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Lesson Two

Lesson Two. The Big Picture. Outline. A tutorial about adjoint Tomographic reconstruction formulated as an optimization problem Fréchet derivative Second-order Fréchet derivative Iterative reconstruction based on Netwon’s algorithm steepest descent algorithm

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Lesson Two

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  1. Lesson Two

  2. The Big Picture

  3. Outline • A tutorial about adjoint • Tomographic reconstruction formulated as an optimization problem • Fréchet derivative • Second-order Fréchet derivative • Iterative reconstruction based on Netwon’s algorithm steepest descent algorithm • Case study: reconstruction for problems whose forward modeling operators are Radon transforms • Projection-Slice Theorem • Adjoint and Inverse of the Radon transform • Filtered back-projection

  4. What is an adjoint? Inner Product: Operator:

  5. Adjoint:

  6. Why is the adjoint important? Image (model) Observation parameter, (xr, xs, t, ϑ, ρ, τ, p, etc…) Misfit measurements Misfit functional Size of the misfit,

  7. Misfit Measurements Input image, our guess for true image

  8. Steepest Descent and Newton’s Method For quadratic objective functions, we need only one Newton iteration, but more than one steepest descent iterations. minimum ε=0.01 minimum Starting value Starting value

  9. Fréchet Derivative At fixed reference point x0 At fixed reference image I0(x)

  10. Second-order Fréchet Derivative Hessian with respect to model

  11. Fréchet Derivative Simple Derivative: Fréchet Derivative: Gradient with respect to model

  12. Iterative Tomographic Reconstruction

  13. Adjoint

  14. Adjoint & Fréchet Derivative Adjoint of the Fréchet derivative of the forward modeling operator Fréchet derivative of χw.r.t. I

  15. Adjoint of Generalized Radon Transform

  16. Adjoint of Radon Transform & Back Projection Two different interpretations: Which observations contribute to a given imaging point (x, y)? Sum up all Radon transforms (observations) along all angles passing through the same imaging point (x, y). Which imaging points are affected by a given observation? Smear the Radon transform back along the line from which the projection is made.

  17. Preconditioner Radon windowing Filtering after Back-projection Convolution between Radon transforms equals Radon transform of convolution Filtered Back-projection (back-projection of the filtered Radon projection) C approximates the inverse of the Hessian

  18. Projection-Slice Theorem The 1D Fourier transform of the Radon projection function is equal to the 2D Fourier transform of the image evaluated on the line that the projection was taken on. ρ ky kx

  19. Inverting Radon Transform by Projection-Slice Theorem • (Step-1) Filling 2-D FT with 1-D FT of Radon along different angles • (Step-2) Polar-to-Cartesian grid conversion for discrete implementation • (Step-3) 2-D IFT

  20. Filtered Back-Projection Projection-Slice

  21. Filtered Back-Projection 4 8 1 15 60

  22. Filtering After Back-Projection Hessian Blurring function Gradient De-blurring

  23. Summary • For Radon transform, Newton optimization is equivalent to filtered back-projection. • The adjoint operator is equivalent to the back-projection operator. • The adjoint operator gives a general method for constructing Fréchet derivatives. • A Radon projection is a slice of the spectrum of the imaging object. • Adjoint is equivalent to transpose, time-reversal is a consequence of causality.

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