Haar Basis Wavelets and Morlet Wavelets
|
|
- Wesley Barber
- 5 years ago
- Views:
Transcription
1 Haar Basis Wavelets and Morlet Wavelets September 9 th, 05 Professor Davi Geiger. The Haar transform, which is one of the earliest transform functions proposed, was proposed in 90 by a Hungarian mathematician Alfred Haar. It is found effective as it provides a simple approach for analyzing the local aspects of a signal. Say we start with an image slice (one dimensional) of size NN, so we can write the image as II = {II(), II(),, II( NN )} oooo {II(ii); ii =,, NN } () Recursive Process of Decomposing an Image in terms of Sums and Differences Let us now group the image by consecutive pairs, i.e., {II(ii); ii =,, NN } {[II(), II()], [II(3), II(4)],, [II(ii ), II(ii)],, [II( NN ), II( NN )]} () We can apply the following linear transformation for each pair [II(ii ), II(ii)], i.e., ss (ii) dd (ii) = HH II(ii ) = II(ii) ) II(ii (3) II(ii) So ss (ii) stores the sum of the two elements in a pair and dd (ii) stores the difference of the two elements in a pair. The upper index refers to this transformation being the first step in a process that follows. Of course, this is invertible, i.e., II(ii ) = II(ii) (ii) ss dd (4) (ii) So we can get back the original set of pairs, the original image {II(ii); ii =,, NN }. Thus, we can represent the image II, of size NN, as NN pairs of the form II(ii) {[ss (), dd ()], [ss (), dd ()],, [ss (ii), dd (ii)],, [ss ( NN ), dd ( NN )]} (5) We can then store the set DD = {dd (), dd (), dd (ii),, dd ( NN )} of size NN and we decompose further the complementary set SS = {ss (), ss (), ss (ii),, ss ( NN )} (6) We interpret SS as a new image of half the size of the original one and we represent the set of pairs
2 SS = {[ss (), ss ()],, [ss (ii ), ss (ii)],, [ss ( NN ), ss ( NN )]} (7) and we apply the reversible linear transformation HH to each pair [ss (ii ), ss (ii)], i.e., ss (ii) dd (ii) = (ii ) ss ss (8) (ii) The upper index refers to the second step in the process. We can then represent the image SS, of size NN, as SS (ii) {[ss (), dd ()], [ss (), dd ()],, [ss (ii), dd (ii)],, [ss ( NN ), dd ( NN )]} (9) Likewise in the first step, we can store the set DD = {dd (), dd (), dd (ii),, dd ( NN )} of size NN and decompose further the set SS = {ss (), ss (), ss (ii),, ss ( NN )} (0) The process can be illustrated for an image of size 4 = 6 pixels as follows Image Step Step II(ii) II II II3 II4 II5 II6 II7 II8 II9 II0 II II II3 II4 II5 II6 ss ss ss 3 ss 4 ss 5 ss 6 ss 7 ss 8 dd dd dd 3 dd 4 dd 5 dd 6 dd 7 dd 8 SS DD ss ss ss 3 ss 4 dd dd dd 3 dd 4 SS DD II = {SS, DD, DD } We now apply the third step of the process on the image set SS, i.e., for each pair [ss (ii ), ss (ii)] we apply the linear transformation ss3 (ii) dd 3 (ii) = (ii ) ss ss () (ii) Leading to two sets DD 3 = {dd 3 (), dd 3 (), dd 3 (ii),, dd 3 ( NN 3 )} ; SS 3 = {ss 3 (), ss 3 (), ss 3 (ii),, ss 3 ( NN 3 )} () Up to the third layer of this process we are left with the following new representation for the image
3 {II(ii); ii =,, NN } {DD, DD, DD 3, SS 3 } (3) where DD is of size NN, DD of size NN, and DD 3, SS 3 of sizes NN 3 each. All together, of course, there are NN numbers. Every step is reversible, so we can quickly recover the image II from {DD, DD, DD 3, SS 3 }. We could repeat this process up to N steps to obtain DD, DD,, DD ii,, DD NN, SS NN, where SS NN would be one number, representing the sum of all the image values. The Haar Transformation Reviewing our process, we started with equation (), applying a linear transformation HH to each pair of data, or ss (ii) dd (ii) = HH II(ii ) = II(ii) ) II(ii (4) II(ii) We then described the next step as the same linear transformation, HH, applied to the image SS of the first layer, i.e., ss (ii) dd (ii) = HH ss (ii ) ss. We can represent the first and second layer (ii) processes as a linear transformation directly on each four elements of the image (concatenating the two steps and concatenating two pairs of image data), i.e., ss (ii) dd II( ii 3) (ii) dd = HH (ii) 4 II( ii ) II( = II( ii 3) ii ) 4 II( ii ) 0 0 II( ii ) dd (ii + ) II( ii) 0 0 II( ii) (5) In this way, we keep simultaneously both layers (both steps) of the representation of the image. Repeating this process towards the third layer, we apply the linear transformation, HH, to the image SS of the second layer, i.e., ss3 (ii) dd 3 (ii) = (ii ) ss ss. We can (ii) represent this third layer process as a linear transformation directly on each eight elements of the image (concatenating the three steps and concatenating four pairs of image data), i.e.,
4 ss 3 (ii) dd 3 II( ii 7) (ii) II( ii 6) dd (ii) II( ii 5) dd (ii + ) II( dd (ii) = HH ii 4) 8 II( ii 3) dd (ii + ) II( ii ) dd (ii + ) II( ii ) dd (ii + 3) II( ii) II( ii 7) II( ii 6) = II( ii 5) II( ii 4) II( ii 3) (6) II( ii ) II( ii ) II( ii) More generally one can write the linear transformation HH ii that is applied to each image subset data of size ii, to be HH ii = HH ii [ ] II ii [ ] where II ii is the identify matrix of size i and is the Kronecker product. If AA is an m x n matrix and BB is a p x q matrix, the Kronecker product is given by the mp x nq matrix aa BB aa nn BB AA BB = aa mm BB aa mmmm BB The output of applying HH kk to the full set of image windows of size kk is a representation of the image in the form {DD, DD,, DD kk, SS kk }. The Haar Basis (the function approach) So far we have represented an image as a vector and the wavelet transformation as a matrix multiplication to the vector. We now consider the image as a function of the pixels. The wavelets are functions of the pixels. The result of applying the wavelet transform to the image is then described by the set of convolutions of two functions, the image and each of the wavelets. Let us be more precise. Defining the Haar wavelet basis as a mother wavelet and scaling it by σσ and translating it by b we have
5 ψψ(xx) = 0 < xx < xx < 0 otherwise ψψ σσ (xx bb) = σσ bb ψψ xx σσ Haar basis ψ(x) Two wavelet properties are required : average is zero and normalization 0 = ψψ σσ (xx bb) dddd and = ψψ σσ (xx bb) dddd For discrete Wavelets is common to assign σσ = jj. Examples with b=0, for σσ = = 0 < xx < oooo 0 < xx < 4 ψψ /(xx) = ψψ(xx) = xx < oooo 4 xx < 0 otherwise σσ = = ψψ (xx) = 0 < xx ψψ xx = < oooo 0 < xx < xx < oooo xx < 0 otherwise σσ = 3 = 8 ψψ (xx) = 0 < xx 8 ψψ xx 8 = 8 8 < oooo 0 < xx < 4 8 xx < oooo 4 xx < otherwise We can write a row associated to "dd jj (ii)" at (4), (5), (6) as the Haar basis convolution at x=0, i.e., ψψ σσ II (xx) = ψψ σσ (xx ) II(xx xx ) dddd for σσ = jj, jj =, 3,,, 0,,, 3,, respectively (and bb = 0). The Haar wavelet transform is then
6 WW ψψ ff(σσ, bb) = ψψ σσ (xx bb) II(xx) dddd = σσ ψψ xx bb II(xx) dddd σσ Where ψψ σσ (xx bb) is the complex conjugate of ψψ σσ (xx bb), and for the discrete case, σσ = jj is called the dyadic dilation and bb = kk jj is the dyadic position. So the basis is described by integers j,k. The Morlet Wavelet Basis (a function approach) ψψ DD σσ (xx xx 0 ) = σσ ψψ σσ DD xx xx 0 σσ = CC σσ eeii ππ ξξ σσ (xx xx 0 ) CC ee (xx xx 0 ) σσ And we can plot the real and complex functions (e.g. for σ=, ξ=4 centered in x0=48 pixels) If we convolve the Morlet with a step edge image, I, i.e., (II ψψ σσ DD )(yy) = xx II(yy xx) ψψ σσ DD (xx) Step Edge Real part of the Convolution Imaginary part The zeros of the real part help detect edges as well as the maximum of the complex part.
7 The Morlet in D (two dimensions), assuming σ=σx=σy, is defined as ψψ σσ,θθ (uu) = CC ππ σσ eeii ξξ σσ (uu.ee θθ ) CC ee uu where uu = (xx, yy), ee θθ = (cos θθ, sin θθ) and ee iiππ ξξ (uu.ee θθ ) = cccccc ππ ξξ σσ (uu. ee θθ) + ii ssssss ππ ξξ σσ (uu. ee θθ). So uu = xx + yy. Let us work with ξξ = 4 (just one peak), so ψψ σσ,θθ (uu) = CC σσ eeii σσ ππ σσ (uu.ee θθ ) CC ee uu σσ Two wavelet properties are required : average is zero and normalization 0 = ψψ σσ (xx bb) dddd Which allow us to obtain CC and CC as follows: First, we obtain CC, real variable, by adjusting it so that and = ψψ σσ (xx bb) dddd 0 = ψψ(uu ) xx yy 0 NN xx NN yy = ee ii ππ σσ (uu.ee θθ ) CC ee uu xx= yy= NN xx NN yy ππ σσ (uu.ee θθ ) ee uu σσ σσ CC = xx= yy= eeii NN NN ee uu xx yy xx= yy= σσ and CC is adjusted so that NN xx NN yy = ψψ(uu )ψψ (uu ) xx= yy= where ψψ (uu ) = CC σσ ee ii ππ ξξ σσ (uu.ee θθ ) CC ee uu σσ is the conjugate of ψψ(uu ).
Wave Motion. Chapter 14 of Essential University Physics, Richard Wolfson, 3 rd Edition
Wave Motion Chapter 14 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 Waves: propagation of energy, not particles 2 Longitudinal Waves: disturbance is along the direction of wave propagation
More informationVariations. ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra
Variations ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra Last Time Probability Density Functions Normal Distribution Expectation / Expectation of a function Independence Uncorrelated
More informationTime Domain Analysis of Linear Systems Ch2. University of Central Oklahoma Dr. Mohamed Bingabr
Time Domain Analysis of Linear Systems Ch2 University of Central Oklahoma Dr. Mohamed Bingabr Outline Zero-input Response Impulse Response h(t) Convolution Zero-State Response System Stability System Response
More informationChapter 22 : Electric potential
Chapter 22 : Electric potential What is electric potential? How does it relate to potential energy? How does it relate to electric field? Some simple applications What does it mean when it says 1.5 Volts
More informationSECTION 8: ROOT-LOCUS ANALYSIS. ESE 499 Feedback Control Systems
SECTION 8: ROOT-LOCUS ANALYSIS ESE 499 Feedback Control Systems 2 Introduction Introduction 3 Consider a general feedback system: Closed-loop transfer function is KKKK ss TT ss = 1 + KKKK ss HH ss GG ss
More informationRotational Motion. Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition
Rotational Motion Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 We ll look for a way to describe the combined (rotational) motion 2 Angle Measurements θθ ss rr rrrrrrrrrrrrrr
More informationWork, Energy, and Power. Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition
Work, Energy, and Power Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 With the knowledge we got so far, we can handle the situation on the left but not the one on the right.
More information(2) Orbital angular momentum
(2) Orbital angular momentum Consider SS = 0 and LL = rr pp, where pp is the canonical momentum Note: SS and LL are generators for different parts of the wave function. Note: from AA BB ii = εε iiiiii
More informationCHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 1 5.1 X-Ray Scattering 5.2 De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles 5.6 Uncertainty Principle Topics 5.7
More information(1) Introduction: a new basis set
() Introduction: a new basis set In scattering, we are solving the S eq. for arbitrary VV in integral form We look for solutions to unbound states: certain boundary conditions (EE > 0, plane and spherical
More informationGrover s algorithm. We want to find aa. Search in an unordered database. QC oracle (as usual) Usual trick
Grover s algorithm Search in an unordered database Example: phonebook, need to find a person from a phone number Actually, something else, like hard (e.g., NP-complete) problem 0, xx aa Black box ff xx
More informationWorksheets for GCSE Mathematics. Quadratics. mr-mathematics.com Maths Resources for Teachers. Algebra
Worksheets for GCSE Mathematics Quadratics mr-mathematics.com Maths Resources for Teachers Algebra Quadratics Worksheets Contents Differentiated Independent Learning Worksheets Solving x + bx + c by factorisation
More informationQuantum Mechanics. An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc.
Quantum Mechanics An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc. Fall 2018 Prof. Sergio B. Mendes 1 CHAPTER 3 Experimental Basis of
More informationMathematics Ext 2. HSC 2014 Solutions. Suite 403, 410 Elizabeth St, Surry Hills NSW 2010 keystoneeducation.com.
Mathematics Ext HSC 4 Solutions Suite 43, 4 Elizabeth St, Surry Hills NSW info@keystoneeducation.com.au keystoneeducation.com.au Mathematics Extension : HSC 4 Solutions Contents Multiple Choice... 3 Question...
More informationA Posteriori Error Estimates For Discontinuous Galerkin Methods Using Non-polynomial Basis Functions
Lin Lin A Posteriori DG using Non-Polynomial Basis 1 A Posteriori Error Estimates For Discontinuous Galerkin Methods Using Non-polynomial Basis Functions Lin Lin Department of Mathematics, UC Berkeley;
More informationNumerical Solution of Fredholm and Volterra Integral Equations of the First Kind Using Wavelets bases
The Journal of Mathematics and Computer Science Available online at http://www.tjmcs.com The Journal of Mathematics and Computer Science Vol.5 No.4 (22) 337-345 Numerical Solution of Fredholm and Volterra
More informationPhysics 371 Spring 2017 Prof. Anlage Review
Physics 71 Spring 2017 Prof. Anlage Review Special Relativity Inertial vs. non-inertial reference frames Galilean relativity: Galilean transformation for relative motion along the xx xx direction: xx =
More informationMath 171 Spring 2017 Final Exam. Problem Worth
Math 171 Spring 2017 Final Exam Problem 1 2 3 4 5 6 7 8 9 10 11 Worth 9 6 6 5 9 8 5 8 8 8 10 12 13 14 15 16 17 18 19 20 21 22 Total 8 5 5 6 6 8 6 6 6 6 6 150 Last Name: First Name: Student ID: Section:
More informationWorksheets for GCSE Mathematics. Algebraic Expressions. Mr Black 's Maths Resources for Teachers GCSE 1-9. Algebra
Worksheets for GCSE Mathematics Algebraic Expressions Mr Black 's Maths Resources for Teachers GCSE 1-9 Algebra Algebraic Expressions Worksheets Contents Differentiated Independent Learning Worksheets
More informationLesson 7: Linear Transformations Applied to Cubes
Classwork Opening Exercise Consider the following matrices: AA = 1 2 0 2, BB = 2, and CC = 2 2 4 0 0 2 2 a. Compute the following determinants. i. det(aa) ii. det(bb) iii. det(cc) b. Sketch the image of
More informationTECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMN-FOOTING PROPERTIES
COMPUTERS AND STRUCTURES, INC., FEBRUARY 2016 TECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMN-FOOTING PROPERTIES Introduction This technical note
More informationModule 7:Data Representation Lecture 35: Wavelets. The Lecture Contains: Wavelets. Discrete Wavelet Transform (DWT) Haar wavelets: Example
The Lecture Contains: Wavelets Discrete Wavelet Transform (DWT) Haar wavelets: Example Haar wavelets: Theory Matrix form Haar wavelet matrices Dimensionality reduction using Haar wavelets file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture35/35_1.htm[6/14/2012
More informationQuantum state measurement
Quantum state measurement Introduction The rotation properties of light fields of spin are described by the 3 3 representation of the 0 0 SO(3) group, with the generators JJ ii we found in class, for instance
More informationGradient expansion formalism for generic spin torques
Gradient expansion formalism for generic spin torques Atsuo Shitade RIKEN Center for Emergent Matter Science Atsuo Shitade, arxiv:1708.03424. Outline 1. Spintronics a. Magnetoresistance and spin torques
More informationEstimate by the L 2 Norm of a Parameter Poisson Intensity Discontinuous
Research Journal of Mathematics and Statistics 6: -5, 24 ISSN: 242-224, e-issn: 24-755 Maxwell Scientific Organization, 24 Submied: September 8, 23 Accepted: November 23, 23 Published: February 25, 24
More informationAngular Momentum, Electromagnetic Waves
Angular Momentum, Electromagnetic Waves Lecture33: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay As before, we keep in view the four Maxwell s equations for all our discussions.
More informationGeneral Strong Polarization
General Strong Polarization Madhu Sudan Harvard University Joint work with Jaroslaw Blasiok (Harvard), Venkatesan Gurswami (CMU), Preetum Nakkiran (Harvard) and Atri Rudra (Buffalo) December 4, 2017 IAS:
More informationCharge carrier density in metals and semiconductors
Charge carrier density in metals and semiconductors 1. Introduction The Hall Effect Particles must overlap for the permutation symmetry to be relevant. We saw examples of this in the exchange energy in
More informationSecondary 3H Unit = 1 = 7. Lesson 3.3 Worksheet. Simplify: Lesson 3.6 Worksheet
Secondary H Unit Lesson Worksheet Simplify: mm + 2 mm 2 4 mm+6 mm + 2 mm 2 mm 20 mm+4 5 2 9+20 2 0+25 4 +2 2 + 2 8 2 6 5. 2 yy 2 + yy 6. +2 + 5 2 2 2 0 Lesson 6 Worksheet List all asymptotes, holes and
More informationLecture 22 Highlights Phys 402
Lecture 22 Highlights Phys 402 Scattering experiments are one of the most important ways to gain an understanding of the microscopic world that is described by quantum mechanics. The idea is to take a
More informationLesson 2. Homework Problem Set Sample Solutions S.19
Homework Problem Set Sample Solutions S.9. Below are formulas Britney Gallivan created when she was doing her paper-folding extra credit assignment. his formula determines the minimum width, WW, of a square
More information10.4 The Cross Product
Math 172 Chapter 10B notes Page 1 of 9 10.4 The Cross Product The cross product, or vector product, is defined in 3 dimensions only. Let aa = aa 1, aa 2, aa 3 bb = bb 1, bb 2, bb 3 then aa bb = aa 2 bb
More informationBIOE 198MI Biomedical Data Analysis. Spring Semester Lab 4: Introduction to Probability and Random Data
BIOE 98MI Biomedical Data Analysis. Spring Semester 209. Lab 4: Introduction to Probability and Random Data A. Random Variables Randomness is a component of all measurement data, either from acquisition
More informationRadiation. Lecture40: Electromagnetic Theory. Professor D. K. Ghosh, Physics Department, I.I.T., Bombay
Radiation Zone Approximation We had seen that the expression for the vector potential for a localized cuent distribution is given by AA (xx, tt) = μμ 4ππ ee iiiiii dd xx eeiiii xx xx xx xx JJ (xx ) In
More information1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2,
1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. x = 4, x = 3, x = 2, x = 1, x = 1, x = 2, x = 3, x = 4, x = 5 b. Find the value(s)
More informationProf. Dr.-Ing. Armin Dekorsy Department of Communications Engineering. Stochastic Processes and Linear Algebra Recap Slides
Prof. Dr.-Ing. Armin Dekorsy Department of Communications Engineering Stochastic Processes and Linear Algebra Recap Slides Stochastic processes and variables XX tt 0 = XX xx nn (tt) xx 2 (tt) XX tt XX
More informationModule 7 (Lecture 25) RETAINING WALLS
Module 7 (Lecture 25) RETAINING WALLS Topics Check for Bearing Capacity Failure Example Factor of Safety Against Overturning Factor of Safety Against Sliding Factor of Safety Against Bearing Capacity Failure
More informationKinetic Model Parameter Estimation for Product Stability: Non-uniform Finite Elements and Convexity Analysis
Kinetic Model Parameter Estimation for Product Stability: Non-uniform Finite Elements and Convexity Analysis Mark Daichendt, Lorenz Biegler Carnegie Mellon University Pittsburgh, PA 15217 Ben Weinstein,
More informationGravitation. Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition
Gravitation Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 What you are about to learn: Newton's law of universal gravitation About motion in circular and other orbits How to
More informationLecture No. 1 Introduction to Method of Weighted Residuals. Solve the differential equation L (u) = p(x) in V where L is a differential operator
Lecture No. 1 Introduction to Method of Weighted Residuals Solve the differential equation L (u) = p(x) in V where L is a differential operator with boundary conditions S(u) = g(x) on Γ where S is a differential
More informationSpecialist Mathematics 2019 v1.2
181314 Mensuration circumference of a circle area of a parallelogram CC = ππππ area of a circle AA = ππrr AA = h area of a trapezium AA = 1 ( + )h area of a triangle AA = 1 h total surface area of a cone
More informationDiscrete Structures Lecture Solving Congruences. mathematician of the eighteenth century). Also, the equation gggggg(aa, bb) =
First Introduction Our goal is to solve equations having the form aaaa bb (mmmmmm mm). However, first we must discuss the last part of the previous section titled gcds as Linear Combinations THEOREM 6
More informationLarge Scale Data Analysis Using Deep Learning
Large Scale Data Analysis Using Deep Learning Linear Algebra U Kang Seoul National University U Kang 1 In This Lecture Overview of linear algebra (but, not a comprehensive survey) Focused on the subset
More information(1) Correspondence of the density matrix to traditional method
(1) Correspondence of the density matrix to traditional method New method (with the density matrix) Traditional method (from thermal physics courses) ZZ = TTTT ρρ = EE ρρ EE = dddd xx ρρ xx ii FF = UU
More informationYang-Hwan Ahn Based on arxiv:
Yang-Hwan Ahn (CTPU@IBS) Based on arxiv: 1611.08359 1 Introduction Now that the Higgs boson has been discovered at 126 GeV, assuming that it is indeed exactly the one predicted by the SM, there are several
More informationThermodynamic Cycles
Thermodynamic Cycles Content Thermodynamic Cycles Carnot Cycle Otto Cycle Rankine Cycle Refrigeration Cycle Thermodynamic Cycles Carnot Cycle Derivation of the Carnot Cycle Efficiency Otto Cycle Otto Cycle
More informationLesson 3: Networks and Matrix Arithmetic
Opening Exercise Suppose a subway line also connects the four cities. Here is the subway and bus line network. The bus routes connecting the cities are represented by solid lines, and the subway routes
More informationFirst Semester Topics:
First Semester Topics: NONCALCULATOR: 1. Determine if the following equations are polynomials. If they are polynomials, determine the degree, leading coefficient and constant term. a. ff(xx) = 3xx b. gg(xx)
More informationSupport Vector Machines. CSE 4309 Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington
Support Vector Machines CSE 4309 Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington 1 A Linearly Separable Problem Consider the binary classification
More informationTerms of Use. Copyright Embark on the Journey
Terms of Use All rights reserved. No part of this packet may be reproduced, stored in a retrieval system, or transmitted in any form by any means - electronic, mechanical, photo-copies, recording, or otherwise
More informationChem 263 Winter 2018 Problem Set #2 Due: February 16
Chem 263 Winter 2018 Problem Set #2 Due: February 16 1. Use size considerations to predict the crystal structures of PbF2, CoF2, and BeF2. Do your predictions agree with the actual structures of these
More informationHopfield Network for Associative Memory
CSE 5526: Introduction to Neural Networks Hopfield Network for Associative Memory 1 The next few units cover unsupervised models Goal: learn the distribution of a set of observations Some observations
More informationConstitutive Relations of Stress and Strain in Stochastic Finite Element Method
American Journal of Computational and Applied Mathematics 15, 5(6): 164-173 DOI: 1.593/j.ajcam.1556. Constitutive Relations of Stress and Strain in Stochastic Finite Element Method Drakos Stefanos International
More informationThe Elasticity of Quantum Spacetime Fabric. Viorel Laurentiu Cartas
The Elasticity of Quantum Spacetime Fabric Viorel Laurentiu Cartas The notion of gravitational force is replaced by the curvature of the spacetime fabric. The mass tells to spacetime how to curve and the
More informationEPR paradox, Bell inequality, etc.
EPR paradox, Bell inequality, etc. Compatible and incompatible observables AA, BB = 0, then compatible, can measure simultaneously, can diagonalize in one basis commutator, AA, BB AAAA BBBB If we project
More informationCHAPTER 2 Special Theory of Relativity
CHAPTER 2 Special Theory of Relativity Fall 2018 Prof. Sergio B. Mendes 1 Topics 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 Inertial Frames of Reference Conceptual and Experimental
More informationCHAPTER 4 Structure of the Atom
CHAPTER 4 Structure of the Atom Fall 2018 Prof. Sergio B. Mendes 1 Topics 4.1 The Atomic Models of Thomson and Rutherford 4.2 Rutherford Scattering 4.3 The Classic Atomic Model 4.4 The Bohr Model of the
More informationWorksheets for GCSE Mathematics. Solving Equations. Mr Black's Maths Resources for Teachers GCSE 1-9. Algebra
Worksheets for GCSE Mathematics Solving Equations Mr Black's Maths Resources for Teachers GCSE 1-9 Algebra Equations Worksheets Contents Differentiated Independent Learning Worksheets Solving Equations
More informationNow, suppose that the signal is of finite duration (length) NN. Specifically, the signal is zero outside the range 0 nn < NN. Then
EE 464 Discrete Fourier Transform Fall 2018 Read Text, Chapter 4. Recall that for a complex-valued discrete-time signal, xx(nn), we can compute the Z-transform, XX(zz) = nn= xx(nn)zz nn. Evaluating on
More informationUniversity of North Georgia Department of Mathematics
Instructor: Berhanu Kidane Course: Precalculus Math 13 University of North Georgia Department of Mathematics Text Books: For this course we use free online resources: See the folder Educational Resources
More informationECE 6540, Lecture 06 Sufficient Statistics & Complete Statistics Variations
ECE 6540, Lecture 06 Sufficient Statistics & Complete Statistics Variations Last Time Minimum Variance Unbiased Estimators Sufficient Statistics Proving t = T(x) is sufficient Neyman-Fischer Factorization
More informationIntegrating Rational functions by the Method of Partial fraction Decomposition. Antony L. Foster
Integrating Rational functions by the Method of Partial fraction Decomposition By Antony L. Foster At times, especially in calculus, it is necessary, it is necessary to express a fraction as the sum of
More informationModule 7 (Lecture 27) RETAINING WALLS
Module 7 (Lecture 27) RETAINING WALLS Topics 1.1 RETAINING WALLS WITH METALLIC STRIP REINFORCEMENT Calculation of Active Horizontal and vertical Pressure Tie Force Factor of Safety Against Tie Failure
More informationPower of Adiabatic Quantum Computation
Power of Adiabatic Quantum Computation Itay Hen Information Sciences Institute, USC Physics Colloquium, USC February 10, 2014 arxiv preprints: 1207.1712, 1301.4956, 1307.6538, 1401.5172 Motivation in what
More informationBHASVIC MαTHS. Skills 1
PART A: Integrate the following functions with respect to x: (a) cos 2 2xx (b) tan 2 xx (c) (d) 2 PART B: Find: (a) (b) (c) xx 1 2 cosec 2 2xx 2 cot 2xx (d) 2cccccccccc2 2xx 2 ccccccccc 5 dddd Skills 1
More informationLesson 8: Complex Number Division
Student Outcomes Students determine the modulus and conjugate of a complex number. Students use the concept of conjugate to divide complex numbers. Lesson Notes This is the second day of a two-day lesson
More informationNational 5 Mathematics. Practice Paper E. Worked Solutions
National 5 Mathematics Practice Paper E Worked Solutions Paper One: Non-Calculator Copyright www.national5maths.co.uk 2015. All rights reserved. SQA Past Papers & Specimen Papers Working through SQA Past
More informationReview for Exam Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa
Review for Exam2 11. 13. 2015 Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa Assistant Research Scientist IIHR-Hydroscience & Engineering, University
More informationINTRODUCTION TO QUANTUM MECHANICS
4 CHAPTER INTRODUCTION TO QUANTUM MECHANICS 4.1 Preliminaries: Wave Motion and Light 4.2 Evidence for Energy Quantization in Atoms 4.3 The Bohr Model: Predicting Discrete Energy Levels in Atoms 4.4 Evidence
More informationModeling of a non-physical fish barrier
University of Massachusetts - Amherst ScholarWorks@UMass Amherst International Conference on Engineering and Ecohydrology for Fish Passage International Conference on Engineering and Ecohydrology for Fish
More informationUnit WorkBook 4 Level 4 ENG U8 Mechanical Principles 2018 UniCourse Ltd. All Rights Reserved. Sample
2018 UniCourse Ltd. A Rights Reserved. Pearson BTEC Levels 4 Higher Nationals in Engineering (RQF) Unit 8: Mechanical Principles Unit Workbook 4 in a series of 4 for this unit Learning Outcome 4 Translational
More informationElastic light scattering
Elastic light scattering 1. Introduction Elastic light scattering in quantum mechanics Elastic scattering is described in quantum mechanics by the Kramers Heisenberg formula for the differential cross
More informationOn The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time Delays
Journal of Mathematics and System Science 6 (216) 194-199 doi: 1.17265/2159-5291/216.5.3 D DAVID PUBLISHING On The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time
More informationClassical RSA algorithm
Classical RSA algorithm We need to discuss some mathematics (number theory) first Modulo-NN arithmetic (modular arithmetic, clock arithmetic) 9 (mod 7) 4 3 5 (mod 7) congruent (I will also use = instead
More informationGeneral Strong Polarization
General Strong Polarization Madhu Sudan Harvard University Joint work with Jaroslaw Blasiok (Harvard), Venkatesan Gurswami (CMU), Preetum Nakkiran (Harvard) and Atri Rudra (Buffalo) May 1, 018 G.Tech:
More information10.1 Three Dimensional Space
Math 172 Chapter 10A notes Page 1 of 12 10.1 Three Dimensional Space 2D space 0 xx.. xx-, 0 yy yy-, PP(xx, yy) [Fig. 1] Point PP represented by (xx, yy), an ordered pair of real nos. Set of all ordered
More informationSECTION 7: FAULT ANALYSIS. ESE 470 Energy Distribution Systems
SECTION 7: FAULT ANALYSIS ESE 470 Energy Distribution Systems 2 Introduction Power System Faults 3 Faults in three-phase power systems are short circuits Line-to-ground Line-to-line Result in the flow
More informationLecture 3. Logic Predicates and Quantified Statements Statements with Multiple Quantifiers. Introduction to Proofs. Reading (Epp s textbook)
Lecture 3 Logic Predicates and Quantified Statements Statements with Multiple Quantifiers Reading (Epp s textbook) 3.1-3.3 Introduction to Proofs Reading (Epp s textbook) 4.1-4.2 1 Propositional Functions
More informationABSTRACT OF THE DISSERTATION
A Study of an N Molecule-Quantized Radiation Field-Hamiltonian BY MICHAEL THOMAS TAVIS DOCTOR OF PHILOSOPHY, GRADUATE PROGRAM IN PHYSICS UNIVERSITY OF CALIFORNIA, RIVERSIDE, DECEMBER 1968 PROFESSOR FREDERICK
More informationCrash course Verification of Finite Automata Binary Decision Diagrams
Crash course Verification of Finite Automata Binary Decision Diagrams Exercise session 10 Xiaoxi He 1 Equivalence of representations E Sets A B A B Set algebra,, ψψ EE = 1 ψψ AA = ff ψψ BB = gg ψψ AA BB
More informationT.4 Applications of Right Angle Trigonometry
424 section T4 T.4 Applications of Right Angle Trigonometry Solving Right Triangles Geometry of right triangles has many applications in the real world. It is often used by carpenters, surveyors, engineers,
More informationA Step Towards the Cognitive Radar: Target Detection under Nonstationary Clutter
A Step Towards the Cognitive Radar: Target Detection under Nonstationary Clutter Murat Akcakaya Department of Electrical and Computer Engineering University of Pittsburgh Email: akcakaya@pitt.edu Satyabrata
More informationLecture 3. STAT161/261 Introduction to Pattern Recognition and Machine Learning Spring 2018 Prof. Allie Fletcher
Lecture 3 STAT161/261 Introduction to Pattern Recognition and Machine Learning Spring 2018 Prof. Allie Fletcher Previous lectures What is machine learning? Objectives of machine learning Supervised and
More informationDiscrete scale invariance and Efimov bound states in Weyl systems with coexistence of electron and hole carriers
Institute of Advanced Study, Tsinghua, April 19, 017 Discrete scale invariance and Efimov bound states in Weyl systems with coexistence of electron and hole carriers Haiwen Liu ( 刘海文 ) Beijing Normal University
More informationMultiple Regression Analysis: Inference ECONOMETRICS (ECON 360) BEN VAN KAMMEN, PHD
Multiple Regression Analysis: Inference ECONOMETRICS (ECON 360) BEN VAN KAMMEN, PHD Introduction When you perform statistical inference, you are primarily doing one of two things: Estimating the boundaries
More informationLesson 24: True and False Number Sentences
NYS COMMON CE MATHEMATICS CURRICULUM Lesson 24 6 4 Student Outcomes Students identify values for the variables in equations and inequalities that result in true number sentences. Students identify values
More informationCHEMISTRY Spring, 2018 QUIZ 1 February 16, NAME: HANS ZIMMER Score /20
QUIZ 1 February 16, 018 NAME: HANS ZIMMER Score /0 [Numbers without decimal points are to be considered infinitely precise. Show reasonable significant figures and proper units. In particular, use generally
More informationGraduate School of Engineering, Kyoto University, Kyoto daigaku-katsura, Nishikyo-ku, Kyoto, Japan.
On relationship between contact surface rigidity and harmonic generation behavior in composite materials with mechanical nonlinearity at fiber-matrix interface (Singapore November 2017) N. Matsuda, K.
More informationLecture 3 Transport in Semiconductors
EE 471: Transport Phenomena in Solid State Devices Spring 2018 Lecture 3 Transport in Semiconductors Bryan Ackland Department of Electrical and Computer Engineering Stevens Institute of Technology Hoboken,
More informationLesson 1: Successive Differences in Polynomials
Lesson 1 Lesson 1: Successive Differences in Polynomials Classwork Opening Exercise John noticed patterns in the arrangement of numbers in the table below. 2.4 3.4 4.4 5.4 6.4 5.76 11.56 19.36 29.16 40.96
More informationA A A A A A A A A A A A. a a a a a a a a a a a a a a a. Apples taste amazingly good.
Victorian Handwriting Sheet Aa A A A A A A A A A A A A Aa Aa Aa Aa Aa Aa Aa a a a a a a a a a a a a a a a Apples taste amazingly good. Apples taste amazingly good. Now make up a sentence of your own using
More informationLesson 7: Algebraic Expressions The Commutative and Associative Properties
Algebraic Expressions The Commutative and Associative Properties Classwork Exercise 1 Suzy draws the following picture to represent the sum 3 + 4: Ben looks at this picture from the opposite side of the
More informationData Preprocessing. Jilles Vreeken IRDM 15/ Oct 2015
Data Preprocessing Jilles Vreeken 22 Oct 2015 So, how do you pronounce Jilles Yill-less Vreeken Fray-can Okay, now we can talk. Questions of the day How do we preprocess data before we can extract anything
More informationPHY103A: Lecture # 9
Semester II, 2017-18 Department of Physics, IIT Kanpur PHY103A: Lecture # 9 (Text Book: Intro to Electrodynamics by Griffiths, 3 rd Ed.) Anand Kumar Jha 20-Jan-2018 Summary of Lecture # 8: Force per unit
More informationLecture 15: Scattering Rutherford scattering Nuclear elastic scattering Nuclear inelastic scattering Quantum description The optical model
Lecture 15: Scattering Rutherford scattering Nuclear elastic scattering Nuclear inelastic scattering Quantum description The optical model Lecture 15: Ohio University PHYS7501, Fall 017, Z. Meisel (meisel@ohio.edu)
More informationCDS 101/110: Lecture 6.2 Transfer Functions
CDS 11/11: Lecture 6.2 Transfer Functions November 2, 216 Goals: Continued study of Transfer functions Review Laplace Transform Block Diagram Algebra Bode Plot Intro Reading: Åström and Murray, Feedback
More informationChapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a)
Chapter 6 Chapter 6 opener A. B. C. D. 6 E. 5 F. 8 G. H. I. J.. 7. 8 5. 6 6. 7. y 8. n 9. w z. 5cd.. xy z 5r s t. (x y). (a b) 5. (a) (x y) = x (y) = x 6y x 6y = x (y) = (x y) 6. (a) a (5 a+ b) = a (5
More informationSUPPLEMENTARY INFORMATION
An atomically thin matter-wave beam splitter Christian Brand, Michele Sclafani, Christian Knobloch, Yigal Lilach, Thomas Juffmann, Jani Kotakoski, Clemens Mangler, Andreas Winter, Andrey Turchanin, Jannik
More informationEECE 2150 Circuits and Signals Final Exam Fall 2016 Dec 9
EECE 2150 Circuits and Signals Final Exam Fall 2016 Dec 9 Name: Instructions: Write your name and section number on all pages Closed book, closed notes; Computers and cell phones are not allowed You can
More informationExam Programme VWO Mathematics A
Exam Programme VWO Mathematics A The exam. The exam programme recognizes the following domains: Domain A Domain B Domain C Domain D Domain E Mathematical skills Algebra and systematic counting Relationships
More information