Multi-Scale/Multi-Resolution: Wavelet Transform
|
|
- Christopher Underwood
- 6 years ago
- Views:
Transcription
1 Multi-Scale/Multi-Resolution: Wavelet Transfor
2 Proble with Fourier Fourier analysis -- breaks down a signal into constituent sinusoids of different frequencies. A serious drawback in transforing to the frequency doain, tie inforation is lost. When looking at a Fourier transfor of a signal, it is ipossible to tell when a particular event took place.
3 Gabor s Proposal: Short Tie Fourier Transfor STFT x ' ' πft ' ' t, f [ x t g t t] e dt Requireents: Signal in tie doain: requires short tie window to depict features of signal. Signal in frequency doain: requires short frequency window long tie window to depict features of signal. 3
4 Fourier Gabor Wavelet 4
5 What are Wavelets? Wavelets are atheatical functions that cut up data into different frequency coponents, and then study each coponent with a resolution atched to its scale. They have advantages over traditional Fourier ethods in analyzing physical situations where the signal contains discontinuities and sharp spikes. 5
6 Wavelets were developed independently in the fields of atheatics, quantu physics, electrical engineering, and seisic geology. Interchanges between these fields during the last ten years have led to any new wavelet applications such as iage copression, turbulence, huan vision, radar, and earthquake prediction. 6
7 Wavelet algoriths process data at different scales or resolutions. If we look at a signal with a large window, we would notice gross features. Siilarly, if we look at a signal with a sall window, we would notice sall features. 7
8 The wavelet analysis procedure is to adopt a wavelet prototype function, called an analyzing wavelet or other wavelet. Teporal analysis is perfored with a contracted, highfrequency version of the prototype wavelet, while frequency analysis is perfored with a dilated, low-frequency version of the sae wavelet. 8
9 Because the original signal or function can be represented in ters of a wavelet expansion using coefficients in a linear cobination of the wavelet functions, data operations can be perfored using ust the corresponding wavelet coefficients. And if we further choose the best wavelets adapted to the data, or truncate the coefficients below a threshold, the data is sparsely represented. This sparse coding akes wavelets an excellent tool in the field of data copression. 9
10 Another View: Wavelets are functions defined over a finite interval and having an average value of zero. Haar wavelet The first ention of wavelets appeared in an appendix of the thesis of A. Haar 909. One property of the Haar wavelet is that it has copact support, which eans that it vanishes outside of a finite interval. Unfortunately, Haar wavelets are not continuously differentiable which soewhat liits their applications. 0
11 What is wavelet transfor? The wavelet transfor is a tool for carving up functions, operators, or data into coponents of different frequency, allowing one to study each coponent separately. The basic idea of the wavelet transfor is to represent any arbitrary function ƒt as a superposition of a set of such wavelets or basis functions. These basis functions or baby wavelets are obtained fro a single prototype wavelet called the other wavelet, by dilations or contractions scaling and translations shifts.
12 The continuous wavelet transfor CWT Fourier Transfor + F ω f t e ωt FT is the su over all the tie of signal ft ultiplied by a coplex exponential. dt
13 Continuous Wavelet transfor for each Scale for each Position Coefficient S,P Signal x Wavelet S,P all tie end end Coefficient Scale 3
14 Siilarly, the Continuous Wavelet Transfor CWT is defined as the su over all tie of the signal ultiplied by a scaled and shifted version of the wavelet function Ψ s, τ t : * γ s, τ f t Ψs t dt,τ where * denotes coplex conugation. This equation shows how a function ƒt is decoposed into a set of basis functions Ψ s, τ t, called the wavelets. The variables s and τ are the new diensions, scale and translation position, after the wavelet transfor. 4
15 The results of the CWT are any wavelet coefficients, which are a function of scale and position 5
16 The wavelets are generated fro a single basic wavelet Ψt, the so-called other wavelet, by scaling and translation: Ψ s, τ t t τ ψ s s s is the scale factor, τ is the translation factor and the factor s -/ is for energy noralization across the different scales. It is iportant to note that in the above transfors the wavelet basis functions are not specified. This is one of the differences between the wavelet transfor and the Fourier transfor, or other transfors. 6
17 Scale Scaling a wavelet siply eans stretching or copressing it. 7
18 Scale and Frequency Low scale a Copressed wavelet Rapidly changing details High frequency ω High scale a stretched wavelet slowly changing details low frequency ω Translation shift Translating a wavelet siply eans delaying its onset. 8
19 Wavelet Properties ψ ω ω < ω adissibility condition: d + ψ ω stands for the Fourier transfor of ψ t The adissibility condition iplies that the Fourier transfor of ω vanishes at the zero frequency, i.e. ψ ψ ω 0 This eans that wavelets ust have a band-pass like spectru. This is a very iportant observation, which can be used to build an efficient wavelet transfor. A zero at the zero frequency DC coponent also eans that the average value of the wavelet in the tie doain ust be zero, ψ t dt 0 ψ t ust be a wave. ω 0 9
20 0
21
22
23 3
24 4
25 Discrete Wavelets Discrete wavelet is written as ψ, k t t kτ 0s ψ s0 s0 and k are integers and s 0 > is a fixed dilation step. The translation factor τ 0 depends on the dilation step. The effect of discretizing the wavelet is that the tie-scale space is now sapled at discrete intervals. We often choose s 0 0 * ψ, t ψ, t dt k n 0 If and kn others 5
26 A band-pass filter The wavelet has a band-pass like spectru Fro Fourier theory we know that copression in tie is equivalent to stretching the spectru and shifting it upwards: F a ω a { f at } F Suppose a This eans that a tie copression of the wavelet by a factor of will stretch the frequency spectru of the wavelet by a factor of and also shift all frequency coponents up by a factor of. 6
27 Scaling-- value of stretch Scaling a wavelet siply eans stretching or copressing it. ft sint scale factor ft sint scale factor ft sin3t scale factor 3 7
28 To get a good coverage of the signal spectru the stretched wavelet spectra should touch each other. Touching wavelet spectra resulting fro scaling of the other wavelet in the tie doain. Suarizing, if one wavelet can be seen as a band-pass filter, then a series of dilated wavelets can be seen as a bank of band-pass filters. 8
29 The scaling function Ηow to cover the spectru all the way down to zero? The solution is not to try to cover the spectru all the way down to zero with wavelet spectra, but to use a cork to plug the hole when it is sall enough. This cork then is a low-pass spectru and it belongs to the so-called scaling function. 9
30 If we look at the scaling function as being ust a signal with a lowpass spectru, then we can decopose it in wavelet coponents and express it like ϕ t γ, k ψ, k, k t adissibility condition for scaling functions ϕ t dt Suarizing once ore, if one wavelet can be seen as a bandpass filter and a scaling function is a low-pass filter, then a series of dilated wavelets together with a scaling function can be seen as a filter bank. 30
31 Results of wavelet transfor: approxiation and details Low frequency: approxiation a High frequency Details d Decoposition can be perfored iteratively 3
32 Levels of decoposition Successively decopose the approxiation Level 5 decoposition a5 + d5 + d4 + d3 + d + d No liit to the nuber of decopositions perfored 3
33 Wavelet synthesis Re-creates signal fro coefficients Up-sapling required 33
34 Multi-level Wavelet Analysis Multi-level wavelet decoposition tree Reassebling original signal 34
35 Discrete Wavelet transfor signal lowpass highpass filters Approxiation a Details d 35
36 Non-stationary Property of Natural Iage 36
37 The Discrete Wavelet Transfor Calculating wavelet coefficients at every possible scale is a fair aount of work, and it generates an awful lot of data. What if we choose only a subset of scales and positions at which to ake our calculations? It turns out, rather rearkably, that if we choose scales and positions based on powers of two -- so-called dyadic scales and positions -- then our analysis will be uch ore efficient and ust as accurate. We obtain ust such an analysis fro the discrete wavelet transfor DWT. 37
38 Approxiations and Details The approxiations are the high-scale, low-frequency coponents of the signal. The details are the low-scale, high-frequency coponents. The filtering process, at its ost basic level, looks like this: The original signal, S, passes through two copleentary filters and eerges as two signals. 38
39 Downsapling Unfortunately, if we actually perfor this operation on a real digital signal, we wind up with twice as uch data as we started with. Suppose, for instance, that the original signal S consists of 000 saples of data. Then the approxiation and the detail will each have 000 saples, for a total of 000. To correct this proble, we introduce the notion of downsapling. This siply eans throwing away every second data point. 39
40 An exaple: 40
41 Reconstructing Approxiation and Details Upsapling 4
42 Pyraidal Iage Structure 4
43 Iage Pyraids Original iage, the base of the pyraid, in the level J log N, Norally truncated to P+ levels. COMPONENTS: Approxiation pyraids, predication residual pyraids Steps:. Copute a reduced-resolution approxiation fro to - level by downsapling;. Upsaple the output of step, get predication iage; 3. Difference between the predication of step and the input of step. 43
44 Subband coding Here an iage is decoposed into a set of band liited Coponents, called subbands. Subbands are reassebled to reconstruct the original Iage without ERROR. 44
45 Subband Coding 45
46 Subband Coding Filters h n and h n are half-band digital filters, their transfer characteristics H 0 -low pass filter, output is an approxiation of xn and H -high pass filter, output is the high frequency or detail part of xn Criteria: h 0 n, h n, g 0 n, g n are selected to reconstruct the input perfectly. 46
47 Splitting the signal spectru with an iterated filter bank. 8B LP 4B HP 4B f f LP B HP B 4B f LP B HP B B 4B f Suarizing, if we ipleent the wavelet transfor as an iterated filter bank, we do not have to specify the wavelets explicitly! This is a rearkable result. 47
48 48 Scaling function two-scale relation k t k h t k + + ϕ ϕ Wavelet k t k g t k + + ϕ ψ The signal ft can be expressed as k t k k t k t f k k + ψ γ ϕ k h k DWT k g k γ
49 49 k h k k g k γ ] [ k k h k h k ] [ k k g k g k γ
50 How to calculate DWT given g, h and a signal f? Initialization: f Exaple: f{,4,-3,0}; h n {/,/ } g n { /,/ } f {,4, 3,0} h n {/,/ } g n {/, / } k h k γ k g k h h0 0 + h 0 3 h 0 h0 + h γ γ 3 0 g 0 g0 0 + g g g0 + g
51 h h h g g g γ
52 5 Wavelet Reconstruction Synthesis Perfect reconstruction : ' ' + H H G G ' ' n g n h n h n g n n + +
53 γ,0 4,0 x y 0 / 0 0 / ' ' ' ' ' ' + x h h h x h h x k h k x { } 3/, 5/ { } 3/, 3/ γ 0 4 / / ' ' ' ' ' ' + y g g g y g g y k g k y γ γ γ γ γ γ
54 -D Discrete Wavelet Transfor A -D DWT can be done as follows: Step : Replace each row with its -D DWT; Step : Replace each colun with its -D DWT; Step 3: repeat steps and on the lowest subband for the next scale Step 4: repeat steps 3 until as any scales as desired have been copleted L H LL LH HL HH LH HL HH original One scale two scales 54
55 -D 4-band filter bank Approxiation Vertical detail Horizontal detail Diagonal details 55
56 Subband Exaple 56
57 Lossy Copression Based on spatial redundancy Measure of spatial redundancy: D covariance Cov X i, σ e -α i*i+* Vertical correlation ρ Horizontal correlation ρ E[Xi,Xi-,] E[X i,] E[Xi,Xi,-] E[X i,] For iages we assue equal correlations Typically e -α ρ ρ 0.95 Measure of loss or distortion: MSE between encoded and decoded iage 57
58 Rate-Distortion Function Tradeoff between bit rate R of copressed iage and distortion D R easured in its per encoder output sybol Copression ratio encoder input bits/r D noralized by the variance of the encoder input Possible SNR definition 0 log 0 D - For iages that can be odeled as uncorrelated Gaussian RD0.5log D - 58
59 Saple vs. Block-based Coding Saple-based In spatial or frequency doain Like the JPEG-LS Make a predictor function often weighted su Copute and quantize residual Encode Block-based Spatial: group pixels into blocks, copress blocks Transfor: group into blocks, transfor, encode 59
60 Copression and Subband Coding Pass an iage through an n-band filter bank Possibly subsaple each filtered output Encode each subband separately Copression ay be achieved by discarding uniportant bands Advantages Fewer artifacts than block-coded copression More robust under transission errors Selective encoding/decoding possible More expensive 60
Feature Extraction Techniques
Feature Extraction Techniques Unsupervised Learning II Feature Extraction Unsupervised ethods can also be used to find features which can be useful for categorization. There are unsupervised ethods that
More informationProblem with Fourier. Wavelets: a preview. Fourier Gabor Wavelet. Gabor s proposal. in the transform domain. Sinusoid with a small discontinuity
Problem with Fourier Wavelets: a preview February 6, 2003 Acknowledgements: Material compiled from the MATLAB Wavelet Toolbox UG. Fourier analysis -- breaks down a signal into constituent sinusoids of
More informationWavelets: a preview. February 6, 2003 Acknowledgements: Material compiled from the MATLAB Wavelet Toolbox UG.
Wavelets: a preview February 6, 2003 Acknowledgements: Material compiled from the MATLAB Wavelet Toolbox UG. Problem with Fourier Fourier analysis -- breaks down a signal into constituent sinusoids of
More informationFourier Series Summary (From Salivahanan et al, 2002)
Fourier Series Suary (Fro Salivahanan et al, ) A periodic continuous signal f(t), - < t
More informationModule 4. Multi-Resolution Analysis. Version 2 ECE IIT, Kharagpur
Module 4 Multi-Resolution Analysis Lesson Multi-resolution Analysis: Discrete avelet Transforms Instructional Objectives At the end of this lesson, the students should be able to:. Define Discrete avelet
More informationFundamentals of Image Compression
Fundaentals of Iage Copression Iage Copression reduce the size of iage data file while retaining necessary inforation Original uncopressed Iage Copression (encoding) 01101 Decopression (decoding) Copressed
More informationA Simple Regression Problem
A Siple Regression Proble R. M. Castro March 23, 2 In this brief note a siple regression proble will be introduced, illustrating clearly the bias-variance tradeoff. Let Y i f(x i ) + W i, i,..., n, where
More informationPattern Recognition and Machine Learning. Artificial Neural networks
Pattern Recognition and Machine Learning Jaes L. Crowley ENSIMAG 3 - MMIS Fall Seester 2016/2017 Lessons 9 11 Jan 2017 Outline Artificial Neural networks Notation...2 Convolutional Neural Networks...3
More informationThis model assumes that the probability of a gap has size i is proportional to 1/i. i.e., i log m e. j=1. E[gap size] = i P r(i) = N f t.
CS 493: Algoriths for Massive Data Sets Feb 2, 2002 Local Models, Bloo Filter Scribe: Qin Lv Local Models In global odels, every inverted file entry is copressed with the sae odel. This work wells when
More informationBootstrapping Dependent Data
Bootstrapping Dependent Data One of the key issues confronting bootstrap resapling approxiations is how to deal with dependent data. Consider a sequence fx t g n t= of dependent rando variables. Clearly
More information3.3 Variational Characterization of Singular Values
3.3. Variational Characterization of Singular Values 61 3.3 Variational Characterization of Singular Values Since the singular values are square roots of the eigenvalues of the Heritian atrices A A and
More informationDigital Image Processing Lectures 15 & 16
Lectures 15 & 16, Professor Department of Electrical and Computer Engineering Colorado State University CWT and Multi-Resolution Signal Analysis Wavelet transform offers multi-resolution by allowing for
More informationList Scheduling and LPT Oliver Braun (09/05/2017)
List Scheduling and LPT Oliver Braun (09/05/207) We investigate the classical scheduling proble P ax where a set of n independent jobs has to be processed on 2 parallel and identical processors (achines)
More informationTracking using CONDENSATION: Conditional Density Propagation
Tracking using CONDENSATION: Conditional Density Propagation Goal Model-based visual tracking in dense clutter at near video frae rates M. Isard and A. Blake, CONDENSATION Conditional density propagation
More informationWavelets and Multiresolution Processing
Wavelets and Multiresolution Processing Wavelets Fourier transform has it basis functions in sinusoids Wavelets based on small waves of varying frequency and limited duration In addition to frequency,
More informationQ ESTIMATION WITHIN A FORMATION PROGRAM q_estimation
Foration Attributes Progra q_estiation Q ESTIMATION WITHIN A FOMATION POGAM q_estiation Estiating Q between stratal slices Progra q_estiation estiate seisic attenuation (1/Q) on coplex stratal slices using
More informationEfficient Filter Banks And Interpolators
Efficient Filter Banks And Interpolators A. G. DEMPSTER AND N. P. MURPHY Departent of Electronic Systes University of Westinster 115 New Cavendish St, London W1M 8JS United Kingdo Abstract: - Graphical
More informationASSUME a source over an alphabet size m, from which a sequence of n independent samples are drawn. The classical
IEEE TRANSACTIONS ON INFORMATION THEORY Large Alphabet Source Coding using Independent Coponent Analysis Aichai Painsky, Meber, IEEE, Saharon Rosset and Meir Feder, Fellow, IEEE arxiv:67.7v [cs.it] Jul
More information13.2 Fully Polynomial Randomized Approximation Scheme for Permanent of Random 0-1 Matrices
CS71 Randoness & Coputation Spring 018 Instructor: Alistair Sinclair Lecture 13: February 7 Disclaier: These notes have not been subjected to the usual scrutiny accorded to foral publications. They ay
More informationA remark on a success rate model for DPA and CPA
A reark on a success rate odel for DPA and CPA A. Wieers, BSI Version 0.5 andreas.wieers@bsi.bund.de Septeber 5, 2018 Abstract The success rate is the ost coon evaluation etric for easuring the perforance
More information1 The Continuous Wavelet Transform The continuous wavelet transform (CWT) Discretisation of the CWT... 2
Contents 1 The Continuous Wavelet Transform 1 1.1 The continuous wavelet transform (CWT)............. 1 1. Discretisation of the CWT...................... Stationary wavelet transform or redundant wavelet
More informationBlock designs and statistics
Bloc designs and statistics Notes for Math 447 May 3, 2011 The ain paraeters of a bloc design are nuber of varieties v, bloc size, nuber of blocs b. A design is built on a set of v eleents. Each eleent
More informationPrincipal Components Analysis
Principal Coponents Analysis Cheng Li, Bingyu Wang Noveber 3, 204 What s PCA Principal coponent analysis (PCA) is a statistical procedure that uses an orthogonal transforation to convert a set of observations
More informationPattern Recognition and Machine Learning. Learning and Evaluation for Pattern Recognition
Pattern Recognition and Machine Learning Jaes L. Crowley ENSIMAG 3 - MMIS Fall Seester 2017 Lesson 1 4 October 2017 Outline Learning and Evaluation for Pattern Recognition Notation...2 1. The Pattern Recognition
More informationChapter 6 1-D Continuous Groups
Chapter 6 1-D Continuous Groups Continuous groups consist of group eleents labelled by one or ore continuous variables, say a 1, a 2,, a r, where each variable has a well- defined range. This chapter explores:
More informationINTRODUCTION TO. Adapted from CS474/674 Prof. George Bebis Department of Computer Science & Engineering University of Nevada (UNR)
INTRODUCTION TO WAVELETS Adapted from CS474/674 Prof. George Bebis Department of Computer Science & Engineering University of Nevada (UNR) CRITICISM OF FOURIER SPECTRUM It gives us the spectrum of the
More informationA New Algorithm for Reactive Electric Power Measurement
A. Abiyev, GAU J. Soc. & Appl. Sci., 2(4), 7-25, 27 A ew Algorith for Reactive Electric Power Measureent Adalet Abiyev Girne Aerican University, Departernt of Electrical Electronics Engineering, Mersin,
More information1 Bounding the Margin
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #12 Scribe: Jian Min Si March 14, 2013 1 Bounding the Margin We are continuing the proof of a bound on the generalization error of AdaBoost
More informationNon-Parametric Non-Line-of-Sight Identification 1
Non-Paraetric Non-Line-of-Sight Identification Sinan Gezici, Hisashi Kobayashi and H. Vincent Poor Departent of Electrical Engineering School of Engineering and Applied Science Princeton University, Princeton,
More informationma x = -bv x + F rod.
Notes on Dynaical Systes Dynaics is the study of change. The priary ingredients of a dynaical syste are its state and its rule of change (also soeties called the dynaic). Dynaical systes can be continuous
More informationPhysics 215 Winter The Density Matrix
Physics 215 Winter 2018 The Density Matrix The quantu space of states is a Hilbert space H. Any state vector ψ H is a pure state. Since any linear cobination of eleents of H are also an eleent of H, it
More informationIntelligent Systems: Reasoning and Recognition. Perceptrons and Support Vector Machines
Intelligent Systes: Reasoning and Recognition Jaes L. Crowley osig 1 Winter Seester 2018 Lesson 6 27 February 2018 Outline Perceptrons and Support Vector achines Notation...2 Linear odels...3 Lines, Planes
More informationEE5900 Spring Lecture 4 IC interconnect modeling methods Zhuo Feng
EE59 Spring Parallel LSI AD Algoriths Lecture I interconnect odeling ethods Zhuo Feng. Z. Feng MTU EE59 So far we ve considered only tie doain analyses We ll soon see that it is soeties preferable to odel
More informationINTRODUCTION. Residual migration has proved to be a useful tool in imaging and in velocity analysis.
Stanford Exploration Project, Report, June 3, 999, pages 5 59 Short Note On Stolt prestack residual igration Paul Sava keywords: Stolt, residual igration INTRODUCTION Residual igration has proved to be
More informationSome Perspective. Forces and Newton s Laws
Soe Perspective The language of Kineatics provides us with an efficient ethod for describing the otion of aterial objects, and we ll continue to ake refineents to it as we introduce additional types of
More informationOBJECTIVES INTRODUCTION
M7 Chapter 3 Section 1 OBJECTIVES Suarize data using easures of central tendency, such as the ean, edian, ode, and idrange. Describe data using the easures of variation, such as the range, variance, and
More informationIntroduction to Wavelet. Based on A. Mukherjee s lecture notes
Introduction to Wavelet Based on A. Mukherjee s lecture notes Contents History of Wavelet Problems of Fourier Transform Uncertainty Principle The Short-time Fourier Transform Continuous Wavelet Transform
More informationData-Driven Imaging in Anisotropic Media
18 th World Conference on Non destructive Testing, 16- April 1, Durban, South Africa Data-Driven Iaging in Anisotropic Media Arno VOLKER 1 and Alan HUNTER 1 TNO Stieltjesweg 1, 6 AD, Delft, The Netherlands
More informationPERIODIC STEADY STATE ANALYSIS, EFFECTIVE VALUE,
PERIODIC SEADY SAE ANALYSIS, EFFECIVE VALUE, DISORSION FACOR, POWER OF PERIODIC CURRENS t + Effective value of current (general definition) IRMS i () t dt Root Mean Square, in Czech boo denoted I he value
More informationLec 05 Arithmetic Coding
Outline CS/EE 5590 / ENG 40 Special Topics (7804, 785, 7803 Lec 05 Arithetic Coding Lecture 04 ReCap Arithetic Coding About Hoework- and Lab Zhu Li Course Web: http://l.web.ukc.edu/lizhu/teaching/06sp.video-counication/ain.htl
More informationlecture 37: Linear Multistep Methods: Absolute Stability, Part I lecture 38: Linear Multistep Methods: Absolute Stability, Part II
lecture 37: Linear Multistep Methods: Absolute Stability, Part I lecture 3: Linear Multistep Methods: Absolute Stability, Part II 5.7 Linear ultistep ethods: absolute stability At this point, it ay well
More information2D Wavelets. Hints on advanced Concepts
2D Wavelets Hints on advanced Concepts 1 Advanced concepts Wavelet packets Laplacian pyramid Overcomplete bases Discrete wavelet frames (DWF) Algorithme à trous Discrete dyadic wavelet frames (DDWF) Overview
More informationBoosting with log-loss
Boosting with log-loss Marco Cusuano-Towner Septeber 2, 202 The proble Suppose we have data exaples {x i, y i ) i =... } for a two-class proble with y i {, }. Let F x) be the predictor function with the
More informationlecture 36: Linear Multistep Mehods: Zero Stability
95 lecture 36: Linear Multistep Mehods: Zero Stability 5.6 Linear ultistep ethods: zero stability Does consistency iply convergence for linear ultistep ethods? This is always the case for one-step ethods,
More informationModel Fitting. CURM Background Material, Fall 2014 Dr. Doreen De Leon
Model Fitting CURM Background Material, Fall 014 Dr. Doreen De Leon 1 Introduction Given a set of data points, we often want to fit a selected odel or type to the data (e.g., we suspect an exponential
More informationOptimal Jamming Over Additive Noise: Vector Source-Channel Case
Fifty-first Annual Allerton Conference Allerton House, UIUC, Illinois, USA October 2-3, 2013 Optial Jaing Over Additive Noise: Vector Source-Channel Case Erah Akyol and Kenneth Rose Abstract This paper
More informationAVOIDING PITFALLS IN MEASUREMENT UNCERTAINTY ANALYSIS
VOIDING ITFLLS IN ESREENT NERTINTY NLYSIS Benny R. Sith Inchwor Solutions Santa Rosa, Suary: itfalls, both subtle and obvious, await the new or casual practitioner of easureent uncertainty analysis. This
More informationMachine Learning Basics: Estimators, Bias and Variance
Machine Learning Basics: Estiators, Bias and Variance Sargur N. srihari@cedar.buffalo.edu This is part of lecture slides on Deep Learning: http://www.cedar.buffalo.edu/~srihari/cse676 1 Topics in Basics
More information3D acoustic wave modeling with a time-space domain dispersion-relation-based Finite-difference scheme
P-8 3D acoustic wave odeling with a tie-space doain dispersion-relation-based Finite-difference schee Yang Liu * and rinal K. Sen State Key Laboratory of Petroleu Resource and Prospecting (China University
More informationCOS 424: Interacting with Data. Written Exercises
COS 424: Interacting with Data Hoework #4 Spring 2007 Regression Due: Wednesday, April 18 Written Exercises See the course website for iportant inforation about collaboration and late policies, as well
More informationEstimating Parameters for a Gaussian pdf
Pattern Recognition and achine Learning Jaes L. Crowley ENSIAG 3 IS First Seester 00/0 Lesson 5 7 Noveber 00 Contents Estiating Paraeters for a Gaussian pdf Notation... The Pattern Recognition Proble...3
More informationPolygonal Designs: Existence and Construction
Polygonal Designs: Existence and Construction John Hegean Departent of Matheatics, Stanford University, Stanford, CA 9405 Jeff Langford Departent of Matheatics, Drake University, Des Moines, IA 5011 G
More informationa a a a a a a m a b a b
Algebra / Trig Final Exa Study Guide (Fall Seester) Moncada/Dunphy Inforation About the Final Exa The final exa is cuulative, covering Appendix A (A.1-A.5) and Chapter 1. All probles will be ultiple choice
More informationCS Lecture 13. More Maximum Likelihood
CS 6347 Lecture 13 More Maxiu Likelihood Recap Last tie: Introduction to axiu likelihood estiation MLE for Bayesian networks Optial CPTs correspond to epirical counts Today: MLE for CRFs 2 Maxiu Likelihood
More informationThe Wilson Model of Cortical Neurons Richard B. Wells
The Wilson Model of Cortical Neurons Richard B. Wells I. Refineents on the odgkin-uxley Model The years since odgkin s and uxley s pioneering work have produced a nuber of derivative odgkin-uxley-like
More informationDigital Image Processing
Digital Image Processing, 2nd ed. Digital Image Processing Chapter 7 Wavelets and Multiresolution Processing Dr. Kai Shuang Department of Electronic Engineering China University of Petroleum shuangkai@cup.edu.cn
More informationChaotic Coupled Map Lattices
Chaotic Coupled Map Lattices Author: Dustin Keys Advisors: Dr. Robert Indik, Dr. Kevin Lin 1 Introduction When a syste of chaotic aps is coupled in a way that allows the to share inforation about each
More informationThe Weierstrass Approximation Theorem
36 The Weierstrass Approxiation Theore Recall that the fundaental idea underlying the construction of the real nubers is approxiation by the sipler rational nubers. Firstly, nubers are often deterined
More informationWavelet Bases Made of Piecewise Polynomial Functions: Theory and Applications *
Applied atheatics 96-6 doi:.436/a.. Published Online February (http://www.scirp.org/ournal/a) Wavelet Bases ade of Piecewise Polynoial Functions: Theory and Applications * Abstract Lorella Fatone aria
More informationPhysically Based Modeling CS Notes Spring 1997 Particle Collision and Contact
Physically Based Modeling CS 15-863 Notes Spring 1997 Particle Collision and Contact 1 Collisions with Springs Suppose we wanted to ipleent a particle siulator with a floor : a solid horizontal plane which
More informationEE67I Multimedia Communication Systems
EE67I Multimedia Communication Systems Lecture 5: LOSSY COMPRESSION In these schemes, we tradeoff error for bitrate leading to distortion. Lossy compression represents a close approximation of an original
More informationIntroduction to Machine Learning. Recitation 11
Introduction to Machine Learning Lecturer: Regev Schweiger Recitation Fall Seester Scribe: Regev Schweiger. Kernel Ridge Regression We now take on the task of kernel-izing ridge regression. Let x,...,
More informationPh 20.3 Numerical Solution of Ordinary Differential Equations
Ph 20.3 Nuerical Solution of Ordinary Differential Equations Due: Week 5 -v20170314- This Assignent So far, your assignents have tried to failiarize you with the hardware and software in the Physics Coputing
More information2 Q 10. Likewise, in case of multiple particles, the corresponding density in 2 must be averaged over all
Lecture 6 Introduction to kinetic theory of plasa waves Introduction to kinetic theory So far we have been odeling plasa dynaics using fluid equations. The assuption has been that the pressure can be either
More informationIn this chapter, we consider several graph-theoretic and probabilistic models
THREE ONE GRAPH-THEORETIC AND STATISTICAL MODELS 3.1 INTRODUCTION In this chapter, we consider several graph-theoretic and probabilistic odels for a social network, which we do under different assuptions
More informationIN modern society that various systems have become more
Developent of Reliability Function in -Coponent Standby Redundant Syste with Priority Based on Maxiu Entropy Principle Ryosuke Hirata, Ikuo Arizono, Ryosuke Toohiro, Satoshi Oigawa, and Yasuhiko Takeoto
More informationIntelligent Systems: Reasoning and Recognition. Artificial Neural Networks
Intelligent Systes: Reasoning and Recognition Jaes L. Crowley MOSIG M1 Winter Seester 2018 Lesson 7 1 March 2018 Outline Artificial Neural Networks Notation...2 Introduction...3 Key Equations... 3 Artificial
More informationKeywords: Estimator, Bias, Mean-squared error, normality, generalized Pareto distribution
Testing approxiate norality of an estiator using the estiated MSE and bias with an application to the shape paraeter of the generalized Pareto distribution J. Martin van Zyl Abstract In this work the norality
More informationThe Transactional Nature of Quantum Information
The Transactional Nature of Quantu Inforation Subhash Kak Departent of Coputer Science Oklahoa State University Stillwater, OK 7478 ABSTRACT Inforation, in its counications sense, is a transactional property.
More informationCausality and the Kramers Kronig relations
Causality and the Kraers Kronig relations Causality describes the teporal relationship between cause and effect. A bell rings after you strike it, not before you strike it. This eans that the function
More informationLecture 9 November 23, 2015
CSC244: Discrepancy Theory in Coputer Science Fall 25 Aleksandar Nikolov Lecture 9 Noveber 23, 25 Scribe: Nick Spooner Properties of γ 2 Recall that γ 2 (A) is defined for A R n as follows: γ 2 (A) = in{r(u)
More informationMeasuring orbital angular momentum superpositions of light by mode transformation
CHAPTER 7 Measuring orbital angular oentu superpositions of light by ode transforation In chapter 6 we reported on a ethod for easuring orbital angular oentu (OAM) states of light based on the transforation
More informationA DISCRETE ZAK TRANSFORM. Christopher Heil. The MITRE Corporation McLean, Virginia Technical Report MTR-89W00128.
A DISCRETE ZAK TRANSFORM Christopher Heil The MITRE Corporation McLean, Virginia 22102 Technical Report MTR-89W00128 August 1989 Abstract. A discrete version of the Zak transfor is defined and used to
More informationChapter 1: Basics of Vibrations for Simple Mechanical Systems
Chapter 1: Basics of Vibrations for Siple Mechanical Systes Introduction: The fundaentals of Sound and Vibrations are part of the broader field of echanics, with strong connections to classical echanics,
More informationCh 12: Variations on Backpropagation
Ch 2: Variations on Backpropagation The basic backpropagation algorith is too slow for ost practical applications. It ay take days or weeks of coputer tie. We deonstrate why the backpropagation algorith
More informationPattern Recognition and Machine Learning. Artificial Neural networks
Pattern Recognition and Machine Learning Jaes L. Crowley ENSIMAG 3 - MMIS Fall Seester 2017 Lessons 7 20 Dec 2017 Outline Artificial Neural networks Notation...2 Introduction...3 Key Equations... 3 Artificial
More informationMULTIRATE DIGITAL SIGNAL PROCESSING
MULTIRATE DIGITAL SIGNAL PROCESSING Signal processing can be enhanced by changing sampling rate: Up-sampling before D/A conversion in order to relax requirements of analog antialiasing filter. Cf. audio
More informationCombining Classifiers
Cobining Classifiers Generic ethods of generating and cobining ultiple classifiers Bagging Boosting References: Duda, Hart & Stork, pg 475-480. Hastie, Tibsharini, Friedan, pg 246-256 and Chapter 10. http://www.boosting.org/
More informationExtension of CSRSM for the Parametric Study of the Face Stability of Pressurized Tunnels
Extension of CSRSM for the Paraetric Study of the Face Stability of Pressurized Tunnels Guilhe Mollon 1, Daniel Dias 2, and Abdul-Haid Soubra 3, M.ASCE 1 LGCIE, INSA Lyon, Université de Lyon, Doaine scientifique
More informationProc. of the IEEE/OES Seventh Working Conference on Current Measurement Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES
Proc. of the IEEE/OES Seventh Working Conference on Current Measureent Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES Belinda Lipa Codar Ocean Sensors 15 La Sandra Way, Portola Valley, CA 98 blipa@pogo.co
More informationForce and dynamics with a spring, analytic approach
Force and dynaics with a spring, analytic approach It ay strie you as strange that the first force we will discuss will be that of a spring. It is not one of the four Universal forces and we don t use
More informationESTIMATING AND FORMING CONFIDENCE INTERVALS FOR EXTREMA OF RANDOM POLYNOMIALS. A Thesis. Presented to. The Faculty of the Department of Mathematics
ESTIMATING AND FORMING CONFIDENCE INTERVALS FOR EXTREMA OF RANDOM POLYNOMIALS A Thesis Presented to The Faculty of the Departent of Matheatics San Jose State University In Partial Fulfillent of the Requireents
More informationUsing a De-Convolution Window for Operating Modal Analysis
Using a De-Convolution Window for Operating Modal Analysis Brian Schwarz Vibrant Technology, Inc. Scotts Valley, CA Mark Richardson Vibrant Technology, Inc. Scotts Valley, CA Abstract Operating Modal Analysis
More information12 Towards hydrodynamic equations J Nonlinear Dynamics II: Continuum Systems Lecture 12 Spring 2015
18.354J Nonlinear Dynaics II: Continuu Systes Lecture 12 Spring 2015 12 Towards hydrodynaic equations The previous classes focussed on the continuu description of static (tie-independent) elastic systes.
More informationOn Conditions for Linearity of Optimal Estimation
On Conditions for Linearity of Optial Estiation Erah Akyol, Kuar Viswanatha and Kenneth Rose {eakyol, kuar, rose}@ece.ucsb.edu Departent of Electrical and Coputer Engineering University of California at
More informationBEE604 Digital Signal Processing
BEE64 Digital Signal Processing Copiled by, Mrs.S.Sherine Assistant Professor Departent of EEE BIHER. COTETS Sapling Discrete Tie Fourier Transfor Properties of DTFT Discrete Fourier Transfor Inverse Discrete
More informationGeneral Properties of Radiation Detectors Supplements
Phys. 649: Nuclear Techniques Physics Departent Yarouk University Chapter 4: General Properties of Radiation Detectors Suppleents Dr. Nidal M. Ershaidat Overview Phys. 649: Nuclear Techniques Physics Departent
More informationSupplementary Material for Fast and Provable Algorithms for Spectrally Sparse Signal Reconstruction via Low-Rank Hankel Matrix Completion
Suppleentary Material for Fast and Provable Algoriths for Spectrally Sparse Signal Reconstruction via Low-Ran Hanel Matrix Copletion Jian-Feng Cai Tianing Wang Ke Wei March 1, 017 Abstract We establish
More informationRecovering Data from Underdetermined Quadratic Measurements (CS 229a Project: Final Writeup)
Recovering Data fro Underdeterined Quadratic Measureents (CS 229a Project: Final Writeup) Mahdi Soltanolkotabi Deceber 16, 2011 1 Introduction Data that arises fro engineering applications often contains
More informationEstimation of ADC Nonlinearities from the Measurement in Input Voltage Intervals
Estiation of ADC Nonlinearities fro the Measureent in Input Voltage Intervals M. Godla, L. Michaeli, 3 J. Šaliga, 4 R. Palenčár,,3 Deptartent of Electronics and Multiedia Counications, FEI TU of Košice,
More informationarxiv: v1 [math.nt] 14 Sep 2014
ROTATION REMAINDERS P. JAMESON GRABER, WASHINGTON AND LEE UNIVERSITY 08 arxiv:1409.411v1 [ath.nt] 14 Sep 014 Abstract. We study properties of an array of nubers, called the triangle, in which each row
More informationQuantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search
Quantu algoriths (CO 781, Winter 2008) Prof Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search ow we begin to discuss applications of quantu walks to search algoriths
More informationA Study on B-Spline Wavelets and Wavelet Packets
Applied Matheatics 4 5 3-3 Published Online Noveber 4 in SciRes. http://www.scirp.org/ournal/a http://dx.doi.org/.436/a.4.5987 A Study on B-Spline Wavelets and Wavelet Pacets Sana Khan Mohaad Kaliuddin
More informationSharp Time Data Tradeoffs for Linear Inverse Problems
Sharp Tie Data Tradeoffs for Linear Inverse Probles Saet Oyak Benjain Recht Mahdi Soltanolkotabi January 016 Abstract In this paper we characterize sharp tie-data tradeoffs for optiization probles used
More informationKernel Methods and Support Vector Machines
Intelligent Systes: Reasoning and Recognition Jaes L. Crowley ENSIAG 2 / osig 1 Second Seester 2012/2013 Lesson 20 2 ay 2013 Kernel ethods and Support Vector achines Contents Kernel Functions...2 Quadratic
More informationSymbolic Analysis as Universal Tool for Deriving Properties of Non-linear Algorithms Case study of EM Algorithm
Acta Polytechnica Hungarica Vol., No., 04 Sybolic Analysis as Universal Tool for Deriving Properties of Non-linear Algoriths Case study of EM Algorith Vladiir Mladenović, Miroslav Lutovac, Dana Porrat
More informationSolving initial value problems by residual power series method
Theoretical Matheatics & Applications, vol.3, no.1, 13, 199-1 ISSN: 179-9687 (print), 179-979 (online) Scienpress Ltd, 13 Solving initial value probles by residual power series ethod Mohaed H. Al-Sadi
More informationEnsemble Based on Data Envelopment Analysis
Enseble Based on Data Envelopent Analysis So Young Sohn & Hong Choi Departent of Coputer Science & Industrial Systes Engineering, Yonsei University, Seoul, Korea Tel) 82-2-223-404, Fax) 82-2- 364-7807
More informationSlide10. Haykin Chapter 8: Principal Components Analysis. Motivation. Principal Component Analysis: Variance Probe
Slide10 Motivation Haykin Chapter 8: Principal Coponents Analysis 1.6 1.4 1.2 1 0.8 cloud.dat 0.6 CPSC 636-600 Instructor: Yoonsuck Choe Spring 2015 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 How can we
More informationOn Constant Power Water-filling
On Constant Power Water-filling Wei Yu and John M. Cioffi Electrical Engineering Departent Stanford University, Stanford, CA94305, U.S.A. eails: {weiyu,cioffi}@stanford.edu Abstract This paper derives
More information