1. Introduction to Diffraction Tomography
Anthony J. Devaney
Department of Electrical and Computer Engineering
Northeastern University
Boston, MA 02115
Rytov Approximation
Accuracy compared with Born
Propagation and Backpropagation
Inversion Algorithms
Filtered Backpropagation
Pseudo-inverse for finite view data
Iterative Algorithms
Examples
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A.J. Devaney Stanford Lectures--Lecture II

2. Historical survey Diffraction Tomography X-ray crystallography
Fourier based
Born/Rytov inversion
Computed
tomography
Conventional
diffraction tomography
Statistical based
methods
Diffraction Tomography
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A.J. Devaney Stanford Lectures--Lecture II

3. Complex Phase Representation
Ricatti Equation
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A.J. Devaney Stanford Lectures--Lecture II

4. Rytov Approximation Rytov Model Rytov approximation
Perturbation introduced
by the object profile
Rytov approximation
Rytov Model
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A.J. Devaney Stanford Lectures--Lecture II

5. Short Wavelength Limit
Classical Tomographic Model
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6. Free Space Propagation of Rytov Phase
Within Rytov approximation phase of field satisfies linear PDE
Rytov transformation
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A.J. Devaney Stanford Lectures--Lecture II

7. Solution to Rytov Model
Rytov transformation
Connection with Born approximation
Mathematical structure of models identical
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A.J. Devaney Stanford Lectures--Lecture II

8. Degradation of the Rytov Model with Propagation Distance
Rytov and Born approximations become identical in far field (David Colton)
Experiments and computer simulations have shown Rytov to be much
superior to Born for large objects--Backpropagate field then use Rytov--Hybrid Model
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A.J. Devaney Stanford Lectures--Lecture II

9. A.J. Devaney Stanford Lectures--Lecture II
Experimental Tests
Sensor system
Hybrid approximation:
Exact from measurement plane to near field
Rytov from near field to object
Incident wave
Rytov
Simulation and experiment:
optical fiber illuminated by red laser
ray trace followed by free space propagation
Rytov
Hybrid
Experiment
Measurement plane
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A.J. Devaney Stanford Lectures--Lecture II
Angular spectrum

10. Generalized Tomographic Model Diffraction Tomography
For the remainder of this lecture we will work in two space dimensions
Generalized Projection (Propagation)
Diffraction tomography is generalization of conventional tomography
to incorporate wave (diffraction effects)
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A.J. Devaney Stanford Lectures--Lecture II

11. A.J. Devaney Stanford Lectures--Lecture II
Classical Geometry y
Rotating coordinate system
Fixed coordinate system x
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12. Weyl Expansion for Classical Geometry in R2
Homogeneous Waves
Evanescent Waves
Dirichlet Green Function
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13. Propagation of Rytov Phase in Free Space
Angular Spectrum Representation of free space propagation of Rytov phase
propagation
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A.J. Devaney Stanford Lectures--Lecture II

14. Propagation in Fourier Space --Backpropagation--
Free space propagation ( > 0) corresponds to low pass filtering of the field data
Backpropagation ( < 0) requires high pass filtering and is unstable (not well posed)
Propagation and Backpropagation of bandlimited phase perturbations
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A.J. Devaney Stanford Lectures--Lecture II

15. Propagation Operator in Classical Geometryx
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16. Spectral Representation of Propagation Operation
Weyl Expansion in 2D
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17. Generalized Projection-Slice TheoremKy y Kx x
Ewald sphere
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18. Short Wavelength Limit
Projection-Slice Theorem
Diffraction tomography Conventional tomography as 0
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A.J. Devaney Stanford Lectures--Lecture II

19. Backpropagation OperatorS1 S0
Incoming Wave
Condition in l.h.s.
Dirichlet or Neumann
on bounding surface S1 +
Backpropagated Phase
Backpropagation
Operator
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A.J. Devaney Stanford Lectures--Lecture II

20. Approximate Equivalence of Two Forms of Backpropagation
Form based on using conjugate Green function
Spectral representation of conjugate Green function form
A.S.E. Form for bandlimited phase perturbations
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A.J. Devaney Stanford Lectures--Lecture II

21. Adjoint of Propagation Operator
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22. Relationship Between Adjoint and Backpropagation Operators
Spectral Representations
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A.J. Devaney Stanford Lectures--Lecture II

23. Reconstruction from Complete Data
Angles defined relative to the fixed (x,y) system
Redefine  to be relative to (,) coordinate system
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A.J. Devaney Stanford Lectures--Lecture II

24. Filtered Backpropagation Algorithm
Convolutional filtering followed by backpropagation and sum over views
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25. A.J. Devaney Stanford Lectures--Lecture II
FPB Algorithm
Filtering:
Backpropagation
Sum over the filtered and backpropagated partial images
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A.J. Devaney Stanford Lectures--Lecture II

26. Filtered backpropagation algorithm
Scattered Field
Filtering
Filtered Scattered Field
Backpropagation
Scattering object
Sum over view angles
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27. A.J. Devaney Stanford Lectures--Lecture II
Simulations
2D objects: objects composed of superposition of cylinders
Single view as function of wavelength
multiple view at fixed wavelength
Comparison of CT versus DT with DT data
multiple view as function of wavelength
Simulations test DT algorithms and not Rytov model
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28. A.J. Devaney Stanford Lectures--Lecture II
Limited View Problem
Generate a reconstruction given data for limited number of view angles
Non-unique
Ghost Objects: objects contained in the null space of the propagation transform
Pseudo-inverse: object function having minimum L2 norm
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A.J. Devaney Stanford Lectures--Lecture II

29. A.J. Devaney Stanford Lectures--Lecture II
Pseudo-Inverse
Re-define the generalized projection operator
Masking Operator
Insures that the adjoint maps ; i.e.,
Form Normal Equations:
Solve using the pseudo-inverse
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30. Interpretation of the Pseudo-Inverse
Solve integral equation in R3
Filtered Backpropagation Algorithm
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31. Computing the Pseudo-Inverse via the FBP Algorithm
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32. A.J. Devaney Stanford Lectures--Lecture II
SIRT Algorithm
Other algorithms include ART and various variants
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