This page contains the computational Matlab files related to the book

*Linear and Nonlinear Inverse Problems wit=
h Practical Applications*

written by Jennifer Mueller and Samul=
i Siltanen and published by SIAM in 2012.

You can order the book at the SIAM webshop.

=20 =20Here we construct the measurement matrix A related to tomographic imagin= g and solve the inverse problem using several regularized methods. These ro= utines make heavy use of radon.m and phantom.m files that are available onl= y in Matlab's Image processing toolbox, not in the basic version of Matlab.=

=20=20

This Matlab routine builds the tomographic measurement matrix for NxN im= ages and N uniformly distributed parallel-beam projections:

=20 =20You can choose your favorite value of N inside the above m-file. Note th= at taking N greater than 64 will choke most personal computers either here = or later when computing the singular value decomposition of A! The examples= on this page are meant to be studied at low dimensions, such as N between = 16 and 50, say.

=20The matrix is saved to your working directory as a .mat file. The number= N is part of the filename, as you can see in these example files:

=20RadonMatrix16.mat, RadonMatrix32.mat

=20=20

You can try out naive inversion involving inverse crime with the followi= ng routines:

=20XRMB_naive_comp.m, XRMB_naive_plot.m

=20=20

These routines interpolate the data from a higher-dimensional model.

= =20XRMC_NoCrimeData_comp.m, XRMC_NoCri= meData_plot.m

=20You can go ahead and see how naive inversion fails again:

=20XRMD_naive_comp.m, XRMD_naive_plot.m

=20=20

Compute the singular value decomposition:

=20XRME_SVD_comp.m, XRME_SVD_plot.m

=20Now we can use the truncated SVD to achieve robust reconstructions:

= =20 =20=20

Compute Tikhonov regularized reconstructions with these routines:

=20XRMG_Tikhonov_comp.m, XRMG_Tikhonov_plot.m=

=20You can experiment by varying the regularization parameter.

=20=20

Here we reconstruct the attenuation coefficient using total variation re= gularization. We smooth the non-differentiable corner of the absolute value= function appearing in the total variation norm, resulting in a smooth obje= ctive functional to be minimized. The optimization method we use is the Bar= zilai-Borwein method.

=20XRMH_aTV_comp.m, XRMH_aTV_plot.m.

=20You will need all of the files below to run the above reconstruction rou= tine:

=20XRMH_aTV_feval.m, XRMH_aTV_fgrad.m, XRMH_aTV_grad.m, XRMH_aTV.m, XRMH_misfit_grad.m, XRMH_misfit.m