years ago
- Views:
Transcription
1 Optimal Data Structures for Spherical Multiresolution Analysis and Synthesis J. A. Rod Blais Dept. of Geomatics Engineering Pacific Institute for the Mathematical Sciences University of Calgary, Calgary, AB ca/
2 Introduction Geopotential and related fields are global with continuous spectra Multiresolution analysis and synthesis require regular array data Spherical Quad Tree (SQT) data structures are most appropriate Spherical Harmonic Wavelets (SHWs) using Transforms (SHTs) Equiangular discretizations are most common in applications Equitriangular (near equiareal) are often desirable in practice Other equidistribution strategies are much less appropriate Simulations with Geopotential Models (GMs) are discussed Concluding Remarks
3 Multiresolution Analysis on Sphere (S 2 ) A multiresolution analysis of L 2 (S 2 ) is a nested sequence of subspaces {V n : n = 0, ± 1, ± 2, } such that 2 2 L (S )... V1 V0 V 1... f(t) V n f(st) V n-1, s > 1 (dilations) f(t) V n f(t-k) V n (translations) a scaling or smoothing function (t) s g (st k) such hthat t{ {(t-k)} is i an orthonormal l basis b i of f V 0 a detail or wavelet function (t) s h k(st k) that is a quadratic conjugate of the scaling function (t) k k k
4 Multiresolution Analysis of GMs GM GM GM GM N N/2 0 M M M N N/2 0 using the lowpass filter kernel with decimation (binary scaling) K : GM GM and k :u (, ) u (, ) L N N/2 L N N/2 and highpass filter kernel with decimation (binary scaling) K : GM M and k :u (, ) v (, ) H N N/2 H N N/2 satisfying Bezout s theorem: K K K K I L L H H for reconstruction purposes: KLGMN/2 KHMN/2 GMN
5 Example: SHT of Order N 2B CS c s s s s s c c s s s s c c c s s s c c c c s s cb1,0 cb1,1 cb1,b1 cb c s cn1,0 cn1,1 cn1,b1 c c c B1,1 B1,1 B1,1 N1, B 1,2 B 1,2 B 1,2 N 1,2 B 1,0 B11 1,1 B 1,B 1 B,B B1 B 1,B 1 N 1,B 1 B,0 B,1 B,B 1 B,B B 1,B N 1,B 1,B B1,B1 N1,B1 N1,B N1,B1 N1,N1
6 Multiresolution Synthesis of GMs For a potential function u(, ) in GM u(, ) u (, ) k v (, ) k v (, ) 0 H 0 H N u N(, ) and with more observational information, u(, ) u (, ) k v (, ) k v (, ) N H N H 2N Recall that with the usual (geodetic) conventions, N 1 m N n,m n n0 m n u (, ) u Y (, ) N1 n c cosm s sinm P (cos ) n0 m0 nm nm nm
7 Estimation of Spectral Coefficients Chebychev Quadrature (CQ) Methods: Equiangular grids of e.g. 2Nx4N points for Δθ = Δλ With 2N equispaced parallels and at least N equispaced meridians Usually excluding the poles for numerical stability» Providing least degree N per data (N 2 coefts, O(N 3 ) oper ns) Least-Squares (LS) Methods: Equiangular grids of e.g. Nx2N points for Δθ = Δλ With at least N parallels and at least N equispaced meridians Usually excluding the poles for numerical stability Multiple least-squares estimation per order» Requiring least data per degree N (N 2 coefts, O(N 4 ) oper ns)
8 SHT Using Chebychev Quadrature SHT: with and z11 z z1k c00 s s 1 z z... z c c... s K z z... z c c... c J1 J2 JK 0 1 c cosm qz P (cos ) J K nm k jk j nm j s nm j1 k1 sinm k n z (c cosm s sinm ) P (cos ) jk nm k nm k nm j n0 m0 in which q j denote the Chebychev quadrature weights [Blais, 2011].
9 SHT Using Least Squares From the previous synthesis formulation z "const " jk const. IDFT (cnm is nm)p nm (cos j ) k1,k nm one has for unknown coefficients and c nm s nm DFT[z jk ] "const." (cnm i s nm )P nm(cos j) k1,k nm implying a least-squares problem per order m, with θ j = j π/n, j = 0, 1,, N-1 λ k = k π/n, k = 0, 1,, 2N-1 c for and with m n, n = 0, 1,, N-1 [Blais, 2011]. nm s nm
10 SHT Computations for Degree N For gridded ddata with at tleast t2n equispaced disolatitude data DFT IDFT z u iv zˆ jk per row jh jh per row jk For gridded data with 2N data per column with constant Δθ Chebychev c is u iv cˆ isˆ nm nm jh jh Quadrature nm nm For gridded data with N data per column with variable Δθ Least c i s u i v cˆ i sˆ nm nm jh jh Squares nm nm Note that only DFTs are generally invertible above.
11 Equidistribution on the Sphere In general for an arbitrary partition of the sphere S 2, N C( n)/n n 1 Discrepancy sup N 2 all CS lim C( n) / N N in which C( k) denotes the characteristic function for the cell C. n 1 With pseudo-random numbers, Disc. O((log log N) 1/2 /N 1/2 ) With quasi-random numbers, Disc. O((log N) s /N ) for dim. s Hence quasi-random (deterministic) sequences are advantageous in higher dimensional simulations [Morokoff & Caflisch, 1994]
12 Quad Tree Data Structures Planar Quad Trees Two-dimensional pyramidal binary data structures Planar domains can be square, rectangular or triangular Most often used for multiresolution analysis/synthesis Spherical Quad Trees Spherical pyramidal binary data structures Spherical partitions can be equiangular or equitriangular Octahedrons and icosahedrons have equitriangular faces Densification can only be near equiareal in general
13 Near Equiareal Strategies Platonic Solids (for exact regularity) Tetrahedron ( 4 vertices, 4 edges, 4 faces) Cube ( 8 vertices, 12 edges, 6 faces) Octahedron ( 6 vertices, 12 edges, 8 faces) Dodecahedron (20 vertices, 30 edges, 12 faces) Icosahedron (12 vertices, 30 edges, 20 faces) Dualities (by interchanging faces and vertices) Tetrahedron Tetrahedron Cube Octahedron Dd Dodecahedron hd Icosahedron
14
15 Cube and Octahedron For the octahedron, radially projected face centres are equispaced on parallels enabling the use of FFTs in SHTs.
16 Othd Octahedron Based dexamples Level One: x x x x x x x x Level Two: x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Level Three: x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Data matrix: 2 L x 4(2 L -1) for level L=1, 2, 3,
17 Octahedron Based EGM 2008 Level: Grid CQ SHT without MASK CQ SHT with MASK Spectral RMS Spatial RMS Spectral RMS Spatial RMS 1: E E E E+00 2: E E E E E E E E+00 3: E E E E-07 4: E E E E-07 5: E E E E-08 6: E E E E-08 7: E E E E-08 8: E E E E-09 9: E E E E-09 10: E E E E-09 11: E E E E-10 12: E E E E-11
18 Octahedron Based EGM 2008 Level: Grid LS SHT without MASK LS SHT with MASK Spectral RMS Spatial RMS Spectral RMS Spatial RMS 1: E E E E+00 2: E E E E-06 3: E E E E-06 4: E E E E-06 5: E E E E E-06 6: E E E E-06 7: E E E E-06 8: E E E E-06 9: E E E E-06 10: E E E E-06 11: E E E E-06
19 Dodecahedron and Icosahedron For the icosahedron, radially projected face centres are equispaced on parallels enabling the use of FFTs in SHTs.
20 Icosahedron Based Examples Level One: x x x x x x x x x x x x x x x x x x x x Level Two : x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Level Three: x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Data matrix: 3 2 L-1 x 10 2 L-1 for level L=1, 2, 3,
21 Icosahedron Based EGM 2008 Level: Grid CQ SHT without MASK CQ SHT with MASK Spectral RMS Spatial RMS Spectral RMS Spatial RMS 2: E E E E-08 3: E E E E-07 4: E E E E-08 5: E E E E-08 6: E E E E-08 7: E E E E-09 8: E E E E-09 9: E E E E-09 10: E E E E-10 11: E E E E-10 10
22 Icosahedron Based EGM 2008 Level: Grid LS SHT without MASK LS SHT with MASK Spectral RMS Spatial RMS Spectral RMS Spatial RMS 1: E E E E-22 2: E E E E-06 3: E E E E-06 4: E E E E-06 5: E E E E-06 6: E E E E-06 7: E E E E-06 8: E E E E-06 9: E E E E-06 10: E E E E-06 06
23 Reuter Grids Homogeneous point distributions ib ti include e.g. Reuter grids Fast computations can be performed using panel clustering New grids are required for each level of decomposition Not a pyramidal data structure for analysis and synthesis Latitude (de eg) 0 eg) Latitude (d Longitude (deg) Longitude (deg)
24 Concluding Remarks Multiresolution analysis/synthesis can be done on 2 or S 2 Equiangular grids are common but not always desirable Equitriangular (near equiareal) are generally convenient Other equidistributed ib t d point sets often have drawbacks SHTs (CQ & LS) analysis/synthesis give comparable RMS With EGM and like data, equitriangular grids work well With noise like data, some analysis problems may arise
Spherical Microphone Arrays
Spherical Microphone Arrays Acoustic Wave Equation Helmholtz Equation Assuming the solutions of wave equation are time harmonic waves of frequency ω satisfies the homogeneous Helmholtz equation: Boundary
More information7. The Gauss-Bonnet theorem
7. The Gauss-Bonnet theorem 7.1 Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed
More informationImage Transformation Techniques Dr. Rajeev Srivastava Dept. of Computer Engineering, ITBHU, Varanasi
Image Transformation Techniques Dr. Rajeev Srivastava Dept. of Computer Engineering, ITBHU, Varanasi 1. Introduction The choice of a particular transform in a given application depends on the amount of
More informationMath 311. Polyhedra Name: A Candel CSUN Math
1. A polygon may be described as a finite region of the plane enclosed by a finite number of segments, arranged in such a way that (a) exactly two segments meets at every vertex, and (b) it is possible
More informationWe have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance.
Solid geometry We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance. First, note that everything we have proven for the
More informationCHAPTER 3 DIFFERENT DOMAINS OF WATERMARKING. domain. In spatial domain the watermark bits directly added to the pixels of the cover
38 CHAPTER 3 DIFFERENT DOMAINS OF WATERMARKING Digital image watermarking can be done in both spatial domain and transform domain. In spatial domain the watermark bits directly added to the pixels of the
More informationPlatonic Solids and the Euler Characteristic
Platonic Solids and the Euler Characteristic Keith Jones Sanford Society, SUNY Oneonta September 2013 What is a Platonic Solid? A Platonic Solid is a 3-dimensional object with flat faces and straight edges
More informationNon-extendible finite polycycles
Izvestiya: Mathematics 70:3 1 18 Izvestiya RAN : Ser. Mat. 70:3 3 22 c 2006 RAS(DoM) and LMS DOI 10.1070/IM2006v170n01ABEH002301 Non-extendible finite polycycles M. Deza, S. V. Shpectorov, M. I. Shtogrin
More informationApplication of optimal sampling lattices on CT image reconstruction and segmentation or three dimensional printing
Application of optimal sampling lattices on CT image reconstruction and segmentation or three dimensional printing XIQIANG ZHENG Division of Health and Natural Sciences, Voorhees College, Denmark, SC 29042
More informationSection 9.4. Volume and Surface Area. Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 9.4 Volume and Surface Area What You Will Learn Volume Surface Area 9.4-2 Volume Volume is the measure of the capacity of a three-dimensional figure. It is the amount of material you can put inside
More informationClassifying 3D Shapes
Classifying 3D Shapes Middle School Texas Essential Knowledge and Skills (TEKS) Math 5.4B Algebraic reasoning The student applies mathematical process standards to develop concepts of expressions and equations.
More informationUtilizing a Discrete Global Grid System For Handling Point Clouds With Varying Location, Time, and Scale
Utilizing a Discrete Global Grid System For Handling Point Clouds With Varying Location, Time, and Scale MSc Thesis Project, Fugro Supervisors: EDWARD VERBREE PETER VAN OOSTEROM Neeraj Sirdeshmukh Delft
More informationLecture 19: Introduction To Topology
Chris Tralie, Duke University 3/24/2016 Announcements Group Assignment 2 Due Wednesday 3/30 First project milestone Friday 4/8/2016 Welcome to unit 3! Table of Contents The Euler Characteristic Spherical
More informationFive Platonic Solids: Three Proofs
Five Platonic Solids: Three Proofs Vincent J. Matsko IMSA, Dodecahedron Day Workshop 18 November 2011 Convex Polygons convex polygons nonconvex polygons Euler s Formula If V denotes the number of vertices
More information(x, y, z) m 2. (x, y, z) ...] T. m 2. m = [m 1. m 3. Φ = r T V 1 r + λ 1. m T Wm. m T L T Lm + λ 2. m T Hm + λ 3. t(x, y, z) = m 1
Class 1: Joint Geophysical Inversions Wed, December 1, 29 Invert multiple types of data residuals simultaneously Apply soft mutual constraints: empirical, physical, statistical Deal with data in the same
More informationTiling of Sphere by Congruent Pentagons
Tiling of Sphere by Congruent Pentagons Min Yan September 9, 2017 webpage for further reading: http://www.math.ust.hk/ mamyan/research/urop.shtml We consider tilings of the sphere by congruent pentagons.
More informationDigital Image Processing
Digital Image Processing Third Edition Rafael C. Gonzalez University of Tennessee Richard E. Woods MedData Interactive PEARSON Prentice Hall Pearson Education International Contents Preface xv Acknowledgments
More informationPÓLYA S THEORY OF COUNTING
PÓLYA S THEORY OF COUNTING MANJIL P. SAIKIA Abstract. In this report, we shall study Pólya s Theory of Counting and apply it to solve some questions related to rotations of the platonic solids. Key Words:
More informationMath 489 Project 1: Explore a Math Problem L. Hogben 1 Possible Topics for Project 1: Explore a Math Problem draft 1/13/03
Math 489 Project 1: Explore a Math Problem L. Hogben 1 Possible Topics for Project 1: Explore a Math Problem draft 1/13/03 Number Base and Regularity We use base 10. The Babylonians used base 60. Discuss
More informationExample: The following is an example of a polyhedron. Fill the blanks with the appropriate answer. Vertices:
11.1: Space Figures and Cross Sections Polyhedron: solid that is bounded by polygons Faces: polygons that enclose a polyhedron Edge: line segment that faces meet and form Vertex: point or corner where
More informationDiffusion Wavelets for Natural Image Analysis
Diffusion Wavelets for Natural Image Analysis Tyrus Berry December 16, 2011 Contents 1 Project Description 2 2 Introduction to Diffusion Wavelets 2 2.1 Diffusion Multiresolution............................
More information11.4 Three-Dimensional Figures
11. Three-Dimensional Figures Essential Question What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron? A polyhedron is a solid that is bounded by polygons, called
More informationTwo Dimensional Wavelet and its Application
RESEARCH CENTRE FOR INTEGRATED MICROSYSTEMS - UNIVERSITY OF WINDSOR Two Dimensional Wavelet and its Application Iman Makaremi 1 2 RESEARCH CENTRE FOR INTEGRATED MICROSYSTEMS - UNIVERSITY OF WINDSOR Outline
More informationSparse Reconstruction / Compressive Sensing
Sparse Reconstruction / Compressive Sensing Namrata Vaswani Department of Electrical and Computer Engineering Iowa State University Namrata Vaswani Sparse Reconstruction / Compressive Sensing 1/ 20 The
More informationA Study of the Rigidity of Regular Polytopes
A Study of the Rigidity of Regular Polytopes A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Helene
More information1 Appendix to notes 2, on Hyperbolic geometry:
1230, notes 3 1 Appendix to notes 2, on Hyperbolic geometry: The axioms of hyperbolic geometry are axioms 1-4 of Euclid, plus an alternative to axiom 5: Axiom 5-h: Given a line l and a point p not on l,
More informationEquipartite polytopes and graphs
Equipartite polytopes and graphs Branko Grünbaum Tomáš Kaiser Daniel Král Moshe Rosenfeld Abstract A graph G of even order is weakly equipartite if for any partition of its vertex set into subsets V 1
More informationIntermediate Math Circles Fall 2018 Patterns & Counting
Intermediate Math Circles Fall 2018 Patterns & Counting Michael Miniou The Centre for Education in Mathematics and Computing Faculty of Mathematics University of Waterloo December 5, 2018 Michael Miniou
More informationComputer Graphics. P08 Texture Synthesis. Aleksandra Pizurica Ghent University
Computer Graphics P08 Texture Synthesis Aleksandra Pizurica Ghent University Telecommunications and Information Processing Image Processing and Interpretation Group Applications of texture synthesis Computer
More informationRight-angled hyperbolic polyhedra
Right-angled hyperbolic polyhedra Andrei Vesnin Sobolev Institute of Mathematics, Novosibirsk, Russia The Fourth Geometry Meeting dedicated to the centenary of A. D. Alexandrov August 20 24, 2012 A.D.Alexandrov:
More informationGEOPHYS 242: Near Surface Geophysical Imaging. Class 8: Joint Geophysical Inversions Wed, April 20, 2011
GEOPHYS 4: Near Surface Geophysical Imaging Class 8: Joint Geophysical Inversions Wed, April, 11 Invert multiple types of data residuals simultaneously Apply soft mutual constraints: empirical, physical,
More informationModule 9 : Numerical Relaying II : DSP Perspective
Module 9 : Numerical Relaying II : DSP Perspective Lecture 36 : Fast Fourier Transform Objectives In this lecture, We will introduce Fast Fourier Transform (FFT). We will show equivalence between FFT and
More informationThe University of Sydney MATH2008 Introduction to Modern Algebra
2 Semester 2, 2003 The University of Sydney MATH2008 Introduction to Modern Algebra (http://www.maths.usyd.edu.au/u/ug/im/math2008/) Lecturer: R. Howlett Computer Tutorial 2 This tutorial explores the
More informationMultiple Integrals. max x i 0
Multiple Integrals 1 Double Integrals Definite integrals appear when one solves Area problem. Find the area A of the region bounded above by the curve y = f(x), below by the x-axis, and on the sides by
More informationDesign of Orthogonal Graph Wavelet Filter Banks
Design of Orthogonal Graph Wavelet Filter Banks Xi ZHANG Department of Communication Engineering and Informatics The University of Electro-Communications Chofu-shi, Tokyo, 182-8585 JAPAN E-mail: zhangxi@uec.ac.jp
More informationVisual Recognition: Image Formation
Visual Recognition: Image Formation Raquel Urtasun TTI Chicago Jan 5, 2012 Raquel Urtasun (TTI-C) Visual Recognition Jan 5, 2012 1 / 61 Today s lecture... Fundamentals of image formation You should know
More informationTomaz Pisanski, University of Ljubljana, Slovenia. Thomas W. Tucker, Colgate University. Arjana Zitnik, University of Ljubljana, Slovenia
Eulerian Embeddings of Graphs Tomaz Pisanski, University of Ljubljana, Slovenia Thomas W. Tucker, Colgate University Arjana Zitnik, University of Ljubljana, Slovenia Abstract A straight-ahead walk in an
More informationEquipartite polytopes and graphs
Equipartite polytopes and graphs Branko Grünbaum Tomáš Kaiser Daniel Král Moshe Rosenfeld Abstract A graph G of even order is weakly equipartite if for any partition of its vertex set into subsets V 1
More informationQuestion. Why is the third shape not convex?
1. CONVEX POLYGONS Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D. Convex 6 gon Another convex 6 gon Not convex Question. Why is the third
More informationOutline Introduction Problem Formulation Proposed Solution Applications Conclusion. Compressed Sensing. David L Donoho Presented by: Nitesh Shroff
Compressed Sensing David L Donoho Presented by: Nitesh Shroff University of Maryland Outline 1 Introduction Compressed Sensing 2 Problem Formulation Sparse Signal Problem Statement 3 Proposed Solution
More informationModelling, migration, and inversion using Linear Algebra
Modelling, mid., and inv. using Linear Algebra Modelling, migration, and inversion using Linear Algebra John C. Bancroft and Rod Blais ABSRAC Modelling migration and inversion can all be accomplished using
More informationOptimum Array Processing
Optimum Array Processing Part IV of Detection, Estimation, and Modulation Theory Harry L. Van Trees WILEY- INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION Preface xix 1 Introduction 1 1.1 Array Processing
More informationWeek 7 Convex Hulls in 3D
1 Week 7 Convex Hulls in 3D 2 Polyhedra A polyhedron is the natural generalization of a 2D polygon to 3D 3 Closed Polyhedral Surface A closed polyhedral surface is a finite set of interior disjoint polygons
More informationINSTRUCTIONS FOR THE USE OF THE SUPER RULE TM
INSTRUCTIONS FOR THE USE OF THE SUPER RULE TM NOTE: All images in this booklet are scale drawings only of template shapes and scales. Preparation: Your SUPER RULE TM is a valuable acquisition for classroom
More informationFourier Transformation Methods in the Field of Gamma Spectrometry
International Journal of Pure and Applied Physics ISSN 0973-1776 Volume 3 Number 1 (2007) pp. 132 141 Research India Publications http://www.ripublication.com/ijpap.htm Fourier Transformation Methods in
More informationDSP-CIS. Part-IV : Filter Banks & Subband Systems. Chapter-10 : Filter Bank Preliminaries. Marc Moonen
DSP-CIS Part-IV Filter Banks & Subband Systems Chapter-0 Filter Bank Preliminaries Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be www.esat.kuleuven.be/stadius/ Part-III Filter
More informationEXPLORATIONS ON THE DIMENSION OF A GRAPH
EXPLORATIONS ON THE DIMENSION OF A GRAPH RYAN KAVANAGH Abstract. Erdős, Harary and Tutte first defined the dimension of a graph G as the minimum number n such that G can be embedded into the Euclidean
More information4. Image Retrieval using Transformed Image Content
4. Image Retrieval using Transformed Image Content The desire of better and faster retrieval techniques has always fuelled to the research in content based image retrieval (CBIR). A class of unitary matrices
More informationRectangular prism. The two bases of a prism. bases
Page 1 of 8 9.1 Solid Figures Goal Identify and name solid figures. Key Words solid polyhedron base face edge The three-dimensional shapes on this page are examples of solid figures, or solids. When a
More informationTHE BASIC TOPOLOGY MODEL OF SPHERICAL SURFACE DIGITAL SPACE
HE BASIC OPOLOGY MODEL OF SPHERICAL SURFACE DIGIAL SPACE HOU Miao-le 1 1 Department of Surveying & Land Science, China University of Mining and echnology (Beijing), D11 Xueyuan Road, Beijing, China, 100083
More informationMr. Whelan Name: Block:
Mr. Whelan Name: Block: Geometry/Trig Unit 10 Area and Volume of Solids Notes Packet Day 1 Notes - Prisms Rectangular Prism: How do we find Total Area? Example 1 6cm Find the area of each face: Front:
More informationMa/CS 6b Class 9: Euler s Formula
Ma/CS 6b Class 9: Euler s Formula By Adam Sheffer Recall: Plane Graphs A plane graph is a drawing of a graph in the plane such that the edges are noncrossing curves. 1 Recall: Planar Graphs The drawing
More informationDUAL TREE COMPLEX WAVELETS Part 1
DUAL TREE COMPLEX WAVELETS Part 1 Signal Processing Group, Dept. of Engineering University of Cambridge, Cambridge CB2 1PZ, UK. ngk@eng.cam.ac.uk www.eng.cam.ac.uk/~ngk February 2005 UNIVERSITY OF CAMBRIDGE
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 informationRecent Developments in Model-based Derivative-free Optimization
Recent Developments in Model-based Derivative-free Optimization Seppo Pulkkinen April 23, 2010 Introduction Problem definition The problem we are considering is a nonlinear optimization problem with constraints:
More informationTHE BASIC TOPOLOGY ISSUES ON SPHERICAL SURFACE QUTERNARY TRANGULAR MESH
THE BASIC TOPOLOGY ISSUES ON SPHERICAL SURFACE QUTERNARY TRANGULAR MESH HOU Miaole a, ZHAO Xuesheng b, c, CHEN Jun c a Department of Surveying & Spatial Science, BICEA (Beijing), 1 Zhanlanguan Road, Beijing,
More informationComputer Vision and Graphics (ee2031) Digital Image Processing I
Computer Vision and Graphics (ee203) Digital Image Processing I Dr John Collomosse J.Collomosse@surrey.ac.uk Centre for Vision, Speech and Signal Processing University of Surrey Learning Outcomes After
More informationINTRODUCTION TO GRAPH THEORY. 1. Definitions
INTRODUCTION TO GRAPH THEORY D. JAKOBSON 1. Definitions A graph G consists of vertices {v 1, v 2,..., v n } and edges {e 1, e 2,..., e m } connecting pairs of vertices. An edge e = (uv) is incident with
More informationMath 462: Review questions
Math 462: Review questions Paul Hacking 4/22/10 (1) What is the angle between two interior diagonals of a cube joining opposite vertices? [Hint: It is probably quickest to use a description of the cube
More informationSparse wavelet expansions for seismic tomography: Methods and algorithms
Sparse wavelet expansions for seismic tomography: Methods and algorithms Ignace Loris Université Libre de Bruxelles International symposium on geophysical imaging with localized waves 24 28 July 2011 (Joint
More informationDD2429 Computational Photography :00-19:00
. Examination: DD2429 Computational Photography 202-0-8 4:00-9:00 Each problem gives max 5 points. In order to pass you need about 0-5 points. You are allowed to use the lecture notes and standard list
More informationIll-Posed Problems with A Priori Information
INVERSE AND ILL-POSED PROBLEMS SERIES Ill-Posed Problems with A Priori Information V.V.Vasin andalageev HIV SPIII Utrecht, The Netherlands, 1995 CONTENTS Introduction 1 CHAPTER 1. UNSTABLE PROBLEMS 1 Base
More informationLecture 3: Camera Calibration, DLT, SVD
Computer Vision Lecture 3 23--28 Lecture 3: Camera Calibration, DL, SVD he Inner Parameters In this section we will introduce the inner parameters of the cameras Recall from the camera equations λx = P
More informationPlatonic Solids. Jennie Sköld. January 21, Karlstad University. Symmetries: Groups Algebras and Tensor Calculus FYAD08
Platonic Solids Jennie Sköld January 21, 2015 Symmetries: Groups Algebras and Tensor Calculus FYAD08 Karlstad University 1 Contents 1 What are Platonic Solids? 3 2 Symmetries in 3-Space 5 2.1 Isometries
More informationAPPLYING EXTRAPOLATION AND INTERPOLATION METHODS TO MEASURED AND SIMULATED HRTF DATA USING SPHERICAL HARMONIC DECOMPOSITION.
APPLYING EXTRAPOLATION AND INTERPOLATION METHODS TO MEASURED AND SIMULATED HRTF DATA USING SPHERICAL HARMONIC DECOMPOSITION Martin Pollow Institute of Technical Acoustics RWTH Aachen University Neustraße
More informationMath 170, Section 002 Spring 2012 Practice Exam 2 with Solutions
Math 170, Section 002 Spring 2012 Practice Exam 2 with Solutions Contents 1 Problems 2 2 Solution key 10 3 Solutions 11 1 1 Problems Question 1: A right triangle has hypothenuse of length 25 in and an
More informationUnit Maps: Grade 2 Math
Place Value and Comparing Numbers 2.3 Place value. The student understands how to represent and compare whole numbers, the relative position and magnitude of whole numbers, and relationships within the
More informationCS354 Computer Graphics Rotations and Quaternions
Slide Credit: Don Fussell CS354 Computer Graphics Rotations and Quaternions Qixing Huang April 4th 2018 Orientation Position and Orientation The position of an object can be represented as a translation
More informationWeek 9: Planar and non-planar graphs. 7 and 9 November, 2018
(1/27) MA284 : Discrete Mathematics Week 9: Planar and non-planar graphs http://www.maths.nuigalway.ie/ niall/ma284/ 7 and 9 November, 2018 1 Planar graphs and Euler s formula 2 Non-planar graphs K 5 K
More informationChapter 12 and 11.1 Planar graphs, regular polyhedra, and graph colorings
Chapter 12 and 11.1 Planar graphs, regular polyhedra, and graph colorings Prof. Tesler Math 184A Fall 2017 Prof. Tesler Ch. 12: Planar Graphs Math 184A / Fall 2017 1 / 45 12.1 12.2. Planar graphs Definition
More informationCSE 252B: Computer Vision II
CSE 252B: Computer Vision II Lecturer: Serge Belongie Scribe : Martin Stiaszny and Dana Qu LECTURE 0 Camera Calibration 0.. Introduction Just like the mythical frictionless plane, in real life we will
More informationBilevel Sparse Coding
Adobe Research 345 Park Ave, San Jose, CA Mar 15, 2013 Outline 1 2 The learning model The learning algorithm 3 4 Sparse Modeling Many types of sensory data, e.g., images and audio, are in high-dimensional
More information2 Geometry Solutions
2 Geometry Solutions jacques@ucsd.edu Here is give problems and solutions in increasing order of difficulty. 2.1 Easier problems Problem 1. What is the minimum number of hyperplanar slices to make a d-dimensional
More informationKeywords - DWT, Lifting Scheme, DWT Processor.
Lifting Based 2D DWT Processor for Image Compression A. F. Mulla, Dr.R. S. Patil aieshamulla@yahoo.com Abstract - Digital images play an important role both in daily life applications as well as in areas
More informationIterative methods for use with the Fast Multipole Method
Iterative methods for use with the Fast Multipole Method Ramani Duraiswami Perceptual Interfaces and Reality Lab. Computer Science & UMIACS University of Maryland, College Park, MD Joint work with Nail
More informationMajor Facilities for Mathematical Thinking and Understanding. (2) Vision, spatial sense and kinesthetic (motion) sense.
Major Facilities for Mathematical Thinking and Understanding. (2) Vision, spatial sense and kinesthetic (motion) sense. Left brain Right brain Hear what you see. See what you hear. Mobius Strip http://www.metacafe.com/watch/331665/
More informationExplore Solids
1212.1 Explore Solids Surface Area and Volume of Solids 12.2 Surface Area of Prisms and Cylinders 12.3 Surface Area of Pyramids and Cones 12.4 Volume of Prisms and Cylinders 12.5 Volume of Pyramids and
More informationParametric Texture Model based on Joint Statistics
Parametric Texture Model based on Joint Statistics Gowtham Bellala, Kumar Sricharan, Jayanth Srinivasa Department of Electrical Engineering, University of Michigan, Ann Arbor 1. INTRODUCTION Texture images
More informationName: Target 12.2: Find and apply surface of Spheres and Composites 12.2a: Surface Area of Spheres 12.2b: Surface Area of Composites Solids
Unit 12: Surface Area and Volume of Solids Target 12.0: Euler s Formula and Introduction to Solids Target 12.1: Find and apply surface area of solids 12.1a: Surface Area of Prisms and Cylinders 12.1b:
More informationMath 213 Student Note Outlines. Sections
Math 213 Student Note Outlines Sections 9.2 9.3 11.1 11.2 11.3 9.2 KEY IDEAS, page 1 of 2 Polygon Vertex Angles Sum Regular Polygons: Vertex Angles Congruence Definition Regular Polygons Definition Tessellation
More informationCompared with the Fast Fourier Transform algorithm, this algorithm is easier to implement including the far zone contribution evaluation that can be d
An alternative algorithm to FFT for the numerical evaluation of Stokes's integral J. Huang Λ P. Van»cek P. Novák Department of Geodesy and Geomatics Engineering, University of New Brunswick, P. O. Box
More informationHOUGH TRANSFORM CS 6350 C V
HOUGH TRANSFORM CS 6350 C V HOUGH TRANSFORM The problem: Given a set of points in 2-D, find if a sub-set of these points, fall on a LINE. Hough Transform One powerful global method for detecting edges
More informationThe radius for a regular polygon is the same as the radius of the circumscribed circle.
Perimeter and Area The perimeter and area of geometric shapes are basic properties that we need to know. The more complex a shape is, the more complex the process can be in finding its perimeter and area.
More informationConstant Time Queries on Uniformly Distributed Points on a Hemisphere
Constant Time Queries on Uniformly Distributed Points on a Hemisphere Mel Slater Department of Computer Science University College London, UK Abstract A set of uniformly distributed points on a hemisphere
More informationGeometry AP Book 8, Part 2: Unit 7
Geometry P ook 8, Part 2: Unit 7 P ook G8-7 page 168 1. base # s V F 6 9 5 4 8 12 6 C 5 10 15 7 6 12 18 8 8 16 24 10 n n-agon n 2n n n + 2 2. 4; 5; 8; 5; No. a) 4 6 6 4 = 24 8 e) ii) top, and faces iii)
More informationZIQIAO YE August 2010
Beteckning: Akademin för teknik och miljö Tessellation of the sphere with different compression techniques ZIQIAO YE August 2010 Bachelor Thesis, 15 hp, C Computer Science Computer Science program Examiner:
More informationUNIT 11 VOLUME AND THE PYTHAGOREAN THEOREM
UNIT 11 VOLUME AND THE PYTHAGOREAN THEOREM INTRODUCTION In this Unit, we will use the idea of measuring volume that we studied to find the volume of various 3 dimensional figures. We will also learn about
More informationLambertian model of reflectance II: harmonic analysis. Ronen Basri Weizmann Institute of Science
Lambertian model of reflectance II: harmonic analysis Ronen Basri Weizmann Institute of Science Illumination cone What is the set of images of an object under different lighting, with any number of sources?
More informationAn Equal Area Quad-tree Global Discrete Grid Subdivision Algorithm Based on Latitude and Longitude Lines
Send Orders for Reprints to reprints@benthamscience.ae The Open Cybernetics & Systemics Journal, 04, 8, 889-895 889 Open Access An Equal Area Quad-tree Global Discrete Grid Subdivision Algorithm Based
More informationOn the Number of Tilings of a Square by Rectangles
University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange University of Tennessee Honors Thesis Projects University of Tennessee Honors Program 5-2012 On the Number of Tilings
More informationDigital Image Processing. Lecture 6
Digital Image Processing Lecture 6 (Enhancement in the Frequency domain) Bu-Ali Sina University Computer Engineering Dep. Fall 2016 Image Enhancement In The Frequency Domain Outline Jean Baptiste Joseph
More informationAdvanced phase retrieval: maximum likelihood technique with sparse regularization of phase and amplitude
Advanced phase retrieval: maximum likelihood technique with sparse regularization of phase and amplitude A. Migukin *, V. atkovnik and J. Astola Department of Signal Processing, Tampere University of Technology,
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 informationWeek 9: Planar and non-planar graphs. 1st and 3rd of November, 2017
(1/26) MA284 : Discrete Mathematics Week 9: Planar and non-planar graphs http://www.maths.nuigalway.ie/~niall/ma284/ 1st and 3rd of November, 2017 1 Recall... planar graphs and Euler s formula 2 Non-planar
More information3.1. Solution for white Gaussian noise
Low complexity M-hypotheses detection: M vectors case Mohammed Nae and Ahmed H. Tewk Dept. of Electrical Engineering University of Minnesota, Minneapolis, MN 55455 mnae,tewk@ece.umn.edu Abstract Low complexity
More informationSubspace Clustering with Global Dimension Minimization And Application to Motion Segmentation
Subspace Clustering with Global Dimension Minimization And Application to Motion Segmentation Bryan Poling University of Minnesota Joint work with Gilad Lerman University of Minnesota The Problem of Subspace
More informationComputer Vision: Lecture 3
Computer Vision: Lecture 3 Carl Olsson 2019-01-29 Carl Olsson Computer Vision: Lecture 3 2019-01-29 1 / 28 Todays Lecture Camera Calibration The inner parameters - K. Projective vs. Euclidean Reconstruction.
More informationApproximate Nearest Neighbors. CS 510 Lecture #24 April 18 th, 2014
Approximate Nearest Neighbors CS 510 Lecture #24 April 18 th, 2014 How is the assignment going? 2 hcp://breckon.eu/toby/demos/videovolumes/ Review: Video as Data Cube Mathematically, the cube can be viewed
More informationREGULAR POLYTOPES REALIZED OVER Q
REGULAR POLYTOPES REALIZED OVER Q TREVOR HYDE A regular polytope is a d-dimensional generalization of a regular polygon and a Platonic solid. Roughly, they are convex geometric objects with maximal rotational
More informationThe Volume of a Platonic Solid
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-007 The Volume of a Platonic Solid Cindy Steinkruger
More information