Electron Tomography

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Electron Tomography

• Introduction
• Presentation of the Problem
• Distance and Alignment
• Classification Problem
• Experiments
• Future work

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Electron Tomography

• Why?
• High resolution
• Allows to see details of organelle structure at
  level of macromolecular dimensions
• Cryo Electron Tomography
• Rapid freezing
• Specimen embedded in water
• Representation of the living state
• Low tolerable electron dose
• Low signal to noise ratio

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Electron Tomography

• 3D structure from 2D projections
• Acquisition of tilt series
• Alignment of tilt series
• Reconstruction (Backprojection)
• Visualization and Analysis

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Line Integrals and Projections
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Line Integrals and Projections
A projection is formed by combining a set of line
integrals.
A simple diagram showing the fan beam projection
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Electron Tomography

• Wedge effect
• Object gets tilted rather than the illumination
  and detector
• Incomplete tilt series, -60 to 60 degrees
• The thickness doubles when tilted 60 degrees

0
60
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The Fourier Slice Theorem - 2D
In 2D The Fourier Slice theorem relates the
Fourier transform of the object along a radial
line.
t
Fourier transform
?
Space Domain
Frequency Domain
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Electron Tomography

• Wedge effect
• Fourier interpretation
• Fourier Slice Theorem

• Radon Fourier relationship
• A slice extracted from the frequency domain
  representation of a volume is equal to the
  Fourier transform of a projection of the volume
  in a direction perpendicular to the slice

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Electron Tomography

• Tomogram
• 3D block of data is represented as a volume
• Voxels 1-4nm per side
• Grayscale value corresponds to the mass density
  of the specimen in that region
• Sub volumes
• Sub parts of the tomogram
• Different wedge effect, due to the relative
  position in the tomogram

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Problem

• Statement
• Given a big set of sub volumes our goal is
• to determine how many different classes of sub
  volumes exist
• to get a model for each one.
• Considerations
• No prior information used in the classification
• The number of classes is unknown
• Some of the volumes may not belong to any of the
  real classes

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Distance and 3D Alignment

• Definition of a distance
• Implies the alignment of the sub volumes
• Take in account wedge
• Band pass filter due to the very low SNR
• Spherical geometry

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Distance and 3D Alignment
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Distance and 3D Alignment

• Spherical harmonics
• function defined on the sphere
• As in Fourier theory can be expanded in
  a basis of spherical harmonics
• The l-th frequency has dimension
• The basis is

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Distance and 3D Alignment

• We have to compute the value of H for every R
• Can be defined a Fourier Transform on SO(3)
• Can be easily compute given

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Distance and 3D Alignment

• Dealing with the wedge
• Window approach
• The wedge is modeled by binary window or mask

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FFT
L candidatos
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FFT
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Distance and 3D Alignment

• Problem

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Classification

• Goal
• Determine a good sub set off sub volumes of
  each class
• Algorithm
• Based in experience of Single Particle
• Iterative procedure, K-means like
• Mean images have more information
• Higher SNR
• Smaller wedge
• Reference-free initialization

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Classification
Initialization Step Reference-free classification
Classification Compute the distance form each
image to all the references. Volumes are
associated with the class of the closest
reference.
Compare References Possibly more than one
reference represents the same real class
of Volumes.
Re-compute references The new average is
computed with a good subset of the volumes
associated with each class.
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Classification

• Initialization step
• Hierarchical clustering
• Algorithm
• Each is considered a group
• Merge the two closest classes
• Iterate till only one class is remaining
• There are different criteria for merging
• Single link
• Complete link
• Wedge overlap Threshold
• Single link
• Complete link
• Others

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Classification

• Result
• Tree with the hierarchical classification
• Where to cut the tree?
• Several criteria
• Distance between groups
• Overlap
• Number of sub volumes
• Inter class variance

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Experiments

• Artificial data
• Data with noise

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Experiments

• Results of the initialization step

D 0,15
D 0,2
D 0,25
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Experiments

• Results after loop

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Future work

• Hierarchical and Spectral clustering
• Polarization Theorem
• Affinity taking in account wedge overlap
• etc
• Tests with all the framework
• Real Data!