DOCSLIB.ORG

EE290T: Advanced Reconstruction Methods for Magnetic Resonance Imaging

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
EE290T: Advanced Reconstruction Methods for Magnetic Resonance Imaging EE290T: Advanced Reconstruction Methods for Magnetic Resonance Imaging Martin Uecker Introduction Topics: I Image Reconstruction as Inverse Problem I Parallel Imaging I Non-Cartestian MRI I Subspace Methods I Model-based Reconstruction I Compressed Sensing Tentative Syllabus I 01: Jan 27 Introduction I 02: Feb 03 Parallel Imaging as Inverse Problem I 03: Feb 10 Iterative Reconstruction Algorithms I {: Feb 17 (holiday) I 04: Feb 24 Non-Cartesian MRI I 05: Mar 03 Nonlinear Inverse Reconstruction I 06: Mar 10 Reconstruction in k-space I 07: Mar 17 Reconstruction in k-space I {: Mar 24 (spring recess) I 08: Mar 31 Subspace methods I 09: Apr 07 Model-based Reconstruction I 10: Apr 14 Compressed Sensing I 11: Apr 21 Compressed Sensing I 12: Apr 28 TBA Today I Review I Non-Cartesian MRI I Non-uniform FFT I Project 2 Direct Image Reconstruction I Assumption: Signal is Fourier transform of the image: Z ~ s(t) = d~x ρ(~x)e−i2π~x·k(t) I Image reconstruction with inverse DFT ky kx ) ) sampling iDFT ~ R t ~ image k(t) = γ 0 dτ G(τ) k-space Requirements I Short readout (signal equation holds for short time spans only) I Sampling on a Cartesian grid ) Line-by-line scanning α echo radio measurement time: slice TR ≥ 2ms read 2D: N ≈ 256 ) seconds phase 3D: N ≈ 256 × 256 ) minutes TR ×N 2D MRI sequence Poisson Summation Formula Periodic summation of a signal f from discrete samples of its Fourier transform f^: 1 1 X X 1 l 2πi l t f (t + nP) = f^( )e P P P n=−∞ l=−∞ No aliasing ) Nyquist criterium l For MRI: P = FOV ) k-space samples are at k = FOV I Periodic summation from discrete Fourier samples on grid I Other positions can be recovered by sinc interpolation (Whittaker-Shannon formula) image k-space I Other positions can be recovered by sinc interpolation (Whittaker-Shannon formula) image k-space I Periodic summation from discrete Fourier samples on grid image k-space I Periodic summation from discrete Fourier samples on grid I Other positions can be recovered by sinc interpolation (Whittaker-Shannon formula) Nyquist-Shannon Sampling Theorem Theorem 1: If a function f (t) contains no frequencies higher than W cps, it is completely determined by giving its ordinates at a series of points spaced 1=2W seconds apart.1 I Band-limited function I Regular sampling I Linear sinc-interpolation 1. CE Shannon. Communication in the presence of noise. Proc Institute of Radio Engineers; 37:10{21 (1949) Fast Fourier Transform (FFT) Algorithms Discrete Fourier Transform (DFT): N−1 N−1 X 2π N ^ i N kn DFTk ffngn=0 := fk = e fn for k = 0; ··· ; N − 1 n=0 Matrix multiplication: O(N2) Fast Fourier Transform Algorithms: O(N log N) Cooley-Tukey Algorithm Decomposition of a DFT: N−1 N N−1 X kn DFTk ffngn=0 = (ξN ) fn n=0 N1−1 N2−1 X k1n1 k2n1 X k2n2 = (ξN1 ) (ξN ) (ξN2 ) fn1+n2N1 n1=0 n2=0 n oN1−1 = DFT N1 (ξ )k2n1 DFT N2 ff gN2−1 k1 N k2 n1+n2N1 n2=0 n1=0 I Recursive decomposition to prime-sized DFTs ) Efficient computation for smooth sizes (efficient algorithms available for all other sizes too) Parallel MRI: Iterative Algorithms Signal equation: Z ~ −i2π~x·~k si (k) = d~x ρ(~x) ci (~x)e V | {z } encoding functions Discretization: A = Pk FC A has size 2562 × (8 × 2562) ) Iterative methods built from matrix-vector products sensitivities Ax, AH y Landweber Iteration Gradient descent: 1 φ(x) = kAx − yk2 2 2 rφ(x) = AH (Ax − y) Iteration rule: H H xn+1 = xn − µA (Axn − y) with µkA Ak ≤ 1 H H = (I − µA A)xn − A y Explicit formula for x0 = 0: n−1 X H j H xn = (I − µA A) µA y j=0 Projection Onto Convex Sets (POCS) Iterative projection onto convex sets: yn+1 = PB PAyn A B Problems: slow, not stable, A \ B = ? POCSENSE Iteration: y^n+1 = PC Py y^n (multi-coil k-spacey ^) Projection onto sensitivities: H −1 PC = FCC F (normalized sensitivities) Projection onto data y: Py y^ = My + (1 − M)^y AA Samsonov, EG Kholmovski, DL Parker, and CR Johnson. POCSENSE: POCS-based reconstruction for sensitivity encoded magnetic resonance imaging. Magn Reson Med, 52:1397{1406, 2004. Krylov Subspace Methods Krylov subspace: n Kn = spani=0···n fT bg ) Repeated application of T . Landweber, Arnoldi, Lanczos, Conjugate gradients, ... Conjugate Gradients T symmetric (or Hermitian) Initialization: r0 = b − Tx0 d0 = r0 Iteration: qi ( Tdi 2 jri j α = H R ri qi xi+1 ( xi + αdi ri+1 ( ri − αqi 2 jri+1j β = 2 jri j di+1 ( ri+1 + βdi Conjugate Gradients on Normal Equations Normal equations: H H 2 A Ax = A y , x = argminx kAx − yk2 Tikhonov: H H 2 2 A A + αI x = A y , x = argminx kAx − yk2 + αkxk2 CGNE: Tx = b with: T = AH A b = AH y Implementation repeat FFT P IFFT FFT P IFFT sum channels point-wise mult. with conjugate sensitivities data flow inverse FFT sampling mask or projection onto data multi-dimensional FFT point-wise multiplication with sensitivities distribute image Conjugate Gradient Algorithm vs Landweber conjugate gradients Landweber 2 1 1: 1:414 1:732 0 2: Project 1: Iterative SENSE Project: Implement and study Cartesian iterative SENSE I Tools: Matlab, reconstruction toolbox, python, ... I Deadline: Feb 24 I Hand in: Working code and plots/results with description. I See website for data and instructions. Project 1: Iterative SENSE Step 1: Implement Model A = PFC AH = C H F −1PH Hints: I Use unitary and centered (fftshift) FFT kFxk2 = kxk2 I Implement undersampling as a mask, store data with zero H I Check hx; Ayi = A x; y for random vectors x; y Project 1: Iterative SENSE Step 2: Implement Reconstruction Landweber (gradient descent)1 H xn+1 = xn + αA (y − Axn) Conjugate gradient algorithm2 Step 3: Experiments I Noise, errors, and convergence speed I Different sampling I Regularization 1. L Landweber. An iteration formula for Fredholm integral equations of the first kind. Amer J Math; 73:615{624 (1951) 2. MR Hestenes and E Stiefel. Methods of conjugate gradients for solving linear systems. J. Ref. N.B.S. 49:409{436 (1952) Today I Review I Non-Cartesian MRI I Non-uniform FFT I Project 2 Parallel Imaging as Inverse Problem Model: Signal from multiple coils (image ρ, sensitivities cj ): Z −i2π~x·~k(t) sj (t) = d~x ρ(~x)cj (~x)e + nj (t) Ω Assumptions: I Image is square-integrable function ρ 2 L2(Ω; C) I Additive multi-variate Gaussian white noise n Problem: Find best approximate/regularized solution in L2(Ω; C). Ω Non-Cartesian Trajectories Examples: Cartesian radial spiral Non-Cartesian Trajectories Practical issues: I Long readout: blurring, artifacts I Imperfect gradient waveforms (e.g. delays, eddy currents) I Efficient implementation of the reconstruction algorithm Spiral Long readout: I Efficient! ? I T2 -Relaxation ) blurring I Off-resonance Spiral Signal equation: Z itγ∆B (~x) −t=T ?(~x) −2πi~k~x y(t) = dx ρ(~x)e 0 e 2 e Ω Off-resonance: itγ∆B0(~x) jt − tTE j ∆γB0 ) e ≈ const Relaxation: ? ? −t=T2 jt − tTE j T2 ) e ≈ const Spiral Signal equation: Z itγ∆B (~x) −t=T ?(~x) −2πi~k~x y(t) = dx ρ(~x)e 0 e 2 e Ω Off-resonance: itγ∆B0(~x) jt − tTE j ∆γB0 ) e ≈ const Relaxation: ? ? −t=T2 jt − tTE j T2 ) e ≈ const Radial π I Not efficient (Nyquist: 2 × spokes vs. Cartesian lines) I Motion robustness I High tolerance to undersampling I Continuous update possible Radial - Undersampling Artifacts Left to right: 96, 48, 24 radial spokes Forward Problem Signal equation: Z ~ −2πikl~x yl = d~x ρ(~x)e Ω Non-Cartesian sample positions: kl Discretization: ρ(~x) pixel values at grid positions ~x ~ X −2πikl~x yl = ρ(~x)e ~x ) non-uniform DFT from grid to non-Cartesian positions. Inverse Non-Uniform Fourier Transform Forward model: Z ~ ikl~x yl = d~x ρ(~x)e Inverse problem: y = Pk Fρ = Aρ Best-approximation (least-squares + minimum-norm): −1 H H x = limα!0A AA + αI y Non-uniform FFT Fourier transform at arbitray (non-Cartesian) positions kl Z −2πikl x yl = dx f (x)e DFT at arbitray (non-Cartesian) positions kl N−1 k n X −2πi l yl = xne N n=0 Fast computation: nuFFT Interpolation to Non-Cartesian Data I Periodic extension: FFT yields Fourier transform on grid I Values at k-space positions recovered by sinc interpolation Whittaker-Shannon formula: 1 X t − nT sin(πx) f (t) = f (nT ) sinc( ) sinc(x) = T πx n=−∞ Approximation of the Interpolation Computation: N=2−1 X t − nT f (t) = f (nT ) kern( ) T n=−N=2 I Sinc interpolation: not practical I Compactly supported kernels: truncated Gaussian, Kaiser-Bessel Kernel with compact support: kern(x) = 0 for jxj >= wT ) Sum has only 2 ∗ w terms ) Efficient computation Approximation of the Interpolation Computation (2D): N=2−1 M=2−1 X X s − nS t − nT f (s; t) = f (nS; mT ) kern( ) kern( ) S T n=−N=2 m=−M=2 Approximation of the Interpolation Computation (2D): N=2−1 M=2−1 X X s − nS t − nT f (nS; mT ) kern( ) kern( ) S T n=−N=2 m=−M=2 Isotropic kernel: 0 1 N=2−1 M=2−1 s 2 2 X X s − nS t − nT f (nS; mT ) kern + @ S T A n=−N=2 m=−M=2 Approximation of the Interpolation I Sinc interpolation: not practical I Finite kernel: Kaiser-Bessel interpolation I Oversampling to reduce approximation errors stop pass stop Approximation of the Interpolation I Sinc interpolation: not practical I Finite kernel: Kaiser-Bessel interpolation I Oversampling to reduce approximation errors stop pass stop Approximation of the Interpolation I Sinc interpolation: not practical I Finite kernel: Kaiser-Bessel interpolation I Oversampling to reduce approximation errors stop pass stop Approximation of the Interpolation I Sinc interpolation: not practical I Finite kernel: Kaiser-Bessel interpolation I Oversampling to reduce approximation errors stop pass stop Approximation of the Interpolation I Sinc interpolation: not practical I Finite kernel: Kaiser-Bessel interpolation I Oversampling to reduce approximation
Load more

Recommended publications
  • Arxiv:1712.04732V2 [Math.NA] 17 Jan 2021 New Window Function, Runtimes for the SE Method Can Be Further Reduced
    Fast Ewald summation for electrostatic potentials with arbitrary periodicity D. S. Shamshirgar,a) J. Bagge,b) and A.-K. Tornbergc) KTH Mathematics, Swedish e-Science Research Centre, 100 44 Stockholm, Sweden. A unified treatment for fast and spectrally accurate evaluation of electrostatic po- tentials subject to periodic boundary conditions in any or none of the three spatial dimensions is presented. Ewald decomposition is used to split the problem into a real- space and a Fourier-space part, and the FFT-based Spectral Ewald (SE) method is used to accelerate the computation of the latter. A key component in the unified treatment is an FFT-based solution technique for the free-space Poisson problem in three, two or one dimensions, depending on the number of non-periodic directions. The computational cost is furthermore reduced by employing an adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling factors. The SE method will always be most efficient for the triply periodic case as the cost of computing FFTs will then be the smallest, whereas the computational cost of the rest of the algorithm is essentially independent of periodicity. We show that the cost of removing periodic boundary conditions from one or two directions out of three will only moderately increase the total runtime. Our comparisons also show that the computational cost of the SE method in the free-space case is around four times that of the triply periodic case. The Gaussian window function previously used in the SE method, is here compared to a piecewise polynomial approximation of the Kaiser-Bessel window function.
    [Show full text]
  • Fourier Analysis of Discrete-Time Signals
    Fourier analysis of discrete-time signals (Lathi Chapt. 10 and these slides) Towards the discrete-time Fourier transform • How we will get there? • Periodic discrete-time signal representation by Discrete-time Fourier series • Extension to non-periodic DT signals using the “periodization trick” • Derivation of the Discrete Time Fourier Transform (DTFT) • Discrete Fourier Transform Discrete-time periodic signals • A periodic DT signal of period N0 is called N0-periodic signal f[n + kN0]=f[n] f[n] n N0 • For the frequency it is customary to use a different notation: the frequency of a DT sinusoid with period N0 is 2⇡ ⌦0 = N0 Fourier series representation of DT periodic signals • DT N0-periodic signals can be represented by DTFS with 2⇡ fundamental frequency ⌦ 0 = and its multiples N0 • The exponential DT exponential basis functions are 0k j⌦ k j2⌦ k jn⌦ k e ,e± 0 ,e± 0 ,...,e± 0 Discrete time 0k j! t j2! t jn! t e ,e± 0 ,e± 0 ,...,e± 0 Continuous time • Important difference with respect to the continuous case: only a finite number of exponentials are different! • This is because the DT exponential series is periodic of period 2⇡ j(⌦ 2⇡)k j⌦k e± ± = e± Increasing the frequency: continuous time • Consider a continuous time sinusoid with increasing frequency: the number of oscillations per unit time increases with frequency Increasing the frequency: discrete time • Discrete-time sinusoid s[n]=sin(⌦0n) • Changing the frequency by 2pi leaves the signal unchanged s[n]=sin((⌦0 +2⇡)n)=sin(⌦0n +2⇡n)=sin(⌦0n) • Thus when the frequency increases from zero, the number of oscillations per unit time increase until the frequency reaches pi, then decreases again towards the value that it had in zero.
    [Show full text]
  • Fractional Calculus and Generalised Norms in Condition Monitoring of a Load Haul Dumper
    Fractional calculus and generalised norms in condition monitoring of a load haul dumper Master’s thesis Juhani Nissilä 1975600 Department of Mathematical Sciences University of Oulu Autumn 2014 Contents Introduction 4 1 Classical analysis 6 1.1 Norms and function spaces . 6 1.2 Convergence of function sequences . 9 1.3 Complex analysis . 10 1.4 Important theorems of analysis . 11 1.5 Distributions . 12 1.6 Iterated differences and derivatives . 16 1.7 Iterated integrals . 18 2 Mathematical background for fractional calculus 20 2.1 Gamma function . 20 2.2 Fourier series . 23 2.2.1 Series in L1;loc and L2;loc . 23 2.2.2 Pointwise convergence . 24 2.3 Fourier transform . 26 2.3.1 Transform in S and L1 . 26 2.3.2 Transform in L2 ...................... 28 2.3.3 Transform in S0 ...................... 29 2.3.4 Properties . 31 2.4 Poisson summation formula and the sampling theorem . 34 2.5 Discrete Fourier transform . 36 2.6 Window functions . 37 3 Fractional calculus in time domain 39 3.1 Riemann-Liouville fractional integral and derivative . 39 3.2 Caputo fractional derivative . 45 3.3 Grünwald-Letnikov fractional derivative and integral . 46 3.4 Equivalences of definitions . 47 4 Fractional calculus in frequency domain 49 4.1 Fourier differintegrals . 49 4.2 Weyl differintegrals . 52 4.3 Equivalences of definitions . 53 4.4 Numerical algorithm . 56 5 Norms, means and other features of signals 60 5.1 Generalised lp norms and Hölder means . 60 5.2 Measurement index . 64 1 6 Condition monitoring of a load haul dumper front axle 65 6.1 Data acquisition and acceleration signals .
    [Show full text]
  • Lecture 5. FFT and Divide and Conquer
    CS711008Z Algorithm Design and Analysis Lecture 5. FFT and Divide and Conquer Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China . 1 / 57 Outline DFT: evaluate a polynomial at n special points; FFT: an efficient implementation of DFT; Applications of FFT: multiplying two polynomials (and multiplying two n-bits integers); time-frequency transform; solving partial differential equations; Appendix: relationship between continuous and discrete Fourier transforms. 2 / 57 DFT: Discrete Fourier Transform n−1 DFT evaluates a polynomial A(x) = a0 + a1x + ::: + an−1x 2 n−1 2π i at n distinct points 1; !; ! ; :::; ! , where ! = e n is the n-th complex root of unity. Thus, it transforms the complex vector a0; a1; :::; an−1 into k another complex vector y0; y1; :::; yn−1, where yk = A(w ), i.e., y0 = a0 + a1 + a2 ::: + an−1 1 2 n−1 y1 = a0 + a1! + a2! ::: + an−1! ::: ::: ::: :::::: n−1 2(n−1) (n−1)2 yn−1 = a0 + a1! + a2! ::: + an−1! Matrix form: 2 3 2 3 2 3 1 1 1 ::: 1 y0 a0 6 7 6 1 !1 !2 :::!n−1 7 6 7 6 y1 7 6 7 6 a1 7 6 7 = 6 1 !2 !4 :::!2(n−1)7 6 7 4 . 5 6 7 4 . 5 . 4::::::::::::::: 5 . 2 yn−1 1 !n−1 !2(n−1) :::!(n−1) an−1 . 3 / 57 FFT: a fast way to implement DFT [Cooley and Tukey, 1965] Direct matrix-vector multiplication requires O(n2) operations when using the Horner’s method, i.e., A(x) = a0 + x(a1 + x(a2 + ::: + xan−1)).
    [Show full text]
  • Digital Signal Processing & Its Applications
    SCHOOL OF BIO AND CHEMICAL ENGINEERING DEPARTMENT OF BIOMEDICAL ENGINEERING UNIT – I – Digital Signal Processing & Its Applications – SEC1324 1 INTRODUCTION TO DISCRETE TIME SIGNALS AND SYSTEMS A signal is a function of independent variables such as time, distance, position, temperature and pressure. A signal carries information, and the objective of signal processing is to extract useful information carried by the signal. Signal processing is concerned with the mathematical representation of the signal and the algorithmic operation carried out on it to extract the information present. For most purposes of description and analysis, a signal can be defined simply as a mathematical function, y where x is the independent variable . y = f (x) signal e.g.: y=sin(ωt) is a function of a variable in the time domain and is thus a time signal X(ω)=1/(-mω2+icω+k) is a frequency domain signal; An image I(x,y) is in the spatial domain Fig.1:Classification 2 at t=0, will have the same motions at all time. There is no place for uncertainty here. If we can uniquely specify the value of θ for all time, i.e., we know the underlying functional relationship between t andθ, the motion is deterministic or predictable. In other words, a signal that can be uniquely determined by a well defined process such as a mathematical expression or rule is called a deterministic signal. The opposite situation occurs if we know all the physics there is to know, but still cannot say what the signal will be at the next time instant-then the signal is random or probabilistic.
    [Show full text]
  • Digital Signal Processing EEE,EE,CSE Analog Electronics
    Module – I 1.1 WHAT IS A SIGNAL We are all immersed in a sea of signals. All of us from the smallest living unit, a cell, to the most complex living organism (humans) are all time receiving signals and are processing them. Survival of any living organism depends upon processing the signals appropriately. What is signal? To define this precisely is a difficult task. Anything which carries information is a signal. In this course we will learn some of the mathematical representations of the signals, which has been found very useful in making information processing systems. Examples of signals are human voice, chirping of birds, smoke signals, gestures (sign language), fragrances of the flowers. Many of our body functions are regulated by chemical signals, blind people use sense of touch. Bees communicate by their dancing pattern. Some examples of modern high speed signals are the voltage charger in a telephone wire, the electromagnetic field emanating from a transmitting antenna, variation of light intensity in an optical fiber. Thus we see that there is an almost endless variety of signals and a large number of ways in which signals are carried from on place to another place. In this course we will adopt the following definition for the signal: A signal is a real (or complex) valued function of one or more real variable(s).When the function depends on a single variable, the signal is said to be one dimensional. A speech signal, daily maximum temperature, annual rainfall at a place, are all examples of a one dimensional signal.
    [Show full text]
  • Review of Signal Processing Sampling and Reconstruction
    Review of Signal Processing This contains a brief review of • Sampling and Reconstruction • Decimation and Interpolation and Resampling Sampling and Reconstruction • I give a review of important facts about 1D theory, the 2D theory is analogous and available in the textbook (AK Jain, Chapter 2, 4). • Some references – The review below is based on the book: Signals and Systems, by Oppenheim, Wilsky and Yoeng. – Also see an excellent tutorial on the web (for figures, examples, and many more details): http://ccrma.stanford.edu/∼jos/mdft/mdft.html – Perfect interpolation of discrete periodic signals is possible to do. See S. Schanze, Sinc interpolation of discrete periodic signals, IEEE Trans Signal Processing, June 1995. Available at: http://ieeexplore.ieee.org/iel4/78/8826/00388863.pdf?arnumber=388863 • Disclaimer: This is a first draft, may have errors and typos, please email me ([email protected]) if you find them. ∞ • R without limits denotes R−∞. Similarly for Pn. • Let fs is sampling frequency, let Ts = 1/fs is the sampling interval. • Know/revise the following 1. Definition of CTFT (Continuous time Fourier transform) 2. Definition of DTFT (Discrete time Fourier transform) 3. Definition of DFT (Discrete Fourier transform) 4. Fourier transform properties and know common signal and CTFT/DTFT/DFT pairs. 5. Nyquist’s theorem, concept of aliasing. • To sample a signal at frequency fs, without aliasing (i.e. to be able to perfectly reconstruct it), it should be bandlimited to fs/2. If not, apply an appropriate Low Pass Filter. • Some important FT pairs 1. Comb function: x(t)= Pn δ(t − nTs), then X(f)= fs Pk δ(f − kfs) sin(2πWt) 2.
    [Show full text]
  • A Generalized Fourier Domain Signal Processing Framework And
    Signal Processing 93 (2013) 1259–1267 Contents lists available at SciVerse ScienceDirect Signal Processing journal homepage: www.elsevier.com/locate/sigpro A generalized Fourier domain: Signal processing framework and applications Jorge Martinez n, Richard Heusdens, Richard C. Hendriks Signal and Information Processing (SIP) Lab. Delft University of Technology, Mekelweg 4, 2628CD Delft, Netherlands article info abstract Article history: In this paper, a signal processing framework in a generalized Fourier domain (GFD) is Received 9 August 2011 introduced. In this newly proposed domain, a parametric form of control on the periodic Received in revised form repetitions that occur due to sampling in the reciprocal domain is possible, without the 18 October 2012 need to increase the sampling rate. This characteristic and the connections of the Accepted 24 October 2012 generalized Fourier transform to analyticity and to the z-transform are investigated. Core Available online 2 November 2012 properties of the generalized discrete Fourier transform (GDFT) such as a weighted Keywords: circular correlation property and Parseval’s relation are derived. We show the benefits of Generalized Fourier domain using the novel framework in a spatial-audio application, specifically the simulation of Generalized discrete Fourier transform room impulse responses for auralization purposes in e.g. virtual reality systems. Weighted circular convolution & 2012 Elsevier B.V. All rights reserved. 1. Introduction are derived such as the weighted circular correlation property and Parseval’s energy relation for the GFD that, In this paper we introduce a framework for signal together with the previously introduced weighted circular processing in a generalized Fourier domain (GFD). In this convolution theorem for the GDFT, are fundamental to domain a special form of control on the periodic repeti- build a general-purpose GFD-based signal processing tions that occur due to sampling in the reciprocal domain framework.
    [Show full text]
  • A Journey in Signal Processing with Jupyter
    A Journey in Signal Processing with Jupyter JEAN-FRANÇOIS BERCHER ESIEE-PARIS 2 Page 2/255 Contents 1 A basic introduction to signals and systems 9 1.1 Effects of delays and scaling on signals ............................ 9 1.2 A basic introduction to filtering ................................. 12 1.2.1 Transformations of signals - Examples of difference equations ............ 12 1.2.2 Filters ......................................... 19 2 Introduction to the Fourier representation 23 2.1 Simple examples ........................................ 23 2.1.1 Decomposition on basis - scalar producs ....................... 26 2.2 Decomposition of periodic functions – Fourier series ..................... 27 2.3 Complex Fourier series ..................................... 29 2.3.1 Introduction ....................................... 29 2.3.2 Computer experiment ................................. 30 3 From Fourier Series to Fourier transforms 39 3.1 Introduction and definitions ................................... 39 3.2 Examples ............................................ 41 3.2.1 The Fourier transform of a rectangular window .................... 42 3.2.2 Fourier transform of a sine wave ............................ 45 3.3 Symmetries of the Fourier transform. .............................. 49 3.4 Table of Fourier transform properties .............................. 51 4 Filters and convolution 53 4.1 Representation formula ..................................... 53 4.2 The convolution operation ................................... 54 4.2.1 Definition .......................................
    [Show full text]
  • C2-Fourier Transform
    Fourier Transform 1 The idea A signal can be interpreted as en electromagnetic wave. This consists of lights of different “color”, or frequency, that can be split apart usign an optic prism. Each component is a “monochromatic” light with sinusoidal shape. Following this analogy, each signal can be decomposed into its “sinusoidal” components which represent its “colors”. Of course these components in general do not correspond to visible monochromatic light. However, they give an idea of how fast are the changes of the signal. 2 Contents • Signals as functions (1D, 2D) – Tools • Continuous Time Fourier Transform (CTFT) • Discrete Time Fourier Transform (DTFT) • Discrete Fourier Transform (DFT) • Discrete Cosine Transform (DCT) • Sampling theorem 3 Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous/ discrete time and frequency variables. – Notations: • CTFT: continuous time FT: t is real and f real (f=ω) (CT, CF) • DTFT: Discrete Time FT: t is discrete (t=n), f is real (f=ω) (DT, CF) • CTFS: CT Fourier Series (summation synthesis): t is real AND the function is periodic, f is discrete (f=k), (CT, DF) • DTFS: DT Fourier Series (summation synthesis): t=n AND the function is periodic, f discrete (f=k), (DT, DF) • P: periodical signals • T: sampling period • ωs: sampling frequency (ωs=2π/T) • For DTFT: T=1 → ωs=2π • This is a hint for those who are interested in a more exhaustive theoretical approach 4 Images as functions • Gray scale images:
    [Show full text]
  • A FFT-Based Convolution Method for Mixed-Periodic Boundary Condition
    A FFT-based convolution method for mixed-periodic Boundary Condition Zhe Chen∗1 1Courant Institute of Mathematical Sciences, New York University April 2020 1 Introduction Convolutions are widely used to simulate colloidal system. The interactions between colloidal particles in Stokes flow can be described by their mobility, which is a function of the displacement between them. If the force applied on particles are given, we can compute velocity of particles by convolution of the mobility kernel and the force. This convolution is expensive and critical for simulating dynamics of large scale colloidal particles, which makes it essential that we have a cheap and accurate numerical method for computing them. Let's assume we compute a discrete convolution of two periodic sequences with length N in each direction in d dimensions. If we directly compute the convolution, it would cost O(N 2d), which is generally not affordable in high dimensions. One of the most popular and promising method that we could use to compute them faster is the Fast Fourier Transform (FFT), which cuts down complexity dramatically to O(N d log(N)), since a convolution becomes dot production in Fourier space. In most colloidal system, nevertheless, it's not trivial to use finite sequence convolutions to represent convolutions of the force and the mobility kernel. For example, the force field is usually considered as finitely supported or infinitely periodic, but the periodicity can be different in different directions, which we call mixed-periodicity, and the kernel can be long-ranged, which means slow-decaying. Moreover, sample points of velocity might be non-uniform since we only need velocity of the particles in most cases.
    [Show full text]
  • Convolution and Applications of Convolution
    © 2014 IJIRT | Volume 1 Issue 6 | ISSN : 2349-6002 CONVOLUTION AND APPLICATIONS OF CONVOLUTION Akshay Behl, Akash Bhatia, Avril Puri Dronacharya College Of Engineering, GGN Abstract-We introduce an integral transform related to typically viewed as a modified version of one of the a Fourier sine-Fourier - Fourier cosine generalized original functions, giving the area overlap between the convolution and prove a Watson type theorem for the two functions as a function of the amount that one of transform. As applications we obtain solutions of some the original functions is translated. Convolution is integral equations in closed form.. Convolution describes similar to cross-correlation. It has applications that the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI include probability, statistics, computer vision, image system theory for a derivation of convolution as the and signal processing, electrical engineering, and result of LTI constraints. In terms of the Fourier differential equations. transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms. Index Terms-Convolution, Watson theorem, Fourier sine transform, fourier cosine transform, Integral equation, Hölder inequality I. INTRODUCTION In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is The convolution of f and g is written f∗g, using an asterisk For functions f, g supported on only (i.e., zero for or star.
    [Show full text]