1. Introduction to Diffraction Tomography Anthony J. Devaney Department of Electrical and Computer Engineering Northeastern University Boston, MA 02115 email: tonydev2@aol.com • Rytov Approximation • Accuracy compared with Born • Propagation and Backpropagation • Inversion Algorithms • Filtered Backpropagation • Pseudo-inverse for finite view data • Iterative Algorithms • Examples A.J. Devaney Stanford Lectures--Lecture II

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

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

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

5. Short Wavelength Limit Classical Tomographic Model A.J. Devaney Stanford Lectures--Lecture II

6. Free Space Propagation of Rytov Phase propagation Within Rytov approximation phase of field satisfies linear PDE Rytov transformation A.J. Devaney Stanford Lectures--Lecture II

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

9. 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 Angular spectrum A.J. Devaney Stanford Lectures--Lecture II

10. Generalized Tomographic ModelDiffraction 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) A.J. Devaney Stanford Lectures--Lecture II

11. Classical Geometry y Rotating coordinate system Fixed coordinate system x A.J. Devaney Stanford Lectures--Lecture II

12. Weyl Expansion for Classical Geometry in R2 Homogeneous Waves Evanescent Waves Dirichlet Green Function A.J. Devaney Stanford Lectures--Lecture II

13. propagation Propagation of Rytov Phase in Free Space Angular Spectrum Representation of free space propagation of Rytov phase 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 A.J. Devaney Stanford Lectures--Lecture II

15. Propagation Operator in Classical Geometry y x A.J. Devaney Stanford Lectures--Lecture II

16. Spectral Representation of Propagation Operation Weyl Expansion in 2D A.J. Devaney Stanford Lectures--Lecture II

17. Generalized Projection-Slice Theorem Ky y Kx x Ewald sphere A.J. Devaney Stanford Lectures--Lecture II

18. Short Wavelength Limit Projection-Slice Theorem Diffraction tomographyConventional tomography as 0 A.J. Devaney Stanford Lectures--Lecture II

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

21. Adjoint of Propagation Operator Adjoint of Propagation Operator A.J. Devaney Stanford Lectures--Lecture II

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

24. Filtered Backpropagation Algorithm Convolutional filtering followed by backpropagation and sum over views A.J. Devaney Stanford Lectures--Lecture II

25. FPB Algorithm Filtering: Backpropagation Sum over the filtered and backpropagated partial images A.J. Devaney Stanford Lectures--Lecture II

26. Filtered backpropagation algorithm Scattered Field Filtering Filtered Scattered Field Backpropagation Scattering object Sum over view angles A.J. Devaney Stanford Lectures--Lecture II

27. 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 A.J. Devaney Stanford Lectures--Lecture II

28. 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 A.J. Devaney Stanford Lectures--Lecture II

29. Insures that the adjoint maps ; i.e., Pseudo-Inverse Re-define the generalized projection operator Masking Operator Form Normal Equations: Solve using the pseudo-inverse A.J. Devaney Stanford Lectures--Lecture II

30. Interpretation of the Pseudo-Inverse Solve integral equation in R3 Filtered Backpropagation Algorithm A.J. Devaney Stanford Lectures--Lecture II

31. Computing the Pseudo-Inverse via the FBP Algorithm A.J. Devaney Stanford Lectures--Lecture II

32. SIRT Algorithm Other algorithms include ART and various variants A.J. Devaney Stanford Lectures--Lecture II