Lesson Two

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# Lesson Two - PowerPoint PPT Presentation

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

Outline
• 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
• Filtered back-projection

Inner Product:

Operator:

Image (model)

Observation parameter, (xr, xs, t, ϑ, ρ, τ, p, etc…)

Misfit measurements

Misfit functional

Size of the misfit,

Misfit Measurements

Input image, our guess for true image

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

Fréchet Derivative

At fixed reference point x0

At fixed reference image I0(x)

Second-order Fréchet Derivative

Hessian with respect to model

Fréchet Derivative

Simple Derivative:

Fréchet Derivative:

Adjoint of the Fréchet derivative of the forward modeling operator

Fréchet derivative of χw.r.t. I

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

windowing

Filtering after Back-projection

Filtered Back-projection (back-projection of the filtered Radon projection)

C approximates the inverse of the Hessian

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
• (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

Projection-Slice

Filtering After Back-Projection

Hessian

Blurring function