Iterative Methods for Phase Retrieval and Xray Crystallography

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Iterative Methods for Phase Retrieval and X-ray
Crystallography
Kerkil Choi

• Advisor Prof. Aaron D. Lanterman
• Date November 9, 2005
• Center for Signal and Image Processing
• School of Electrical and Computer Engineering
• Georgia Institute of Technology

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Outline

• Phase retrieval
• X-ray crystallography
• Iterative solution methods
• Conclusions

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Where Are We?

• Phase retrieval
• X-ray crystallography
• Iterative solution methods
• Conclusions

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Phase Retrieval
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Phase Dominance
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Is Solution Unique?
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Constraints
Known support
Nonnegativity
Feasible value range
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Where Are We?

• Phase retrieval
• X-ray crystallography
• Iterative solution methods
• Conclusions

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X-ray Crystallography
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Undersampling
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Unaliased vs. Aliased Autocorrelations
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Is Solution Unique?
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Constraints in Crystallography
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Where Are We?

• Phase retrieval
• X-ray crystallography
• Iterative solution methods
• Conclusions

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Do Magical Phase Retrieval Algorithms Exist?
No constraints
Constraints
Phase retrieval algorithm
Phase retrieval algorithm
??
?!
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The Most Popular Phase Retrieval Algorithm
Fienups Algorithm
Nonnegativity
Are estimates always correct?
Fienups phase retrieval algorithm
Measured Fourier magnitudes
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Minimum I-divergence Methods
Autocorrelation operation
Iterative algorithm

Goal
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I-divergence

• I-divergence?

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Why I-divergence?

• Why not the squared-error measure?
• - Csiszár showed the following theoretical
  results

All functions involved are nonnegative.
Minimizing Csiszárs I-divergence
A set of postulates intuitively desirable for
deterministic estimation problems
Functions involved are real valued.
Minimizing the squared-error measure
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Minimization of I-divergence
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Penalized Minimum I-divergence Algorithms
Maximizing Penalized Poisson likelihood
Data size 8
Minimizing Penalized I-divergence
Asymptotically equivalent
Penalized EM algorithms for Poisson data models
Asymptotic versions
Greens one-step-late
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Schulz-Snyder Phase Retrieval Algorithm
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Convergence of the Schulz-Snyder Algorithm to
Local Minima
Local minimum
True image
Schulz-Snyder phase retrieval algorithm
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Support constraint (1)
Local minimum
True image
Schulz-Snyder phase retrieval algorithm
Loosely known support
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Support constraint (2)
Exactly known support
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Penalized Minimum I-divergence Algorithm for
Avoiding Local Minima
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Penalty Methods vs. Local Minima

Are penalty methods always helpful?
Of course not!!!
When are they helpful???
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Future Work

Global optimization
Avoiding local minima problem
Penalty methods
Support estimation from autocorrelation
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Minimum I-divergence Methods for X-ray
Crystallography
Estimates?
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I-divergence vs. R-factor
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Future Work

Global optimization
Avoiding local minima problem
Penalty methods
Incorporating various constraints
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Where Are We?

• Phase retrieval
• X-ray crystallography
• Iterative solution methods
• Conclusions

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Conclusions

• Minimum I-divergence methods possess good
  potential for phase retrieval and x-ray
  crystallography.
• For phase retrieval, our current minimum
  I-divergence algorithms suffer from becoming
  trapped in local minima.
• Avoiding this problem is an important avenue for
  future work.

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QA
Thank you!!!