DOCSLIB.ORG

The Fourier Transform

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Fourier Transform The Fourier Transform As we have seen, any (sufficiently smooth) function f(t) that is periodic can be built out of sin’s and cos’s. We have also seen that complex exponentials may be used in place of sin’s and cos’s. We shall now use complex exponentials because they lead to less writing and simpler computations, but yet can easily be converted into sin’s and cos’s. If f(t) has period 2ℓ, its (complex) Fourier series expansion is ℓ ∞ π π ik ℓ t 1 ik ℓ t f(t)= cke with ck = 2ℓ f(t)e− dt (1) k= ℓ X−∞ Z− ik π t ik π (t+2ℓ) Not surprisingly, each term cke ℓ in this expansion also has period 2ℓ, because cke ℓ = ik π t i2kπ ik π t cke ℓ e = cke ℓ . We now develop an expansion for non-periodic functions, by allowing complex exponentials (or equivalently sin’s and cos’s) of all possible periods, not just 2ℓ, for some fixed ℓ. So, from now on, do not assume that f(t) is periodic. For simplicity we’ll only develop the expansions for functions that are zero for all sufficiently large t . | | With a little more work, one can show that our conclusions apply to a much broader class of functions. Let L > 0 be sufficiently large that f(t) = 0 for all t L. We can get a Fourier series expansion for the part | | ≥ of f(t) with L<t<L by using the periodic extension trick. Define F (t) to be the unique function − L determined by the requirements that i) F (t)= f(t) for L<t L L − ≤ ii) FL(t) is periodic of period 2L Then, for L<t<L, − L ∞ π π ik L t 1 ik L t f(t)= FL(t)= ck(L)e where ck(L)= 2L f(t)e− dt (2) k= L X−∞ Z− If we can somehow take the limit L , we will get a representation of f that is is valid for all t’s , not → ∞ just those in some finite interval L<t<L. This is exactly what we shall do, by the simple expedient − of interpreting the sum in (2) as a Riemann sum approximation to a certain integral. For each integer k, th π π define the k frequency to be ωk = k L and denote by ∆ω = L the spacing, ωk+1 ωk, between any two iωt − successive frequencies. Also define fˆ(ω)= ∞ f(t)e− dt. Since f(t) = 0 for all t L −∞ | |≥ L π ∞ R π ∞ 1 ik L t 1 i(k L )t 1 iωk t 1 ˆ 1 ˆ ck(L)= 2L f(t)e− dt = 2L f(t)e− dt = 2L f(t)e− dt = 2L f(ωk)= 2π f(ωk) ∆ω L Z− Z−∞ Z−∞ In this notation, ∞ 1 ˆ iωkt f(t)= FL(t)= 2π f(ωk)e ∆ω (3) k= X−∞ for any L<t<L. As we let L , the restriction L<t<L disappears, and the right hand − → ∞ − 1 ∞ ˆ iωt side converges exactly to the integral 2π f(ω)e dω. To see this, cut the domain of integration into −∞ y R 1 ˆ iωt y = 2π f(ω)e ω 0 ω1 ω2 ω3 ∆ω c Joel Feldman. 2007. All rights reserved. March 1, 2007 The Fourier Transform 1 small slices of width ∆ω and approximate, as in the above figure, the area under the part of the graph of 1 ˆ iωt 1 ˆ iωk t 2π f(ω)e with ω between ωk and ωk + ∆ω by the area of a rectangle of base ∆ω and height 2π f(ωk)e . 1 iωt 1 iωk t ∞ ˆ ∞ ˆ This approximates the integral 2π f(ω)e dω by the sum 2π k= f(ωk)e ∆ω. As L the −∞ → ∞ approximation gets better and better−∞ so that the sum approaches the integral. So taking the limit of (3) as R P L gives → ∞ 1 ∞ ˆ iωt ˆ ∞ iωt f(t)= 2π f(ω)e dω where f(ω)= f(t)e− dt (4) Z−∞ Z−∞ The function fˆ is called the Fourier transform of f. It is to be thought of as the frequency profile of the signal f(t). Example 1 Suppose that a signal gets turned on at t = 0 and then decays exponentially, so that at f(t)= e− if t 0 0 if t<≥ 0 for some a> 0. The Fourier transform of this signal is t(a+iω) ∞ iωt ∞ at iωt ∞ t(a+iω) e− ∞ 1 fˆ(ω)= f(t)e− dt = e− e− dt = e− dt = = 0 0 (a + iω) 0 a + iω Z−∞ Z Z − √ 2 2 1 ω ˆ iφ(ω) Since a + iω has modulus a + ω and phase tan− a , we have that f(ω) = A(ω)e with A(ω) = 1 = 1 and φ(ω)= tan 1 ω . The amplitude A(ω) and phase φ(ω) are plotted below. a+iω √a2+ω2 − a | | − A(ω) a a ω ϕ(ω) a a ω Example 2 Suppose that a signal consists of a single rectangular pulse of width 1 and height 1. Let’s say that it gets turned on at t = 1 and turned off at t = 1 . The standard name for this “normalized” − 2 2 rectangular pulse is 1 1 if 1 <t< 1 rect(t)= 2 2 0 otherwise− 1 1 t − 2 2 It is also called a normalized boxcar function. The Fourier transform of this signal is 1/2 iωt 1/2 iω/2 iω/2 ∞ iωt iωt e− e e− 2 ω rect(ω)= rect(t) e− dt = e− dt = = − = ω sin 2 1/2 iω 1/2 iω Z−∞ Z− − − d c Joel Feldman. 2007. All rights reserved. March 1, 2007 The Fourier Transform 2 1/2 when ω = 0. When ω = 0, rect(0) = 1/2 dt = 1. By l’Hˆopital’s rule 6 − R d sin ω 1 cos ω lim rect(ω) = lim 2 2 = lim 2 2 2 =1= rect(0) ω 0 ω 0 ω 0 → → ω → 1 d d so rect(ω) is continuous at ω = 0. There is a standard function called “sinc” that is defined(1) by sinc ω = sin ω ω ω . In this notation rect(ω) = sinc 2 . Here is a graph of rect(ω). d d 1 d rect(ω) d ω 2π 2π − Properties of the Fourier Transform Linearity If α and β are any constants and we build a new function h(t)= αf(t)+ βg(t) as a linear combination of two old functions f(t) and g(t), then the Fourier transform of h is ∞ iωt ∞ iωt ∞ iωt ∞ iωt hˆ(ω)= h(t)e− dt = αf(t)+ βg(t) e− dt = α f(t)e− dt + β g(t)e− dt (L) Z−∞ Z−∞ Z−∞ Z−∞ = αfˆ(ω)+ βgˆ(ω) Time Shifting Suppose that we build a new function h(t)= f(t t ) by time shifting a function f(t) by t . The easy − 0 0 way to check the direction of the shift is to note that if the original signal f(t) has a jump when its argument t = a, then the new signal h(t) = f(t t ) has a jump when t t = a, which is at t = a + t , t units to − 0 − 0 0 0 the right of the original jump. t0 f(t) h(t)= f(t t ) − 0 t a a + t0 The Fourier transform of h is ′ ∞ iωt ∞ iωt ∞ iω(t +t0) hˆ(ω)= h(t)e− dt = f(t t )e− dt = f(t′)e− dt′ where t = t′ + t , dt = dt′ − 0 0 Z−∞ Z−∞ Z−∞ (T) ′ iωt0 ∞ iωt iωt0 = e− f(t′)e− dt′ = e− fˆ(ω) Z−∞ (1) sin ω There are actually two different commonly used definitions. The first, which we shall use, is sinc ω = ω . The second is sin πω sinc ω = πω . It is sometimes called the normalized sinc function. c Joel Feldman. 2007. All rights reserved. March 1, 2007 The Fourier Transform 3 Scaling t If we build a new function h(t)= f α by scaling time by a factor of α> 0, then the Fourier transform of h is ′ ∞ iωt ∞ t iωt ∞ iωαt hˆ(ω)= h(t)e− dt = f e− dt = α f(t′)e− dt′ where t = αt′, dt = αdt′ α (S) Z−∞ Z−∞ Z−∞ = αfˆ(αω) Example 3 Now consider a signal that consists of a single rectangular pulse of height H, width W and centre C. y H y = rHWC (t) t C W C + W − 2 2 t C The function rHWC (t) = H rect −W (with rect the normalized rectangular pulse of Example 2) has height H and jumps when t C = 1 , i.e. t = C 1 W and so is the specified signal. By combining properties (L), −W 2 2 ± ± t C (T) and (S), we can determine the Fourier transform of rHWC (t) = H rect −W for any H, C and W . We build it up in three steps. First we consider the special case in which H = 1 and C = 0. Then we have R (t) = rect t . So, ◦ 1 W according to the scaling property (S) with t t t f(t) = rect(t), h(t)= R1(t) = rect W = f W = f α with α = W we have ˆ ˆ ˆ 2 Wω 2 Wω R1(ω)= h(ω)= αf(αω)= W rect(Wω)= W Wω sin 2 = ω sin 2 t C Next we allow a nonzero C too and consider R d(t) = rect − = R (t C).
Load more

Recommended publications
  • Convolution! (CDT-14) Luciano Da Fontoura Costa
    Convolution! (CDT-14) Luciano da Fontoura Costa To cite this version: Luciano da Fontoura Costa. Convolution! (CDT-14). 2019. hal-02334910 HAL Id: hal-02334910 https://hal.archives-ouvertes.fr/hal-02334910 Preprint submitted on 27 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Convolution! (CDT-14) Luciano da Fontoura Costa [email protected] S~aoCarlos Institute of Physics { DFCM/USP October 22, 2019 Abstract The convolution between two functions, yielding a third function, is a particularly important concept in several areas including physics, engineering, statistics, and mathematics, to name but a few. Yet, it is not often so easy to be conceptually understood, as a consequence of its seemingly intricate definition. In this text, we develop a conceptual framework aimed at hopefully providing a more complete and integrated conceptual understanding of this important operation. In particular, we adopt an alternative graphical interpretation in the time domain, allowing the shift implied in the convolution to proceed over free variable instead of the additive inverse of this variable. In addition, we discuss two possible conceptual interpretations of the convolution of two functions as: (i) the `blending' of these functions, and (ii) as a quantification of `matching' between those functions.
    [Show full text]
  • Eliminating Aliasing Caused by Discontinuities Using Integrals of the Sinc Function
    Sound production – Sound synthesis: Paper ISMR2016-48 Eliminating aliasing caused by discontinuities using integrals of the sinc function Fabián Esqueda(a), Stefan Bilbao(b), Vesa Välimäki(c) (a)Aalto University, Dept. of Signal Processing and Acoustics, Espoo, Finland, fabian.esqueda@aalto.fi (b)Acoustics and Audio Group, University of Edinburgh, United Kingdom, [email protected] (c)Aalto University, Dept. of Signal Processing and Acoustics, Espoo, Finland, vesa.valimaki@aalto.fi Abstract A study on the limits of bandlimited correction functions used to eliminate aliasing in audio signals with discontinuities is presented. Trivial sampling of signals with discontinuities in their waveform or their derivatives causes high levels of aliasing distortion due to the infinite bandwidth of these discontinuities. Geometrical oscillator waveforms used in subtractive synthesis are a common example of audio signals with these characteristics. However, discontinuities may also be in- troduced in arbitrary signals during operations such as signal clipping and rectification. Several existing techniques aim to increase the perceived quality of oscillators by attenuating aliasing suf- ficiently to be inaudible. One family of these techniques consists on using the bandlimited step (BLEP) and ramp (BLAMP) functions to quasi-bandlimit discontinuities. Recent work on antialias- ing clipped audio signals has demonstrated the suitability of the BLAMP method in this context. This work evaluates the performance of the BLEP, BLAMP, and integrated BLAMP functions by testing whether they can be used to fully bandlimit aliased signals. Of particular interest are cases where discontinuities appear past the first derivative of a signal, like in hard clipping. These cases require more than one correction function to be applied at every discontinuity.
    [Show full text]
  • The Best Approximation of the Sinc Function by a Polynomial of Degree N with the Square Norm
    Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 307892, 12 pages doi:10.1155/2010/307892 Research Article The Best Approximation of the Sinc Function by a Polynomial of Degree n with the Square Norm Yuyang Qiu and Ling Zhu College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China Correspondence should be addressed to Yuyang Qiu, [email protected] Received 9 April 2010; Accepted 31 August 2010 Academic Editor: Wing-Sum Cheung Copyright q 2010 Y. Qiu and L. Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The polynomial of degree n which is the best approximation of the sinc function on the interval 0, π/2 with the square norm is considered. By using Lagrange’s method of multipliers, we construct the polynomial explicitly. This method is also generalized to the continuous function on the closed interval a, b. Numerical examples are given to show the effectiveness. 1. Introduction Let sin cxsin x/x be the sinc function; the following result is known as Jordan inequality 1: 2 π ≤ sin cx < 1, 0 <x≤ , 1.1 π 2 where the left-handed equality holds if and only if x π/2. This inequality has been further refined by many scholars in the past few years 2–30. Ozban¨ 12 presented a new lower bound for the sinc function and obtained the following inequality: 2 1 4π − 3 π 2 π2 − 4x2 x − ≤ sin cx.
    [Show full text]
  • Oversampling of Stochastic Processes
    OVERSAMPLING OF STOCHASTIC PROCESSES by D.S.G. POLLOCK University of Leicester In the theory of stochastic differential equations, it is commonly assumed that the forcing function is a Wiener process. Such a process has an infinite bandwidth in the frequency domain. In practice, however, all stochastic processes have a limited bandwidth. A theory of band-limited linear stochastic processes is described that reflects this reality, and it is shown how the corresponding ARMA models can be estimated. By ignoring the limitation on the frequencies of the forcing function, in the process of fitting a conventional ARMA model, one is liable to derive estimates that are severely biased. If the data are sampled too rapidly, then maximum frequency in the sampled data will be less than the Nyquist value. However, the underlying continuous function can be reconstituted by sinc function or Fourier interpolation; and it can be resampled at a lesser rate corresponding to the maximum frequency of the forcing function. Then, there will be a direct correspondence between the parameters of the band-limited ARMA model and those of an equivalent continuous-time process; and the estimation biases can be avoided. 1 POLLOCK: Band-Limited Processes 1. Time-Limited versus Band-Limited Processes Stochastic processes in continuous time are usually modelled by filtered versions of Wiener processes which have infinite bandwidth. This seems inappropriate for modelling the slowly evolving trajectories of macroeconomic data. Therefore, we shall model these as processes that are limited in frequency. A function cannot be simultaneously limited in frequency and limited in time. One must choose either a band-limited function, which extends infinitely in time, or a time-limited function, which extents over an infinite range of frequencies.
    [Show full text]
  • Shannon Wavelets Theory
    Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2008, Article ID 164808, 24 pages doi:10.1155/2008/164808 Research Article Shannon Wavelets Theory Carlo Cattani Department of Pharmaceutical Sciences (DiFarma), University of Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy Correspondence should be addressed to Carlo Cattani, [email protected] Received 30 May 2008; Accepted 13 June 2008 Recommended by Cristian Toma Shannon wavelets are studied together with their differential properties known as connection coefficients. It is shown that the Shannon sampling theorem can be considered in a more general approach suitable for analyzing functions ranging in multifrequency bands. This generalization R ff coincides with the Shannon wavelet reconstruction of L2 functions. The di erential properties of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon ∞ wavelets are C -functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction of the derivatives of the C -functions. Copyright q 2008 Carlo Cattani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Wavelets 1 are localized functions which are a very useful tool in many different applications: signal analysis, data compression, operator analysis, and PDE solving see, e.g., 2 and references therein. The main feature of wavelets is their natural splitting of objects into different scale components 1, 3 according to the multiscale resolution analysis.
    [Show full text]
  • Chapter 11: Fourier Transform Pairs
    CHAPTER 11 Fourier Transform Pairs For every time domain waveform there is a corresponding frequency domain waveform, and vice versa. For example, a rectangular pulse in the time domain coincides with a sinc function [i.e., sin(x)/x] in the frequency domain. Duality provides that the reverse is also true; a rectangular pulse in the frequency domain matches a sinc function in the time domain. Waveforms that correspond to each other in this manner are called Fourier transform pairs. Several common pairs are presented in this chapter. Delta Function Pairs For discrete signals, the delta function is a simple waveform, and has an equally simple Fourier transform pair. Figure 11-1a shows a delta function in the time domain, with its frequency spectrum in (b) and (c). The magnitude is a constant value, while the phase is entirely zero. As discussed in the last chapter, this can be understood by using the expansion/compression property. When the time domain is compressed until it becomes an impulse, the frequency domain is expanded until it becomes a constant value. In (d) and (g), the time domain waveform is shifted four and eight samples to the right, respectively. As expected from the properties in the last chapter, shifting the time domain waveform does not affect the magnitude, but adds a linear component to the phase. The phase signals in this figure have not been unwrapped, and thus extend only from -B to B. Also notice that the horizontal axes in the frequency domain run from -0.5 to 0.5. That is, they show the negative frequencies in the spectrum, as well as the positive ones.
    [Show full text]
  • Lecture 6 Basic Signal Processing
    Lecture 6 Basic Signal Processing Copyright c 1996, 1997 by Pat Hanrahan Motivation Many aspects of computer graphics and computer imagery differ from aspects of conven- tional graphics and imagery because computer representations are digital and discrete, whereas natural representations are continuous. In a previous lecture we discussed the implications of quantizing continuous or high precision intensity values to discrete or lower precision val- ues. In this sequence of lectures we discuss the implications of sampling a continuous image at a discrete set of locations (usually a regular lattice). The implications of the sampling pro- cess are quite subtle, and to understand them fully requires a basic understanding of signal processing. These notes are meant to serve as a concise summary of signal processing for computer graphics. Reconstruction Recall that a framebuffer holds a 2D array of numbers representing intensities. The display creates a continuous light image from these discrete digital values. We say that the discrete image is reconstructed to form a continuous image. Although it is often convenient to think of each 2D pixel as a little square that abuts its neighbors to fill the image plane, this view of reconstruction is not very general. Instead it is better to think of each pixel as a point sample. Imagine an image as a surface whose height at a point is equal to the intensity of the image at that point. A single sample is then a “spike;” the spike is located at the position of the sample and its height is equal to the intensity asso- ciated with that sample.
    [Show full text]
  • Chapter 1 the Fourier Transform
    Chapter 1 The Fourier Transform 1.1 Fourier transforms as integrals There are several ways to define the Fourier transform of a function f : R ! C. In this section, we define it using an integral representation and state some basic uniqueness and inversion properties, without proof. Thereafter, we will consider the transform as being defined as a suitable limit of Fourier series, and will prove the results stated here. Definition 1 Let f : R ! R. The Fourier transform of f 2 L1(R), denoted by F[f](:), is given by the integral: 1 Z 1 F[f](x) := p f(t) exp(−ixt)dt 2π −∞ for x 2 R for which the integral exists. ∗ We have the Dirichlet condition for inversion of Fourier integrals. R 1 Theorem 1 Let f : R ! R. Suppose that (1) −∞ jfj dt converges and (2) in any finite interval, f,f 0 are piecewise continuous with at most finitely many maxima/minima/discontinuities. Let F = F[f]. Then if f is continuous at t 2 R, we have 1 Z 1 f(t) = p F (x) exp(itx)dx: 2π −∞ ∗This definition also makes sense for complex valued f but we stick here to real valued f 1 Moreover, if f is discontinuous at t 2 R and f(t + 0) and f(t − 0) denote the right and left limits of f at t, then 1 1 Z 1 [f(t + 0) + f(t − 0)] = p F (x) exp(itx)dx: 2 2π 1 From the above, we deduce a uniqueness result: Theorem 2 Let f; g : R ! R be continuous, f 0; g0 piecewise continuous.
    [Show full text]
  • A Rational Approximation of the Sinc Function Based on Sampling and the Fourier Transforms
    A rational approximation of the sinc function based on sampling and the Fourier transforms S. M. Abrarov∗ and B. M. Quine∗† October 22, 2019 Abstract In our previous publications we have introduced the cosine product- to-sum identity [17] M 2M−1 Y t 1 X 2m − 1 cos = cos t 2m 2M−1 2M m=1 m=1 and applied it for sampling [1, 2] as an incomplete cosine expansion of the sinc function in order to obtain a rational approximation of the Voigt/complex error function that with only 16 summation terms can provide accuracy ∼ 10−14. In this work we generalize this approach and show as an example how a rational approximation of the sinc function can be derived. A MATLAB code validating these results is presented. Keywords: sinc function; sampling; Fourier transform; rational ap- arXiv:1812.10884v3 [math.NA] 22 Oct 2019 proximation; numerical integration ∗Dept. Earth and Space Science and Engineering, York University, Toronto, Canada, M3J 1P3. †Dept. Physics and Astronomy, York University, Toronto, Canada, M3J 1P3. 1 1 Introduction Sampling is a powerful mathematical tool that can be utilized in many fields of Applied Mathematics and Computational Physics [18{20]. One of the popular methods of sampling is based on the sinc function that can be defined as [9, 11] 8 sin t < ; t 6= 0 sinc (t) = t : 1; t = 0: In particular, any function f (t) defined in some interval t 2 [a; b] can be approximated by the following equation (see, for example, equation (3) in [18]) N X π f (t) = f (t ) sinc (t − t ) + " (t) ; (1) n h n n=−N where tn are the sampling points, h is the small adjustable parameter and " (t) is the error term.
    [Show full text]
  • 19. the Fourier Transform in Optics
    19. The Fourier Transform in optics What is the Fourier Transform? Anharmonic waves The spectrum of a light wave Fourier transform of an exponential The Dirac delta function The Fourier transform of ejt, cos(t) 1 f ()tF ( )expjtd Fftjtdt() ()exp 2 There’s no such thing as exp(jt) All semester long, we’ve described electromagnetic waves like this: jt Et Re Ee0 What’s wrong with this description? Well, what is its behavior after a long time? limEt ??? t In the real world, signals never last forever. We must always expect that: limEt 0 t This means that no wave can be perfectly “monochromatic”. All waves must be a superposition of different frequencies. Jean Baptiste Joseph Fourier, our hero Fourier went to Egypt with Napoleon’s army in 1798, and was made Governor of Lower Egypt. Later, he was concerned with the physics of heat and developed the Fourier series and transform to model heat-flow problems. He is also generally credited with the first discussion of the greenhouse effect as a Joseph Fourier, 1768 - 1830 source of planetary warming. “Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics.” Lord Kelvin What do we hope to achieve with the Fourier Transform? We desire a measure of the frequencies present in a wave. This will lead to a definition of the term, the “spectrum.” Monochromatic waves have only one frequency, . Light electric field electric Light Time This light wave has many frequencies.
    [Show full text]
  • Introduction to Image Processing #5/7
    Outline Introduction to Image Processing #5/7 Thierry Geraud´ EPITA Research and Development Laboratory (LRDE) 2006 Thierry Geraud´ Introduction to Image Processing #5/7 Outline Outline 1 Introduction 2 Distributions About the Dirac Delta Function Some Useful Functions and Distributions 3 Fourier and Convolution 4 Sampling 5 Convolution and Linear Filtering 6 Some 2D Linear Filters Gradients Laplacian Thierry Geraud´ Introduction to Image Processing #5/7 Outline Outline 1 Introduction 2 Distributions About the Dirac Delta Function Some Useful Functions and Distributions 3 Fourier and Convolution 4 Sampling 5 Convolution and Linear Filtering 6 Some 2D Linear Filters Gradients Laplacian Thierry Geraud´ Introduction to Image Processing #5/7 Outline Outline 1 Introduction 2 Distributions About the Dirac Delta Function Some Useful Functions and Distributions 3 Fourier and Convolution 4 Sampling 5 Convolution and Linear Filtering 6 Some 2D Linear Filters Gradients Laplacian Thierry Geraud´ Introduction to Image Processing #5/7 Outline Outline 1 Introduction 2 Distributions About the Dirac Delta Function Some Useful Functions and Distributions 3 Fourier and Convolution 4 Sampling 5 Convolution and Linear Filtering 6 Some 2D Linear Filters Gradients Laplacian Thierry Geraud´ Introduction to Image Processing #5/7 Outline Outline 1 Introduction 2 Distributions About the Dirac Delta Function Some Useful Functions and Distributions 3 Fourier and Convolution 4 Sampling 5 Convolution and Linear Filtering 6 Some 2D Linear Filters Gradients Laplacian
    [Show full text]
  • Fourier Transform and Image Filtering
    Fourier Transform and Image Filtering CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010 The Fourier Transform Fourier Transform • Forward, mapping to frequency domain: • Backward, inverse mapping to time domain: Fourier Series • Projection or change of basis • Coordinates in Fourier basis: • Rewrite f as: Example: Step Function Step function as sum of infinite sine waves Discrete Fourier Transform Fourier Basis • Why Fourier basis? • Orthonormal in [-pi, pi] • Periodic • Continuous, differentiable basis FT Properties Common Transform Pairs Dirac delta - constant Common Transform Pairs Rectangle – sinc sinc(x) = sin(x) / x Common Transform Pairs Two symmetric Diracs - cosine Common Transform Pairs Comb – comb (inverse width) Common Transform Pairs Gaussian – Gaussian (inverse variance) Common Transform Pairs Summary Quiz What is the FT of a triangle function? Hint: how do you get triangle function from the functions shown so far? Triangle Function FT Triangle = box convolved with box So its FT is sinc * sinc Fourier Transform of Images 2D Fourier Transform • Forward transform: • Backward transform: • Forward transform to freq. yields complex values (magnitude and phase): 2D Fourier Transform Fourier Spectrum Image Fourier spectrum Origin in corners Log of spectrum Retiled with origin In center Fourier Spectrum–Rotation Phase vs Spectrum Image Reconstruction from Reconstruction from phase map spectrum Fourier Spectrum Demo http://bigwww.epfl.ch/demo/basisfft/demo.html Low-Pass Filter • Reduce/eliminate high frequencies • Applications – Noise
    [Show full text]