| CODE | MAT5715 | ||||||||
| TITLE | Inverse Problems and Applications to Medical Imaging | ||||||||
| UM LEVEL | 05 - Postgraduate Modular Diploma or Degree Course | ||||||||
| MQF LEVEL | 7 | ||||||||
| ECTS CREDITS | 15 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | Inverse problems arise from the need to interpret indirect and incomplete measurements. As an area of contemporary mathematics, the field of inverse problems is strongly driven by applications and has been growing steadily in the past 30 years. This growth has been fostered both by advances in computation and by theoretical breakthroughs. Modern digital sensors provide vast amounts of data related to diverse areas including engineering, geophysics, medicine, biology, physics, chemistry, and finance. As a result, the need for inversion techniques can be expected to increase in the future. The main goal of this study-unit is to provide a solid background to inverse problems from both a computational and theoretical perspective. The study-unit will cover: Na茂ve reconstructions and inverse crimes. Ill-posedness in inverse problems. Forward map and Hadamard鈥檚 conditions. Regularized inversion. Singular value decomposition for matrices. Minimum norm solutions. Truncated singular value decomposition. Tikhonov regularization. Mollifiers. Choosing the regularization parameter: Morozov鈥檚 discrepancy principle, the L-curve method. Linear and nonlinear inverse problems and their applications to Medical Imaging (e.g. thermograhy, electrical impedance tomography, X-ray tomography). Study-Unit Aims: This study-unit provides depth of knowledge in both theoretical and computational techniques for solving inverse problems by virtue of their applications to industry, geophysics and medicine. In this study-unit, students will learn the theoretical basis behind linear inverse and ill-posed problems including some of the present classical concepts of regularization. Some important examples of more modern linear and nonlinear inverse problems which arise in the area of Medical Imaging will also be studied. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - Demonstrate an understanding of the nature and setup of inverse and ill-posed problems for typical practical applications and of the mathematical conditions characterizing their solutions. - Demonstrate depth of awareness of a range of theoretical and computational inversion techniques, and critically analyse and compare these methodologies with regard to their suitability for solving different inverse problems in the area of Medical Imaging. 2. Skills: By the end of the study-unit the student will be able to: - Analyse published methods of solution to inverse and ill- posed problems in Medical Imaging and evaluate critically their accuracy and appropriateness. - Select regularization methods for ill-posed problems, and make informed judgements about the issues involved in the choice of regularization parameters. - Demonstrate enhanced transferable and professional skills such as problem solving, information technology and self-direction and originality in solving inverse and ill-posed problems, in general, and in Medical Imaging, in particular. Main Text/s and any supplementary readings: - Mueller J L & Siltanen S (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM Series on Computational Science & Engineering. Philadelphia: SIAM. - Kirsch A (2011). An Introduction to the Mathematical theory of Inverse Problems. 2nd ed. Applied Mathematical Sciences 120. Springer. - Hansen P C (2010). Discrete Inverse Problems: Insight and Algorithms (Fundamentals of Algorithms). SIAM Series on Fundamentals of Algorithms. SIAM. - Isakov V (2006). Inverse Problems for Partial differential Equations. 2nd ed. Applied Mathematical sciences 127. Springer. - Vogel C R (2002). Computational Methods for Inverse Problems. SIAM Series on Frontiers in Applied Mathematics. Philadelphia: SIAM. - Engl H W, Hanke M & Neubauer A (2000). Regularization of Inverse Problems. Kluwer Academic Publishers. - Bertero M & Boccacci P (1998). Inverse Problems in Imaging. CRC Press. - Kervokian J (2000). Partial differential equations: Analytical Solutions Techniques. 2nd ed. Texts in Applied Mathematics 35. New York: Springer-Verlag. - Kress R (1999). Linear Integral Equations. 2nd ed. Applied Mathematical Sciences 82. New York: Springer-Verlag. |
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| ADDITIONAL NOTES | Pre-requisite Qualifications: B.Sc. (Mathematics) Follows from: MAT3755 |
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| STUDY-UNIT TYPE | Lecture and Independent Study | ||||||||
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| LECTURER/S | Cristiana Sebu |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2025/6. It may be subject to change in subsequent years. |
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