AIR Tools II: algebraic iterative reconstruction methods, improved implementation

Abstract

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The block-wise circumcentered–reflection method

“…Finally, we can gather Cases 1 and 2. From (20), (21) and (24), we have G(z) −ẑ ≤ r V ∩W z −ẑ for all z ∈ R n , with r V ∩W ∈ [0, 1) given by r V ∩W max {r 1 , r W , r 2 } = max {r 1 , r 2 } .…”

“…This is a standard synthetic image that serves as the model of a human head and is used for testing image reconstruction algorithms. The data for the matrix A and the vector b were generated using AIR Tools II, a package by Hansen and Jørgensen [20], and imported to be used in the Julia implementation. In this case, A has 5732 rows and 2500 columns.…”

Perturbed fixed point iterative methods based on pattern structure

“…In this section, we give an example from the field of tomography (image reconstruction from projections) and show that the AS Algorithm is able to have more improvement in relative error compare to string‐averaging method. We used the (matlab‐based) AIR tools software, see also recently published paper . The AIR tools package produces the phantom, the matrix and the right‐hand side.…”

A note on the minimization of a Tikhonov functional with ℓ1-penalty

“…In order to obtain a discretized version of problem (6.1), we make use of the toolbox AIR TOOLS II by Hansen and Jorgensen [18]. Therein, the density function f is considered as a piecewise constant function on an m × m pixel grid (see Figure 6.3 for examples).…”