Image analysis IV
C9940 3-Dimensional Transmission Electron Microscopy
S1007 Doing structural biology with the electron microscope
April 24, 2017
Outline
Image analysis III
More on FFTs
Classification
• Review of multivariate data analysis
• Classification in 2D
• Classification in 3D
Resolution estimation
• Fourier Shell Correlation
• Expectation value of noise
• “Gold standard” resolution
Some simple 2D Fourier transforms:
a row of points
Some simple 2D Fourier transforms:
a series of lines
Some simple 2D Fourier transforms:
a 2D lattice
Single point
If the point was infinitely sharp,
the FFT would be flat.
Some simple 1D transforms:
a sharp point (Dirac delta function)
http://en.labs.wikimedia.org/wiki/Basic_Physics_of_Nuclear_Medicine/Fourier_Methods
Single point
If the point was infinitely sharp,
the FFT would be flat.
Two points
Three points
Five points
One row
Two rows
Three rows
Five rows
Full lattice
Animation
What if?
?
Molecule g(x)
lattice: f(x) Set a molecule down at every
lattice point.
Adapted from David DeRosier
Convolution: a review
Cross-correlation: F*(X) G(X)
f(x)
F(X) F(X) G(X)
f(x)•g(x)
G(X)
g(x)
What if?
?
Hint
What if?
f(x)
F(X)
f(x)•g(x)
G(X)
g(x)
F(X) G(X)
?
Classification
Reiteration of the problem
8 classes of faces, 64x64 pixels
With noise added
Before we can average the data, we first should find homogeneous subsets.
Average:
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
Multivariate data analysis (MDA)
1 2 3 4 9 10 11 125 6 7 8 13 14 15 16
Multivariate data analysis (MDA), or
Multivariate statistical analysis (MSA)
Our 16-pixel image can be reorganized into a 16-coordinate vector.
MDA: Reconstituted images
Linear combinations of these images will give us
approximations of the images that make up the data.
Average Eigenimage #1 Eigenimage #2 Eigenimage #3
c0 + c1
+ c2
+ c3
+ ...
Phantom images of worm hemoglobin
MDA of worm hemoglobin
Average:
+c0
-c0
+c1
+c2
+c3
+c4
+c5
-c1
-c2
-c3
-c4
-c5
1 2 3 4 9 10 11 125 6 7 8 13 14 15 16
Classification
How do we categorize/classify the images?
K-means classification
A number K of images are chosen as seeds.
BAD: Some clusters may be overrepresented/underrepresented.
Diday's method of moving centers
Diday's method of moving centers
Diday's method of moving centers
Diday's method of moving centers
We will note the images that always “travel” together, and will call them a class.
Dendrogram
Dendrogram
Hierarchical ascendant classification
“Images”
Hierarchical Ascendant Classification
All images are represented.
The dendrogram will be too heavily branched to interpret without truncation.
Binary-tree viewer
BAD: Information about the height of the branch is lost.
Classification in 3D
Classification:
Reference-based classification vs.
Maximum likelihood (ML3D)
Reference-based classification:
• Possible conformations must be
known.
• The combination of parameters
(shift, rotation, class) is chosen
from the highest correlation
value.
• Possible reference bias
ML3D
• Possible conformations are
not known.
• The probability of the
occurrence of the
parameters (shift, rotation,
class) is maximized.
• Random, data-dependent
RELION is a variation of maximum likelihood.
Seeding ML3D classification
There will be slight differences in the reconstructions.
We will iteratively maximize the likelihood of a
particle belonging to a particular class.
images
We split the data set into K classes at random.
How good is our reconstruction?
images
“odd” reconstruction
“even” reconstruction
We split the data set into halves and compare them.
How do we evaluate the quality of a reconstruction?
Fourier Shell Correlation (FSC)
Properties:
- Fourier terms have amplitude + phase.
- Correlation values range from -1 to +1.
- Noise should give an average of 0.
- The comparison is done as a function of spatial frequency (or “resolution”)
Reconstruction1
Reconstruction2
term 1 term 2
Fourier Shell Correlation curve
FSC curve with expectation value of noise
Why does σ vary with spatial frequency?
Random walks:
Why signal-to-noise improves with √N
The “Drunkard's walk”
Let's conduct an experiment.
The “Drunkard's walk”
0 1 2 3 4-1-2-3-4
We're going to assume that each step is random and independent of previous steps.
The “Drunkard's walk”
0 1 2 3 4-1-2-3-4
t=1
t=2
t=3
t=4
t=5
t=6
The teetotaler's walk
0 1 2 3 4-1-2-3-4
t=1
t=2
t=3
t=4
Expectation value
The expected distance that “noise” travels increases with √N.
However, it is not as fast as the distance that “signal” travels.
Thus, as we collect more data, the SNR increase by N/√N = √N
Random walks: more information
Expectation values
and how they related to resolution criteria
With small N, behavior is more unpredictable
One resolution criterion was to compare the FSC to, say, 3*σ.
BUT:
The σ value describes the behavior of unaligned noise.
Review: model bias
N = 128 N = 256 N = 512
N = 1024 N = 2048 original
The model bias can yields false correlations in real space
is equivalent to false correlations in Fourier space.
images
“odd” reconstruction
“even” reconstruction
Refinement: classical and “gold standard”
+
OLD STRATEGY
merge & refine orientations
“GOLD STANDARD”
refinement1 refinement2
Different resolution criteria
FSC=0.5
FSC=0.143
FSC=0.333