Date: Mon, 8 Aug 2022 10:11:57 +0300 (EEST) Message-ID: <1013656938.17326.1659942717387@wiki-1.it.helsinki.fi> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_Part_17325_954576766.1659942717387" ------=_Part_17325_954576766.1659942717387 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html X-ray tomography with matrices

# X-ray tomography with matrices

## X= -ray tomography with matrices

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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.

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You can order the book at the SIAM webshop.

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Here 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.=

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### Con= structing the measurement matrix A

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

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XRMA_matrix_comp.m

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You 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.

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The 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:

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### = Creating data that contains inverse crime

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

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### Creat= ing data without inverse crime

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

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

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### Regularized inversion using truncated SVD

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Compute the singular value decomposition:

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

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XRMF_truncSVD_comp.m

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### Tikhonov regul= arization

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Compute Tikhonov regularized reconstructions with these routines:

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You can experiment by varying the regularization parameter.

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### Approximate total variation regularization

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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.

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

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