Constructing Approximation Kernels for Non-Harmonic Fourier Data
|
|
- Melvyn Hunter
- 5 years ago
- Views:
Transcription
1 Constructing Approximation Kernels for Non-Harmonic Fourier Data Aditya Viswanathan California Institute of Technology SIAM Annual Meeting 2013 July / 19
2 Joint work with Anne Gelb Sidi Kaber Research supported in part by National Science Foundation grants CNS , DMS and DMS (FRG). 0 / 19
3 Motivating Application Magnetic Resonance Imaging Physics of MRI dictates that the MR scanner collect samples of the Fourier transform of the specimen being imaged. 1 / 19
4 Motivating Application Magnetic Resonance Imaging Collecting non-uniform measurements has certain advantages; for example, they are easier and faster to collect, and, aliased images retain diagnostic qualities. Reconstructing images from such measurements accurately and efficiently is, however, challenging. 1 / 19
5 Model Problem Let f be defined in R with support in [ π, π). Given ˆf(ω k ) = f, e iω kx, k = N,, N, ω k not necessarily Z, compute an approximation to the underlying (possibly piecewise-smooth) function f, an approximation to the locations and values of jumps in the underlying function; i.e., [f](x) := f(x + ) f(x ). Issues Sparse sampling of the high frequencies, i.e., ω k k > 1 for k large. The DFT is not defined for ω k k; the FFT is not directly applicable. 2 / 19
6 Outline Introduction Motivating Application Simplified Model Problem Non-Harmonic Fourier Reconstruction Uniform Re-sampling Convolutional Gridding Harmonic and Non-Harmonic Kernels Designing Convolutional Gridding Kernels Edge Detection Concentration Method Design of Non-Harmonic Edge Detection Kernels 2 / 19
7 Uniform Re-sampling (Rosenfeld) We consider direct methods of recovering f and [f] from ˆf(ω k ). Due to our familiarity with harmonic Fourier reconstructions and the applicability of FFTs, we will consider a two step process: 1. Approximate the Fourier coefficients at equispaced modes 2. Compute a standard (filtered) Fourier partial sum Basic Premise f is compactly supported in physical space. Hence, the Shannon sampling theorem is applicable in Fourier space; i.e., ˆf(ω) = k= sinc(ω k) ˆf k, ω R. 3 / 19
8 Uniform Re-sampling Implementation We truncate the problem as follows ˆf(ω k ) sinc(ω k l) ˆf }{{} l, l M A R 2N+1 2M+1 k = N,, N The equispaced coefficients are approximated using f l = A ˆf(ω k ), where A is the Moore-Penrose pseudo-inverse of A. A and its properties characterize the resulting approximation. Regularization may be used (truncated SVD, Tikhonov regularization) in the presence of noise. A is (unfortunately) a dense matrix in general, with the computation of f requiring O ( N 2) operations. 4 / 19
9 Uniform Re-sampling Sampling Patterns Consider the sampling pattern ω k = k ± U[0, µ], k = N,, N where U[a, b] denotes an iid uniform distribution in [a, b] with equiprobable positive/negative jitter. Jitter µ κ(a) / 19
10 Uniform Re-sampling An Example Reconstruction using jittered samples (µ = 0.5). Error Fourier Modes Recontruction 6 / 19
11 Uniform Re-sampling An Example Reconstruction using jittered samples (µ = 0.5). Error Fourier Modes Reconstruction 6 / 19
12 From Uniform Re-sampling to Convolutional Gridding Recall that for uniform re-sampling, we use the relation ˆf(ω) = k sinc(ω k) ˆf k = ( ˆf sinc)(ω) Since the Fourier transform pair of the sinc function is the box/rect function (of width 2π and centered at zero), we have f Π ˆf sinc Now consider replacing the sinc function by a bandlimited function ˆφ such that ˆφ( ω ) = 0 for ω > q (typically a few modes wide). We now have f φ ˆf ˆφ 7 / 19
13 Convolutional Gridding (Jackson/Meyer/Nishimura and others) Gridding is an inexpensive direct approximation to the uniform re-sampling procedure. Given measurements ˆf(ω k ), we compute an approximation to ˆf ˆφ at the equispaced modes using ( ˆf ˆφ)(l) l ω k q α k ˆf(ωk ) ˆφ(l ω k ), l = M,..., M. Now that we are on equispaced modes, use a (F)DFT to reconstruct an approximation to f φ in physical space. Recover f by dividing out φ. α k are density compensation factors (DCFs) and determine the accuracy of the reconstruction. 8 / 19
14 Analysis of the Convolution Gridding Sum The gridding approximation can be written as α k ˆf(ωk ) ˆφ(l ω k ) e ilx f cg (x) = = l M l ω k q φ(x) k l α ( k f(ξ)e iω k ξ dξ ) ( φ(η)e i(l ωk)η dη ) e ilx φ(x) ( ) ( ) f(ξ)φ(η) α k e iω k(η ξ) e il(x η) dξdη k }{{}}{{} A = α N (η ξ) D N (x η) φ(x) (f A α = N )(η) φ(η) D N (x η)dη φ(x) = ([{f Aα N } φ] D N) (x) φ(x) l 9 / 19
15 The Dirichlet Kernel A Review Given ˆf k := f, e ikx, k = N,, N, a periodic repetition of f may be reconstructed using the Fourier partial sum P N f(x) = ˆf k e ikx = (f D N )(x), where k N D N (x) = k N e ikx is the Dirichlet kernel. D N is the bandlimited (2N + 1 mode) approximation of the Dirac delta distribution. 10 / 19
16 The Dirichlet Kernel A Review D N completely characterizes the Fourier approximation P N f. Filtered and jump approximations are similarly characterized by equivalent filtered and (filtered) conjugate Dirichlet kernels. 10 / 19
17 The Non-Harmonic Kernel Consider the non-harmonic kernel A N (x) = k N e iω kx A N is non-periodic. The non-harmonic kernel is a poor approximation to the Dirac delta distribution. Depending on the mode distribution, A N may be non-decaying. Filtering is of no help under these circumstances. 11 / 19
18 The Non-Harmonic Kernel Jittered Modes ω k = k ± U[0, µ], µ = / 19
19 The Non-Harmonic Kernel Log Modes ω k logarithmically spaced 11 / 19
20 Outline Introduction Motivating Application Simplified Model Problem Non-Harmonic Fourier Reconstruction Uniform Re-sampling Convolutional Gridding Harmonic and Non-Harmonic Kernels Designing Convolutional Gridding Kernels Edge Detection Concentration Method Design of Non-Harmonic Edge Detection Kernels 11 / 19
21 Designing Gridding Kernels Recall that the convolutional gridding sum can be written in the form f cg (x) = ([{f Aα N } φ] D N) (x) φ(x) where the (weighted) non-harmonic kernel A α N(x) = k N α k e iω kx. α k are free design parameters which we choose such that A α N is compactly supported and a good reconstruction kernel (such as the Dirichlet kernel) in the interval of interest. 12 / 19
22 Design Problem Formulation Choose α = {α k } N N such that α k e iωkx k N l M e ilx x π 0 else Discretizing on an equispaced grid, we obtain the linear system of equations Dα = b, where D l,j = e iω lx j denotes the (non-harmonic) DFT matrix, and b p = sin(m+1/2)xp sin(x p/2 ) Π are the values of the Dirichlet kernel on the equispaced grid. 13 / 19
23 Design Problem Formulation Choose α = {α k } N N such that α k e iωkx k N l M σ l e ilx x π 0 else Discretizing on an equispaced grid, we obtain the linear system of equations Dα = b, where D l,j = e iω lx j denotes the (non-harmonic) DFT matrix, and b p are the values of the (filtered) Dirichlet kernel on the equispaced grid. 13 / 19
24 Numerical Results Reconstruction ω k logarithmically spaced N = 256 measurements Iterative weights solved using LSQR 14 / 19
25 Numerical Results Reconstruction Error ω k logarithmically spaced N = 256 measurements Iterative weights solved using LSQR 14 / 19
26 Numerical Results DCF weights α ω k logarithmically spaced N = 256 measurements Iterative weights solved using LSQR 14 / 19
27 Outline Introduction Motivating Application Simplified Model Problem Non-Harmonic Fourier Reconstruction Uniform Re-sampling Convolutional Gridding Harmonic and Non-Harmonic Kernels Designing Convolutional Gridding Kernels Edge Detection Concentration Method Design of Non-Harmonic Edge Detection Kernels 14 / 19
28 Concentration Method (Gelb, Tadmor) Approximate the singular support of f using the generalized conjugate partial Fourier sum S σ N[f](x) = i N k= N ˆf(k) sgn(k) σ ( ) k e ikx N σ k,n (η) = σ( k N ) are known as concentration factors which are required to satisfy certain admissibility conditions. Under these conditions, S σ N[f](x) = [f](x) + O(ɛ), ɛ = ɛ(n) > 0 being small i.e., SN σ [f] concentrates at the singular support of f. 15 / 19
29 Concentration Factors Factor Expression π sin(α η) Trigonometric σ T (η) = Si(α) α sin(x) Si(α) = dx 0 x Polynomial σ P (η) = p π η p p is the order of [ the factor ] Exponential σ E (η) = Cη exp 1 α η (η 1) C - normalizing constant α - order π C = 1 1 N 1 N [ exp 1 α τ (τ 1) ] dτ Table: Examples of concentration factors Figure: Envelopes of Factors in k-space 16 / 19
30 Some Examples Jump Approximation Trigonometric Factor Jump Approximation Exponential Factor f S N [f] f(x) 0 f(x) f S N [f] x x (a) Trigonometric Factor (b) Exponential Factor Figure: Jump Function Approximation, N = / 19
31 Designing Non-Harmonic Edge Detection Kernels Choose α = {α k } N N such that i sgn(l)σ( l /N)e ilx x π α k e iωkx l M k N 0 else Discretizing on an equispaced grid, we obtain the linear system of equations Dα = b, where D l,j = e iω lx j denotes the (non-harmonic) DFT matrix, and bp are the values of the generalized conjugate Dirichlet kernel on the equispaced grid. 18 / 19
32 Numerical Results Jump Approximation and Corresponding Weights ω k logarithmically spaced N = 256 measurements Iterative weights solved using LSQR 19 / 19
33 Summary and Future Directions 1. Applications such as MR imaging require reconstruction from non-harmonic Fourier measurements. 2. Assuming the function of interest is compactly supported, the underlying relation between non-harmonic and harmonic Fourier data is the Shannon sampling theorem (sinc interpolation). 3. Convolutional gridding is an efficient approximation to sinc-based resampling. 4. A set of free parameters known as the density compensation factors (DCFs) allow us to design gridding kernels with favorable characteristics. 5. To do compare results with frame theoretic approaches, use banded DCFs to obtain better gridding approximations. 19 / 19
34 Selected References 1. D. Rosenfeld, An Optimal and Efficient New Gridding Algorithm Using Singular Value Decomposition, in Magn. Reson. Med., Vol. 40, No. 1 (1998), pp J. Jackson, C. Meyer and D. Nishimura, Selection of a Convolution Function for Fourier Inversion Using Gridding, in IEEE Trans. Med. Img., Vol. 10, No. 3 (1991), pp J. Fessler and B. Sutton, Nonuniform Fast Fourier Transforms Using Min-Max Interpolation, in IEEE Trans. Sig. Proc., Vol. 51, No. 2 (2003), pp J. Pipe and P. Menon, Sampling Density Compensation in MRI: Rationale and an Iterative Numerical Solution, in Magn. Reson. Med., Vol. 41, No. 1 (1999), pp A. Gelb and E. Tadmor, Detection of Edges in Spectral Data, in Appl. Comp. Harmonic Anal., 7 (1999), pp / 19
Numerical Approximation Methods for Non-Uniform Fourier Data
Numerical Approximation Methods for Non-Uniform Fourier Data Aditya Viswanathan aditya@math.msu.edu 2014 Joint Mathematics Meetings January 18 2014 0 / 16 Joint work with Anne Gelb (Arizona State) Guohui
More informationDirect Methods for Reconstruction of Functions and their Edges from Non-Uniform Fourier Data
Direct Methods for Reconstruction of Functions and their Edges from Non-Uniform Fourier Data Aditya Viswanathan aditya@math.msu.edu ICERM Research Cluster Computational Challenges in Sparse and Redundant
More informationFourier Reconstruction from Non-Uniform Spectral Data
School of Electrical, Computer and Energy Engineering, Arizona State University aditya.v@asu.edu With Profs. Anne Gelb, Doug Cochran and Rosemary Renaut Research supported in part by National Science Foundation
More informationCompensation. June 29, Arizona State University. Edge Detection from Nonuniform Data via. Convolutional Gridding with Density.
Arizona State University June 29, 2012 Overview gridding with density compensation is a common method for function reconstruction nonuniform Fourier data. It is computationally inexpensive compared with
More informationFourier reconstruction of univariate piecewise-smooth functions from non-uniform spectral data with exponential convergence rates
Fourier reconstruction of univariate piecewise-smooth functions from non-uniform spectral data with exponential convergence rates Rodrigo B. Platte a,, Alexander J. Gutierrez b, Anne Gelb a a School of
More informationAPPROXIMATING THE INVERSE FRAME OPERATOR FROM LOCALIZED FRAMES
APPROXIMATING THE INVERSE FRAME OPERATOR FROM LOCALIZED FRAMES GUOHUI SONG AND ANNE GELB Abstract. This investigation seeks to establish the practicality of numerical frame approximations. Specifically,
More informationRecovery of Compactly Supported Functions from Spectrogram Measurements via Lifting
Recovery of Compactly Supported Functions from Spectrogram Measurements via Lifting Mark Iwen markiwen@math.msu.edu 2017 Friday, July 7 th, 2017 Joint work with... Sami Merhi (Michigan State University)
More informationPolynomial Approximation: The Fourier System
Polynomial Approximation: The Fourier System Charles B. I. Chilaka CASA Seminar 17th October, 2007 Outline 1 Introduction and problem formulation 2 The continuous Fourier expansion 3 The discrete Fourier
More informationIMAGE RECONSTRUCTION FROM UNDERSAMPLED FOURIER DATA USING THE POLYNOMIAL ANNIHILATION TRANSFORM
IMAGE RECONSTRUCTION FROM UNDERSAMPLED FOURIER DATA USING THE POLYNOMIAL ANNIHILATION TRANSFORM Anne Gelb, Rodrigo B. Platte, Rosie Renaut Development and Analysis of Non-Classical Numerical Approximation
More informationJim Lambers ENERGY 281 Spring Quarter Lecture 5 Notes
Jim ambers ENERGY 28 Spring Quarter 27-8 ecture 5 Notes These notes are based on Rosalind Archer s PE28 lecture notes, with some revisions by Jim ambers. Fourier Series Recall that in ecture 2, when we
More informationMGA Tutorial, September 08, 2004 Construction of Wavelets. Jan-Olov Strömberg
MGA Tutorial, September 08, 2004 Construction of Wavelets Jan-Olov Strömberg Department of Mathematics Royal Institute of Technology (KTH) Stockholm, Sweden Department of Numerical Analysis and Computer
More informationFourier Series. (Com S 477/577 Notes) Yan-Bin Jia. Nov 29, 2016
Fourier Series (Com S 477/577 otes) Yan-Bin Jia ov 9, 016 1 Introduction Many functions in nature are periodic, that is, f(x+τ) = f(x), for some fixed τ, which is called the period of f. Though function
More informationFast and Robust Phase Retrieval
Fast and Robust Phase Retrieval Aditya Viswanathan aditya@math.msu.edu CCAM Lunch Seminar Purdue University April 18 2014 0 / 27 Joint work with Yang Wang Mark Iwen Research supported in part by National
More informationAdaptive Edge Detectors for Piecewise Smooth Data Based on the minmod Limiter
Journal of Scientific Computing, Vol. 8, os. /3, September 6 ( 6) DOI:.7/s95-988 Adaptive Edge Detectors for Piecewise Smooth Data Based on the minmod Limiter A. Gelb and E. Tadmor Received April 9, 5;
More informationNumerical Methods I Orthogonal Polynomials
Numerical Methods I Orthogonal Polynomials Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001, Fall 2010 Nov. 4th and 11th, 2010 A. Donev (Courant Institute)
More informationApplied and Computational Harmonic Analysis
Appl. Comput. Harmon. Anal. 27 (2009) 351 366 Contents lists available at ScienceDirect Applied and Computational Harmonic Analysis www.elsevier.com/locate/acha Letter to the Editor Nonlinear inversion
More informationTopics in Fourier analysis - Lecture 2.
Topics in Fourier analysis - Lecture 2. Akos Magyar 1 Infinite Fourier series. In this section we develop the basic theory of Fourier series of periodic functions of one variable, but only to the extent
More informationFast Angular Synchronization for Phase Retrieval via Incomplete Information
Fast Angular Synchronization for Phase Retrieval via Incomplete Information Aditya Viswanathan a and Mark Iwen b a Department of Mathematics, Michigan State University; b Department of Mathematics & Department
More informationInverse problems Total Variation Regularization Mark van Kraaij Casa seminar 23 May 2007 Technische Universiteit Eindh ove n University of Technology
Inverse problems Total Variation Regularization Mark van Kraaij Casa seminar 23 May 27 Introduction Fredholm first kind integral equation of convolution type in one space dimension: g(x) = 1 k(x x )f(x
More informationChapter 6: Fast Fourier Transform and Applications
Chapter 6: Fast Fourier Transform and Applications Michael Hanke Mathematical Models, Analysis and Simulation, Part I Read: Strang, Ch. 4. Fourier Sine Series In the following, every function f : [,π]
More informationECG782: Multidimensional Digital Signal Processing
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu ECG782: Multidimensional Digital Signal Processing Spring 2014 TTh 14:30-15:45 CBC C313 Lecture 05 Image Processing Basics 13/02/04 http://www.ee.unlv.edu/~b1morris/ecg782/
More informationNov : Lecture 18: The Fourier Transform and its Interpretations
3 Nov. 04 2005: Lecture 8: The Fourier Transform and its Interpretations Reading: Kreyszig Sections: 0.5 (pp:547 49), 0.8 (pp:557 63), 0.9 (pp:564 68), 0.0 (pp:569 75) Fourier Transforms Expansion of a
More informationbe the set of complex valued 2π-periodic functions f on R such that
. Fourier series. Definition.. Given a real number P, we say a complex valued function f on R is P -periodic if f(x + P ) f(x) for all x R. We let be the set of complex valued -periodic functions f on
More informationDiscrete Fourier Transform
Discrete Fourier Transform DD2423 Image Analysis and Computer Vision Mårten Björkman Computational Vision and Active Perception School of Computer Science and Communication November 13, 2013 Mårten Björkman
More informationLecture Notes 5: Multiresolution Analysis
Optimization-based data analysis Fall 2017 Lecture Notes 5: Multiresolution Analysis 1 Frames A frame is a generalization of an orthonormal basis. The inner products between the vectors in a frame and
More informationDISCRETE FOURIER TRANSFORM
DD2423 Image Processing and Computer Vision DISCRETE FOURIER TRANSFORM Mårten Björkman Computer Vision and Active Perception School of Computer Science and Communication November 1, 2012 1 Terminology:
More informationRecovery of high order accuracy in radial basis function approximation for discontinuous problems
Recovery of high order accuracy in radial basis function approximation for discontinuous problems Chris L. Bresten, Sigal Gottlieb 1, Daniel Higgs, Jae-Hun Jung* 2 Abstract Radial basis function(rbf) methods
More informationThe Resolution of the Gibbs Phenomenon for Fourier Spectral Methods
The Resolution of the Gibbs Phenomenon for Fourier Spectral Methods Anne Gelb Sigal Gottlieb December 5, 6 Introduction Fourier spectral methods have emerged as powerful computational techniques for the
More informationAccelerated MRI Image Reconstruction
IMAGING DATA EVALUATION AND ANALYTICS LAB (IDEAL) CS5540: Computational Techniques for Analyzing Clinical Data Lecture 15: Accelerated MRI Image Reconstruction Ashish Raj, PhD Image Data Evaluation and
More informationSuper-resolution by means of Beurling minimal extrapolation
Super-resolution by means of Beurling minimal extrapolation Norbert Wiener Center Department of Mathematics University of Maryland, College Park http://www.norbertwiener.umd.edu Acknowledgements ARO W911NF-16-1-0008
More informationLecture 7 January 26, 2016
MATH 262/CME 372: Applied Fourier Analysis and Winter 26 Elements of Modern Signal Processing Lecture 7 January 26, 26 Prof Emmanuel Candes Scribe: Carlos A Sing-Long, Edited by E Bates Outline Agenda:
More informationPoisson Summation, Sampling and Nyquist s Theorem
Chapter 8 Poisson Summation, Sampling and Nyquist s Theorem See: A.6.1, A.5.2. In Chapters 4 through 7, we developed the mathematical tools needed to describe functions of continuous variables and methods
More informationUncertainty principles for far field patterns and applications to inverse source problems
for far field patterns and applications to inverse source problems roland.griesmaier@uni-wuerzburg.de (joint work with J. Sylvester) Paris, September 17 Outline Source problems and far field patterns AregularizedPicardcriterion
More informationWavelets: Theory and Applications. Somdatt Sharma
Wavelets: Theory and Applications Somdatt Sharma Department of Mathematics, Central University of Jammu, Jammu and Kashmir, India Email:somdattjammu@gmail.com Contents I 1 Representation of Functions 2
More informationRecovering Exponential Accuracy From Non-Harmonic Fourier Data Through Spectral Reprojection
Recovering Eponential Accuracy From Non-Harmonic Fourier Data Through Spectral Reprojection Anne Gelb Taylor Hines November 5, 2 Abstract Spectral reprojection techniques make possible the recovery of
More informationThe discrete and fast Fourier transforms
The discrete and fast Fourier transforms Marcel Oliver Revised April 7, 1 1 Fourier series We begin by recalling the familiar definition of the Fourier series. For a periodic function u: [, π] C, we define
More informationComplex geometrical optics solutions for Lipschitz conductivities
Rev. Mat. Iberoamericana 19 (2003), 57 72 Complex geometrical optics solutions for Lipschitz conductivities Lassi Päivärinta, Alexander Panchenko and Gunther Uhlmann Abstract We prove the existence of
More informationCONTINUOUS IMAGE MATHEMATICAL CHARACTERIZATION
c01.fm Page 3 Friday, December 8, 2006 10:08 AM 1 CONTINUOUS IMAGE MATHEMATICAL CHARACTERIZATION In the design and analysis of image processing systems, it is convenient and often necessary mathematically
More informationNon-Uniform Fast Fourier Transformation (NUFFT) and Magnetic Resonace Imaging 2008 년 11 월 27 일. Lee, June-Yub (Ewha Womans University)
Non-Uniform Fast Fourier Transformation (NUFFT) and Magnetic Resonace Imaging 2008 년 11 월 27 일 Lee, June-Yub (Ewha Womans University) 1 NUFFT and MRI Nonuniform Fast Fourier Transformation (NUFFT) Basics
More informationA practical guide to the recovery of wavelet coefficients from Fourier measurements
A practical guide to the recovery of wavelet coefficients from Fourier measurements Milana Gataric Clarice Poon April 05; Revised March 06 Abstract In a series of recent papers Adcock, Hansen and Poon,
More informationLecture 6 January 21, 2016
MATH 6/CME 37: Applied Fourier Analysis and Winter 06 Elements of Modern Signal Processing Lecture 6 January, 06 Prof. Emmanuel Candes Scribe: Carlos A. Sing-Long, Edited by E. Bates Outline Agenda: Fourier
More informationDeep Learning: Approximation of Functions by Composition
Deep Learning: Approximation of Functions by Composition Zuowei Shen Department of Mathematics National University of Singapore Outline 1 A brief introduction of approximation theory 2 Deep learning: approximation
More informationSparse Sampling: Theory and Applications
November 24, 29 Outline Problem Statement Signals with Finite Rate of Innovation Sampling Kernels: E-splines and -splines Sparse Sampling: the asic Set-up and Extensions The Noisy Scenario Applications
More informationFOURIER TRANSFORMS. 1. Fourier series 1.1. The trigonometric system. The sequence of functions
FOURIER TRANSFORMS. Fourier series.. The trigonometric system. The sequence of functions, cos x, sin x,..., cos nx, sin nx,... is called the trigonometric system. These functions have period π. The trigonometric
More informationSuper-resolution via Convex Programming
Super-resolution via Convex Programming Carlos Fernandez-Granda (Joint work with Emmanuel Candès) Structure and Randomness in System Identication and Learning, IPAM 1/17/2013 1/17/2013 1 / 44 Index 1 Motivation
More informationProblem descriptions: Non-uniform fast Fourier transform
Problem descriptions: Non-uniform fast Fourier transform Lukas Exl 1 1 University of Vienna, Institut für Mathematik, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria December 11, 2015 Contents 0.1 Software
More informationFirst-order overdetermined systems. for elliptic problems. John Strain Mathematics Department UC Berkeley July 2012
First-order overdetermined systems for elliptic problems John Strain Mathematics Department UC Berkeley July 2012 1 OVERVIEW Convert elliptic problems to first-order overdetermined form Control error via
More informationStructured Linear Algebra Problems in Adaptive Optics Imaging
Structured Linear Algebra Problems in Adaptive Optics Imaging Johnathan M. Bardsley, Sarah Knepper, and James Nagy Abstract A main problem in adaptive optics is to reconstruct the phase spectrum given
More informationReconstructing conductivities with boundary corrected D-bar method
Reconstructing conductivities with boundary corrected D-bar method Janne Tamminen June 24, 2011 Short introduction to EIT The Boundary correction procedure The D-bar method Simulation of measurement data,
More informationOPTIMAL FILTER AND MOLLIFIER FOR PIECEWISE SMOOTH SPECTRAL DATA
MATHEMATICS OF COMPUTATIO Volume, umber, Pages S 5-578(XX)- OPTIMAL FILTER AD MOLLIFIER FOR PIECEWISE SMOOTH SPECTRAL DATA JARED TAER This paper is dedicated to Eitan Tadmor for his direction. Abstract.
More informationSEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis
SEISMIC WAVE PROPAGATION Lecture 2: Fourier Analysis Fourier Series & Fourier Transforms Fourier Series Review of trigonometric identities Analysing the square wave Fourier Transform Transforms of some
More informationA Wavelet-based Method for Overcoming the Gibbs Phenomenon
AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '08), Harvard, Massachusetts, USA, March 4-6, 008 A Wavelet-based Method for Overcoming the Gibbs Phenomenon NATANIEL GREENE Department of Mathematics Computer
More informationPhase Retrieval from Local Measurements: Deterministic Measurement Constructions and Efficient Recovery Algorithms
Phase Retrieval from Local Measurements: Deterministic Measurement Constructions and Efficient Recovery Algorithms Aditya Viswanathan Department of Mathematics and Statistics adityavv@umich.edu www-personal.umich.edu/
More informationElliptic Problems for Pseudo Differential Equations in a Polyhedral Cone
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 9, Number 2, pp. 227 237 (2014) http://campus.mst.edu/adsa Elliptic Problems for Pseudo Differential Equations in a Polyhedral Cone
More informationA Singular Integral Transform for the Gibbs-Wilbraham Effect in Inverse Fourier Transforms
Universal Journal of Integral Equations 4 (2016), 54-62 www.papersciences.com A Singular Integral Transform for the Gibbs-Wilbraham Effect in Inverse Fourier Transforms N. H. S. Haidar CRAMS: Center for
More informationFourier Series. ,..., e ixn ). Conversely, each 2π-periodic function φ : R n C induces a unique φ : T n C for which φ(e ix 1
Fourier Series Let {e j : 1 j n} be the standard basis in R n. We say f : R n C is π-periodic in each variable if f(x + πe j ) = f(x) x R n, 1 j n. We can identify π-periodic functions with functions on
More informationThe Poisson summation formula, the sampling theorem, and Dirac combs
The Poisson summation ormula, the sampling theorem, and Dirac combs Jordan Bell jordanbell@gmailcom Department o Mathematics, University o Toronto April 3, 24 Poisson summation ormula et S be the set o
More informationAPPROXIMATING A BANDLIMITED FUNCTION USING VERY COARSELY QUANTIZED DATA: IMPROVED ERROR ESTIMATES IN SIGMA-DELTA MODULATION
APPROXIMATING A BANDLIMITED FUNCTION USING VERY COARSELY QUANTIZED DATA: IMPROVED ERROR ESTIMATES IN SIGMA-DELTA MODULATION C. SİNAN GÜNTÜRK Abstract. Sigma-delta quantization is a method of representing
More informationGenerated sets as sampling schemes for hyperbolic cross trigonometric polynomials
Generated sets as sampling schemes for hyperbolic cross trigonometric polynomials Lutz Kämmerer Chemnitz University of Technology joint work with S. Kunis and D. Potts supported by DFG-SPP 1324 1 / 16
More informationTwo-parameter regularization method for determining the heat source
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 8 (017), pp. 3937-3950 Research India Publications http://www.ripublication.com Two-parameter regularization method for
More informationMathematics for Chemists 2 Lecture 14: Fourier analysis. Fourier series, Fourier transform, DFT/FFT
Mathematics for Chemists 2 Lecture 14: Fourier analysis Fourier series, Fourier transform, DFT/FFT Johannes Kepler University Summer semester 2012 Lecturer: David Sevilla Fourier analysis 1/25 Remembering
More informationProceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005
Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER
More informationFast Algorithms for the Computation of Oscillatory Integrals
Fast Algorithms for the Computation of Oscillatory Integrals Emmanuel Candès California Institute of Technology EPSRC Symposium Capstone Conference Warwick Mathematics Institute, July 2009 Collaborators
More informationQuality Improves with More Rays
Recap Quality Improves with More Rays Area Area 1 shadow ray 16 shadow rays CS348b Lecture 8 Pat Hanrahan / Matt Pharr, Spring 2018 pixelsamples = 1 jaggies pixelsamples = 16 anti-aliased Sampling and
More informationWavelet Bi-frames with Uniform Symmetry for Curve Multiresolution Processing
Wavelet Bi-frames with Uniform Symmetry for Curve Multiresolution Processing Qingtang Jiang Abstract This paper is about the construction of univariate wavelet bi-frames with each framelet being symmetric.
More informationSpectral Methods and Inverse Problems
Spectral Methods and Inverse Problems Omid Khanmohamadi Department of Mathematics Florida State University Outline Outline 1 Fourier Spectral Methods Fourier Transforms Trigonometric Polynomial Interpolants
More informationApplied and Computational Harmonic Analysis 7, (1999) Article ID acha , available online at
Applied and Computational Harmonic Analysis 7, 0 35 (999) Article ID acha.999.06, available online at http://www.idealibrary.com on Detection of Edges in Spectral Data Anne Gelb Department of Mathematics,
More informationarxiv: v1 [eess.sp] 4 Nov 2018
Estimating the Signal Reconstruction Error from Threshold-Based Sampling Without Knowing the Original Signal arxiv:1811.01447v1 [eess.sp] 4 Nov 2018 Bernhard A. Moser SCCH, Austria Email: bernhard.moser@scch.at
More informationBeyond incoherence and beyond sparsity: compressed sensing in the real world
Beyond incoherence and beyond sparsity: compressed sensing in the real world Clarice Poon 1st November 2013 University of Cambridge, UK Applied Functional and Harmonic Analysis Group Head of Group Anders
More informationFourier transform of tempered distributions
Fourier transform of tempered distributions 1 Test functions and distributions As we have seen before, many functions are not classical in the sense that they cannot be evaluated at any point. For eample,
More informationLinear Operators and Fourier Transform
Linear Operators and Fourier Transform DD2423 Image Analysis and Computer Vision Mårten Björkman Computational Vision and Active Perception School of Computer Science and Communication November 13, 2013
More informationPreconditioning. Noisy, Ill-Conditioned Linear Systems
Preconditioning Noisy, Ill-Conditioned Linear Systems James G. Nagy Emory University Atlanta, GA Outline 1. The Basic Problem 2. Regularization / Iterative Methods 3. Preconditioning 4. Example: Image
More informationDETECTION OF EDGES IN SPECTRAL DATA II. NONLINEAR ENHANCEMENT
SIAM J. UMER. AAL. Vol. 38, o. 4, pp. 389 48 c 2 Society for Industrial and Applied Mathematics DETECTIO OF EDGES I SPECTRAL DATA II. OLIEAR EHACEMET AE GELB AD EITA TADMOR Abstract. We discuss a general
More informationBoundedness of Fourier integral operators on Fourier Lebesgue spaces and related topics
Boundedness of Fourier integral operators on Fourier Lebesgue spaces and related topics Fabio Nicola (joint work with Elena Cordero and Luigi Rodino) Dipartimento di Matematica Politecnico di Torino Applied
More informationSemi-Lagrangian Formulations for Linear Advection Equations and Applications to Kinetic Equations
Semi-Lagrangian Formulations for Linear Advection and Applications to Kinetic Department of Mathematical and Computer Science Colorado School of Mines joint work w/ Chi-Wang Shu Supported by NSF and AFOSR.
More informationWe have to prove now that (3.38) defines an orthonormal wavelet. It belongs to W 0 by Lemma and (3.55) with j = 1. We can write any f W 1 as
88 CHAPTER 3. WAVELETS AND APPLICATIONS We have to prove now that (3.38) defines an orthonormal wavelet. It belongs to W 0 by Lemma 3..7 and (3.55) with j =. We can write any f W as (3.58) f(ξ) = p(2ξ)ν(2ξ)
More informationAssignment 2 Solutions
EE225E/BIOE265 Spring 2012 Principles of MRI Miki Lustig Assignment 2 Solutions 1. Read Nishimura Ch. 3 2. Non-Uniform Sampling. A student has an assignment to monitor the level of Hetch-Hetchi reservoir
More informationMATH 220 solution to homework 4
MATH 22 solution to homework 4 Problem. Define v(t, x) = u(t, x + bt), then v t (t, x) a(x + u bt) 2 (t, x) =, t >, x R, x2 v(, x) = f(x). It suffices to show that v(t, x) F = max y R f(y). We consider
More informationContributors and Resources
for Introduction to Numerical Methods for d-bar Problems Jennifer Mueller, presented by Andreas Stahel Colorado State University Bern University of Applied Sciences, Switzerland Lexington, Kentucky, May
More informationStatistics of Stochastic Processes
Prof. Dr. J. Franke All of Statistics 4.1 Statistics of Stochastic Processes discrete time: sequence of r.v...., X 1, X 0, X 1, X 2,... X t R d in general. Here: d = 1. continuous time: random function
More informationWAVELETS, SHEARLETS AND GEOMETRIC FRAMES: PART II
WAVELETS, SHEARLETS AND GEOMETRIC FRAMES: PART II Philipp Grohs 1 and Axel Obermeier 2 October 22, 2014 1 ETH Zürich 2 ETH Zürich, supported by SNF grant 146356 OUTLINE 3. Curvelets, shearlets and parabolic
More informationA fast method for solving the Heat equation by Layer Potentials
A fast method for solving the Heat equation by Layer Potentials Johannes Tausch Abstract Boundary integral formulations of the heat equation involve time convolutions in addition to surface potentials.
More informationarxiv: v3 [math.na] 6 Sep 2015
Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples Ben Adcock Milana Gataric Anders C. Hansen April 23, 2018 arxiv:1405.3111v3 [math.na] 6
More informationIntroduction to Fourier Analysis
Lecture Introduction to Fourier Analysis Jan 7, 2005 Lecturer: Nati Linial Notes: Atri Rudra & Ashish Sabharwal. ext he main text for the first part of this course would be. W. Körner, Fourier Analysis
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 12 Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial,
More informationPreconditioning. Noisy, Ill-Conditioned Linear Systems
Preconditioning Noisy, Ill-Conditioned Linear Systems James G. Nagy Emory University Atlanta, GA Outline 1. The Basic Problem 2. Regularization / Iterative Methods 3. Preconditioning 4. Example: Image
More informationWe denote the space of distributions on Ω by D ( Ω) 2.
Sep. 1 0, 008 Distributions Distributions are generalized functions. Some familiarity with the theory of distributions helps understanding of various function spaces which play important roles in the study
More informationMATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 34: Improving the Condition Number of the Interpolation Matrix Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu
More informationLecture 9 February 2, 2016
MATH 262/CME 372: Applied Fourier Analysis and Winter 26 Elements of Modern Signal Processing Lecture 9 February 2, 26 Prof. Emmanuel Candes Scribe: Carlos A. Sing-Long, Edited by E. Bates Outline Agenda:
More informationApproximation by Conditionally Positive Definite Functions with Finitely Many Centers
Approximation by Conditionally Positive Definite Functions with Finitely Many Centers Jungho Yoon Abstract. The theory of interpolation by using conditionally positive definite function provides optimal
More informationA Tutorial on Wavelets and their Applications. Martin J. Mohlenkamp
A Tutorial on Wavelets and their Applications Martin J. Mohlenkamp University of Colorado at Boulder Department of Applied Mathematics mjm@colorado.edu This tutorial is designed for people with little
More informationECG782: Multidimensional Digital Signal Processing
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu ECG782: Multidimensional Digital Signal Processing Filtering in the Frequency Domain http://www.ee.unlv.edu/~b1morris/ecg782/ 2 Outline Background
More informationOn the coherence barrier and analogue problems in compressed sensing
On the coherence barrier and analogue problems in compressed sensing Clarice Poon University of Cambridge June 1, 2017 Joint work with: Ben Adcock Anders Hansen Bogdan Roman (Simon Fraser) (Cambridge)
More informationWavelets and Image Compression Augusta State University April, 27, Joe Lakey. Department of Mathematical Sciences. New Mexico State University
Wavelets and Image Compression Augusta State University April, 27, 6 Joe Lakey Department of Mathematical Sciences New Mexico State University 1 Signals and Images Goal Reduce image complexity with little
More informationInverse Theory Methods in Experimental Physics
Inverse Theory Methods in Experimental Physics Edward Sternin PHYS 5P10: 2018-02-26 Edward Sternin Inverse Theory Methods PHYS 5P10: 2018-02-26 1 / 29 1 Introduction Indirectly observed data A new research
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationDiscrete Fourier transform (DFT)
Discrete Fourier transform (DFT) Signal Processing 2008/9 LEA Instituto Superior Técnico Signal Processing LEA (IST) Discrete Fourier transform 1 / 34 Periodic signals Consider a periodic signal x[n] with
More informationNEW BOUNDS FOR TRUNCATION-TYPE ERRORS ON REGULAR SAMPLING EXPANSIONS
NEW BOUNDS FOR TRUNCATION-TYPE ERRORS ON REGULAR SAMPLING EXPANSIONS Nikolaos D. Atreas Department of Mathematics, Aristotle University of Thessaloniki, 54006, Greece, e-mail:natreas@auth.gr Abstract We
More informationIntroduction to the Mathematics of Medical Imaging
Introduction to the Mathematics of Medical Imaging Second Edition Charles L. Epstein University of Pennsylvania Philadelphia, Pennsylvania EiaJTL Society for Industrial and Applied Mathematics Philadelphia
More informationLinear Inverse Problems
Linear Inverse Problems Ajinkya Kadu Utrecht University, The Netherlands February 26, 2018 Outline Introduction Least-squares Reconstruction Methods Examples Summary Introduction 2 What are inverse problems?
More information