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