Mathematical Foundations of Imaging

IIDS67521 Mathematical Foundations of Imaging 15 Semester 1 Postgraduate Taught Level 7 Timothy Cootes Unit coordinator Division of Informatics, Imaging and Data Sciences 1 FHEQ level (Framework for Higher Education Qualifications) ' Masters/Integrated Masters P4 ' European Credit Transfer & Accumulation System Rating : 7.5 The course unit will use lectures, ‘pencil and paper’ exercises and some basic programming to introduce students to a number of fundamental mathematical and scientific concepts that underpin the majority of the imaging course units that form the remainder of the programme.  These are:Statistics and Probability Summary statistics, hypothesis testing, multivariate statistics, normal, binomial, Poisson, Chi-square distributions. Discrete and continuous probability, probability density, conditional probability, Bayes theorem.Matrix methods Linear algebra, transition matrices, geometric transforms, eigenvalue methods, tensors, model fitting (least squares)Fourier Methods Fourier Series, Fourier Transform, convolution, Linear Time/Space Invariant Systems point spread function, modulation transfer function, image reconstruction.Introduction to programming: Basic numerical programming in Python (NumPy/SciPy)  The course unit will use lectures, ‘pencil and paper’ exercises and some basic programming to introduce students to a number of fundamental mathematical and scientific concepts that underpin the majority of the imaging course units that form the remainder of the programme.  These are:Statistics and Probability Summary statistics, hypothesis testing, multivariate statistics, normal, binomial, Poisson, Chi-square distributions. Discrete and continuous probability, probability density, conditional probability, Bayes theorem.Matrix methods Linear algebra, transition matrices, geometric transforms, eigenvalue methods, tensors, model fitting (least squares)Fourier Methods Fourier Series, Fourier Transform, convolution, Linear Time/Space Invariant Systems point spread function, modulation transfer function, image reconstruction.Introduction to programming: Basic numerical programming in Python (NumPy/SciPy)  The unit aims to: Provide students with the mathematical and scientific foundation necessary to undertake the imaging course units.The material in this course unit provides a common knowledge and skill base for the remainder of the programme.  Students from a physical science background may have some experience in some of this material.  However, few will have developed the full skill set required in these specific topics.  Most of the material will be new to students from a non-physical science background, but the introductory maths and physics in the Basic Skills course unit should provide them with a basis to advance to this content.  Students should be able to: <ul><li>Calculate summary statistics from experimental data Taught: Statistics sessions  Assessed: Statistics exam</li><li>Calculate estimates of probabilities of events based on available knowledge Taught: Statistics sessions  Assessed: Statistics exam</li><li>Choose appropriate methods of determining whether results are statistically significant Taught: Statistics sessions  Assessed: Statistics exam</li><li>Recognise problems that can be solved using matrix methods Taught: Linear algebra sessions  Assessed: Final exam</li><li>Represent geometric relationships using linear algebra Taught: Linear algebra sessions  Assessed: Final exam</li><li>Interpret Fourier transforms of images Taught: Fourier analysis sessions  Assessed: Final exam </li></ul> Students should be able to:<ul><li>Apply mathematical and statistical knowledge to the understanding of imaging techniques and more widely in scientific investigations. Taught: All sessions Assessed: Final dissertation </li></ul> Students should be able to:<ul><li>Solve moderately complex mathematical and numerical problems. Taught: Linear algebra sessions  Assessed: Dissertation, formative in-class exercises, formative programming assignment</li><li>Apply mathematical and statistical techniques to analysing scientific data. Taught: Algebra & stat.s sessions  Assessed: Dissertation and formative programming assignment</li><li>Write short Python scripts to solve numerical problems (such as linear equations) Taught: Python programming sessions  Assessed: In-class exercises </li></ul> Students should be able to: <ul><li>Have increased confidence in dealing with statistical and mathematical concepts Assessed: Dissertation </li></ul> The course unit will consist of 12 sessions, most of which will consist of a lecture together with written individual and group exercises.  On-line formative quizzes will be used as appropriate.  Some concepts will be reinforced using MATLAB exercises, and basic techniques for setting up and solving linear equations using Python will be introduced.Sessions 1-4: Solving problems in linear algebra. Introductions to geometry and basic PythonSessions 5-8: Statistical summaries of data, probability and measurement error, statistical validity.Sessions 9-12: Representation of images, Fourier methods, Image reconstruction, and Inverse problems  <figure class="table"><table><tbody><tr><td>Assessment task</td><td>Length</td><td>How and when feedback is provided</td><td>Weighting within unit</td></tr><tr><td>Formative: Problem sheets after each lecture</td><td>20 minutes each</td><td>In class/On canvasComments back within 3 weeks </td><td>N/A</td></tr><tr><td>Formative: Computer based assignment</td><td>500 words</td><td>In class/On canvasComments back within 3 weeks </td><td>N/A</td></tr><tr><td>Summative: MCQ exam during the course</td><td>40 minutes</td><td>In class/On canvasComments back within 3 weeks </td><td>33%</td></tr><tr><td>Summative: Final Exam</td><td>80 minutes</td><td>In class/On canvasComments back within 3 weeks </td><td>67%</td></tr></tbody></table></figure> In class/On canvasComments back within 3 weeks  MSc Applied AI for Med Imaging MSc Applied AI for Med Imaging Not Set Mandatory N “Numerical Recipes: The Art of Scientific Computing”, W.H.Press et al. Cambridge University Press (<a href="https://numerical.recipes/" target="_blank">https://numerical.recipes/</a>).“Introduction to Linear Algebra” (6th Edition, 2023) Gilbert Strang<a href="https://www.google.co.uk/search?q=Statistics+at+square+one&sca_esv=b6c66537db370904&sxsrf=ANbL-n4H1fiWxNjlTlXVcc3Ep90_PLBDyw%3A1769176793600&ei=2X5zad-1JOrRhbIPxpq3oAM&ved=2ahUKEwiZ5cC56aGSAxU8QkEAHZ2VNhcQgK4QegQIAxAB&uact=5&oq=Statistics+at+Square+one+how+to+cite&gs_lp=Egxnd3Mtd2l6LXNlcnAiJFN0YXRpc3RpY3MgYXQgU3F1YXJlIG9uZSBob3cgdG8gY2l0ZTIFECEYoAFIyyJQgAJYyBtwAXgAkAEAmAFYoAGEB6oBAjEzuAEDyAEA-AEBmAINoAL9BsICChAAGEcY1gQYsAPCAgUQABiABMICBhAAGBYYHsICBBAhGBXCAgUQIRifBcICCxAAGIAEGIoFGIYDwgIIEAAYgAQYogTCAgUQABjvBcICBxAhGAoYoAGYAwCIBgGQBgiSBwIxM6AHzjGyBwIxMrgH-AbCBwU0LjYuM8gHGIAIAQ&sclient=gws-wiz-serp" target="_blank">Statistics at square one</a> (10th ed.). Campbell, M. J., & Swinscow, T. D. V. (2002). BMJ Books.“Head First Statistics”, Griffiths, D. (2008).. O'Reilly MediaUseful tutorials for learning to use the NumPy mathematical libraries: numpy.org/learn  Lectures 36 114