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 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Historical survey Diffraction Tomography X-ray crystallography Fourier based Born/Rytov inversion Computed tomography Conventional diffraction tomography Statistical based methods Diffraction Tomography 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Complex Phase Representation Ricatti Equation 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Rytov Approximation Rytov Model Rytov approximation Perturbation introduced by the object profile Rytov approximation Rytov Model 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Short Wavelength Limit Classical Tomographic Model 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Free Space Propagation of Rytov Phase Within Rytov approximation phase of field satisfies linear PDE Rytov transformation 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Solution to Rytov Model Rytov transformation Connection with Born approximation Mathematical structure of models identical 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

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 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

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 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II Angular spectrum

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) 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

A.J. Devaney Stanford Lectures--Lecture II Classical Geometry y Rotating coordinate system Fixed coordinate system x 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Weyl Expansion for Classical Geometry in R2 Homogeneous Waves Evanescent Waves Dirichlet Green Function 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Propagation of Rytov Phase in Free Space Angular Spectrum Representation of free space propagation of Rytov phase propagation 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

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 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Propagation Operator in Classical Geometry x 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Spectral Representation of Propagation Operation Weyl Expansion in 2D 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Generalized Projection-Slice Theorem Ky y Kx x Ewald sphere 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Short Wavelength Limit Projection-Slice Theorem Diffraction tomography Conventional tomography as 0 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Backpropagation Operator S1 S0 Incoming Wave Condition in l.h.s. Dirichlet or Neumann on bounding surface S1 + Backpropagated Phase Backpropagation Operator 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

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 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Adjoint of Propagation Operator 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Relationship Between Adjoint and Backpropagation Operators Spectral Representations 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Reconstruction from Complete Data Angles defined relative to the fixed (x,y) system Redefine  to be relative to (,) coordinate system 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Filtered Backpropagation Algorithm Convolutional filtering followed by backpropagation and sum over views 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

A.J. Devaney Stanford Lectures--Lecture II FPB Algorithm Filtering: Backpropagation Sum over the filtered and backpropagated partial images 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Filtered backpropagation algorithm Scattered Field Filtering Filtered Scattered Field Backpropagation Scattering object Sum over view angles 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

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 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

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 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

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 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Interpretation of the Pseudo-Inverse Solve integral equation in R3 Filtered Backpropagation Algorithm 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

Computing the Pseudo-Inverse via the FBP Algorithm 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II

A.J. Devaney Stanford Lectures--Lecture II SIRT Algorithm Other algorithms include ART and various variants 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II