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
Observation parameter, (xr, xs, t, ϑ, ρ, τ, p, etc…)
Size of the misfit,
Input image, our guess for true image
For quadratic objective functions, we need only one Newton iteration, but more than one steepest descent iterations.
At fixed reference point x0
At fixed reference image I0(x)
Hessian with respect to model
Gradient with respect to model
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.
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
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.