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Figure: high-resolution X-ray tomographic slice through a walnut (left), reconstruction from few data using an old method (middle), and reconstruction from few data using a modern method (right).

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## Lecturers

Main lecturer: Samuli Siltanen

Guest lecturers: Paola Elefante, Andreas Hauptmann, Alexander Meaney, and Zenith Purisha

## Scope

15 sp.

## Type

Advanced studies

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Matlab resources: DC_convmtx.m, deconv1_cont_comp.m, deconv1_cont_plot.m,

deconv2_discretedata_comp.m, deconv2_discretedata_plot.m, deconv3_naive_comp.m, deconv3_naive_plot.m

**Lecture 4 (January 20, 2015)**

Computational studies of the one-dimensional deconvolution. Simulation of crime-free data. Book sections 2.1.3 and 2.1.4.

**Lecture 5 (January 21, 2015)**

. Book chapter 3.5.

Singular value decomposition of matrices and how it explains the instability of naive inversionMatlab resources: deconv4_SVD_comp.m, deconv4_SVD_plot.m, deconv5_truncSVD_comp.m, deconv5_truncSVD_plot.m

**Lecture 6 (January 23, 2015)**

Well-posedness and ill-posedness in the sense of Hadamard. Quick introduction to the concept of regularization. Minimum norm solution and the Moore-Penrose pseudoinverse. Book sections 3.1 and 3.4. and 4.1. (No computational stuff this time.)

## Exams

There will be an exam after the lecture part of the course.

**Lecture 7 (January 27, 2015)**

Introduction to X-ray measurements and their mathematical modelling. Book sections 2.3.1, 2.3.2 and 2.3.4.

Videos: Recording a projection image by scanning, Collection of tomographic data

**Lecture 8 (January 28, 2015)**

Construction of X-ray measurement matrices using Matlab's built-in function radon.m. Naive reconstruction from data with inverse crime. Also: truncated SVD reconstructions from data with inverse crime. Book sections 2.3.4, 2.3.5, 4.2 and 4.4.3.

Matlab resources: Radon_demo.m, XR01_buildA.m, XR02_naivetest.m, XR03_truncSVD.m

**Lecture 9 (January 30, 2015)**

**Paola Elefante:** *Dynamic X-ray sparse tomography*

Motivation for sparse tomography: hints on how cancer is formed and X-rays as environmental factors, possible improvements for medical care. Motivation for dynamic tomography: veterinary applications, material testing, monitoring the effects of cancer medications, and angiography (with video). The level set method, brief introduction. Presenting simulated data, comparison with FBP and Tikhonov. State of the art of local research and upcoming challenges. Slides available here.

**Zenith Purisha:** *X-ray tomography using MCMC and NURBS*

1. Motivation for using NURBS, the building blocks in CAD modelling. Why does CAD use NURBS: it involves a few parameters only, making computation efficient. Taking sparse data from a homogenous object to have quick measurement process. In medical environment, sparse data will reduce radiation dose. The idea is implementing Bayesian Inversion and NURBS to recover the parameters.

2. Introduction about NURBS (one slide, a video), Tomographic Measurement model (Included Basic Xray measurement, Radon transform, Discrete tomographic Data, and of course NURBS-based tomographic model), Bayesian Inversion and MCMC.

3. Results of reconstruction from real data

**Lecture 10 (February 3, 2015)**

Construction of tomographic data without inverse crime. The trick is to simulate the sinogram at higher resolution and then interpolate from that the sinogram corresponding to lower resolution.

Matlab resources: XR04_NoCrimeData_comp.m, XR04_NoCrimeData_plot.m

**Lecture 11 (February 4, 2015)**

Introduction to Tikhonov regularization. Finding the minimum of the Tikhonov functional using the singular value decomposition.

Book section 5.1.

Matlab resources: deconv6_Tikhonov_comp.m, deconv6_Tikhonov_plot.m

**Lecture 12 (February 6, 2015)**

Introduction to the generation of X-rays. Tomography lab visit.

**Lecture 13 (February 10, 2015)**

Introduction to normal equations and the stacked form method for computing Tikhonov regularized solutions.

Matlab resources: deconv7_genTikhonov_comp.m, deconv7_genTikhonov_plot.m

Book section 5.2.

**Lecture 14 (February 13, 2015)**

Large-scale implementation. Matrix free formulation and the conjugate gradient method. Book section 5.5

Matlab resources: XR05_Tikhonov_comp.m, XR05_Tikhonov_plot.m, XRMC_NoCrime_bigData256_50.mat

Further reading material on optimization and conjugate gradient can be found here: http://www4.ncsu.edu/~ctk/lv/lvpub.html

**Lecture 15 (February 17, 2015)**

Sparsity-promoting reconstruction for the deconvolution problem using L1-norm regularization. Book sections 6.1 and 6.2.

**Lecture 16 (February 18, 2015)**

Review of four variational regularization methods in the case of 1D deconvolution:

(1) Classical Tikhonov regularization (penalty: L2 norm of f),

(2) Generalized Tikhonov regularization with derivative penalty (penalty: L2 norm of Lf),

(3) Sparsity-promoting regularization (penalty: L1 norm of f),

(4) Total variation (TV) regularization (penalty: L1 norm of Lf).

Implementation of TV regularization for the 2D tomography case using quadratic programming. Book section 6.

Matlab resources for 1D deconvolution: all files updated, please download them from this Dropbox directory.

Matlab resources for 2D tomography: XR09_TV_comp.m (it works!)

** **

**Lecture 17 (February 20, 2015)**

Closer study of TV-based tomography using the routine XR09_TV_comp.m. We added comparison to filtered back-projection, as you can see by running XR09_TV_plot.m. You will need the additional files MyDS2.m and MyDScol.m. Discussion of large-scale TV computation based on smoothing out the absolute value function and applying the Barzilai-Borwein minimization method for the resulting smooth objective functional.

Book section 6.

For a basic introduction to optimization, see Chapter 10 of the classical book Numerical Recipes.

** **

**Lecture 18 (February 24, 2015)**

Wavelet-based regularization and Besov norm penalties. See this document.

Book chapter 7.

Matlab resources: BesovNormMatrixDiag.m, HaarTransformMatrix.m, deconv10_B111_comp.m, deconv10_B111_plot.m.

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**Lecture 19 (February 25, 2015)**** **

Mathematical derivation of the Filtered Back-Projection method. Here are details: FBP.pdf. Also, see Chapter 3 of the book by Kak and Slaney.

Book chapter 2.3.3.

**Lecture 20 (Last lecture! February 27, 2015)**

Introduction to the course exam. It is recommended to take part in this session if you plan to pass the course.

Discussion of project work topics, division to project groups. It is recommended to take part in this session if you plan to make the project work.

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## Matlab resources

You can always find the latest set of Matlab files in this Dropbox directory.

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## Exam

The home exam is here. Please return your answers at latest on March 16 by email attachment to Andreas Hauptmann. Specific instructions are given in the exam itself.

Exam files: 2015_home_exam.pdf, 2015_home_exam.tex, Hadamard2.jpg

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## Bibliography

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Points for all completed exercises will be accumulated and part of the final grade.

**NOTE!** It has been decided that we will not publish the solutions for future exercises. Please attend the exercise sessions and/or ask your fellow students.

Exercise 1 (January 20-22, 2015)

Exercise 2 (January 27-29, 2015)

Exercise 3 (February 3-5, 2015)

Exercise 4 (February 10-12, 2015)

Exercise 5 (February 17-19, 2015) - M1 & M2 corrected

Exercise 6 (February 24-26, 2015)

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## Project work

Project work assistants: Alexander Meaney and Andreas Hauptmann

The idea is to study an inverse problem both theoretically and computationally in **teams of two students**. The end product is a scientific poster that the team will present in a poster session on

**Thursday, May 7, 2015 at 14:15-16:00 **in the Industrial Mathematics Laboratory (Exactum C131).

The poster can be printed using the laboratory's large scale printer. Please send your poster via email as a pdf attachment to Andreas Hauptmann by Tuesday 5th May, 12 pm. Then your poster will be printed by the poster session on May 7.

The idea of the project work is to study an inverse problem both theoretically and computationally in **teams of two students**. The end product is a scientific poster that the team will present in a poster session in the Industrial Mathematics Laboratory. The poster can be printed using the laboratory's large scale printer. The classical table of contents is recommended for structuring the poster:

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The project is about **X-ray tomography**. You can measure a dataset yourself in the X-ray facility of the Industrial Mathematics Laboratory:

You can choose your own object to image in the X-ray lab. Examples of good objects include a capasitor, excised tooth, or other objects of roughly the size of a fingertip. Please contact Alexander Meaney to find out if your object is good for imaging.

In the project you are supposed to take a subset of the data with only few projections, such as 20 projection directions. Use It's a good idea to use one of these methods to recover the walnut slice from sparse data:

- Tikhonov regularization based on conjugate gradient method, perhaps with non-negativity constraint,
- approximate total variation regularization implemented iteratively with the Barzilai-Borwein method,some other suitable method.

Optimally, you should have an automatic method for choosing the regularization parameter and an automatic stopping criteria for the iteration. These are both difficult requirements, so have a simple approach as plan B if a more complicated approach does not work.

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Two things will be graded in the meeting about the first goal: (a) the draft of project work and (b) your understanding of the Matlab codes at the above webpage relevant to your topic. The grade represents 30% of the final grade of the project work. Please agree on a meeting time (in the period April 1-4) with the lecturer for reviewing and grading the first goal.

**Second and final goal:** poster is presented in the session on May 7. The poster will be printed in size A1. You may create your own poster (from scratch), or you can use e.g. this template as a starting point and edit its layout, colors, fonts, etc. as much as you like.

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## Results of the poster projects

The posters have been presented in a poster session on May 7. Each group had 5 minutes to present their project and explain what they have done. The resulting posters can be found as PDF files in the following:

X-ray tomography on a Lego block

Sparse X-ray Tomography using Total Variation

X-Ray tomography with sparse data

Dental Imaging using Sparse Angle X-ray CT with Tikhonov Regularization

X-ray Tomography Project Work Inverse Problems Course 2015

Practical X-ray tomography by total variation reconstruction

Inverse Problems Project: Pistachio

CT imaging of hard materials with beam hardening

Dictionary Based JPEG Artefact Removal

Total Variation Regularization with Barzilai and Borwein Optimization Method

Matrix-free X-ray tomography with sparse data

And here are some happy students presenting their work. The pictures have been taken during the poster session.

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## Feedback

Feedback was collected in the middle of the course using this form.

Here are the results:

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