Automatic Alignment of

1
Automatic Alignment of An Application to
Fernando Amat, Farshid Moussavi, Mark Horowitz,
PhD.- Department , Luis Comolli, PhD. - Life
Sciences Division, Lawrence , 4th International
Congress on Electron Poster Session B-Image

• PRECISE AUTO ALIGNMENT
• 3D reconstructions of intact cells in their
  native structure are made using Cryogenic
  Electron Tomography (Cryo-ET),
• gtNeeds precise alignment of tilt series
• Existing techniques have difficulty tracking in
  noisy images of thick samples (which are also not
  flat)
• gt extensive manual intervention becomes
  required (1-2 days hand alignment)

THE ALIGNMENT PROBLEM -Most precise alignment is
obtained by using fiducial markers. -Quality
depends on accurate correspondence and tracking
of markers throughout the tilt series
CORRESPONDENCE WITH GLOBAL INFORMATION

• Solve correspondence for pairs of images.
• Combine results for complete correspondence
• Use redundancy in choosing pairs of images.

For One Pair of Images
PROBLEM Correspondence errors propagate and
alignment may fail to reach desired accuracy
Which marker in right image(s) corresponds to
one from left image?
? ?
Projection Model Estimation
?
Aligned Images
Raw Images
Correspondence
??????
??????
Acquire Tilt Series Images
??????
(2)
Which group of markers in right image(s)
corresponds to group from left image?

• SOLUTION
• Reduce Correspondence errors by using Global
  Information.
• Make Projection Model Estimation more robust to
  Correspondence errors.

(3)
COMPLEXITY is EXPONENTIAL in SIZE OF GROUP!
Correspond/ Align Images
EXPLOIT LOCAL DEPENDENCIES TO REDUCE
COMPLEXITY In general, all correspondence
assignments depend on each other. But the
dependence is strongest in a smaller
neighborhood. Relative locations of nearby pairs
of markers provide the most valuable information
for correspondence.
Projection Model Estimation
3D Reconstruct

• Correspondence May Still Make Errors
• Projection Model Estimation must tolerate
  outliers
• Use of robust convex optimization Huber penalty
  and regularization
• For small errors quadratic penalty (as least
  squares)
• For big errors linear penalty
• Large deviations (outliers) dont dominate the
  trajectories
• Easy to detect and correct outliers automatically
• Minimization problem to find parameters of the
  microscope for each image (G, tx ,ty) and 3D
  location for each fiducial (X,Y,Z)

This allows us to consider pairs of nearby
markers instead of the entire group.
(Mi,Mj near each other)
(1)
which is O(m2) (msize of group)
Definition A Markov Random Field (MRF) is a set
of random variables, in which the variables
depend on each other through their neighbors. The
group of markers MM1,M2,.Mm forms an MRF

• NEW APPROACH
• GLOBAL CORRESPONDENCE based on Markov Random
  Fields (MRFs), combined with ROBUST ESTIMATION
  of Projection Model
• We achieve fully automatic reconstruction from
  extremely difficult datasets in less than 3 hours
  on one machine using this framework.

(4)
Well known approximate methods to estimate
factors (Fs) for MRF (we use Loopy Belief
Propagation, or LBP)