Cryo-electron microscopy Tibor Füzik Lecture 6 Spatial waves, Fourier transform, image formation contrast transfer function Spatial wave • Oscillates in space A -A t=0 s T period wavelenght λ x=0 x=λ 0 2π 0 2 Spatial wave Amplitude Spatial frequency Phase shift http://www.maxmcarter.com/sinewave/generate_sinewave.php -1 1 -2 2 -1 1 -1 1 2π0 π Adding sine waves Every single complex wave we can construct by addition of series of single waves Can we do the opposite way? What sine waves this periodical function consist of? (k=1, A1, P1) (k=2, A2, P2) (k=3, A3, P3) Fourier decomposition Fundamental frequency (k=1, A1, P1) 1st harmonics (k=2, A2, P2) 2nd harmonics (k=3, A3, P3) DC component (Ao) A constant function λ → ∞ => k → 0 A A.sin(0x) → A Square wave – Fourier decomposition A(k=even) = 0 A(k=odd) > 0 Sharp edges require hi frequency waves Step function Step function wikipedia.org Fourier decomposition Every periodical function can be decomposed into sum of infinite number of sine waves Jean-Baptiste Joseph Fourier Fourier transform Inverse Fourier transform Fourier decomposition of spatial waves Acos(ϕ) Asin(ϕ) A How can we store Fourier transform • Need to store waves (parameters of waves) • Reciprocal space • series of wave functions • series of wave vectors • 2 ways of wave vector representation • as amplitudes and corresponding phases • as complex numbers Wave as a vector F from Lecture 2 k A2 Power spectra k A2 k A2 A2 k Reciprocal space – Power spectra 0 2 -1 1 -2 2 -1 1 1 1 -2 10 -1 -1 2 -1 Reciprocal space – clpx function k=1 k=2 k=3 Ao k A2 0 1 2 3 Power spectrum -1-2-3 f(x) F(k) k A2 0 1 2 3 4 5 6 7 Reciprocal space – step function Power spectrumf(x) -1-2-3-4-5-6-7 www.oreilly.com Fourier transform of 1D discrete waves • Sampling cause discretization of the wave • Finite number of Fourier components 0 1 0 -1 0 A0 A1 A2 A3 A4 A5 A6 A7 A8 P0 P1 P2 P3 P4 P5 P6 P7 P8 Nyquist component k A2 1-1 Power spectrum Reciprocal space Detector with discrete pixels 2D waves 1D wave k -> number of wave periods 2D wave h, k -> number of wave periods per x, yh=1, k=0, amp=1, phase=0 x y Profile plot 2D waves 1D wave k -> number of wave periods 2D wave h, k -> number of wave periods per x, y h=0, k=1, amp=1, phase=0 x y h=20, k=0, amp=1, phase=0h=1, k=0, amp=1, phase=90 h=0, k=-1, amp=1, phase=90 2D waves h=1, k=1, amp=1, phase=0 h=1, k=-1, amp=1, phase=0 h=2, k=7, amp=1, phase=0 Combining 2D waves h=4, k=1, amp=2, phi=0 h=2, k=2, amp=1, phi=90o h=20, k=0, amp=1, phi=0 SUM Fourier transform of 2D waves Grant Jensen Inverse FT 10x10 (x,y,z) samples 2D Fourier transform of simple 2D waves h=1, k=0, amp=1, phase=0 f(x,y) F(h,k) h k 1-1 Power spectrum h=1, k=1, amp=1, phase=0 f(x,y) F(h,k) h1-1 1 -1 k 2D Fourier transform of simple 2D waves Power spectrum h=4, k=1, amp=2, phase=0 h=2, k=2, amp=1, phase=90 h=20, k=0, amp=1, phase=0 SUM 2D Fourier transform of simple 2D waves Power spectra 3D waves 1D wave k -> number of wave periods 2D wave h, k -> number of wave periods per x, y 3D wave h, k, l -> number of wave periods per x, y, z h=1; k=0; l=0 h=2; k=3; l=0 3D waves h=2; k=3; l=4 h=2; k=3; l=4 Sum of 3D waves Sum of multiple (3) 3D waves 3D Fourier transform Grant Jensen 10x10x10 (x,y,z) samples 3D reconstruction Reciprocal space Real-space from Lecture 3 Good to know about reciprocal space • Every single point in reciprocal space affects all the points in real- space • Every single point in real-space affects all the points in reciprocal space • More far from the center of the power spectrum – higher the spatial frequency • While only amplitudes are represented in the power spectrum, the underlying phases are equally important Letting the low freq. pass Letting the hi freq. pass Low-pass filter Hi-pass filter Bandpass filter DC component DC component removed Image rotation Real-space Reciprocal-space Real-space Reciprocal-space Reciprocal-space phases Image translation Fourier space cropping, padding Reciprocal-space cropping Downscaling (~lowpass) Reciprocal-space padding Upscaling (without adding information) Convolution • Convolution is a mathematical operation on two functions (f and g) that produces a third function (f*h) that expresses how the shape of one is modified by the other. • f*h ~ “pass the function f over the function g take the area under” • Convolution is commutative operation Convolution https://lpsa.swarthmore.edu/Convolution/CI.html Convolution theorem Gaussian PSF = Point spread function Electron scattering Braggs law escattered eX-ray scattered X-ray Electron scattering – TEM image formation from Lecture 2 Image formation in TEM Image plane back focal plane escattered electrons Fourier transform of image (every point in reciprocal space affects all the point in real-space) Real-space image sample el.mag. lens Bragg plane (scattering center) e- e- escattered escattered escattered eobjective aperture Assumptions - f(x), g(x) waves of same frequency - amplitude of wave f(x) >> wave g(x) - f(x) has phase-shift: phi0 - g(x) has phase-shift: phi0+Δphi Sum of phase shifted waves Argand diagram https://www.desmos.com/calculator/cwc8bn4snj real img 0π/2ππ 3/2π π/2 real img 0π/2ππ 3/2π π/2 real img 0π/2ππ 3/2π π/2 Δphi = 0 Δphi = π/2 Δphi = π Phase change during wave propagation • When a wave propagates in space, it continuously changes its phase Distance 4λ Phase 0 π 2ππ/2 3/2π 2π Distance 4.5λ π Distance 4.25λ π/2 Distance Phase shift n*λ 0, 2π n*λ + λ/4 π/2 n*λ + λ/2 π n*λ + ¾λ 3/2π Scattered electron travels an extra distance Contrast transfer function • detectors detect intensity (Amp2) not phases • when e- scatters π/2 phase-shift is introduced • Un-diffracted beam = non-scattered e• intensity of un-diffracted beam >> diffracted • Increase/decrease of intensity relative to un-diffracted beam = contrast e- (of equal phases) scattered electrons sample el.mag. lens spatial frequency (frequency of the image not the electrons) -1 1 +0λ +λ/4 +¾λ +λ/2 Distance Phase shift n*λ 0, 2π n*λ + λ/4 π/2 n*λ + λ/2 π n*λ + ¾λ 3/2π real img 3/2π π/2 + π/2 real 0π/2ππ 3/2π img π/2 + 0 real 0π/2ππ 3/2π img π/2 + π real img 0π/2ππ 3/2π π/2 π/2 + 3/2π image plane contrast e- (of equal phases) scattered electrons sample el.mag. lens +0λ +λ/4 +¾λ +λ/2 image plane (underfocus) Front focal plane back focal plane Contrast transfer function (CTF) defocus wavelength (e-) spherical aberration spider.wadsworth.org In focus images suffer from low contrast 0 µm = in focus -0.5 µm https://ctf-simulation.herokuapp.com/ Contrast transfer function (CTF) • Electron microscope images are convoluted by a point spread function • Point spread function in EM is represented by CTF in Fourier space • CTF has zero values (information loss) Orlova, Saibil 2011Thuman-Commike and Chiu, Micron -1 µm -2.7 µm Envelope function • Hi frequencies in CTF are damped • Envelope function • Chromatic aberrations • Focus spread • Energy spread • Variance in hi-tension • Defocus • Coherence of the electron beam spider.wadsworth.org Point spread function of TEM Every single point in image is the convolution of PSF and the object Image Object Point spread function 2D point spread function Wikipedia Russo&Henderson, 2018 CTF correction Convolution theorem spider.wadsworth.org object CTF Real-space Reciprocal-space Real-space CTF corrected image What was the shape of the original object represented by the image ? PSF convoluted image Estimation of CTF • CTF function of the image is unknown • Simulate/fit CTF that represents the Amp oscillation of the F(I) • Find the parameters of the CTF curve (mainly defocus) What we have learned….. • Spatial waves: 1D, 2D, 3D • Fourier transform of spatial waves: 1D, 2D, 3D • Inverse Fourier transform • Reciprocal space and its properties • TEM image formation: phase contrast • CTF and its properties • Point spread function and CTF correction Lena Forsén (*31 March 1951) 20191972 “age filter” Age 21 Age 69 “time convolution” The end