FB820
Lecture 6
3DEM methods
Jiri Novacek
Content
- image formation, CTF, image filtering
- image alignment in 2D
- 3D reconstruction
- common lines
- random conical tilt
Image formation
Image formation
A – amplitude contrast
s – spatial frequency
Cs – spherical abberation
λ – electron wavelength
z – defocus
Contrast transfer function
- Finite source size
- Energy spread (defocus)
- MTF of the camera
- Generic envelope (drift, charging, multiple scattering)
Envelope function
Contrast transfer function
Envelope function
kV=300,ac=0.07,cs=2.7,z=-1,apix=1,B=30
kV=300,ac=0.07,cs=2.7,z=-1,apix=1,B=300
Contrast transfer function
Low defocus
High defocus
Contrast transfer function
Image filtering
unfiltered image lowpass filtered (50A) lowpass filtered (250A)
130A
unfiltered image lowpass filtered (50A) bandpass filtered (1000,10A)
130A
Image filtering
John O’Brien (1991). The
New Yorker
Projection theorem
The 2D Fourier transform of the projection of a 3D density is
a central section of the 3D Fourier transform of the density,
perpendicular to the direction of projection.
Projection theorem
- 2D projections of an 3D object (handedness)
- high noise level (low sensitivity)
- convolution with microscope point spread functions
cryo-TEM imaging
- 2D projections of an 3D object (handedness)
- high noise level (low sensitivity)
- convolution with microscope point spread functions
cryo-TEM imaging
n=1
Averaging
n=1 n=2 n=8 n=16 n=64 n=256
Signal to noise ratio increases with square-root of n
Averaging
Sum of unaligned particles
Sum of aligned particles
Image alignment in 2D
In order to align the particles in 2D, we need to determine three parameters:
- two translational
- one rotational (on of the Euler angles)
δx
δy
ψ ψ
δy
δx
ψ
δx
Image alignment in 2D
Cross correlation function in 1D
Cross correlation function in 1D
- measure of similarity of two data series as a function of displacement of these functions
- in 2D optimal overlay of two images
- normalized cross-correlation – ccc = <-1,1>
Cross correlation
Cross correlation function in 1D
Cross correlation function in 1D
Cross correlation function in 1D
Cross correlation function in 1D
Cross correlation function in 1D
Cross correlation function in 1D
Cross correlation function in 1D
Cross correlation function in 1D
ConvolutionCross-correlation
Cross correlation function in 1D
Cross correlation function in 2D
Image alignment in 2D
Image alignment in 2D
Image alignment in 2D
Image alignment in 2D
Convolution
FT(F I) = FT(F) . FT(I)
FT(F I) = FT(F)* . FT(I)
Convolution theorem
Cross-correlation in 2D
Cross-correlation in 2D
Image rotation
- the images contain not only shift but also rotation
- cross-correlation - image sliding over the template (shift)
- (log)-polar transform → image transformation from cartesian to polar coordinates → rotational problem
shifted to translational problem → utilization of similar approaches as for image shift determination
x
y
R
ψ
R
ψ
Image alignment in 2D
We take a series of rings from each image, unravel them,
and compute a series of 1D cross-correlation functions.
Shifts along these unraveled CCFs is equivalent to a rotation in Cartesian
space.
Image alignment in 2D
Image alignment in 2D
Image alignment in 2D
- after rotation
Image alignment in 2D
- rotation and translation are interdependent – (rot→trans) ≠ (trans→rot)
=> order of the operation matters
shift: (25,45), rotation: 60°
shift→rotation
rotation→shift
Image alignment in 2D
- rotation and translation are interdependent – (rot→trans) ≠ (trans→rot)
- define reasonable range of shifts (e.g. (-2;+2)) and perform rotational alignment for each shifted image
Example: for the shift of +/-2 pixels in x and y → 25 alignment rotational alignments → each alignment
results in optimal rotational alignment and ccc → compare ccc and select maximal ccc to determine the
final shift and translation
=> increased complexity
Image alignment in 2D
Suppose we shift the image in x & y.
The new pixels will be weighted averages of the old pixels.
The more the mix the pixels, the worse the result will be.
Interpolation
Image alignment in 2D
Suppose we shift the image in x & y.
The new pixels will be weighted averages of the old pixels.
The more the mix the pixels, the worse the result will be.
Interpolation
Image alignment in 2D
Suppose we shift the image in x & y.
The new pixels will be weighted averages of the old pixels.
The more the mix the pixels, the worse the result will be.
Shift Rotation
Image alignment in 2D
The Fourier transform of noise is noise
- “White” noise is evenly distributed in Fourier space
- “White” means that each pixel is independent
Image alignment in 2D
Interpolation
The Fourier transform of noise is noise
- “White” noise is evenly distributed in Fourier space
- “White” means that each pixel is independent
The degradation of the
images means that we
should minimize the
number of interpolations.
Image alignment in 2D
Interpolation
Image alignment in 3D
Image alignment in 3D
Image alignment in 3D
1. Different orientations
2. Known orientations
3. Many particles
4. CTF parameters
3D reconstruction
Two general ways for 3D reconstruction:
- Real space
- Fourier space
3D reconstruction
We are going to reconstruct a 2D object from 1D projections. The principle is
the similar to, but simpler than, reconstructing a 3D object from 2D projections.
Real space reconstruction
3D reconstruction
Real space reconstruction
3D reconstruction
Real space reconstruction
3D reconstruction
- reconstruction is the inversion of projection
3D reconstruction
- reconstruction is the inversion of projection
3D reconstruction
- reconstruction is the inversion of projection
3D reconstruction
- reconstruction is the inversion of projection
3D reconstruction
The reconstruction does not agree well with the projections
Potential solution: Simultaneous Iterative Reconstruction Technique
Original Reconstructed
3D reconstruction
- simultaneous iterative reconstruction technique
Compute re-projections of your model.
Compare the re-projections to your experimental data.
There will be differences.
Weight the differences by a fudge factor, λ.
Adjust the model by the difference weighted by λ.
Repeat
3D reconstruction
- simultaneous iterative reconstruction technique
3D reconstruction
Fourier space reconstruction
Projection theorem
Central section theorem
3D reconstruction
Fourier space reconstruction
Projection theorem
Central section theorem
3D reconstruction
Converting from polar to Cartesian coordinates
3D reconstruction
3D reconstruction
1. Different orientations
2. Known orientations
3. Many particles
4. CTF parameters
3D reconstruction
3D reconstruction
Tomography
3D reconstruction
Angular Reconstruction
Common lines
Angular Reconstruction
Common lines
Random conical tilt
Random conical tilt
Random conical tilt
Random conical tilt
Random conical tilt
Random conical tilt
- we cannot tilt the stage to 90 deg → “missing cone”
Random conical tilt
- filling the missing cone
Random conical tilt