Introduction to Inverse Problems (2 lectures)


 Annice West
 2 years ago
 Views:
Transcription
1 Introduction to Inverse Problems (2 lectures) Summary Direct and inverse problems Examples of direct (forward) problems Deterministic and statistical points of view Illposed and illconditioned problems An illustrative example: The deconvolution problem Truncated Fourier decomposition (TFD); Tikhonov regularization Generalized Tikhonov regularization; Bayesian perspective. Iterative optimization. IP, José Bioucas Dias, 2007, IST 1
2 Direct/Inverse problems Direct (forward) problem Causes Inverse problem Effects Example: Direct problem: the computation of the trajectories of bodies from the knowledge of the forces. Inverse problem: determination of the forces from the knowledge of the trajectories Newton solved the first direct/inverse problem: the determintion of the gravitation force from the Kepler laws describing the trajectories of planets 2
3 An example: a linear time invariant (LTI) system Direct problem: Fourier domain Inverse problem: Source of difficulties: is unbounded A perturbation on leads to a perturbation on given by high frequencies of the perturbation are amplified, degrading the estimate of f 3
4 Image deblurring Observation model in (linear) image restoration/reconstruction observed image noise Linear operator original image (e.g., blur, tomography, MRI,...) Goal: estimate f from g 4
5 Image deblurring via regularization original Blurred, 9x9 uniform restored 5
6 MRI example Hydrogen density 2D frequency samples (9.4%) 6
7 Compressed Sensing (sparse representation) Sparse vector f Random matrix Observed data y Compressed Sensing N=1000 M=
8 Classes of direct problems Deterministic observation mechanism Original data (image) Operator + Observed data (image) perturbation 8
9 Classes of direct problems (deterministic) Linear spaceinvariant imaging systems Blur (motion, outoffocus, Diffractionlimited imaging atmospheric) Near field acoustic holography Channel equalization Parameter identification Linear spacevariant imaging systems (first kind Fredholm equation) Xray tomography MR imaging Radar imaging Sonar imaging Inverse diffraction Inverse source Linear regression 9
10 Classes of direct problems Statistical observation mechanism Original data (image) Observed data (image) Ex: Linear/nonlinear observations in additive Gaussian noise + 10
11 Classes of direct problems (statistic) Linear/nonlinear observation driven by nonadditive noise Parameters of a distribution Random signal/ image Rayleigh noise in coherent imaging Poisson noise in photoelectric conversion SPET (single photon emission tomography) PET (positron emission tomography) Ex: Amplitude in a coherent imaging system (radar, ultrasound) Terrain reflectance Inphase/quadrature backscattered signal 11
12 Wellposed/illposed inverse problems [Hadamard, 1923] Definition: Let be a (possible nonlinear) operator The inverse problem of solving Hadamard sense if: is wellposed in the 1) A solution exists for any in the observed data space 2) The solution is unique 3) The inverse mapping is continuous An inverse problem that is not wellposed is termed illposed The operator A of an inverse well/illposed problem is termed well/illposed 12
13 Finite/Infinite dimensional linear operators Linear Operators: The linear inverse problem equivalently, is wellposed if 1) and 2) holds or, and If is finitedimensional, the corresponding inverse problem is wellposed iif either one of the properties 1) and 2) holds Example: In infinitedimensional spaces Consider A defined on, If a solution of exists, it is unique since However, there are elements not in Thus, A is illposed (point 1 of the Hadamard conditions does not hold) Stability is also lacking: Take Then, does not converge when 13
14 Illconditioned inverse problems lllposed lllconditioned Many wellposed inverse problems are illconditioned, in the sense that For linear operators (tight bound) 14
15 Example: Discrete deconvolution Cyclic convolution Matrix notation Nperiodic funtions A is cyclic Toeplitz 15
16 Example: Discrete deconvolution 16
17 Eigendecomposition of cyclic matrices (unitary) Eigenvector (Fourier) matrix Eigenvalue matrix (diagonal) is the DFT of at frequency 17
18 Example: Discrete deconvolution (geometric viewpoint) Action of A on f Note: 18
19 Example: Discrete deconvolution (inferring f) Assume that Then is invertible and Thus, assuming the direct model We have error 19
20 Example: cyclic convolution with a Gaussian kernel What went wrong? 20
21 Example: Discrete deconvolution (estimation error) Size of the error Assume that Thus Which is a set enclosed by an ellipsoid with radii 21
22 Example: Discrete deconvolution (estimation error) The estimation error is the vector The components satisfy 22
23 Cyclic convolution with a Gaussian kernel (cont.) (unit impulse function) Noise dominates at high frequencies and is amplified by 23
24 Example: Discrete deconvolution (A is illposed) Assume now that is not invertible and it may happen that i.e, some are zero Leastsquares solution Projection error 24
25 Example: Discrete deconvolution (A is illposed) Leastsquares approach Orthogonal components 25
26 Example: Discrete deconvolution (A is illposed) Invisible objects is the minimum norm solution (related to the MoorePenrose inverse) 26
27 Example: Discrete deconvolution (Regularization) A is illconditioned A is illposed In both cases small eigenvalues are sources of instabilities Often, the smaller the eigenvalue the more oscilating the corresponding eigenvector (high frequences) Regularization by filtereing: shrink/threshold large values of i.e, multiply the eigenvalues by a regularizer filter such that as 27
28 Example: Discrete deconvolution Regularization by filtering (frequency multiplication time convolution) Such that 1) as 2 ) The larger eigenvalues are retained Truncated Fourier Decomposition (TFD) Tikhonov (Wiener) filter 28
29 Example: Discrete deconvolution (Regularization by filtering) TFD Tikhonov Tikhonov regularization Thus Solution of the variational problem 29
30 Example: Discrete deconvolution (1D example) Gaussian shaped of standard deviation = f g frequency
31 Example: Discrete deconvolution (1D example TFD) f f f f 1 f f f f 1 f f 1 f f
32 Example: Discrete deconvolution (2D examplefd) uniform 32
33 Example: Discrete deconvolution (2D exampletfd) 33
34 Curing Illposed/Illconditioned inverse problems Golden rule for solving illposed/illconditioned inverse problems Search for solutions which: are compatible with the observed data satisfy additional constraints (a priori or prior information) coming from the (physics) problem 34
35 Generalized Tikhonov regularization Tikhonov and TFD regularization are not well suited to deal with data Nonhomogeneities, such as edges Generalized Tikhonov regularization Bayesian viewpoint Data Discrepancy term Penalty/ Regularization term Negative loglikelihood Negative logprior 35
36 Dominating approaches to regularization 1) 2) 3) 4) In given circumstances 2), 3), and 4) are equivalent 36
37 Example: Discrete deconvolution (Nonquadratic regularization) penalize oscillatory solutions discontinuity preserving (robust) regularization is nonconvex hard optimization problem nondiscontinuity preserving regularization is convex treatable optimization problem 37
38 Optimization  Quadratic Linear system of equations Large systems require iterative methods  Nonquadratic and smooth Methods: Steepest descent, nonlinear conjugate gradient, Newton, trust regions,  Nonquadratic and nonsmooth Constrained optimization (Linear, quadratic, secondorder cone programs) Methods: Iterative Shrinkage/Thesholding; Coordinate Subspace Optimization; forwardbackward splitting; Primaldual Newton Majorization Minimizaton (MM) class 38
39 Majorization Minorization (MM) Framework Let Majorization Minorization algorithm:...with equality if and only if Easy to prove monotonicity: Notes: should be easy to maximize EM is an algorithm of this type. 39
40 Example: Discrete deconvolution (1D example NQ Regula.) Tikhonov f f f f f f f f f f f f
41 Example: Discrete deconvolution (2D exampletotal Variation) Total variation regularization (TV) where TV regularizer penalizes highly oscilatory solutions, while it preserves the edges 41
42 Bibliography [Ch1. RB2, Ch1. L1] Important topics Euclidian and Hilbert spaces of functions [App. A, RB2] Linear operators in function spaces [App. B, RB2] Euclidian vector spaces and matrices [App. C, RB2] Properties of the DFT and the FFT algorithm [App. B, RB2] Matlab scripts TFD_regularization_1D.m TFD_regularization_2D.m TFD_Error_1D.m TV_regulatization_1D.m 42
ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S.
ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute of Technology, New York, U.S.A. (g) CRC Press Taylor
More informationNumerical Methods For Image Restoration
Numerical Methods For Image Restoration CIRAM Alessandro Lanza University of Bologna, Italy Faculty of Engineering CIRAM Outline 1. Image Restoration as an inverse problem 2. Image degradation models:
More informationComputational Optical Imaging  Optique Numerique.  Deconvolution 
Computational Optical Imaging  Optique Numerique  Deconvolution  Winter 2014 Ivo Ihrke Deconvolution Ivo Ihrke Outline Deconvolution Theory example 1D deconvolution Fourier method Algebraic method
More informationAnother Example: the Hubble Space Telescope
296 DIP Chapter and 2: Introduction and Integral Equations Motivation: Why Inverse Problems? A largescale example, coming from a collaboration with Università degli Studi di Napoli Federico II in Naples.
More informationSparse recovery and compressed sensing in inverse problems
Gerd Teschke (7. Juni 2010) 1/68 Sparse recovery and compressed sensing in inverse problems Gerd Teschke (joint work with Evelyn Herrholz) Institute for Computational Mathematics in Science and Technology
More informationRegularized Matrix Computations
Regularized Matrix Computations Andrew E. Yagle Department of EECS, The University of Michigan, Ann Arbor, MI 489 Abstract We review the basic results on: () the singular value decomposition (SVD); ()
More informationAdvanced Signal Processing and Digital Noise Reduction
Advanced Signal Processing and Digital Noise Reduction Saeed V. Vaseghi Queen's University of Belfast UK WILEY HTEUBNER A Partnership between John Wiley & Sons and B. G. Teubner Publishers Chichester New
More informationROBERTO BATTITI, MAURO BRUNATO. The LION Way: Machine Learning plus Intelligent Optimization. LIONlab, University of Trento, Italy, Apr 2015
ROBERTO BATTITI, MAURO BRUNATO. The LION Way: Machine Learning plus Intelligent Optimization. LIONlab, University of Trento, Italy, Apr 2015 http://intelligentoptimization.org/lionbook Roberto Battiti
More informationEigenvalues of C SMS (16) 1 A. Eigenvalues of A (32) 1 A (16,32) 1 A
Negative Results for Multilevel Preconditioners in Image Deblurring C. R. Vogel Department of Mathematical Sciences Montana State University Bozeman, MT 597170240 USA Abstract. A onedimensional deconvolution
More informationCOMPUTATIONAL INTELLIGENCE (INTRODUCTION TO MACHINE LEARNING) SS16. Lecture 2: Linear Regression Gradient Descent Nonlinear basis functions
COMPUTATIONAL INTELLIGENCE (INTRODUCTION TO MACHINE LEARNING) SS16 Lecture 2: Linear Regression Gradient Descent Nonlinear basis functions LINEAR REGRESSION MOTIVATION Why Linear Regression? Regression
More informationInverse problems. Nikolai Piskunov 2014. Regularization: Example Lecture 4
Inverse problems Nikolai Piskunov 2014 Regularization: Example Lecture 4 Now that we know the theory, let s try an application: Problem: Optimal filtering of 1D and 2D data Solution: formulate an inverse
More informationAN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEYINTERSCIENCE A John Wiley & Sons, Inc.,
More information1. Introduction. Consider the computation of an approximate solution of the minimization problem
A NEW TIKHONOV REGULARIZATION METHOD MARTIN FUHRY AND LOTHAR REICHEL Abstract. The numerical solution of linear discrete illposed problems typically requires regularization, i.e., replacement of the available
More informationNumerical Recipes in C
2008 AGIInformation Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. Numerical Recipes in C The Art of Scientific Computing Second Edition
More informationLecture 14. Point Spread Function (PSF)
Lecture 14 Point Spread Function (PSF), Modulation Transfer Function (MTF), Signaltonoise Ratio (SNR), Contrasttonoise Ratio (CNR), and Receiver Operating Curves (ROC) Point Spread Function (PSF) Recollect
More informationP164 Tomographic Velocity Model Building Using Iterative Eigendecomposition
P164 Tomographic Velocity Model Building Using Iterative Eigendecomposition K. Osypov* (WesternGeco), D. Nichols (WesternGeco), M. Woodward (WesternGeco) & C.E. Yarman (WesternGeco) SUMMARY Tomographic
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 6 Three Approaches to Classification Construct
More informationConvolution. 1D Formula: 2D Formula: Example on the web: http://www.jhu.edu/~signals/convolve/
Basic Filters (7) Convolution/correlation/Linear filtering Gaussian filters Smoothing and noise reduction First derivatives of Gaussian Second derivative of Gaussian: Laplacian Oriented Gaussian filters
More informationLeastSquares Intersection of Lines
LeastSquares Intersection of Lines Johannes Traa  UIUC 2013 This writeup derives the leastsquares solution for the intersection of lines. In the general case, a set of lines will not intersect at a
More informationBindel, Fall 2012 Matrix Computations (CS 6210) Week 8: Friday, Oct 12
Why eigenvalues? Week 8: Friday, Oct 12 I spend a lot of time thinking about eigenvalue problems. In part, this is because I look for problems that can be solved via eigenvalues. But I might have fewer
More informationApplications to Data Smoothing and Image Processing I
Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is
More informationSection for Cognitive Systems DTU Informatics, Technical University of Denmark
Transformation Invariant Sparse Coding Morten Mørup & Mikkel N Schmidt Morten Mørup & Mikkel N. Schmidt Section for Cognitive Systems DTU Informatics, Technical University of Denmark Redundancy Reduction
More informationALGEBRAIC EIGENVALUE PROBLEM
ALGEBRAIC EIGENVALUE PROBLEM BY J. H. WILKINSON, M.A. (Cantab.), Sc.D. Technische Universes! Dsrmstedt FACHBEREICH (NFORMATiK BIBL1OTHEK Sachgebieto:. Standort: CLARENDON PRESS OXFORD 1965 Contents 1.
More informationModern methods for solving inverse problems
Modern methods for solving inverse problems Jürgen Hesser Experimental Radiation Oncology University Medical Center Mannheim University of Heidelberg 1 Motivation Image Guided Radiation Therapy (IGRT)
More informationContents. Gbur, Gregory J. Mathematical methods for optical physics and engineering digitalisiert durch: IDS Basel Bern
Preface page xv 1 Vector algebra 1 1.1 Preliminaries 1 1.2 Coordinate System invariance 4 1.3 Vector multiplication 9 1.4 Useful products of vectors 12 1.5 Linear vector Spaces 13 1.6 Focus: periodic media
More informationSTUDY GUIDE LINEAR ALGEBRA. David C. Lay University of Maryland College Park AND ITS APPLICATIONS THIRD EDITION UPDATE
STUDY GUIDE LINEAR ALGEBRA AND ITS APPLICATIONS THIRD EDITION UPDATE David C. Lay University of Maryland College Park Copyright 2006 Pearson AddisonWesley. All rights reserved. Reproduced by Pearson AddisonWesley
More informationChapter 3. Inverse Problems
20 Chapter 3 Inverse Problems Mathematicians often cannot help but cringe when physicists are doing math, and even more so whenever physicists claim to be rigorous with their math. This chapter is not
More informationBlind Deconvolution of Corrupted Barcode Signals
Blind Deconvolution of Corrupted Barcode Signals Everardo Uribe and Yifan Zhang Advisors: Ernie Esser and Yifei Lou Interdisciplinary Computational and Applied Mathematics Program University of California,
More informationSystem Identification for Acoustic Comms.:
System Identification for Acoustic Comms.: New Insights and Approaches for Tracking Sparse and Rapidly Fluctuating Channels Weichang Li and James Preisig Woods Hole Oceanographic Institution The demodulation
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical
More informationNumerical Methods I Eigenvalue Problems
Numerical Methods I Eigenvalue Problems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420001, Fall 2010 September 30th, 2010 A. Donev (Courant Institute)
More informationLinear Algebra. Introduction
Linear Algebra Caren Diefenderfer, Hollins University, chair David Hill, Temple University, lead writer Sheldon Axler, San Francisco State University Nancy Neudauer, Pacific University David Strong, Pepperdine
More informationBlind Deconvolution of Barcodes via Dictionary Analysis and Wiener Filter of Barcode Subsections
Blind Deconvolution of Barcodes via Dictionary Analysis and Wiener Filter of Barcode Subsections Maximilian Hung, Bohyun B. Kim, Xiling Zhang August 17, 2013 Abstract While current systems already provide
More information2.5 Complex Eigenvalues
1 25 Complex Eigenvalues Real Canonical Form A semisimple matrix with complex conjugate eigenvalues can be diagonalized using the procedure previously described However, the eigenvectors corresponding
More informationBy choosing to view this document, you agree to all provisions of the copyright laws protecting it.
This material is posted here with permission of the IEEE Such permission of the IEEE does not in any way imply IEEE endorsement of any of Helsinki University of Technology's products or services Internal
More informationELECE8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems
Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Minimum Mean Square Error (MMSE) MMSE estimation of Gaussian random vectors Linear MMSE estimator for arbitrarily distributed
More information1646 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 6, NO. 12, DECEMBER 1997
1646 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL 6, NO 12, DECEMBER 1997 Restoration of a Single Superresolution Image from Several Blurred, Noisy, and Undersampled Measured Images Michael Elad and Arie
More informationLatest Results on HighResolution Reconstruction from Video Sequences
Latest Results on HighResolution Reconstruction from Video Sequences S. Lertrattanapanich and N. K. Bose The Spatial and Temporal Signal Processing Center Department of Electrical Engineering The Pennsylvania
More informationNeed for Speed in MRI & Parallel Imaging I
G16.4428 Practical Magnetic Resonance Imaging II Sackler Institute of Biomedical Sciences New York University School of Medicine Need for Speed in MRI & Parallel Imaging I Ricardo Otazo, PhD ricardo.otazo@nyumc.org
More informationChange Point Estimation of Bar Code Signals
1 Change Point Estimation of Bar Code Signals Leming Qu, and YiCheng Tu Abstract Existing methods for bar code signal reconstruction is based on either the local approach or the regularization approach
More informationNumerical Analysis Introduction. Student Audience. Prerequisites. Technology.
Numerical Analysis Douglas Faires, Youngstown State University, (Chair, 20122013) Elizabeth Yanik, Emporia State University, (Chair, 20132015) Graeme Fairweather, Executive Editor, Mathematical Reviews,
More informationMarkov chains and Markov Random Fields (MRFs)
Markov chains and Markov Random Fields (MRFs) 1 Why Markov Models We discuss Markov models now. This is the simplest statistical model in which we don t assume that all variables are independent; we assume
More informationComputational Methods CMSC/AMSC/MAPL 460. Eigenvalues and Eigenvectors. Ramani Duraiswami, Dept. of Computer Science
Computational Methods CMSC/AMSC/MAPL 460 Eigenvalues and Eigenvectors Ramani Duraiswami, Dept. of Computer Science Eigen Values of a Matrix Definition: A N N matrix A has an eigenvector x (nonzero) with
More informationPrincipal Component Analysis Application to images
Principal Component Analysis Application to images Václav Hlaváč Czech Technical University in Prague Faculty of Electrical Engineering, Department of Cybernetics Center for Machine Perception http://cmp.felk.cvut.cz/
More informationSignal Processing for Intelligent Sensor Systems with MATLAB 9
Signal Processing for Intelligent Sensor Systems with MATLAB 9 Second Edition David С Swanson @ CRC Press Taylor & Francis Gro jroup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis
More informationCompressive Acquisition of Sparse Deflectometric Maps
Compressive Acquisition of Sparse Deflectometric Maps Prasad Sudhakar*, Laurent Jacques*, Adriana Gonzalez Gonzalez* Xavier Dubois +, Philippe Antoine +, Luc Joannes + *: Louvain University (UCL), LouvainlaNeuve,
More informationORTHOGONAL PROJECTION REGULARIZATION OPERATORS
ORTHOGONAL PROJECTION REGULARIZATION OPERATORS S. MORIGI, L. REICHEL, AND F. SGALLARI Abstract. Tikhonov regularization often is applied with a finite difference regularization operator that approximates
More informationBildverarbeitung und Mustererkennung Image Processing and Pattern Recognition
Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition 710.080 2VO 710.081 1KU 1 Optical Flow (I) Content Introduction Local approach (Lucas Kanade) Global approaches (HornSchunck,
More informationDigital Image Processing
Digital Image Processing Using MATLAB Second Edition Rafael C. Gonzalez University of Tennessee Richard E. Woods MedData Interactive Steven L. Eddins The MathWorks, Inc. Gatesmark Publishing A Division
More informationInternational Journal of Advanced Research in Computer Science and Software Engineering
Volume 4, Issue 8, August 2014 ISSN: 2277 128X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: www.ijarcsse.com Comparative Study
More informationLecture 9: Continuous
CSC2515 Fall 2007 Introduction to Machine Learning Lecture 9: Continuous Latent Variable Models 1 Example: continuous underlying variables What are the intrinsic latent dimensions in these two datasets?
More informationPTE505: Inverse Modeling for Subsurface Flow Data Integration (3 Units)
PTE505: Inverse Modeling for Subsurface Flow Data Integration (3 Units) Instructor: Behnam Jafarpour, Mork Family Department of Chemical Engineering and Material Science Petroleum Engineering, HED 313,
More informationCurve Fitting. Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization.
Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization Subsections LeastSquares Regression Linear Regression General Linear LeastSquares Nonlinear
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationNew Method for Deconvolving Adaptive Optics Images
New Method for Deconvolving Adaptive Optics Images or: (All you ever wanted to know about...) Myopic maximumlikelihood deconvolution with a spacevarying kernel Ralf Flicker, François Rigaut Gemini Observatory,
More informationDual Methods for Total VariationBased Image Restoration
Dual Methods for Total VariationBased Image Restoration Jamylle Carter Institute for Mathematics and its Applications University of Minnesota, Twin Cities Ph.D. (Mathematics), UCLA, 2001 Advisor: Tony
More informationThe Power Method for Eigenvalues and Eigenvectors
Numerical Analysis Massoud Malek The Power Method for Eigenvalues and Eigenvectors The spectrum of a square matrix A, denoted by σ(a) is the set of all eigenvalues of A. The spectral radius of A, denoted
More informationLinear Regression CS434. Supervised learning
Linear Regression CS434 A regression problem We want to learn to predict a person s height based on his/her knee height and/or arm span This is useful for patients who are bed bound and cannot stand to
More informationPrinciples of Digital Communication
Principles of Digital Communication Robert G. Gallager January 5, 2008 ii Preface: introduction and objectives The digital communication industry is an enormous and rapidly growing industry, roughly comparable
More informationJournal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics 226 (2009) 92 102 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
More informationAdvanced Computational Techniques in Engineering
Course Details Course Code: 5MWCI432 Level: M Tech CSE Semester: Semester 4 Instructor: Prof Dr S P Ghrera Advanced Computational Techniques in Engineering Outline Course Description: Data Distributions,
More informationComputational Methods. Least Squares Approximation/Optimization
Computational Methods Least Squares Approximation/Optimization Manfred Huber 2010 1 Least Squares Least squares methods are aimed at finding approximate solutions when no precise solution exists Find the
More informationOptical Flow. Thomas Pock
Optical Flow Thomas Pock 1 Optical Flow (II) Content Global approaches (HornSchunck, TVL1) Coarsetofine warping 2 The Horn and Schunck (HS) Method 3 The Horn and Schunck (HS) Method Global energy to
More informationRegression Using Support Vector Machines: Basic Foundations
Regression Using Support Vector Machines: Basic Foundations Technical Report December 2004 Aly Farag and Refaat M Mohamed Computer Vision and Image Processing Laboratory Electrical and Computer Engineering
More informationNumerical Recipes in C++
Numerical Recipes in C++ The Art of Scientific Computing Second Edition William H. Press Los Alamos National Laboratory Saul A. Teukolsky Department of Physics, Cornell University William T. Vetterling
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More informationVector Derivatives, Gradients, and Generalized Gradient Descent Algorithms
Vector Derivatives, Gradients, and Generalized Gradient Descent Algorithms ECE 275A Statistical Parameter Estimation Ken KreutzDelgado ECE Department, UC San Diego November 1, 2013 Ken KreutzDelgado
More informationLinear Threshold Units
Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear
More informationSignal Detection. Outline. Detection Theory. Example Applications of Detection Theory
Outline Signal Detection M. Sami Fadali Professor of lectrical ngineering University of Nevada, Reno Hypothesis testing. NeymanPearson (NP) detector for a known signal in white Gaussian noise (WGN). Matched
More informationNumerical Methods I Solving Linear Systems: Sparse Matrices, Iterative Methods and NonSquare Systems
Numerical Methods I Solving Linear Systems: Sparse Matrices, Iterative Methods and NonSquare Systems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420001,
More informationCS 591.03 Introduction to Data Mining Instructor: Abdullah Mueen
CS 591.03 Introduction to Data Mining Instructor: Abdullah Mueen LECTURE 3: DATA TRANSFORMATION AND DIMENSIONALITY REDUCTION Chapter 3: Data Preprocessing Data Preprocessing: An Overview Data Quality Major
More informationLECTURE NOTES: FINITE ELEMENT METHOD
LECTURE NOTES: FINITE ELEMENT METHOD AXEL MÅLQVIST. Motivation The finite element method has two main strengths... Geometry. Very complex geometries can be used. This is probably the main reason why finite
More informationReview Jeopardy. Blue vs. Orange. Review Jeopardy
Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 03 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?
More informationInverse Problems and Regularization An Introduction
Inverse Problems and Regularization An Introduction Stefan Kindermann Industrial Mathematics Institute University of Linz, Austria What are Inverse Problems? One possible definition [Engl, Hanke, Neubauer
More informationNMR Measurement of T1T2 Spectra with Partial Measurements using Compressive Sensing
NMR Measurement of T1T2 Spectra with Partial Measurements using Compressive Sensing Alex Cloninger Norbert Wiener Center Department of Mathematics University of Maryland, College Park http://www.norbertwiener.umd.edu
More informationMean value theorem, Taylors Theorem, Maxima and Minima.
MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and expressions. Permutations and Combinations.
More informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More informationUnderstanding and Applying Kalman Filtering
Understanding and Applying Kalman Filtering Lindsay Kleeman Department of Electrical and Computer Systems Engineering Monash University, Clayton 1 Introduction Objectives: 1. Provide a basic understanding
More information4F7 Adaptive Filters (and Spectrum Estimation) Least Mean Square (LMS) Algorithm Sumeetpal Singh Engineering Department Email : sss40@eng.cam.ac.
4F7 Adaptive Filters (and Spectrum Estimation) Least Mean Square (LMS) Algorithm Sumeetpal Singh Engineering Department Email : sss40@eng.cam.ac.uk 1 1 Outline The LMS algorithm Overview of LMS issues
More informationTotal Variation Regularization in PET Reconstruction
Total ariation Regularization in PET Reconstruction Milán Magdics, Balázs Tóth, Balázs Kovács, and László SzirmayKalos BME IIT, Hungary magdics@iit.bme.hu Abstract. Positron Emission Tomography reconstruction
More information2094 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER InformationTheoretic Image Formation. (Invited Paper)
2094 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998 InformationTheoretic Image Formation Joseph A. O Sullivan, Senior Member, IEEE, Richard E. Blahut, Fellow, IEEE, and Donald L.
More informationADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB
ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute
More informationWAVES AND FIELDS IN INHOMOGENEOUS MEDIA
WAVES AND FIELDS IN INHOMOGENEOUS MEDIA WENG CHO CHEW UNIVERSITY OF ILLINOIS URBANACHAMPAIGN IEEE PRESS Series on Electromagnetic Waves Donald G. Dudley, Series Editor IEEE Antennas and Propagation Society,
More informationTHE PROBLEM OF finding localized energy solutions
600 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 3, MARCH 1997 Sparse Signal Reconstruction from Limited Data Using FOCUSS: A Reweighted Minimum Norm Algorithm Irina F. Gorodnitsky, Member, IEEE,
More informationMeanShift Tracking with Random Sampling
1 MeanShift Tracking with Random Sampling Alex Po Leung, Shaogang Gong Department of Computer Science Queen Mary, University of London, London, E1 4NS Abstract In this work, boosting the efficiency of
More informationLecture 5: Variants of the LMS algorithm
1 Standard LMS Algorithm FIR filters: Lecture 5: Variants of the LMS algorithm y(n) = w 0 (n)u(n)+w 1 (n)u(n 1) +...+ w M 1 (n)u(n M +1) = M 1 k=0 w k (n)u(n k) =w(n) T u(n), Error between filter output
More informationCortical Source Localization of Human Scalp EEG. Kaushik Majumdar Indian Statistical Institute Bangalore Center
Cortical Source Localization of Human Scalp EEG Kaushik Majumdar Indian Statistical Institute Bangalore Center Cortical Basis of Scalp EEG Baillet et al, IEEE Sig Proc Mag, Nov 2001, p 16 Mountcastle,
More informationLecture 3: Linear methods for classification
Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,
More informationLecture Topic: LowRank Approximations
Lecture Topic: LowRank Approximations LowRank Approximations We have seen principal component analysis. The extraction of the first principle eigenvalue could be seen as an approximation of the original
More informationFinite Dimensional Hilbert Spaces and Linear Inverse Problems
Finite Dimensional Hilbert Spaces and Linear Inverse Problems ECE 174 Lecture Supplement Spring 2009 Ken KreutzDelgado Electrical and Computer Engineering Jacobs School of Engineering University of California,
More informationarxiv:math/ v2 [math.fa] 2 Nov 2003
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint Ingrid Daubechies Michel Defrise Christine De Mol February 1, 2008 arxiv:math/0307152v2 [math.fa] 2 Nov 2003 Abstract
More informationDiscrete Fourier Series & Discrete Fourier Transform Chapter Intended Learning Outcomes
Discrete Fourier Series & Discrete Fourier Transform Chapter Intended Learning Outcomes (i) Understanding the relationships between the transform, discretetime Fourier transform (DTFT), discrete Fourier
More informationIntroduction and message of the book
1 Introduction and message of the book 1.1 Why polynomial optimization? Consider the global optimization problem: P : for some feasible set f := inf x { f(x) : x K } (1.1) K := { x R n : g j (x) 0, j =
More informationBayesian Image SuperResolution
Bayesian Image SuperResolution Michael E. Tipping and Christopher M. Bishop Microsoft Research, Cambridge, U.K..................................................................... Published as: Bayesian
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More information168 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 1, JANUARY Yonina C. Eldar, Member, IEEE, Aharon BenTal, and Arkadi Nemirovski
168 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 1, JANUARY 2005 Robust MeanSquared Error Estimation in the Presence of Model Uncertainties Yonina C. Eldar, Member, IEEE, Aharon BenTal, and Arkadi
More informationTime Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication
Time Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication Thomas Reilly Data Physics Corporation 1741 Technology Drive, Suite 260 San Jose, CA 95110 (408) 2168440 This paper
More informationOutline. Random Variables. Examples. Random Variable
Outline Random Variables M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno Random variables. CDF and pdf. Joint random variables. Correlated, independent, orthogonal. Correlation,
More informationADVANCED ALGORITHMS FOR EQUALIZATION ON ADSL CHANNEL
ADVANCED ALGORITHMS FOR EQUALIZATION ON ADSL CHANNEL T. Mazanec Institute of Information Theory and Automation, Dept. of Signal Processing Abstract Digital subscriber line modems equalize response of line
More informationThe Image Deblurring Problem
page 1 Chapter 1 The Image Deblurring Problem You cannot depend on your eyes when your imagination is out of focus. Mark Twain When we use a camera, we want the recorded image to be a faithful representation
More information