Introduction to Diffraction Tomography Anthony J Devaney Department

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Introduction to Diffraction Tomography Anthony J. Devaney Department of Electrical and Computer Engineering Northeastern

Introduction to Diffraction Tomography Anthony J. Devaney Department of Electrical and Computer Engineering Northeastern University Boston, MA 02115 email: tonydev 2@aol. com • Rytov Approximation • Accuracy compared with Born • Propagation and Backpropagation • Inversion Algorithms • Filtered Backpropagation • Pseudo-inverse for finite view data • Iterative Algorithms • Examples 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 1

Historical survey X-ray crystallography Fourier based Born/Rytov inversion Computed tomography Conventional diffraction tomography Statistical

Historical survey X-ray crystallography Fourier based Born/Rytov inversion Computed tomography Conventional diffraction tomography Statistical based methods Diffraction Tomography 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 2

Complex Phase Representation Ricatti Equation 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 3

Complex Phase Representation Ricatti Equation 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 3

Rytov Approximation Perturbation introduced by the object profile Rytov approximation Rytov Model 10/31/2020 A.

Rytov Approximation Perturbation introduced by the object profile Rytov approximation Rytov Model 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 4

Short Wavelength Limit Classical Tomographic Model 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 5

Short Wavelength Limit Classical Tomographic Model 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 5

Free Space Propagation of Rytov Phase Within Rytov approximation phase of field satisfies linear

Free Space Propagation of Rytov Phase Within Rytov approximation phase of field satisfies linear PDE Rytov transformation 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 6

Solution to Rytov Model Rytov transformation Connection with Born approximation Mathematical structure of models

Solution to Rytov Model Rytov transformation Connection with Born approximation Mathematical structure of models identical 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 7

Degradation of the Rytov Model with Propagation Distance Rytov and Born approximations become identical

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 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 8

Experimental Tests Sensor system Incident wave Hybrid approximation: • Exact from measurement plane to

Experimental Tests Sensor system Incident wave Hybrid approximation: • Exact from measurement plane to near field • Rytov from near field to object Rytov Simulation and experiment: • optical fiber illuminated by red laser • ray trace followed by free space propagation • Rytov • Hybrid • Experiment 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II Measurement plane Angular spectrum 9

Generalized Tomographic Model Diffraction Tomography For the remainder of this lecture we will work

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) 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 10

Classical Geometry y Rotating coordinate system Fixed coordinate system x 10/31/2020 A. J. Devaney

Classical Geometry y Rotating coordinate system Fixed coordinate system x 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 11

Weyl Expansion for Classical Geometry in R 2 Homogeneous Waves Evanescent Waves Dirichlet Green

Weyl Expansion for Classical Geometry in R 2 Homogeneous Waves Evanescent Waves Dirichlet Green Function 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 12

Propagation of Rytov Phase in Free Space Angular Spectrum Representation of free space propagation

Propagation of Rytov Phase in Free Space Angular Spectrum Representation of free space propagation of Rytov phase 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 13

Propagation in Fourier Space -Backpropagation-- Free space propagation ( > 0) corresponds to low

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 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 14

Propagation Operator in Classical Geometry y x 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II

Propagation Operator in Classical Geometry y x 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 15

Spectral Representation of Propagation Operation Weyl Expansion in 2 D 10/31/2020 A. J. Devaney

Spectral Representation of Propagation Operation Weyl Expansion in 2 D 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 16

Generalized Projection-Slice Theorem Ky y Kx x Ewald sphere 10/31/2020 A. J. Devaney Stanford

Generalized Projection-Slice Theorem Ky y Kx x Ewald sphere 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 17

Short Wavelength Limit Projection-Slice Theorem Diffraction tomography Conventional tomography as 0 10/31/2020 A. J.

Short Wavelength Limit Projection-Slice Theorem Diffraction tomography Conventional tomography as 0 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 18

Backpropagation Operator S 0 S 1 Incoming Wave + Condition in l. h. s.

Backpropagation Operator S 0 S 1 Incoming Wave + Condition in l. h. s. + Dirichlet or Neumann on bounding surface S 1 Backpropagated Phase Backpropagation Operator 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 19

Approximate Equivalence of Two Forms of Backpropagation Form based on using conjugate Green function

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 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 20

Adjoint of Propagation Operator 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 21

Adjoint of Propagation Operator 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 21

Relationship Between Adjoint and Backpropagation Operators Spectral Representations 10/31/2020 A. J. Devaney Stanford Lectures-Lecture

Relationship Between Adjoint and Backpropagation Operators Spectral Representations 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 22

Reconstruction from Complete Data Angles defined relative to the fixed (x, y) system Redefine

Reconstruction from Complete Data Angles defined relative to the fixed (x, y) system Redefine to be relative to ( , ) coordinate system 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 23

Filtered Backpropagation Algorithm Convolutional filtering followed by backpropagation and sum over views 10/31/2020 A.

Filtered Backpropagation Algorithm Convolutional filtering followed by backpropagation and sum over views 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 24

FPB Algorithm Filtering: Backpropagation Sum over the filtered and backpropagated partial images 10/31/2020 A.

FPB Algorithm Filtering: Backpropagation Sum over the filtered and backpropagated partial images 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 25

Filtered backpropagation algorithm Scattered Field Scattering object Filtering Backpropagation Filtered Scattered Field Sum over

Filtered backpropagation algorithm Scattered Field Scattering object Filtering Backpropagation Filtered Scattered Field Sum over view angles 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 26

Simulations 2 D objects: objects composed of superposition of cylinders • Single view as

Simulations 2 D 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 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 27

Limited View Problem Generate a reconstruction given data for limited number of view angles

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 L 2 norm 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 28

Pseudo-Inverse Re-define the generalized projection operator Masking Operator Insures that the adjoint maps ;

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 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 29

Interpretation of the Pseudo-Inverse Solve integral equation in R 3 Filtered Backpropagation Algorithm 10/31/2020

Interpretation of the Pseudo-Inverse Solve integral equation in R 3 Filtered Backpropagation Algorithm 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 30

Computing the Pseudo-Inverse via the FBP Algorithm 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II

Computing the Pseudo-Inverse via the FBP Algorithm 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 31

SIRT Algorithm Other algorithms include ART and various variants 10/31/2020 A. J. Devaney Stanford

SIRT Algorithm Other algorithms include ART and various variants 10/31/2020 A. J. Devaney Stanford Lectures-Lecture II 32

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