Image analysis I
C9940 3-Dimensional Transmission Electron Microscopy
S1007 Doing structural biology with the electron microscope
March 2, 2015
Syllabus, original
Week Date Instructor Topic
1 02/16 D. Nemecek & T. Shaikh Introduction/Tour/History
2 02/23 D. Nemecek & T. Shaikh Electron Optics
3 03/02 D. Nemecek Specimen preparation
4 03/09 T. Shaikh Image analysis I
5 03/16 T. Shaikh Image analysis II
6 03/23 T. Shaikh 3D reconstruction
7 03/30 T. Shaikh Single-particle reconstruction
(Easter)
8 04/13 D. Nemecek Tomography I
9 04/20 D. Nemecek Tomography II
10 04/27 D. Nemecek Visualization/Segmentation
11 05/04 D. Nemecek Hybrid methods
Syllabus, updated
Week Date Instructor Topic
1 02/16 D. Nemecek & T. Shaikh Introduction/Tour/History
2 02/23 D. Nemecek & T. Shaikh Electron Optics
3 03/02 T. Shaikh Image analysis I
4 03/09 D. Nemecek Specimen preparation
5 03/16 T. Shaikh Image analysis II
6 03/23 T. Shaikh 3D reconstruction
7 03/30 T. Shaikh Single-particle reconstruction
(Easter)
8 04/13 D. Nemecek Tomography I
9 04/20 D. Nemecek Tomography II
10 04/27 D. Nemecek Visualization/Segmentation
11 05/04 D. Nemecek Hybrid methods
Correction/clarification from Feb. 16
http://ernst.ruska.de
First Siemens microscope, 1939
http://emu.msim.org.uk
First commercial EM, 1937
Metropolitan-Vickers EM1
(EM2 shown)
The first commercial
electron microscope
was actually by the
British company
Metropolitan-Vickers
in 1937. However,
the magnification was
worse than for the
light microscope, so
the Siemens is
considered “first.”
Correction/clarification
Metropolitan Vickers
eventually became
AEI, which built a 1.2
million volt EM-7.
http://www.wadsworth.org
Outline
Correction/clarification form Feb. 16
Intro to image analysis
Relationship between imaging and diffraction
Fourier transforms
Theory
Examples in 1D
Examples in 2D
Digitization
Fourier filtration
Contrast transfer function
Resolution
Outline
Correction/clarification form Feb. 16
Intro to image analysis
Relationship between imaging and diffraction
Fourier transforms
Theory
Examples in 1D
Examples in 2D
Digitization
Fourier filtration
Contrast transfer function
Resolution
http://www.microscopy.ethz.ch
http://electron6.phys.utk.edu
Relationship between imaging and diffraction
http://electron6.phys.utk.edu
How do X-ray microscopes work?
http://ssrl.slac.stanford.eduhttp://www.microscopy.ethz.ch
How do X-ray microscopes work?
http://ssrl.slac.stanford.edu
Fabrizio...Barrett, 1999, Nature
Outline
Correction/clarification form Feb. 16
Intro to image analysis
Relationship between imaging and diffraction
Fourier transforms
Theory
Examples in 1D
Examples in 2D
Digitization
Fourier filtration
Contrast transfer function
Resolution
Relevance of Fourier transforms to EM
Fourier transform ~ diffraction pattern
see John Rodenburg's site, http://rodenburg.org
ν=α/λ
Fourier series
A Fourier series is an expansion of a
periodic function f(x) in terms of an infinite
sum of sines and cosines
Fourier transforms: Definition
f: function which we are transforming (1D)
x: axis coordinate
i: √-1
k: spatial frequency
F(k): Fourier coefficient at frequency k
Fourier transforms: Definition
Fourier coefficients, discrete functions
sine and cosine are orthogonal.
The inner product of the two is zero.
A function with the same frequency but
with an offset will have some components
of both sine and cosine.
That is, a and b will be non-zero.
A function with a different frequency will
have coefficients of zero.
http://cnx.org
The higher the
spatial frequencies
(i.e., higher
resolution) that are
included, the more
faithful the
representation of
the original function
will be.
Some properties

As n increases, so does the spatial frequency,
i.e., the “resolution.”
− For example, sin(2x) oscillates faster than sin(x)

Computation of a Fourier transform is a
completely reversible operation.
− There is no loss of information.

Fourier terms (or coefficients) have amplitude
and phase.

The diffraction pattern is the physical
manifestation of the Fourier transform
– Phase information is lost in a diffraction pattern.
– An image contains both phase and amplitude information.
Some simple 1D transforms:
a 1D lattice
Some simple 1D transforms:
a box
http://cnx.org
Some simple 1D transforms:
a Gaussian
Some simple 1D transforms:
a sharp point (Dirac delta function)
http://en.labs.wikimedia.org/wiki/Basic_Physics_of_Nuclear_Medicine/Fourier_Methods
Some simple 2D Fourier
transforms:
a row of points
Some simple 2D Fourier
transforms:
a 2D lattice
Some simple 2D Fourier
transforms:
a sharp disc
Some simple 2D Fourier
transforms:
a series of lines
What do we mean by spatial frequency?
origin
spatial frequency
From Wikipedia
Fourier
filtration
From Wikipedia PowerspectrumProfile
A “high-pass”
filter
A “low-pass” filter
Contrast transfer function
Why do we defocus?
1.0
0.1
Typical amplitude contrast is estimated a 0.08-0.12
(minus noise)
water
macromolecule
Instead of amplitude contrast,
we'll use phase contrast.
Phase contrast in light microscopy
Bright-field image Phase-contrast image
http://www.microbehunter.com
In EM, even with defocus, the contrast is poor.
Signal-to-noise ratio for cryoEM typically given to be between 0.07 and 0.10.
E. coli 70S ribosomes, field width ~1440Å.
unfiltered filtered
Optical path
Specimen
Back focal plane
Image plane
At focus, all we would see is amplitude contrast.
Specimen
Back focal plane
Image plane
Optical path with defocus
Perfect focus
Image plane
Focal plane
O
A B
OA path of unscattered beam
OB path of scattered beam
The length OA is
also the amount of defocus Δf
Optical path with defocus
What is the path difference between the scattered and unscattered beams?
Path difference as a function of Δf
OB = OA/cos(a)
Expressed in the number of wavelengths λ
Phase difference is the sine
O
A B
a
Some typical values
OA = Δf = 10,000 Å
λ = 0.02 Å
a < 0.01
A more precise formulation of the CTF can be found in
Erickson & Klug A (1970). Philosophical Transactions of the Royal Society B. 261:105.
Proper form the CTF
where:
Cs: spherical aberration
k: spatial frequency (resolution)
How does the CTF affect an image?
original
original
combined
Still a zero present
Outline
Correction/clarification form Feb. 16
Intro to image analysis
Relationship between imaging and diffraction
Fourier transforms
Theory
Examples in 1D
Examples in 2D
Digitization
Fourier filtration
Contrast transfer function
Resolution
How do we evaluate the quality of a
reconstruction?
Now, how do we compare the two half-set
reconstructions?
images
“odd” reconstruction
“even” reconstruction
We split the data set into halves and compare them.
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;
A better example
It is controversial what single number to use to describe this curve,
but a common practice is to report the value where the FSC=0.5
as the nominal resolution.