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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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Presentation Transcript
outline
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
what is an adjoint
What is an adjoint?

Inner Product:

Operator:

why is the adjoint important
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,

misfit measurements
Misfit Measurements

Input image, our guess for true image

slide9

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

fr chet derivative
Fréchet Derivative

At fixed reference point x0

At fixed reference image I0(x)

second order fr chet derivative
Second-order Fréchet Derivative

Hessian with respect to model

fr chet derivative1
Fréchet Derivative

Simple Derivative:

Fréchet Derivative:

Gradient with respect to model

adjoint fr chet derivative
Adjoint & Fréchet Derivative

Adjoint of the Fréchet derivative of the forward modeling operator

Fréchet derivative of χw.r.t. I

adjoint of radon transform back projection
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.

preconditioner
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

projection slice theorem
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

inverting radon transform by projection slice theorem
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
filtered back projection
Filtered Back-Projection

Projection-Slice

filtering after back projection
Filtering After Back-Projection

Hessian

Blurring function

Gradient

De-blurring

summary
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.