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Lecture 9 (January 30, 2015) 

Two demonstrations of current research in X-ray tomography reconstruction methods.

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

Slides available here.

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.mXR04_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.mdeconv6_Tikhonov_plot.m

Lecture 12 (February 6, 2015) 

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

Slides available here.


Lecture 13 (February 10, 2015) 

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

Matlab resources: deconv7_genTikhonov_comp.mdeconv7_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:

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.mWe 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.mHaarTransformMatrix.mdeconv10_B111_comp.mdeconv10_B111_plot.m.


Lecture 19 (February 25, 2015)


Mathematical derivation of the Filtered Back-Projection method. Here are details: FBP.pdf

Book chapter 2.3.3.


Image AddedLecture 29 (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.


Matlab resources

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