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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 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II
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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
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Complex Phase Representation
Ricatti Equation 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II
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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
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Short Wavelength Limit
Classical Tomographic Model 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II
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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
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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
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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
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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
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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
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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
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Weyl Expansion for Classical Geometry in R2
Homogeneous Waves Evanescent Waves Dirichlet Green Function 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II
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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
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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
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Propagation Operator in Classical Geometry
x 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II
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Spectral Representation of Propagation Operation
Weyl Expansion in 2D 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II
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Generalized Projection-Slice Theorem
Ky y Kx x Ewald sphere 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II
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Short Wavelength Limit
Projection-Slice Theorem Diffraction tomography Conventional tomography as 0 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II
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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
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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
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Adjoint of Propagation Operator
11/6/2018 A.J. Devaney Stanford Lectures--Lecture II
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Relationship Between Adjoint and Backpropagation Operators
Spectral Representations 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II
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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
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Filtered Backpropagation Algorithm
Convolutional filtering followed by backpropagation and sum over views 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II
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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
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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
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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
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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
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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
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Interpretation of the Pseudo-Inverse
Solve integral equation in R3 Filtered Backpropagation Algorithm 11/6/2018 A.J. Devaney Stanford Lectures--Lecture II
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Computing the Pseudo-Inverse via the FBP Algorithm
11/6/2018 A.J. Devaney Stanford Lectures--Lecture II
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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
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