GBS765 Electron microscopy Lecture 1 Waves and Fourier transforms 10/14/14 9:05 AM
Some fundamental concepts: Periodicity! If there is some a, for a function f(x), such that f(x) = f(x + na) then function is periodic with the period a 0 a 2a 3a
Some fundamental concepts: The sine wave! Fundamental properties of sine waves:!!! Wavelength, λ!!!related property: frequency ν = 1/λ Amplitude, A Related property: Intensity I = A 2 Phase, φ
A cosine wave! f(x) = cos (x) 5 3 1-10 -5-1 0 5 10-3 -5 Amplitude f(x) = 5 cos (x) 5 4 3 2 1 0-10 -5-1 0 5 10-2 -3-4 -5 Phase f(x) = 5 cos (x + 3.14) 5 3 1-10 -5-1 0 5 10-3 -5 Frequency (=1/wavelength) f(x) = 5 cos (3 x + 3.14) 5 3 1-10 -5-1 0 5 10-3 -5
Sine waves can be used to describe the movement of electromagnetic radiation (light, X-rays) and electron waves through time and space! sine wave represents amplitude of radiation at a particular time and/or position!!! Sine waves can also used to describe any periodic phenomenon, including the physical structure of an object!! In which case a combination of sine waves may be used to represent variations in e.g. electron density!
The Fourier series Any (periodic) function f(x) can be constructed as a sum of (co)sine waves: $ % n=1 f (x) = A 0 + A i cos(2"xn /T + # n ) This is called a Fourier summation Breaking down a function into its sine wave components is called Fourier analysis Each wave i has an amplitude A i and a phase φ i (T is the period (=wavelength)) Alternative ( classic ) view: f (x) = A 0 + # $ [ A n cos(2"xn /T) + B n sin(2"xn /T)] n=1 This is the same as the above description because adding a sine and a cosine wave is the same as applying a phase shift
Explore the Fourier series with http://falstad.com/fourier/ Another FT applet: http://www.jhu.edu/~signals/fourier2/ file:///users/dokland/desktop/fourier/index.html
The Fourier series f (x) = A 0 + f (x) = A 0 + " # n=!" # $ n=1 can also be written as [ A n cos(2"xn /T) + B n sin(2"xn /T)] A n e i2! xn/t+" n because e ix = cos x + isin x More generally f (x) = 1 " # A(u)e i2! [ux+" (u)] du = 1 2!!" 2!! " F(u) = f (x)e!i2!ux dx #!" (Euler s formula) " #!"! F(u)e i2!ux du! F(u) = F(u)! i! (u) e F(u) (complex) is the Fourier Transform of f(x) (real)
1.5 y The sine wave Data #3Data #3 Wavelength, " Waves as vectors 1 Column 5 Column 5 0.5 0-0.5-1 -1.5 Represented as vector A: Amplitude, A #/2 # 3#/2 2# Phase,! F sinφ y = A sin (2#x/" +!) Column 4 Column 4 F A sin! Imag F A F A Column 5 Column 5 x A wave of amplitude A and phase φ can also be described as a vector F A wave of phase φ can be considered a combination of a sine wave of 0 phase and a 90 shifted wave The imaginary number i can simply be thought of as a 90 phase shift (there is nothing imaginary about it it is as real as the real part!)! F =! F cos! +i! F sin! = Ae i!! F cosφ A cos! Real A =! F cos! B =! F sin! Real part Imaginary part A + ib A = Acos! + i Asin! = A e i!
Frequency spectra human voice A frequency spectrum = a Fourier transform Spectroscopy = analysis of a phenomenon in terms of its frequency (wavelength) distribution (i.e. spectrum)
The Fourier Transform: real space and reciprocal space When you carry out a Fourier Transform operation you move from real space to reciprocal space! [Also called the spatial domain and the frequency domain]! In real space a coordinate (x) represents a position in space and the function f(x) represents the value at that point! In reciprocal space, the coordinate (u) represents a particular frequency of a (co)sine wave! Higher u = higher frequency! The FT function F(u) then represents the amplitude AND the phase of the wave!!(i.e. it is a complex number, or a vector)! If this seems abstract -- let s look at some transforms!
The Dirac delta (impulse) function { for x = 0 The Dirac delta function δ(x) is defined as 0 for x 0 # "!(x)dx =1!" The Fourier transform of the δ function $!{!(x)} = ˆ!(u) =!(x)e "i2"ux dx =1 # "# The comb function (sampling function, or impulse train) is a series of delta functions
Examples of Fourier transforms sinc function; sinc(x) = sin(x)/x
The Fourier Transform and Imaging Why is the Fourier transform important? because it describes many physical phenomena, including diffraction and image formation by a lens it forms the basis for image processing and 3D reconstruction algorithms it will be essential for understanding X-ray crystallography as well NOTE: You don t have to remember the exact mathematical formulation for the FT, nor will you be required to analytically calculate any FTs!
Fourier transforms in two dimensions You can calculate the FT in any number of dimensions We will be particularly interested in two dimensions (i.e. images) and three dimensions (i.e. structures) F(u,v) = 1 NM N#1 M #1 $ $ x= 0 y= 0 i2" (ux / N + vy / M ) f (x, y)e etc
sinc function NB: Note that this only represents the amplitude part of the FT! The phases are important too!
Note 1: The FT is always centrosymmetric: F(u,v) = F(-u,-v) Note 2: Correspondence in symmetry and rotation between real space and Fourier space
f(x,y) F(u,v) F(u,v) f(x,y) F(u,v) f(x,y)
Filtering
Amplitude and Phase real space f(x,y) y! F(u, v) =!{ f (x, y)} = F(u,! v) e FT i! (u,v) x where F(u,v) is the amplitude and φ(u,v) is the phase reciprocal space (the color represents the phase) v u amplitudes phases reciprocal space IFT real space
The convolution function is given as: f (x) " g(x) = % & $% This is hard to visualize Convolution f (#)g(x $#)d# Think of it as smearing out the function f(x) by sliding g(x) over it and multiplying them together at each point. Convolution with a delta function δ(x) is equivalent to copying the function at the position of the delta function e.g. a crystal is the convolution of a unit cell with a lattice Convolution with a Gaussian is equivalent to a blurring of the function e.g. a blurry image is the convolution of a perfect image with a point spread function
Convolution in Fourier space Convolution in real space is equivalent to multiplication in Fourier space Thus, f (x)! g(x) " F(u)#G(u) F(u)! G(u) " f (x)# g(x) We can calculate convolution of two functions by multiplying their Fourier transforms Now THAT s what I call convenient!
Convolution f(x,y) F(u,v) amplitudes and phases g(x,y) G(u,v) Fourier Transform
! H(u, v) =! F(u, v)!! G(u, v) =! F(u, v)! G(u, v) e i[! F (u,v)+! G (u,v)]
Multiplication in reciprocal space = convolution in real space H(u,v) h(x,y) IFT This is equal to the convolution of f(x,y) with g(x,y)
Convolution used to describe Gaussian blurring (PSF)
Convolution and low-pass filtering
Convolution of a unit cell and a lattice (=crystal) Note how the FT retains some properties of the individual units and some properties of the distribution (lattice)
Definition of correlation: f (x) o g(x) = Convolution and correlation $ % #$ f (")g(x + ")d" Correlation is equivalent to finding the best match between two functions Correlation is usually calculated in Fourier space (much faster): f (x) o g(x) " F(u) # G *(u) F(u) o G(u) " f (x) # g*(x) Correlation in real space is equal to multiplication in Fourier space (and vice versa)
Diffraction from a periodic specimen (double slit) Diffraction of light from a double slit Phase difference 0 Waves reinforce Phase difference!/2 Waves cancel out Phase difference! Waves reinforce Angles θ at which waves reinforce are given by Bragg s law: nλ = 2d sin θ Phase difference 3!/2 Waves cancel out See simulation at http://www.falstad.com/ripple/
Scattering angle and spatial Diffraction of light from a double slit frequency Any periodic function can be mathematically described as a sum of sine waves Phase difference 0 Waves reinforce Each wave has a spatial frequency (=resolution) that corresponds to a particular spacing (ν = 1/d) Phase difference!/2 Waves cancel out NOTE: do not confuse this wave with the electron wave (with wavelength λ) λ d θ Each spatial frequency ν (=spacing d) gives rise to a wave scattered at a specific angle θ: Phase difference! Waves reinforce Phase difference 3!/2 Waves cancel out sin θ θ = λ / d = λν This is equivalent to a Fourier transform of the object function F(θ) = FT {f(x)}
Diffraction and imaging Diffraction of X-rays: Diffraction vs. imaging X-ray detector Object Note: there are no lenses available for X-rays! incident rays X-rays:! = 1.5Å Diffracted rays Peaks in the recorded diffraction pattern Imaging (microscopy): Lens Diffraction plane Image Object Incident rays Electrons:! = 0.03Å Visible light:! = 5000Å Diffracted rays Focused rays
Fourier theory of imaging Incident beam ψ 0 Specimen = ρ xyz Specimen plane ψ s ψ 0 = 1 φ xy = ρ xyz dz ψ s 1 iφ(x,y) BFP = diffraction plane Image plane ψ i ψ f ψ f = F{ψ s } = δ(0) iφ(u,v) CTF = exp[iχ(u,v)] = cosχ(u,v) + isinχ(u,v) ψ f = ψ f x CTF x A(u,v) x E(u,v) δ(0) iφ(u,v)sinχ(u,v) Strictly speaking, cosχ(u,v) also comes in here Also, let s forget about A and E for the time being ψ i = F{ψ f } = 1 iφ(x,y)*f{sinχ(u,v)} I(x,y) = ψ i 2 = [1 iφ(x,y)*f{sinχ(u,v)}] 2 1 2φ(x,y)*F{sinχ(u,v)} F(u,v) = F{I(x,y)} = δ(0) 2Φ(u,v) x sinχ(u,v)